Physiographic and Hydraulic Studies of Rivers, 1961 GEOLOGICAL SURVEY PROFESSIONAL This volume was published as separate chapters A~M PAPE R 42 2Cat. for Earth Sci. Ub. UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director 1 CONTENTS [Letters designate the separately published chapters] (A) Morphology and hydrology of a glacial stream—White River, Mount Rainier, Washington, by Robert K. Fahnestock. (B) Hydraulic geometry of a small tidal estuary, by Robert M. Myrick and Lima B. Leopold. (C) Drainage density and streamflow, by Charles W. Carlston. (D) Channel patterns and terraces of the Loup Rivers in Nebraska, by James O. Brice. (B) Channel geometry of Piedmont streams as related to frequency of floods, by F. H. Kilpatrick and H. H. Barnes, Jr. (F) Sediment yield of the Castaie watershed, Western Los Angeles County, California—a quantitative geomorphic approach, by Lawrence K. Lustig. (G) The distribution of branches in river networks, by Ennio V. Giusti and William J. Schneider. (H) River meanders—theory of minimum variance, by Walter B. Langbein and Luna B. Leopold. (I) An approach to the sediment transport problem from general physics, by R. A. Ragnold. (J) Resistance to flow in alluvial channels, by D. B. Simons and B. V. Richardson. (K) Erosion and deposition produced by the flood of December 1964 on Coffee Creek, Trinity County, California, by John H. Stewart and Valmore C. LaMarche, Jr. (L) River channel bars and dunes—theory of kinematic waves, by Walter B. Langbein. (M) Flood surge on the Rubicon River, California—hydrology, hydraulics, and boulder transport, by Kevin M. Scott and George C. Gravlee, Jr. o 1 f I 7 DAY t EARTH SCIENCES ilBRARV Morphology and Hydrology of a Glacial Stream— White River, Mount Rainier Washington GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-A U.S.S.D. * 'Morphology and Hydrology of a Glacial Stream— White River, Mount Rainier Washington By ROBERT K. FAHNESTOCK PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-A A study of the hydraulic and morphologic processes by which a valley train is formed by a proglacial stream UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C.CONTENTS Page Abstract__________________________________________________ A1 Introduction_______________________________________________ 1 Description of study area__________________________________ 2 Physical features_____________________________________ 2 Climate and vegetation________________________________ 5 Geology_______________________________________________ 5 Emmons Glacier________________________________________ 6 Characteristics of White River channels____________________ 8 Discharge______________________________________________ 10 Width, depth, velocity, and area_______________________ 12 Shape____„-------------------------------------------- 15 Mode of change_________________________________________ 16 Water-surface slope, hydraulic roughness, and flow characteristics____________________-____________ 18 Transportation, erosion, and deposition on the valley train________________________________________________ 19 Character of the source materials______________________ 19 Morainic debris____________________________________ 19 Mudflow deposits___________________________________ 20 Valley-train deposits______________________________ 22 Analyses of particle size - __________________________ 23 Sieve analyses_____________________________________ 23 Pebble counts________,________________________ 24 Transportation of bedload______________________________ 26 Methods of measurement and analysis of data___ 26 Antidunes______________________________________ 27 Analysis of competence of the White River. 29 Transportation of suspended load_______________________ 32 Size of material in suspension_____________________ 32 Concentration and discharge________________________ 32 Page Transportation, erosion, and deposition on the valley train—Continued Valley train elevation change______________________ A33 Methods of measurement__________________________ 33 Computation of net elevation and volume change. 35 Elevation change on cross sections of the valley train_________________________________________ 36 Source of deposits______________________________ 40 Channel pattern__________________________________________ 42 Description of pattern______________________________ 42 Changes during the period of study______________ 42 Features of the White River channel patterns.. 49 Analysis of pattern_________________________________ 51 Channels—number, persistence, and location on the valley train___________________________ 51 Relation of pattern to channel characteristics_ 53 Relation of pattern to elevation change________... 56 Causes of a braided pattern_________________________ 57 Erodible banks__________________________________ 57 Rapid variation in discharge-------------------- 58 Slope___________________________________________ 58 Abundant load___________________________________ 59 Related problems_________________________________________ 58 Longitudinal profile of valley and equilibrium in a regrading glacial stream__________________________ 59 Influence of glacier regimen on development of a valley train______________________________________ 59 Summary__________________________________________________ 60 References cited_________________________________________ 66 Index___________________________________________________- 69 ILLUSTRATIONS Figure 1. Emmons Glacier and the White River study area, Mount Rainier--------------------------------------------- 2. Study area and distribution of active glaciers on Mount Rainier, 1913 and 1958--------------------------- 3. Map showing approximate positions of Emmons Glacier ice front, 1930-58, line of profile, and kettles in 1959. . 4. Profiles of valley train and former glacier surface______________________________________________________ 5. Small kettle_____________________________________________________________________________________________ 6. Diagrammatic cross section through kettle________________________________________________________________ 7. Gage height record of White River------------------------------------------------------------------------ 8. Relation of mean velocity, mean depth, and width to discharge of White River channels-------------------- 9. Relation of average width-depth ratios to discharge------------------------------------------------------ 10. Cross sections of several White River channels showing width-depth ratio, shape factor, and Froude number— 11. Cross sections of White River channels showing adjustment to load and discharge------------------------- 12. Standing waves in a White River channel_________________________________________________________________ 13. Distribution of deposits on and near White River, valley train------------------------------------------ 14. Tills................................................................................................... 15. Cutbank in mudflow sequence----------------------------------------------------------------------------- 16. Detail of cutbank in mudflow sequence___________________________________________________________________ 17. Mudflow deposit_________________________________________________________________________________________ 18. Mudflow truncated in foreground by stream--------------------------------------------------------------- 19. Boulder front on surface of mudflow--------------------------------------------------------------------- 20. Cutbank in valley train_________________________________________________________________________________ 21. Cutbank in valley train_________________________________________________________________________________ Pago A3 4 7 8 9 9 12 13 16 17 18 18 19 20 20 21 21 21 21 22 22 hiIV CONTENTS Page Figure 22. Small stream depositing fines which bury coarser deposits________________________________________________ A23 23. Relation of size of material to distance below glacier_____________________________________________________ 25 24. Relation of size of material to slope______________________________________________________________________ 26 25. Sampling bed load with screen______________________________________________________________________________ 27 26. Large load moving over rough bed in shallow channel________________________________________________________ 27 27. Antidunes__________________________________________________________________________________________________ 28 28. Measurement of amplitude and wavelength of antidunes______________________________________________________ 28 29. Longitudinal cross section and plan view of antidunes______________________________________________________ 28 30. Relation of particle size to velocity______________________________________________________________________ 29 31. Relation of particle size to tractive force________________________________________________________________ 31 32. Diurnal variation in suspended sediment concentration with discharge----------------------------------- 33 33. Relation of suspended sediment concentration to discharge______________________________________________ 34 34. Relation of suspended sediment concentration to discharge 50 feet downstream from the division into two chan- nels______________________________________________________________________________________________________ 34 35. White River valley train showing locations of cross sections, areas for volume estimates, and widening----- 34 36. Erosion and deposition on cross sections___________________________________________________________________ 37 37. Erosion and deposition on cross section 3__________________________________________________________________ 39 38. Valley train near head of West Emmons Creek________________________________________________________________ 41 39. Pattern changes, 1:00 to 2:00 p.m., July 18, 1958__________________________________________________________ 44 40. Panorama of the White River valley train___________________________________________________________________ 46 41. Meanders and nose on White River___________________________________________________________________________ 47 42. Meander pattern of White River_____________________________________________________________________________ 47 43. Upper reach of White River________________________________________________________________________________ 48 44. Relict braided pattern of August 4, 1958___________________________________________________________________ 49 45. White River near cross section 10, July 21, 1959----------------------------------------------------------- 50 46. Relict braided pattern of the White River, September 3, 1958_______________________________________________ 51 47. White River nose___________________________________________________________________________________________ 52 48. A braided reach of White River_____________________________________________________________________________ 53 49. Relation of discharge range to number of channels at cross sections________________________________________ 54 50. Hydrograph and number of channels at selected cross sections_______________________________________________ 55 51. Imbrication of boulders on a bar___________________________________________________________________________ 56 52. Braided pattern during the degradation of the valley train_________________________________________________ 57 TABLES Page Table 1. Channel parameters of the White River and other streams------------------------------------------------------- A14 2. Size distribution of valley-train materials____________________________________________________________________ 24 3. Suspended load of White River, August 1958-------------------------------------------------------------------- 32 4. Cross section surveys of the White River valley train, 1957-60------------------------------------------------- 35 5. Changes in elevation and volume of the valley train of the White River, 1957-60-------------------------------- 36 6. Number of channels at each cross section of the White River, 1958-59------------------------------------------- 43 7. Characteristics of the White River channels, 1958-59----------------------------------------------------------- 62 8. Bedload of the White River, July-August 1958___________________________________________________________________ 64 SYMBOLS a Coefficient in w=aQb k A Area of channel cross section L b Exponent in w=aQb m c Coefficient in d=cQr n' d A Mean depth=— w Q s dmax Maximum depth V D Sediment size, intermediate diameter of particle Dso Median diameter Vo.& F Froude number=-== \!gd Exponent in d=cQf V» W / y 9 Acceleration due to gravity T Coefficient in v—kQm Stream length Exponent in v=kQm Roughness parameter from modified Manning equation Discharge in cubic feet per second Water surface slope Mean velocity Current meter velocity, 0.6 depth Float velocity at surface Channel width at water surface Specific weight of water Tractive force=ydsPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS MORPHOLOGY AND HYDROLOGY OF A GLACIAL STREAM-WHITE RIVER, MOUNT RAINIER, WASHINGTON By Robert K. Fahnestock ABSTRACT This is a study of the processes by which a valley train is formed by a proglaeial stream. The area investigated is the White River valley on the northeast flank of Mount Rainier, between the present terminus of Emmons Glacier and the moraine marking the terminal position in 1913. Five square miles of the 7.5-square-mile drainage basin above this moraine are presently covered by active ice. Measurements of channel characteristics were made in 112 channels developed in noncohesive materials. Channel widths ranged from 0.7 to 60 feet, mean depths from about 0.03 foot to 2.08 feet, and mean velocities from 0.3 to 9 feet per second for discharges of about 0.01 to 430 cfs. The relations between these variables can be expressed by the equations: w=aQb, d = cQf, and v = kQm. The exponents for White River channels were found to be similar to the average of those for streams in the Southwestern States. In contrast, Brandywine Creek, Pa., with cohesive bank materials, had higher velocity exponents and extremely low width exponents. Width and depth of channels in noncohesive materials may change by scour and deposition as well as by flow at different depths in predetermined channels. White River channels, with steep slopes in coarse noncohesive materials, were narrower, slightly shallower, and had much higher flow velocities than the channels of Brandywine Creek in cohesive materials. Slope of the valley train was related to particle size and discharge. Pebble counting demonstrated a systematic decrease of 60 mm in median diameter of the valley train deposits in a distance of 4,200 feet downstream from the source areas. Discharge was essentially constant through this reach, the stream received no major additions. Discharges of 200 to 500 cubic feet per second were capable of transporting almost all sizes of materials present and thus modified the form of the valley train. Data on the velocities required to transport coarse materials in White River showed that a curve in which diameter is proportional to velocity to the 2.6 power approximates the relation better than the traditional sixth power law in which diameter is proportional to velocity to the 2.0 power. The few samples contained suspended-load concentrations up to 17,000 ppm. The most graphic evidence of the large amount of material transported by the White River was the amount eroded and deposited on the valley train itself. Measurements indicated an average net increase in elevation of 1.2 feet during 1958 and a net decrease of 0.12 foot in 1959. Description and analysis of the change in pattern of the White River were difficult at high flows because of the rapidity of the change. However, a marked change from a meandering pattern to a braided pattern took place with the onset of the high summer flows and the pattern returned to meanders with the low flows of fall. Explanations offered in the literature for the cause of braided patterns include erodible banks, rapid and large variation in discharge, slope, and abundant load. The common element in all explanations seems to be a movement of bed load in such quantity or of such coarseness that there is deposition within the channel, causing the diversion of flow from one channel into other channels in a valley wide enough to provide freedom to braid. White River braiding took place most actively at large loads and discharges. Although examples of braiding by an aggrading stream are common, the White River and the Sunwapta River, Alberta, have reaches in which degradation took place while the stream had a braided pattern. The conclusion is reached that both braided and meandering reaches can occur along the same stream, which may be aggrading, poised, or degrading. The pattern alone does not conclusively define the regimen of the stream. The regimen of the glacier has long-term effects in providing debris to the stream; short-term effects of weather and runoff determine the current hydraulic characteristics, rate of deposition and erosion, and channel pattern. INTRODUCTION This is a study of the processes by which a valley train is formed by a proglacial stream. The term valley train, as used here, is an outwash plain laterally constricted by valley walls. Studies of such distinctly different geologic environments improve the understanding of past geologic events and knowledge of the interrelations of hydraulic and morphologic variables. Comparisons of data from distinctly different environments make possible the evaluation of the sensitivity of hydraulic parameters to environment. It is hard to imagine a more radical departure from normal stream regimen than a glacierized drainage basin (fig. 1). Diurnal fluctuations in discharge bring bankfull or overbank flow for brief periods to many short reaches of the stream. As a large part of the precipitation falls as snow, it may be years before heavy accumulations at high elevations are reflected in changes of position of the glacier and in runoff. A1A2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Most of the runoff occurs during the months of June, July, and August, which in many environments would be the period of extreme low water. The presence of the Emmons Glacier makes it possible to study the relation of stream and glacial regimen. The stream pattern changes rapidly in response to changes in discharge which cause much shifting of debris on the valley train. The coarseness of the materials being transported, the high stream gradient, and the rapid flow provide situations in which the influence of these factors can be measured under extreme conditions. Hjulstrom (1935) and Sundborg (1954) in their detailed studies of river systems summarized contemporary knowledge of sediment transport and open-channel hydraulics and applied this knowledge to natural channels, which had low velocities, tranquil or subcritical flow, relatively fine bed materials, and low slopes. Leopold and Maddock (1953) presented a method of quantitative analysis of channel characteristics of natural streams. They limited their discussion to a number of rivers in the Great Plains and the Southwest. Wolman (1955) applied the method to a stream in a more humid region. In a recent investigation where these methods were used, Miller (1958) studied high mountain streams in regions which had at one time supported glaciers. All of these channels differed from laboratory channels and from the rapid streams which issue from glaciers to flow with steep gradients and constantly changing channel patterns to the .sea. Hjulstrom led an expedition during the summers of 1951 and 1952 to study the alluvial outwash plains (sandurs) of Iceland and the mechanics of braided streams. The stream studied by Hjulstrom is much larger and has more gentle gradients than the White River and pattern changes appear to take place much less rapidly than on the White River valley train. The first section of this report describes the regional setting and the recent history of the Emmons Glacier. The second and third sections cover the hydrology and hydraulic characteristics of White River channels and the transportation, erosion, and deposition of materials of the valley train. The fourth section is a description and analysis of the channel pattern change on the valley train. The fifth section is a discussion of the related problems, the application of the concept of equilibrium to valley train formation and the influence of the glacier regime on valley train formation. The present report is a modification of a thesis 1 submitted to Cornell University. Work in the field was performed by the author as a member of the Geological Survey under the general supervision of C. C. McDonald, Chief, General Hydrology Branch. The i Fahnestock, Robert, K., 1960, Morphology and Hydrology of a glacial stream: Ithaca, N.Y., Cornell University, doctoral thesis. field assistants were T. G. Bond in 1958 and P. V. D. Gott in 1959. Cornell University professors Marvin Bogema, R. A. Christman, P. G. Mayer, and C. M. Nevin, graduate committee members; E. H. Muller and L. L. Ray, committee chairmen; and M. G. Wolman, U.S. Geological Survey, made many helpful suggestions in regard to the fieldwork and the manuscript. John Savini, U.S. Geological Survey, was extremely helpful in the field through his aid with problems of stream gaging and in preparation of streamflow records. Ann M. Fahnestock, the author’s wife, provided material support in the fieldwork, maintenance of camp, and preparation of the original manuscript. DESCRIPTION OF STUDY AREA PHYSICAL FEATURES The White River study area lies within Mount Rainier National Park on the northeast flank of Mount Rainier (figs. 1 and 2), west of the crest of the Cascade Mountains, 80 miles south southeast of Seattle, Wash. The area of most intensive study was the 1-mile reach of the stream between the present terminus of the Emmons Glacier and the valley loop moraine, which marks the 1913 terminus. The study area is reached from the White River Campground by a truck trail leading up the left bank of the Inter Fork and by a trail across the moraines. This area was selected because it is accessible and includes both a site for a gage where the stream is confined to one channel and a 7 reach where at times the channel pattern changes rapidly. Measurements to determine the relation of size of material and slope were made in several other areas. An area of 7.5 square miles (M. F. Meier, written communication, 1958) is tributary to the gage at the moraine (fig. 2). Of this area, 5 square miles was covered by active glacier ice, 4.4 square miles by the Emmons Glacier, and 0.6 square mile by that part of the Frying Pan Glacier thought to contribute to the White River above the gage. Flow from the Frying Pan Glacier, which does not pass through the study area, enters the White River about 3 miles below the gage by way of Frying Pan Creek. In the period 1957-59 the Emmons Glacier (fig. 2) extended from its source area on the summit of Mount Rainier at an elevation of more than 14,000 feet to an elevation of about 5,300 feet through a valley carved into the northeast flank of the mountain. The depth of this erosion is suggested by the following data: The surface of the glacier is about 2,000 feet lower than the summit of Little Tahoma Peak (fig. 1) on a line transverse to the valley; and the terminus in 1910 was at an elevation of about 4,700 feet near the present stream-MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A3 Figure 1.—Emmons Glacier and the White River study area, Mount Rainier. Photograph by M. F. Meier, October 1, 1958.physiographic and hydraulic studies of rivers Figure 2.—Study area and distribution of active glaciers on Mount Rainier, 1913 and 1958.MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A5 gaging station, which is some 1,700 feet lower than the adjacent Yakima Park, Baker Point, and Antler Peak (fig. 2). The White River is tributary to Puget Sound through the Stuck and Puyallup Rivers. A stream-gaging station equipped with a water-stage recorder is 1 mile below the glacier at the moraine; another station, White River at Greenwater, Wash., is 28 miles downstream, just above the junction with the Greenwater River and 2 miles below the junction with the West Fork White River. The source of the West Fork White River is the Winthrop Glacier, which emanates from the same icefield as the Emmons Glacier. CLIMATE AND VEGETATION The position of Mount Rainier on the west side of the Cascade crest is of great climatic significance. The prevailing winds, moisture-laden westerlies from the Pacific Ocean, are cooled as they rise to cross the Cascade Mountains, causing heavy precipitation. The proximity of the Pacific Ocean has a moderating effect on the temperature in both summer and winter. On Mount Rainier, approximately 75 percent of the precipitation falls from October through May. Yearly snowfall at the higher elevations of the mountain has not been measured, but records kept on the southwest side of the mountain at Paradise (elev 5,400 ft) indicate an average precipitation of 100 inches (including 50 ft of snow) and at Longmire (elev 2,160 ft) approximately 78 inches (including 15 ft of snow). Brockman (1947, p. 2) noted that “while no records are available for Yakima Park (northeast side of the mountain) . . . the snowfall is considerably less.” The maximum depth of snow on the ground at Yakima Park is from 40 to 50 percent that at Paradise, which is almost 1,000 feet lower. The vegetation of Mount Rainier National Park varies with elevation from that of the humid transitional zone of the Puget Sound Lowland to that of the Arctic-Alpine Zone above 6,500 feet (Brockman, 1947). However, in the area studied there is no vegetation on the valley train owing to the incessant reworking of the surface. Scattered trees as much as 57 (Sigafoos and Hendricks, 1961, p. 4) years in age grow on the adjacent ablation moraine, which is underlain by stagnant ice. Older trees are found on the adjacent lateral moraines. The vegetation is too sparse to have any appreciable effect on the runoff characteristics of the drainage basin. On the valley train below the 1910-13 moraine, however, alder has found a footing and scattered groups of evergreen and poplar trees are growing. Older trees grow on terraces, the lowest of which is about 2 to 5 feet above the present valley train. Some trees on channel islands and banks have been killed where 681-370 0—63-------2 deposition has raised the bed of the stream above its former banks. GEOLOGY The bedrock formations of the drainage basin range widely in appearance, composition, and occurrence. These formations provide the glacier, and thus the stream, with rocks having a wide range of abrasion resistance and specific gravity. The average specific gravity of 29 specimens from the ablation moraine was 2.5, ranging from 1.9 to 3.5 as determined in the field by means of a spring scale. The lavas, agglomerates, and mudflows of the Mount Rainier volcanics of Coombs (1936) lie on a rugged erosion surface cut into the Keechelus andesite series. According to Coomb’s (1936) description of the rocks and their occurrence, the Keechelus andesitic series, composed of massive tuffs, breccias, and porphyries with subordinate andesite flows, felsites, basalts, hornfels, and sediments, was intruded by the Sno-qualmie granodiorite. The reaction zone produced by this intrusion is the source of additional rock types. A more detailed examination of the geology of Mount Rainier National Park is currently (1959) being made by Waters, Hobson, and Fisk of Johns Hopkins University. Emmons Glacier and the White River have cut through the Mount Rainier volcanics into the Keechelus andesitic series, but the distribution of these formations is obscured by the presence of the glacier and its deposits, and by the large areas of talus on the slopes above the most recent lateral moraines. The Snoqualmie granodiorite is not known to crop out upstream from the gaging station, but some of the rocks found in the valley train may be from the reaction zone between the granodiorite and the Keechelus. Apparently the varied lithology of the bedrock does not influence the present longitudinal profile of the stream, for only two small patches of bedrock, neither of which is in the present streambed, are known to crop out within the valley train. Variations in the resistance of the bedrock to erosion by the glacier are apparent in the domes and hollows of the glacier surface and the resulting crevasse pattern of the glacier (fig. 1). An unusual event in the geologic history of the northeast side*of the mountain is the volcanic mudflow (greater than 0.25 cubic mile) described by Crandell and Waldron (1956). It issued from the northeast flank of the mountain and flowed down both forks of the White River, spreading out in the Puget Sound Lowland to points beyond Enumclaw, some 45 miles downstream. Carbon-14 analyses of wood samples from the mudflow have dated it at about 4,800 years ago.A6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS The glacial geology of the White River valley has been studied by D. R. Crandell, R. D. Miller, D. R. Mullineaux, and H. H. Waldron of the U.S. Geological Survey, but their findings are not yet published. Evidence of the extent of the Emmons Glacier in Pleistocene time is the glaciated character of the White River valley for many miles downstream. Crandell and others (oral communication) have found a number of lateral moraines at various elevations within the valley and are attempting to correlate them with deposits in other valleys. The massive plug of debris that blocks the valley about 1 mile below the glacier terminus is evidence of the extent of the glacier at intervals over the last 1,000 years. It is composed of several moraines of different ages. The dating of these moraines by tree-ring count is reported in Sigafoos and Hendricks (1961). Crandell (oral communication) and Sigafoos and Hendricks found that trees about 60 to 80 years old are growing on the innermost stabilized moraine; whereas, less than 100 feet toward the glacier, trees about 50 years old are growing on an ablation moraine on top of stagnant ice. EMMONS GLACIER The study area has been uncovered by the stagnation and melting of the Emmons Glacier since its position was recorded by some of the earliest observers. One of the first authors to record his observations on the “White River Glacier” as it was then called, was S. F. Emmons, who with A. D. Wilson made the second successful ascent of the mountain in 1870. Emmons in a letter to his chief (King, 1871) seems to have been overenthusiastic and exaggerated the glacier’s dimensions, stating: The main White River glacier, the greatest of the whole, pours straight down from the rim of the crater in a northeasterly direction, and pushes its extremity farther out into the valley than any of the others. Its greatest width on the steep slope of the mountain must be four or five miles, narrowing towards its extremity to about a mile and a half; its length can be scarcely less than ten miles. The map of Mount Rainier and its glaciers by Sarvent and Evans dated 1896, (plate 68, in Russell, 1898, p. 363) shows the Emmons Glacier terminus in approximately the same position as shown by the 1913 U.S. Geological Survey map (fig. 2). The 1896 map, however, shows two streams issuing from the glacier—one as shown on the 1913 map and the second flowing into the Inter Fork at the left side of the glacier terminus. F. E. Matthes (1914) stated: The youngest moraine, fresh looking as if deposited only yesterday, lies but 50 feet above the glaciers’ surface and a scant 100 feet distant from its edge; the older ridges subdued in outline and already tinged with verdure lie several hundred feet higher on the slope. Matthes gave the length of the Emmons Glacier as about 5)( miles and its width as 1% miles in its upper half. The area of active ice, measured from the 1913 map as approximately 5.3 square miles, had decreased to 4.4 square miles in 1958 (M. F. Meier, written communication). Most of the 0.9 square mile decrease represents areas of ablation moraine and valley train underlain by slowly melting stagnant ice. Matthes’ (1914) estimate of 8.5 square miles could not be verified by rechecking with the 1913 map. Periodically since 1930 the U.S. National Park Service has studied the position of the glacier terminus; the data show the rate of recession of the point at which the stream emerged from under the glacier to be about 75 feet per year. The positions of the glacier (fig. 3) are sketched from Park Service data and photographs provided by V. R. Bender, park naturalist. In 1943 an ice tunnel was discovered to have caved in upstream from the ice face being measured; the caving produced a second point from which the ice faces receded both upstream and downstream. This second point was near the present junction of East and West Emmons Creeks; thus, an ice mass was left bridging the valley until at least 1953. Rigsby (1951) spent 6 weeks in 1950 studying ice petrofabrics and glacier motion on the Emmons Glacier. He determined the rate of motion to be as much as 0.75 foot per day in the center of the glacier, about half a mile above the position of the terminus in 1958 (fig. 2). The general advance of glaciers of the Cascade Mountains (Hubley, 1956, and Harrison, 1956a and 1956b) had also started on the Emmons Glacier by 1953. The National Park Service has measured this advance periodically since 1953 and has found that it \/averaged about 165 feet per year from 1953 through 1957 and continued in 1962. In 1957 Arthur Johnson, of the U.S. Geological Survey, began taking a yearly series of phototheodolite pictures from which topographic maps of the glacier might be made by terrestrial photogrammetry. Volumetric computations as well as measurements of the change in position of the glacier terminus will be possible when these maps are available. The present location of the valley train is in the approximate position of the clear ice shown by the 1913 U.S. Geological Survey map. The debris-covered ice along the valley train appears to have changed little in elevation since the 1913 map, emphasizing the role played by the debris in insulating the ice along the sides of the glacier tongue. The approximate elevation change of the clear ice is shown in figure 4. Down-wasting is almost overshadowed by the combination of caving and melting, and by erosion and melting from below by streams of water and air that flowed through ice tunnels in this central part of the glacier. In 1959,MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A7 0 500 1000 FEET L—I_1_I_I_I_________I CONTOUR INTERVAL 20 FEET DATUM IS ARBITRARY EXPLANATON Ice tunnel Edge of valley train Kettle Stream flow gaging station Figure 3.^~^4&p showing Bpproxim&tG positions of Emmons Glscler ice front, 1930-58, from N&tionsl P0 3 5 7 3 1 2 24 0745 370 »0 3 2 6 5 4 3 25 0740 370 >0 2 5 6 4 4 2 28 1220 430 4 6 7 5 3 2 2 28 1900 600 4 7 4 5 6 5 2 29 0745 430 3 3 6 4 6 4 1 31. 1030 320 1 2 6 6 2 2 1 Aug: 2 1005 210 1 5 4 5 2 1 1 3 1920 210 1 4 3 3 1 1 1 4 1230 195 2 3 2 3 2 1 1 5 1900 280 1 5 4 4 2 1 1 6 1810 370 1 4 5 5 3 2 3 7 0830 220 1 4 6 5 5 3 2 8 1100 210 2 4 9 7 4 3 2 8 1900 ' 340 2 5 8 9 3 2 2 9 1500 490 1 3 7 10 4 2 2 11 0910 300 1 5 6 6 1 1 2 13 1245 310 1 1 4 6 1 1 1 14 1 4 8 6 4 3 2 15 1815 280 1 3 7 5 1 1 1 18 1720 250 1 1 6 4 3 2 1 19 1820 210 1 2 5 3 2 3 1 20 0835 210 1 1 6 5 4 3 2 22 1755 500 1 2 6 8 5 6 3 25.— 0840 300 1 1 8 7 3 2 1 28 0820 120 2 7 8 2 1 2 29.... 1120 170 1 2 8 7 2 2 2 Sept: 2 0830 170 1 2 7 6 1 1 2 4 1545 180 1 2 9 7 1 1 2 30 1130 100 1 1 2 2 1 1 2 Date Time Discharge Cross section— (cu ft/sec) 3 5 6 7 10 11 12 1959 June: 18 1535 130 1 1 3 3 3 2 2 20 0930 140 2 2 3 3 2 2 2 26 1400 120 2 1 2 4 2 2 3 30. 1000 90 1 1 1 3 2 2 3 July: 7 1815 100 1 1 1 2 2 2 2 9 1830 140 1 1 3 4 10 1415 140 1 2 3 4 11 2 1 3 4 3 3 3 12 1630 190 2 2 4 5 3 3 3 13 1600 150 2 2 4 4 4 3 3 15 1230 160 1 1 3 5 3 3 4 16 1910 230 1 1 3 6 17 2000 230 1 2 5 6 18 1325 250 2 2 3 3 3 3 4 19 0940 220 1 2 2 4 4 3 4 19 2000 420 1 3 4 5 3 4 3 20. 1250 370 2 3 4 4 2 2 2 20 1930 290 2 4 4 5 4 3 3 21 1930 280 2 3 4 6 3 2 3 24 0915 430 1 2 5 5 4 5 2 24... 1940 420 6 4 4 4 3 2 25 1045 490 2 2 3 4 4 3 3 26 1800 380 2 2 5 5 30 1315 230 2 3 5 5 5 4 3 Aug: 1. 1815 280 2 3 3 4 3 1642 170 1 3 3 2 4 3 3 4 1600 180 1 3 2 3 2 2 3 5 1450 230 1 4 2 2 3 3 2 6 1345 220 1 3 3 3 2 3 2 7 1420 240 1 3 2 2 3 3 3 8 1505 250 L_ 3 1 3 3 5 3 9 1520 210 1 3 1 3 4 4 3 10 1430 220 1 3 3 3 3 3 3 11 1725 170 1 3 1 3 2 2 3 13 1710 180 1 2 1 3 3 3 3 14 1530 220 1 3 2 4 2 3 3 17 1445 170 1 3 2 3 2 3 2 18 1505 140 1 3 2 3 2 2 2 20- 0945 100 1 2 2 3 2 2 2 24 1000 150 1 2 2 2 3 2 2 25 1035 120 1 1 3 2 28 1705 160 1 2 2 3 2 2 2 29 1510 180 1 2 2 2 3 2 3 Nov: 27 1200 40 1 2 3 2 3 3 2 1 The ch annel at cross section 3 was flowing through an ice tunnel. taken as frequently as several times per day when channels were changing rapidly and every 3 to 4 days when there was little or no change. Table 6 summarizes the number of channels in the vicinity of each cross section and the estimated total discharge for White River at the time each panorama was photographed. In addition, figure 50 shows the number of channels at the odd numbered cross sections in relation to changes in discharge. Both tranquil and shooting turbulent flow commonly occur in natural channels. Wolman and Brush (1961) have shown that meandering began with the onset of shooting flow in some of their flume channels in non-cohesive sand materials. They suggested that because these meanders occur at much higher Froude numbers than those of major rivers they are not dynamically similar in all respects and, therefore, termed the former 1 ‘pseudomeanders. ’ ’ Photographs of White River taken after extended periods of low flow (less than 200 cfs) show what appears to be a meandering pattern from the junction of West and East Emmons Creeks to the gage. As the flows in these meanders have relatively high Froude numbers (table 7), they are not dynamically similar to common meanderings with much lower Froude numbers. The meanders of the White River shown in figures 41 to 43 may approach the condition of pseudomeanders in a natural channel.A44 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS A. 1:00 p. m. B. 1:15 p.m. Figure 39.—Pattern changes, 1:00 to 2:00 p.m., July 18, 1958. A, 1:00 p.m.; B, 1:15 p.m.; C, 1:30 p.m.; D, 1:45 p.m.; E, 2:00 p.m. Discharge is about 350 cfs. Arrows indicate new channels or bars which have appeared since the previous photograph.MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A45 E. 2:00 p. m.A46 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERSMORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A47 Figure 41.—Meanders and nose on the White River near cross section 5, August 13, 1958. Plane table map by Fahnestock and Bond. Figure 42.—Meander pattern of the White River, September 30, 1958. Discharge is approximately 110 cfs. Photograph by John Savini. During August 1957, East and West Emmons Creeks did not always join near cross section No. 2. At times East Emmons Creek flowed through an ice tunnel along the right valley wall and joined West Emmons Creek through several channels, occupied at different times, which entered the valley train between Nos. 4 and 5. Braiding occurred more frequently below the junction than upstream from it. When first visited in 1958 (June 18), there was a braided pattern with two or more channels (table 6; fig. 49) from the vicinity of cross section No. 5 to the constriction at the moraine. From the junction of East and West Emmons Creeks to No. 5 the flow was along the right side of the valley train in one channel, which received several small tributaries from the ice tunnel noted in 1957. By afternoon on the 18th there were more channels at all cross sections, and some flow had shifted to the left side of the valley train. By June 26, the flow upstream from No. 5 had shifted almost entirely from the right to the left side of the valley train; from Nos. 6 to 10 there were several channels; and farther downstream the flow had shifted to the right, concentrating in one channel. This situation remained, with minor changes, until about July 8, when the number of channels increased at all cross sections. During the next few days the flow was concentrated into fewer channels, and most channels were in the left half of the valley train between Nos. 5 and 7.A48 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 43.—Upper reach of the White River, August 13,1959. Arbitrary datum. Planetable map by Fahnestock and Gott. The number of channels at most cross sections continued to diminish, with minor shifts of position, until the 17th and 18th (figs. 39 and 40), when active braiding took place from cross sections Nos. 5 to 7, and limited braiding between Nos. 10 to 12. By July 21, this activity had decreased; likewise the number of channels had decreased at most cross sections with no apparent change on the surface of the valley from Nos. 2 to 4. The Emmons Creeks, after joining, had been diverted by their deposits into the ice tunnel on the right side of the valley train. The situation changed little except for shifts of the channels and braiding at Nos. 10 to 12. On July 28, the stream had left the ice tunnel, and active braiding occurred from the junction of the two creeks to the gage. The next day the number of channels had diminished (table 6). Between July 29 and August 3, the main channel between Nos. 5 and 7 was diverted from the center of the valley train to the right side and later back to the left. The pattern of August 4 is shown in figure 44. Only minor changes in pattern occurred until August 8 when active braiding took place between Nos. 5 and 7. Photographs of the pattern on August 11 showed a general decrease in the number of channels at most cross sections. The main channel was along the left side of the valley train from cross section No. 2 to the constriction above No. 10, where it changed to the right. This situation persisted until August 22, when active braiding took place from Nos. 5 to 11. From August 25 until early in September, the relict pattern of this braiding persisted between Nos. 5 and 7, whereas the main channel reverted to its right-bank position downstream from No. 7. Photographs taken by John Savini on a visit in late September (fig. 42) show that the stream had a meandering pattern. The pattern on June 18, 1959, was similar to that of the previous September. The main channel was in approximately the same place, but the meandering was not as prominent and there were more channel islands. No major pattern changes took place until July 12, when braiding developed between cross sections Nos. 5 and 7. This braided pattern persisted with minor changes until July 17, when there was renewed braiding in the same reach. No channels were present on the right half of the valley train from No. 3 to the constriction above No. 10, where the stream crossed over to the right side. A small clear stream issued from kettles near No. 10 and flowed down the left side of the valley train to the main channel at No. 12. The active braiding of July 17 continued onMORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A49 Figure 44.—Relict braided pattern of August 4, 1958. Discharge is approximately 220 cfs. Note stream through ice tunnel. the 18th, and pattern changes were rapid between Nos. 10 to 12 on the 19th. Fig. 45 shows changes between July 19 and 21 in channels near No. 10. The braided pattern persisted with shifting channels and bar development in the reach between Nos. 5 and 7, and on July 24 vigorous braiding took place from No. 3 to the gage, although no streams were present on the right half of the valley train above No. 10. This pattern persisted with minor modifications and a decrease in the number of channels until the middle of August, when the gradual development of a meandering pattern became apparent, although a number of islands remained. This meander pattern continued to develop into September. October, November, and December storms may have caused the stream to revert temporarily to a braided pattern after each storm. FEATURES OF THE WHITE RIVER CHANNEL PATTERNS Features of the White River patterns are shown in figures 41 and 46 to 48. Figure 46 shows the braided pattern and water-surface profile near cross section 7 Inspection of the panoramas suggests the flows of August 20-25 were probably responsible. For a detailed map of the feature called topographic nose, see figure 47 (August 26). On that date the nose, which had already developed, was gradually being modified. The depths of water are shown by means of bottom and water-surface contours. Such features appear to be a type of small alluvial fan and are quite common on the valley train. Other examples are shown in figures 48 and 41. In each example there is a relatively deep and swift reach upstream which is confined by either cutbanks or “levees.” At the end of this reach, the water spreads and flows in several directions in sheets or in a number of poorly defined channels. When the feature first develops, boulders rolled through the swift reach are deposited with a great amount of noise and splashing on the bars at the end of the chute. Often the deposition of these bars blocks other channels (right foreground, fig. 39), forming pools. A delta is then built into the pool from the main stream and from the blocked stream with fine material settling out in the deeper and quieter portions of the pool. These deposits are the only concentrations of fine materials on the valley train. The subsequent history of the nose depends on the amount of water and sediment delivered to it. Often a decrease in load occurs at some time after deposition of the nose with a consequent channeling of the depositsA50 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS FEET °n 5- io-J o SECTION A-A' 50 100 FEET ___i ■ i I___________________I VERTICAL EXAGGERATION x 5 B FEET 28—| SECTION OF LEVEE B-B' 0 5 10 1—1—I L 1 I_____I_____________ NO VERTICAL EXAGGERATION Figure 45.—White River near cross section 10, July 21, 1959. Planeiable map by Fahnestock and Gott. and concentration of flow. If sufficient material is available as a result of this scour, the nose may then be left in the form of cut banks and levees and a second nose deposited downstream. The term “nose” has been used here, as these features actually project above the general slope of the valley train. It is possible to sit on the streambank near one of these features and watch the water rolling boulders at eye level only a few feet away across the stream. These noses or alluvial fans may be similar in some respects to the horseshoe bars described by Hjulstrom (1952, p. 340). I Contrary to what might be considered normal behavior for a stream, White River channels are often not in depressions but along the top of a ridge. This ridge is formed by the natural levees like those shown in figures 45 and 48. Striking changes in alinement and pattern are caused by deposition of levees that shift the position of the channel, confine it, or block off a channel entrance, thus diverting the flow to another channel as if a valve had been closed. These levees may be similar to the features described by F. E. Matthes (1930, p. 109). Levees may be formed also by the coalescing of bars in a bed of the river, confiningMORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A51 Figure 46.—Relict braided pattern of the White River, September 3,1958. Planetable map by Fahnestock and Bond. the flow to a narrower and deeper channel. These bars appear to be similar to those described by Hjulstrom (1935, p. 340). Hjulstrom pointed out that the thalweg of a glacial stream is not necessarily on the outside of these bends, as in a normal channel, but may occur in the middle or on the inside of the bend. Bars are formed quite rapidly under favorable conditions (fig. 39). Time-lapse photography is quite useful in recording and analyzing the changes which take place. Figures 41 to 43 show reaches of meandering channel. Water-surface slopes are shown for both figures 41 and 43, and the depth of water is shown along the thalweg in figure 41. It is interesting to note the common riffles or crossovers and pools are missing but there are periodic changes in slope which correspond to bends in the plan view. The water surface shows alternations of steep and gentle slopes as does the channel bottom. The gentle slopes are characterized by swift, relatively smooth flow, whereas that in the steeper part appears to be much more turbulent and the bottom much rougher. The fact that along the thalweg there appears to be little if any shallowing in the steeper reaches may well be misleading, as the mean depth may be less because of the greater irregularity of the bed in these sections. The “meanders” have the peculiar appearance of a series of bends connected by short straight reaches with swift quiet flow. The steeper parts are usually associated with bends. From the sketch maps (figures 41, 43, 45, 46 and 48), it is apparent that no greater sinuosity is associated with the meanders than with the anastomosing channels. As Leopold and Wolman (1957) observed in several rivers, individual anabranches may meander, and meandering and braiding reaches may alternate along the stream channel. ANALYSIS OF PATTERN CHANNELS--NUMBER, PERSISTENCE, AND LOCATION ON THE VALLEY TRAIN Figure 49 shows the number of channels at each cross section for each discharge range. The complexity of the problem of determining the cause of braiding and variation in number of channels is well illustrated by this figure. The direct relation between the number of channels and discharge at each cross section is qualified by the number of times that a particular discharge has occurred, probably because the more frequently a discharge occurs the more likely it is to occur at least once under optimum conditions for braiding, and because a lower discharge tends for a time at least to occupy most if not all of the channels of the preceding high flow. It is also qualified, as shown by figure 50, by the time of occurrence and duration of the particular discharge. If a lower discharge occurs for several consecutive days following a higher discharge, the number of channels gradually diminishes.A52 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS EXPLANATION / / 'FI) / / / / / TTTTTT VERTICAL EXAGGERATION ; Surface contour Interval 0.5 foot Bottom contour Interval I foot o--------------o Line of profile Edge of water DATUM IS ARBITRARY PYgure 47—White River nose. August 26, 1958. For location see figure 46. Planetable map by Fahnestock and Bond. The best correlation in figure 49 is that of number of channels with location on the valley train. This correlation appears to be in part a function of the width of the valley train at that point. The factor of slope must be considered, but its role is obscured by the decrease in number of channels below the constriction between cross sections 7 and 10. One might surmise that optimum conditions exist at cross sections 6 and 7 from the large number of channels. These cross sections have valley slopes of 0.04. This large number of channels occurs less frequently at the cross sections both upstream and downstream from Nos. 5 and 6, which are on higher and lower valley slopes. It is thought that slope may be a dominant factor and that the deposition which took place on Nos. 3 through 7 in 1958 may account for most of the material brought into this reach by the stream. This deposition deprived the stream of a heavy bed load farther downstream and lessened the likelihood of its braiding at Nos. 10, 11 and 12. An alternative explanation to account for the number of bifurcations in a given reach emphasizes the importance of valley train width in conjunction with the distance below the glacier and distance between cross sec- tions. Two cross sections only a few feet apart are likely to have the same number of channels at the same time, and the number of chances a stream has to bifurcate is directly proportional to the length of reach and the number of channels within the reach. A channel along the valley wall is limited in its freedom to divide or migrate in the direction of the valley wall. This control operates where the valley wall is unconsolidated, by heavily loading the impinging stream, and causing the deposition of a bar which diverts the flow from the wall. Thus, the longer the cross section and the farther it is downstream from a constriction, the greater the freedom to braid. Constrictions exist at the junction of the East and West Emmons Creeks and between cross sections Nos. 7 and 10. Although from these considerations Nos. 3 and 11 should have a similar number of channels, No. 11 should have more because there are usually more channels upstream from the constriction. Fewer channels at No. 3 than at No. 11 might also be favored by the higher valley slope (0.075 ft per ft) as the competency and capacity of the stream are likely to be greater than on the lower slopes at No. 11. InspectionMORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A53 25 Figure 48.—A braided reach of the White River between cross sections 5 and 6. Longitudinal cross section of thalweg and cross section of channel surveyed August 8. Planetable map, August 7, 1959, by Fahnestock and Gott. of figure 50 and table 6 verifies the number of channels, for the two cross sections are similar, No. 11 having slightly more channels than No. 3. The relatively large number of channels at No. 10 is due at least in part to the persistence downstream of some of the many channels at No. 7 in spite of the effect of the constriction. RELATION OP PATTERN TO CHANNEL CHARACTERISTICS It appears that channel pattern and the hydraulic characteristics of the channels should be related. Leopold and Wolman (1957, p. 63) found that—- When streams of different patterns are considered in terms of hydraulic variables, braided patterns seem to be differentiated from meandering ones by certain combinations of slope, discharge, and width-to-depth ratio. Straight channels, however, have less diagnostic combinations of these variables. The regular spacing and alternation of shallows and deeps is characteristic, however, of all three patterns. Their plot of the relation between slope and discharge (Leopold and Wolman, 1957, figure 46, p. 59) for the three types of channels—braided, meandering, and straight—showed that the line s=0.06 <20 44 divided meandering channels from braided ones for the streams that they studied. A braided pattern occurred on higher slopes for a given discharge and at a higher discharge for a given slope than did a meandering pattern. No clear-cut relation was found for “straight” channels although most of. them lay above the line. It was hoped that a study of the White River channels might aid in explaining the mechanism which produced this separation. A similar plot of the White River channels for which slope data were available showed, however, no such systematic division. All White River channels lay above the line in the braid region. The “pseudomeanders” of the White River occur on slopes and at discharges similar to those of the other patterns. The description of these “meanders” as straight .channels joined by bends may be quite appropriate. An increase in discharge did produce a pronounced tendency to braid as is shown by the seasonal change from meandering to braiding (figs. 40, 42). Leopold and Wolman (1957, p. 53) described the formation of a braided pattern in this way: A mode of formation of a braided channel was demonstrated by a small stream in the laboratory. The braided pattern developed after deposition of an initial central bar. The bar consisted of coarse particles, which could not' be transported under localDischarge ranges, in Number of photographs taken cubic feet per second at indicated discharge cn _j UJ O cr UJ CD D Z 11 10 9 8 7 6 5 4 3 2 1 0 1---------< 100 cfs-----------------4 1+--------100-149 ----------------- 4 2- ------- 150-199 ---------------- 12 2+-------- 200-249 ---------------- 20 3- ------- 250-299 ----------------- 6 3+------:— 300-349 ---------------- 9 4 --------350-399 ------------------ 5 4+--------400-449 ------------------ 6 5 --------450-499 ------------------ 2 5+-------->500 -------------------- 2 Discharge range Valley slope at section, in feet per foot Pebble count Dso, in millimeters Distance to glacier, in feet Width, in feet 2+ 3+ 4+ '5 + !- 3- 4- 5- 1+ 2 + 3+ 4+ 5+ 1- 2- 3- 4- 5- 1+ 2+ 3+ 4+ 5+ 1- 2- 3- 4- 5- 1+ 2+ 3+ ’+ 5+ 1- 2- 3- 4- 5- 1+ 2+ 3+ 4+ 5+ 1- 2- 3- 4- 5- 1+ 2+ 3+ 4+ 5 + 1- 2- 3- 4- 5- 1+ 2+ 3+ 4+ 1- 2- 3- 4- 5 Section 3 Section 5 Section 6 Section 7 Section 10 Section 11 Section 12 0.075 0.05 0.04 0.04 0.03 0.03 0.03 175 165 145 - 125 - 120 3600 4500 5150 5400 6000 6250 6600 320 425 440 440 290 360 170 Figure 49.—Relation of discharge range to number of channels at cross sections (based on panoramic photographs such as figure 40). See also table 6. A54 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERSDISCHARGE, IN CUBIC NUMBER OF CHANNELS FEET PER SECOND K i •f. T> IS n 13 \ > s/ \ XI \ r \ n V V 4 l \ i V 1 \ \ \ 1 / / __ / Vi A \ i / / V !, I 1 \ \ LV A / i .J V4/ \ i / A 0 t: P £ Cross section 11 £ A z E Cross section 7 L \|| A 7 V \ Cross section 5 /\ / V Cross section 3 / \/V \ ill i i 1 l|l 400 300 200 100 JUNE JULY AUGUST SEPTEMBER JUNE JULY AUGUST SEPTEMBER 1958 1959 Figure 50.—Hydrograph of discharge and number of channels at selected cross sections. MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A55A56 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS conditions existing in that reach, and of finer material trapped among these coarser particles. This coarse fraction became the nucleus of the bar which subsequently grew into an island. Both in the laboratory-river and in its natural counterpart, Horse Creek near Daniel, Wyo., gradual formation of a central bar deflected the main current against the channel banks causing them to erode. Time-lapse photography of White River channels showed the development of similar bars which had similar effects upon adjacent banks. Leopold and Wolman (1957) in their description of the formation of these central bars noted that materials were rolled across the surface of the bar and deposited in the quiet water downstream. Not all boulders in White River channels reached this quiet water, however. Figure 51 shows the “imbricate” structure of boulders which were rolled into position. Deposition of these bars usually occurred at high flows when the abundant bed load, set in motion by the increased velocity, combined with the increased discharge to cause rapid widening and shallowing of the channel and the formation of central bars. These bars became islands due to scour of adjacent channels or to a decrease in discharge. Incipient bars may be seen developing in figure 10 (channel 26), in both channels of figure 11, and in figure 39. An increase in slope occurred in the vicinity of these bars which had divided the channel. A braided pattern may also result from reoccupation of old channel beds. This was an important mechanism in the formation of White River patterns. The major islands of the braided pattern shown in figures 39, 44, and 46 had such an origin. A pattern developed in this manner does not show an appreciable increase in slope for the divided channel over that for a single channel. Leopold and Wolman (1957) and Rubey (1952) cite an increased width-depth ratio with channel di- Figure 51.—Imbrication of boulders on a bar. Man points in the direction of flow. vision. It is possible, however, that this increase may be more apparent than real, owing to the method of computing width-depth ratio.' Channel widths are the measured top-surface width of the channel. The depths used in computing the width-depth ratio are mean hydraulic depths computed by dividing cross-sectional area of flow by top width. Therefore, if several channels are lumped into one computation the resulting width-depth ratio may be several times larger than if the channels were treated individually. White River channels, having a wide range of discharges, have similar shapes. Maximum depths in anabranches may be as great as in the undivided channels. Bifurcation thus has no necessary effect on channel shape. Because a width-depth ratio based on the sum of anabranch widths does not give the same picture of channel shape as a width-depth ratio based on individual channels, it is thought that all channels should be treated as individuals even if located in a braided reach. RELATION OF PATTERN TO ELEVATION CHANGE The concept of a braiding stream as an instrument of aggradation is generally accepted. A braided stream as an agent of degradation may not be quite as familiar. Mackin (1956) mentioned braiding in a “stable or slowly degrading reach.” Leopold and Wolman (1957, p. 53) stated: Braiding is not necessarily an indication of excessive total load. A braided pattern, once established, may be maintained with only slow modifications. The stability of the features in the braided reaches of Horse Creek suggests that rivers with braided patterns may be as close to quasi-equilibrium as are rivers possessing meandering or other patterns. / At two places during the White River study degradation and braiding were closely associated. Degradation took place at cross section 3 during period 2b, July 25 to August 5, 1958 (table 5), when there was a net erosion of 1.9 feet. Figure 52 shows that at least half of the valley train at cross section 3 is covered by water in 3 or 4 channels. This photograph, the only record of the channel pattern for this period, shows that the stream which removed the material was braided. Cross sections Nos. 5, 6, and 7 showed a net elevation loss for period 6, June 20-July 25, 1959, of 0.2, 0.4, and 0.3 foot, respectively (table 5). As this degradation took place over less than half the length of the cross sections and as these calculated values are based on the entire length, the loss in elevation in the vicinity of the stream was at least twice as great. During this same period, there were from 1 to 6 channels at No. 5, 1 to 5 at No. 6, and 2 to 6 at No. 7 (table 6 and fig. 50). The greatest number of channels at each cross sectionMORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A57 Figure 52.— Braided pattern during the degradation of the valley train between cross sections 2 and 4. occurred during the highest discharges of the summer, when one would expect the greatest elevation change to take place. Active braiding was again associated with net degradation. A similar situation appears to exist on the Sunwapta River below Athabaska Glacier, Alberta, Canada. There the recession of the glacier from its terminal moraine has created a small lake which serves as a settling basin to clarify the water of the river as it leaves the glacier. This river, as a result of lack of load, has cut 3 to 4 feet below the level of its former valley train, which now forms a terrace. This mechanism of terrace formation was described by Ray (1935). The terrace along the Sunwapta River is now beginning to develop a sparse vegetation of grasses and small plants. The stream, observed in September 1959 at a low stage, had numerous islands and reaches with multiple channels. It appeared as though it had braided actively with the higher discharges of the summer. These three examples illustrate braiding of a degrading stream. It is concluded that both braided and meandering reaches can occur along the same stream, which may be aggrading, poised, or degrading. Braiding is an indication of channel instability and does not conclusively define the regimen of the stream. Arrows indicate the ends of cross section 3. 4:00 p.m., July 28,1968. CAUSES OF A BRAIDED PATTERN Several explanations have been offered for the phenomenon of braided channels by numerous authors who have dealt with the subject. Explanations are quoted for examination in light of the findings on the White River. At the risk of oversimplification it appears that they can be summarized under the headings: erodible banks, rapid and large variation in discharge, slope, abundant load, and local incompetence. ERODIBLE BANKS Fisk (1943, p. 46) suggested that the character of the Mississippi River reflects that material through which it flows. He stated that the Mississippi has— 1. A tendency to braid where bank caving is active. Bank caving is active where sediments are easliy erodible. Sands are the most easily erodible sediments, less permeable sediments are tougher. 2. Tendency to braid where slope is steep and sediments easily erodible, and where slope is excessively low and load great. Friedkin (1945, p. 16) in his model studies of meandering, produced braiding in one test. He reported: This stream at first developed a series of bends, but erosion of banks was so rapid that the channel in the bendway became as shallow as the [point] bars. The flow overran the bars and dis-A58 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS persed through the channel . . . No sand was fed at the entrance of this stream; braiding resulted solely from excessive bank erosion. As the channel shoaled and the bed of the river raised, the slope became steeper. Mackin (1956) in describing the braiding of the Wood River of Idaho stated that the river . . . meanders in a forest for many miles, braids in a 3-mile segment where the valley floor is prairie, and returns to a meandering habit where the river re-enters a forest. The river is stable or slowly degrading in all three segments. The essential cause of the drastic difference in channel characteristics in adjoining segments is a difference in bank resistance due to presence or absence of bank vegetation. Much of the material in transport by the White River is probably derived from bank erosion; but this factor alone, although it may occasionally develop a braided pattern, is insufficient to cause the active braiding of White River channels. Frequently at the same time and discharge, both braided and meandering reaches were present on the valley train. At times the number of channels was increasing in one area while decreasing in another. Braiding and meandering were not limited to specific parts of the valley train. All such changes on the White River cannot be explained in terms of changing erodibility of banks but must reflect other factors that are more easily changed. It is possible that winnowing by the high flows of 1958 may have slightly decreased the erodibility of the reach of the White River between cross sections 2 and 4, giving it a stability in 1959 relative to 1958. A more probable explanation for this relative “stability” is that in 1959 less bed load was transported through this reach, affording less opportunity for deposition within the channel. RAPID VARIATION IN DISCHARGE Doeglas (1951) stated that none of the frequently mentioned causes of braiding (“greater slope, loose debris, or greater discharge”) accounted for the pattern of some rivers during the Pleistocene. He suggested (p. 297): Large and sudden variations in the runoff seem to form braided rivers. A more regular runoff throughout the year gives a meandering river. The gradient and the available sedimentary material seem to be of little importance. The White River does show a tendency toward adjustment to higher discharges where they are continued for a period of several days. In both 1958 and 1959, a discharge sufficient to produce radical changes in pattern at some profiles during June and early July produced relatively minor changes in August; for example, the number of channels at cross sections 3 and 5 during August 1958 varied from zero to 7 (fig. 50). The change in discharge and previous flow history appear to be almost as important as quantity of water in producing changes in pattern. This may be an illustration of Doeglas’ idea of discharge variation as a cause of braiding. It is a likely mechanism for deriving more load for the same discharge and might work as follows in a White River channel. During a period of low flow, the fine sediments would be winnowed from the bed of the channel in some places and deposited in others, but coarser sediments would be left on the bed. With a significant increase in discharge, these coarser sediments would start to move and new supplies of finer materials would be uncovered. The increased discharge might also reoccupy abandoned channels and set in motion all the materials deposited in them by waning stages of the former stream. This excess of load would cause local deposition, which in turn would result in scour of adjacent banks and an increase in load. A more gradual increase in flow might allow the stream to develop a channel that could transport the increased load without forming bars and developing a braided pattern. The frequency with which braided reaches are interspersed with meandering reaches (Leopold and Wolman, 1957, and Mackin, 1956) and laboratory studies (Friedkin, 1945; and Leopold and Wolman, 1957) during which braiding was produced with no variation in discharge would seem to indicate that rapid discharge variation is eliminated as a cause for braiding in most streams. SLOPE The suggestion that change in slope alone is sufficient to cause braiding does not explain phenomena observed along the White River. Braiding developed on slopes which range from 0.01 to 0.20, but only coincident with bed-load movement of coarse materials. The river frequently braids in one part of the valley train on slopes both higher and lower than slopes of other parts where it has only one channel. In some cases, slope may serve to aid the stream in setting in motion enough material to form a braided pattern. In others, it may serve only to maintain velocities so high that the deposition of bars does not take place. ABUNDANT LOAD Hjulstrom (1952, p. 310) stated: A fundamental fact for understanding the braiding of rivers is the great sedimentary load which they carry, Russell (1939, p. 1200, 1201) stated: Available load seems to be the factor separating meandering from braided streams. A smaller load, in proportion to carrying capacity, at the moment, makes for meandering, a larger load for braiding * * * Those [streams] flowing through sandy or gravelly flood plains are more readily overloaded and therefore tend to anastomose, or display the effects of braiding.MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A59 Rubey (1952) in his report on the Hardin and Brussels quadrangles in Illinois suggested that the form of the Mississippi and Illinois Rivers in that area resembled a braided stream more closely than a meandering one, stating (p. 123)— * * * they follow somewhat crooked courses, it is true, but the curves are due not to meander growth but to diversion of the channel by large alluvial islands. He noted (p. 124)-— It is probably a significant fact that the islands are larger and more numerous at the mouths of tributary streams * * * An alternative interpretation explains most of the islands much more satisfactorily. Tributary streams commonly have steeper gradients and carry more debris per unit volume of water than the river; hence they deposit part of their load as deltas and as submerged bars across the tributary mouths. Some of the sand bars and flats built at times of flood stand well above the water level at low and normal stages of the river * * * some of the willow bars grow larger and higher until they become “timber islands” the surface of which is built to the level of the mainland flood plain and covered with dense growths of large hardwood trees. Once an island reaches this stage it becomes an essentially permanent feature. The pattern changes of the White River in 1958, which were both more frequent and more widespread than those in 1959, illustrate the role that abundant bed load played when it was introduced to the section of valley train under observation. That the material was not derived by bank erosion within the area is evident from the net gain in elevation for the valley train (1.2 ft for the year, table 5). In 1959 the decrease in activity and relatively little braiding in the reach between cross sections 2 and 5 suggest that the braiding that did take place was largely caused by bed load derived within the reaches under observation. The net loss of elevation on cross sections 5 to 7 during the summer of 1959 suggests that material was derived from both bed and banks and that braiding occurred in a degrading reach of the stream. To the writer’s knowledge, it has not been suggested that a braided channel pattern can be developed without an appreciable bed load. The common element in all the above explanations appears to be a movement of bed load exceeding local competence or capacity, and consequent deposition within the channel causing the diversion of flow from one channel into one or more other channels. Observations of braiding by the White River suggest that the rate of change of the pattern in a stream which is not restricted by resistant banks is controlled by the amount of bed load. From observations^ of White River and the suggestions of authors cited, braiding is favored by the presence of erodible banks and fluctuations in discharge. RELATED PROBLEMS LONGITUDINAL PROFILE OF VALLEY AND EQUILIBRIUM IN A REGRADING GLACIAL STREAM Many similarities exist between the White River as it deposits and reworks the materials of the valley train and the graded streams described by Gilbert (1877), Davis (1902), Kesseli (1941), Mackin (1948) and others. Mackin (1948, p. 471) defined a graded stream as— * * * one in which, over a period of years, slope is delicately adjusted to provide, with available discharge and with prevailing channel characteristics, just the velocity required for the transportation of the load supplied from the drainage basin. The graded stream is a system in equilibrium; its diagnostic characteristic is that any change in any of the controlling factors will cause a displacement of the equilibrium in a direction that will tend to absorb the effect of the change. He emphasized the idea of a system in long-term equilibrium while recognizing similar short-term changes in the regimen. Rubey (1952), Leopold and Maddock (1953), and Wolman (1955) have discussed the equilibrium adjustment of the channel cross section. Wolman (1955, p. 41) applied the term “quasi-equilibrium” to adjustments made by streams that were not related to the length of time a set of conditions exists. This term is perhaps unnecessary, as the diagnostic characteristic of a graded stream cited by Mackin above is just this tendency to adjust. Mackin (1948, p. 477) recognized the similarities between streams that were raising the level of their beds and graded streams and suggested (p. 478) that the term “aggrading” be restricted to upbuilding at grade. The White River, during the period of study, would appear to fit this definition in the reach from the junction of the Emmons Creeks to the gage. As it appears to be cutting down upstream from the junction, it then could be described as a regrading stream showing adjustment between the variables of slope, channel, cross section, discharge, bed material size, and load. INFLUENCE OF GLACIER REGIMEN ON THE DEVELOPMENT OF A VALLEY TRAIN Emmons Glacier has retreated about 1 mile since 1910, most of this distance since 1930, and has advanced about 0.1 mile recently as shown in figure 3. The stream in 1958 and 1959 has deposited upon and eroded the valley train, resulting in net deposition in 1958 and net erosion in 1959. This response of the stream is probably related to the difference in weather and availability of material in the two years. It is apparent that the response is not related to the glacial advance of approximately 100 feet during this period. Observations of the stream, at the point at which it issuesA 60 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS from the glacier and downstream, suggest that most of its load is derived by erosion downstream from the glacier. The effect of the stream on the valley train would then seem to be a function of the amount of water provided by glacial melt rather than a function of the load introduced directly to the stream from the glacier. The glacial regimen would seem to be related to the valley-train deposits only through the quantity of water that it makes available to the stream and the debris that it has deposited, subject to erosion by the stream. For example, the melting back of Emmons Glacier has uncovered large quantities of debris which are now available to the stream. Continued advance of the Emmons Glacier may radically alter the availability of this material and thus cause changes downstream. Similarly, changes in position of the termini of continental glaciers of the past must have radically altered the availability of debris to their drainage streams. Such effects are probably restricted to the immediate vicinity of the glacier, as a stream on alluvium will derive a load within a short distance and this load will be determined by its discharge, gradient, and availability of readily erodible material. If periods of glacial advance and retreat can be considered analogous to winter and summer on the White River, advance (winter) would provide much smaller discharges and less debris; and retreat (summer) larger discharges and more debris. This is obviously a great oversimplification, as there were winters and summers, in the glacial climate. An additional assumption would be a constant rate of glacier flow because fluctuation in the glacier flow rate might change the position of the terminus without changing the discharge. The removal of a continental glacier from a drainage basin could well bring a decrease in the discharge of the stream, bringing to a close the deposition of a valley train. While the regimen of Emmons Glacier has long-term effects in providing debris to the stream, the short-term effects of weather and runoff determine the rate of deposition and erosion, the hydraulic characteristics, and the pattern of the stream. SUMMARY 1. Channels of the White River, developed in coarse noncohesive materials, are narrower, slightly shallower, and have higher velocities of flow on higher slopes at discharges of similar magnitude than channels with cohesive bank materials as exemplified by Brandywine Creek. 2. The extreme values of the intercepts (in the equations w=aQ'> and v=kQm) for width and velocity, and the rate of change of width-depth ratio with dis- charge extrapolated for data on Brandywine Creek, may reflect the failure of the channel to adjust to small discharges and indicate a relation of intercept to bank material. 3. Although the stream had some load before it issued from the glacier terminus, most of its load was provided by erosion of morainic debris, mudflow, and valley-train deposits in the reach below the glacier. 4. Analysis of a sample of mudflow and valley-train deposits showed that 40 percent of the mudflow material was less than 0.105 mm in comparison to 10 percent of the valley-train material, indicating removal of the finer fraction from the valley-train materials. Measurements demonstrated a systematic decrease in median diameter of the valley-train materials coarser than 4 mm with distance from the source. The change of 60 mm in median diameter for the valley train materials that took place over a distance of some 4,200 feet is attributed to selective transportation. Increases in median diameter such as those 1.7 and 25 miles below the glacier terminus occur at new sources of coarse material. 5. If one accepts the hypothesis of Wolman and Miller (1960) that the dominant process occurs with sufficient frequency and magnitude to cause most of the changes observed, and is neither the frequent event, which is less than the threshold value, nor the infrequent catastrophic event, then it follows that the slope forming discharge for White River channels in this reach lies between 200 and 500 cubic feet per second. 6. When conditions are such that antidunes are developed, much material is carried through the section without deposition. The rate of movement of antidunes, therefore, cannot be used to estimate the total bed load, as is sometimes done with dunes. 7. Measurements of competency in White River channels as well as data from Hjulstrom (1935) and Nevin (1946) suggest that for coarse materials the “sixth power law” should be the 7.8 power law. Logarithmic plotting shows that a line with a slope of 2.6 fits the White River data better than one with a slope of 2.0 (the “sixth power law” expressed in terms of linear dimensions rather than mass). 8. Evidence of the great amount of material transported by the White River is provided by the amount of erosion and deposition on the valley train. The average net elevation change for all profiles in 1958 was + 1.2 feet and in 1959, —0.1 foot. The abundant discharge and high gradients of both East and West Emmons Creeks, together with the unique conditions described for West Emmons Creek, and the widening of the valley train, provided this abundant load during the summer of 1958. The high gradients of the Emmons Creeks combined with the lower flows of 1959MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A61 supplied materials to the valley train in greatly reduced quantities, so that erosion exceeded deposition. 9. The channel pattern changed from “meandering” to braided with the onset of high summer discharges and returned to “meandering” with the lower discharges of late summer and fall. 10. Flows in the “meanders” of the White River have relatively high Froude numbers. The “meanders” are a series of bends connected by short straight reaches of swift quiet flow; steeper gradients are usually associated with bends. It is apparent that there is no greater sinuosity associated with the meanders than with the anastomosing channels. 11. The wider the valley train and the farther it is downstream from a constriction, as exists at the junction of the Emmons Creeks, the more numerous the channels when braiding occurs. 12. Leopold and Wolman (1957) described the formation of a braided pattern in a flume channel by the deposition of a center bar which subsequently became in island. Observations of the White River show that a braided pattern may also result from the reoccupation of numerous old channels due to deposition within the main channel as well as to high-flows which raise the water surface. This was an important mechanism in the formation and alteration of White River channel patterns. A pattern de- veloped in this manner does not show an appreciable increase in slope for the divided reach as did the channel divided by the deposition of .a center bar. 13. Both braided and meandering reaches can occur along White River in reaches where it is aggrading, poised, or degrading. The pattern alone does not conclusively define the regimen of the stream. 14. Braided channel patterns cannot be developed without bed load. The common element in all explanations of braiding appears to be a movement of bed load with local deposition within the channel, causing the diversion of flow from one channel into one or more other channels, or the deposition of channel bars and the development of islands. The rate of pattern change appears to be directly related to the amount of bed load. 15. The White River is a regrading stream showing adjustment between the variables of slope, channel cross section, discharge, load, and size of the bed and bank materials. 16. Observations of the White River show that although the regimen of the glacier has long-term effects in providing debris to the stream, the short-term effects of weather and runoff determine the rate of deposition and erosion, the hydraulic characteristics, and the pattern of the stream.Table 7.—Characteristics of the White River channels, 1958-59 Hour Channel No. Date From To Discharge (cfs) Width (feet) Mean depth (feet) 1958 1 6/30 1745 1900 134.0 18.0 1.44 2 7/1 0945 1205 124.0 18.0 1.47 3 7/21 1750 1935 183.5 35.0 .88 4 7/21 1750 1935 96.7 25 .87 5 7/21 1750 1935 12.8 13 .45 6 7/13 . 175 1.5 .078 7 8/3 1500 1608 190.0 25 1.31 8 8/3 1500 1608 35.3 18 .58 9 8/7 1145 1230 79.5 22 .94 10 8/7 1145 1230 73.7 21 .84 11 8/7 1145 1230 85.3 18 .98 12 8/15 1648 42.0 15. 5 . 72 13 8/15 1600 8.39 8.5 14 8/18 1845 1930 26.7 10 .96 15 8/18 1845 1930 18.6 10.5 .51 16 8/18 1845 1930 163.0 25 1.10 17 8/19 0900 0920 23.7 15 .64 18 8/19 0900 0920 134.0 25 1.06 19 8/19 1415 1445 49.6 33 .56 20 8/19 1415 1445 211.0 29 1.21 21 8/19 1415 1445 2.1 6 .20 22 8/19 1855 1920 32.9 16 .56 23 8/19 1855 1920 172.0 30 .99 24 8/20 1545 1630 10.4 10 .50 25 8/20 1545 1630 333.0 60 1.01 26 8/21 0855 0915 72.9 32 .637 27 8/21 0855 0915 207.0 31 1.18 28 8/21 1030 1100 69.3 24 .87 29 8/21 1030 1100 209.0 32 1.10 30_... 8/25 .07 1.3 31 8/25 .29 2.0 32 8/25 .02 .78 33 .. 8/25 .045 .95 34 ... 8/25 .23 1. 50 35 — 8/25 . 17 2.75 36 8/25 1230 .22 2 .093 37 8/25 .063 1.10 38 8/25 .005 .63 39 8/25 .074 2.40 40 . 8/25 .25 4.10 41 8/26 72.5 16.8 0 93 42 8/26 199.0 27 1 31 43 8/26 9.6 9 / 40 44. 8/26 2.9 27 45. 8/28 130.0 24 1 13 46 8/28 0925 1010 147.0 28 1.11 47. 8/28 194. 0 28 1 27 48 .. 8/28 183.0 28 49 8/29 1. 50 7.8 50 8/29 1. 79 6 5 51 8/29 . 76 52 9/3 1400 1435 93.7 24 1.09 53 9/3 1445 1520 96.2 24 1.13 54 9/30 1310 1350 108.0 22 1.25 55 9/30 1425 1505 106.0 22 1.26 56 11/26 30.7 15 .65 Area (square feet) Mean velocity (feet per sec.) Water surface slope (feet per foot) Froude num- ber Maxi- mum depth (feet) Shape factor Roughness factor n' Width- depth ratio 25.9 5.17 0.76 2.5 1.7 26.4 4.70 .72 2.5 1.7 30.8 5.95 .90 1.4 1. 6 21.7 4. 45 .84 1.6 5.8 1.93 .51 1.0 2.2 .117 1.5 0.014 « .95 .021 32.7 5.80 .89 1.9 1.4 10.4 3.29 .045 .74 0.8 1.4 • 067 31 20.6 3. 86 .028 .70 1.5 1.6 .062 23 17.7 4.16 .029 .80 1.1 1.3 .054 25 17.7 4. 82 .023 .86 1.6 1.6 .046 18 11.2 3. 76 .018 .78 1.1 1.5 .042 22 3.75 2. 24 .030 .59 .5 1.1 .066 42 9.6 2. 78 .50 1.5 1.6 5.4 3. 46 .85 .8 1.6 27.5 5.93 .020 .98 1.9 1.7 .038 23 9.6 2. 55 .029 .56 1.2 1.9 .074 23 26.4 5.09 .032 .95 2.0 1.9 .054 24 18.6 2. 67 .022 .85 .9 1.6 .056 58 35.0 6.03 .029 .90 1.7 1.4 .047 24 1.2 1.75 .43 .4 2.0 9.0 3. 65 .027 .86 1.0 1.8 .045 29 29.7 5. 77 .031 1.02 1.4 1.4 .045 30 5.0 2.0 .023 .50 .9 1.8 .071 20 60. 7 5. 49 .032 .96 1.8 1.8 .049 60 20.0 3.65 .026 .81 1.4 2.2 .049 50 36.6 5.16 .045 .92 2.1 1.8 .062 26 20.8 3.34 .025 .63 1.2 1.4 .064 28 35.2 5.95 .027 1.00 1.6 1.4 .044 29 .065 1.10 .024 8. 85 .14 2.7 .028 25 .266 1.1 .013 8.53 .16 1.2 .040 15 .032 .5 .026 6. 43 .10 2.4 .056 19 .045 1.0 .055 6. 81 .08 1.7 .045 20 .226 1.0 .0092 «. 46 .18 1.2 .040 10 .333 .5 .004 8.25 .19 1.6 .045 23 .186 1.20 .0025 «. 69 .15 1.6 .013 22 .070 .9 .0057 6.63 .11 1.7 .020 17 .017 .3 .015 6. 32 .05 1.8 .055 23 .246 .3 .0017 6.166 .17 1.7 .045 24 .830 .3 .0017 8.118 .34 1.7 .70 20 15.6 4.67 .022 .85 1.4 1.5 .045 18 35.3 5.67 .032 .88 1.9 1.4 .056 21 3.6 2. 66 .029 .74 .6 1.5 .052 22 1.35 2.15 .039 .72 .5 1.8 .059 18 27.2 4. 78 .79 1.4 1.2 21 31.1 4.73 .79 1.7 1.5 35.5 5. 46 .84 1.7 1.3 22 34.8 5.27 .83 1.8 1.4 23 .883 1.7 .004 .90 .24 2.2 .013 71 1.26 1.3 .030 .53 .29 1.5 .068 34 .76 1.0 .011 3.43 .25 1.5 .048 26 26.1 3.59 .61 1.9 1.5 22 27.2 3.53 .58 1.9 1.5 21 27.5 3.96 .62 1.0 1.5 27.8 3.92 9.8 2.93 .64 1.4 1.2 23 Distance above gage (feet) Bottom condition i Discharge measure- ment No. 5 400 Rough___ ---do___ ---do___ ---do___ Remarks (2). Boulders rolling through section. 1,300 800 3a 4 4 5 5 2,000 1,300 do. 900 900 Small anabranch. Flow increased during measurements (2). Boulders rolling through section.2 Flow decreased after measurement. 1,500 1,000 8 9 9 10 10 Boulders rolling through section. Much material moving along bottom.2« 1,000 ‘"900 Boulders. Cobbles. 0.005-0.04' pavement- 11 11 12 12 Channel 26 loosing flow to channel 27 during measurement.4 Discharge shifting. Small clear stream, little load.3 <.07' pavement. <.05'—....... <.05' pavement. Pavement..... <67o5'~-IIIIIIIIIIIII........... Dune ripples sand . ............ ____do_________________________- V)- Clear water stream.3 (3). Clear water stream.3 (3). (3). (3). (3). (3). 1,500 250 Cobbles, gravel. 13 13 13 13 14 Much material moving, uniform depth. Flow fast, turbulent. Flow fairly uniform. 250 220 220 ____do. ____do. ____dO- Sandy. Rocky. 15 16 17 Flow fast, uniform, slightly muddy. Material moving along bottom. Flow fast and fairly uniform. Flow very smooth. Chute on one side. Sandy, few rocks 220 Cobbles, gravel, small boulders. 18 220 Cobbles, small, boulders. 19 250 Cobbles, gravel, small boulders. 20 250 Gravel, cobbles. 21 50 Cobbles, gravel 21a White water. Do. Flow fairly uniform. Do. Clear, some shore ice, well frozen.3 A62 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 1959 57 5/27 1215 1225 5.39 8 .44 3.5 1.54 .41 8. 58. 5/27 1230 1255 48.82 23 .72 16.5 2.96 .61 1.0 59. 6/18 1830 1845 135.3 19 1.33 25.2 5.37 .83 2.1 60. 6/19 1840 57.48 28 .66 18.4 3.12 .68 1.1 61. 6/19 1100 53. 53 16 .81 13.0 4.12 .81 1.0 62 6/19 1930 89.18 16 1.01 16.2 5. 54 .97 1.5 63. 6/30 1520 1534 100. 30 19 1.15 21.9 4.58 .75 1.6 64.. 6/30 1536 1543 10.2 7 .47 3.3 3.09 .79 .8 65 7/2 0930 1035 109. 96 30 .86 25.85 4.25 .018 .81 1.5 66. 7/2 1040 1050 7. 79 14 .34 4.8 1.62 .49 .6 67 7/2 1100 1110 4.28 13 .34 4. 45 .96 .29 .5 68 7/9 1400 1440 108. 94 26 1.01 26.4 4. 66 .017 .82 1.4 69 7/9 1440 1457 8. 77 14 .41 5.8 1.51 .017 .41 .7 70. 7/9 1505 1512 3.16 6.5 .42 2. 7 1.17 .32 .6 71 7/10 1530 1554 143. 98 39 .94 36.7 3. 93 .022 .71 1.5 72 7/10 1715 1735 173. 30 23 1.38 31.7 5. 47 .023 .82 1.9 73 7/11 1820 1822 .4 2.5 .16 .4 .92 .016 .40 .3 74 7/11 1822 1850 145.9 24.5 1.18 29.0 5. 03 .016 .82 1.8 75 7/11 1850 1920 84.1 31 .68 21.2 3.97 .018 .85 1.1 76 7/13 0934 1603 118.8 20 1.07 21.4 5. 55 .018 .94 1.9 77 7/15 1233 127.8 19.5 1.36 26.4 4.84 .016 . 73 1.8 78.... 7/15 1345 46.9 14.5 .88 12.7 3. 69 .016 .69 1.4 79 7/17 1730 175.9 34 1.00 34.0 5.17 .011 .91 1.6 80.. , 7/17 1833 78.5 23 1.01 23.2 3.38 .011 .59 1.8 81-... 7/19 1435 1453 153.8 24 1.12 26.8 5. 73 .022 .94 1.5 82 7/19 1454 1514 211.9 23 1.18 27.1 7.83 .024 1.27 2.3 83.... 7/21 1530 1548 298.0 29 1.13 32.7 9.12 .014 1.51 1.8 84 7/21 1714 1728 2.5 7 .23 1.6 1.56 .57 .4 85 7/21 1728 1744 16.0 16 . 41 6.5 2.4 .66 .6 86 7/21 1755 1802 18.0 11 .53 5.8 3.05 .016 .74 .9 87... 7/21 1805 1811 26.0 16 .51 8.2 3. 48 .014 .86 .8 88 7/24 1543 1630 432.0 28 2.08 58.2 7. 43 .91 3.2 89 7/30 1617 1655 238.0 28 1.05 29.4 8.10 .058 1.38 2.2 90 7/31 0945 1055 223.0 29 1.47 42.5 5.25 .76 2.8 91... 7/31 1415 1545 281.0 29 1.69 49.1 5.73 .033 .78 3.0 92... 7/31 1640 1715 170.0 30 1.60 48.1 5.61 .033 .78 2.9 93 8/1 1855 1930 273.0 30 1.53 46.0 5.92 .84 2.9 94.. 8/3 1800 1848 182.0 28 1.24 34.6 5.27 .020 .83 2.0 95 8/5 .96 7.2 .211 1. 52 .631 .002 .24 .28 96 8/5 1830 1855 177.0 31 1.08 33.5 5. 21 .027 .88 2.2 97 8/9 1449 1605 201.0 28 1.34 37.5 5. 37 .82 2.0 98 8/13 1135 1200 55.1 11.2 .94 10.5 5. 25 .053 .95 1.7 99 8/13 1300 1330 180.0 24 1.35 32.3 5.23 .062 .79 1.8 100 8/17 1540 1600 170.0 20 1.48 29.6 5. 58 .025 .81 2.5 101 8/24 1022 1038 133.0 24 1.12 26.8 4.96 .017 .83 1.7 102 8/24 1108 1123 8.0 9 .37 3.3 2. 51 .0035 .73 .6 103 8/24 1410 1502 145.0 32.5 .94 30.4 4. 76 .058 .86 1.4 104 8/24 1600 1620 14.0 11 .53 5.8 2.46 .020 .59 .8 105 8/28 1055 1155 149.0 27 1.10 29.8 5.17 .87 2.0 106 8/28 1300 1350 165.0 32 1.00 32.1 5.14 .91 1.9 107 9/22 1230 1305 86.0 20 1.23 24.7 3. 48 .55 2.0 108 9/22 1310 1315 1.7 6 .38 2.3 .74 .47 .5 109.... 11/10 1210 1225 19.0 16 .53 8.6 2.20 . 53 .7 110.. 11/10 1230 1245 7.0 10 .33 3.3 2.11 .65 .5 Ill 11/27 1235 1305 39.0 15 .93 14.0 2.78 .51 1.7 112 11/27 1305 1313 3.0 6 .48 2.9 1.03 .26 .7 1 Largest material present is given as bottom condition. 2 Water temperaturo measured and within 35°-39° range. 3 Pigmy current meter used. 4 Measurement of load made in reach at time of discharge measurement. See table 3 for data on load measurement. 1.8 1.4 1.6 1.7 1.2 1.5 1.4 1.7 1.7 1.8 1.5 1.4 1.7 1.4 1.6 1.4 1.9 1.5 1.6 1.8 1.3 1.6 1.6 1.8 1.3 1.9 1.6 1.7 1.5 1.7 1.6 1.5 2.1 1.9 1.8 1.8 1.9 1.6 1.3 2.0 1.5 1.8 1.3 1.7 1.5 1.6 1.5 1.5 1.8 1.9 1.5 1.3 1.3 1.5 1.8 1.5 18 30 22 32 200 22 14 150 23 42 300 24 20 24 16 24 16 25 15 500 25 .042 35 240 Gravel, cobbles ... 26 41 26 38 26 .042 26 300 27 .071 34 27 16 27 .054 42 28 .051 17 29 .060 16 30 .042 21 30 .039 46 30 .028 19 .048 14 200 31 .047 16 31 .030 34 250 32 .046 23 32 .041 22 250 33 .033 20 33 .021 26 30 39 .040 21 .032 31 13 34 .046 27 Cobbles, gravel, holders. 35 20 36 .067 17 Cobbles, boulders, 37 .066 19 gravel. 38 20 39 .046 25 40 .036 35 .048 29 300 41 21 42 .063 12 .085 18 43 .042 14 44 .045 22 45 .018 24 45 .070 34 .054 21 24 180 46 32 280 47 16 160 48 16 48 30 100 49 30 150 49 16 25 50 12 50 50 2 ft of snow on valley train. Large kettles, water white to muddy. No bed load movement, rough. Flow swift, enlarged during p.m. Flow swift. Boulders rolling. Flow fast and uniform. Chute along left edge draining kettles. Flow swift, 1 in. material moving. Underfit stream. Clear stream. Flow swift. Do. Rocks to 0.5 ft rolling through section. Boulders rolling, shook ground. Flow swift. Flow swift. 6-in. rocks moving, one was 1 ft. Flow shifted somewhat into ch. 82 during measurement. Material 8-10 in. moving. Cobbles and boulders moving along bottom. Rocks rolling bottom. Muddy, boulders and cobbles moving. Flow turbulent. Cobbles rolling, flow turbulent. Sand, dune ripples, wide channel Flow swift. Flow turbulent. Flow fast, turbulent, very tfigh slope. Standing wave in section, flow swift. Nothing moving, flow swift. Nothing moving. Occasional boulder rolling. Channel slightly underfit and spread out. Material moving along bottom. Flow fast. Do. Water milky. Up to 4 ft snow, 2 ft shore ice.8 Do. Flood of Nov. 22 widened control area.8 Do. 8 Water temperature measured and within 32°-34° range. 6 Based on float velocities. MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON A63A64 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Table 8.—Bed load of the White River [Asterisk (*) indicates L, length; W, width; H, height] Channel width (feet) Depth (feet) Trac- Maximum velocity (fps) Mean veloc- Slope (feet Load Discharge Date, time Max. Mean tive force Point Float ity (fps) per foot) Dimensions* 5 largest boulders (feet) (L) (W) (H) (Lbs per min per ft) (cfs) Remarks Screen-Caught Samples 7/8/58,1730 hrs_______ 7/8/58, 1730 hrs...... 7/8/58, 1730 hrs...... 8/3/58, 1500-1600 hrs.. 8/5/58, 1500-1600 hrs.. 8/3/58................ 8/3/58________________ 8/7/58, 1540 hrs...... 8/9/58, 1730-1800 hrs.. 8/15/58, 1648 hrs_____ 8/20/58, 1545-1630 hrs. 8/21/58, 0835-0915 hrs. 8/21/58, 1115 hrs_____ 7/13/59, 0934 hrs..... 7/9/59,1645 hrs....... 7/9/59, 1710 hrs...... 1.1 0.79 » 8.2 9.4 0.012 0.46 0.29 0.21 .32 .29 .18 .32 .25 .21 .32 .25 .18 .29 .25 .16 1.2 .57 i 8.2 9.4 .012 .37 .33 .16 20 .37 .25 .19 .25 .23 .15 .29 .21 .17 .25 .21 .17 .9 .53 » 8.2 9.4 .012 .38 .34 .25 50 .38 .33 .21 .29 .26 .21 .39 .25 .16 .29 .25 .16 18 .8 .58 1.53 4.5 5.0 3.47 .045 .36 .32 .25 35 .27 .22 .25 .22 .31 .18 .13 .10 .40 .37 .17 .16 .13 .07 .20 25 1.7 1.3 2.04 6.8 6.5 5.8 .025 189 .50 .36 .19 .47 .36 .22 .48 .35 .31 .42 .35 .30 1.2 2.7 31.03 4.2 9.0 .023 .35 .28 .20 .36 .25 .11 .29 .25 .16 1.05 2. 66 1.90 4.8 7.1 .029 .32 .28 .22 7 .24 .24 .18 .27 .23 .18 .25 .22 .19 .30 .19 .17 27 1.1 . 7 1.22 1 7.5 8.6 .027 1.3 1.1 1.1 114 1.3 1.1 .9 1.3 .9 .9 1.1 .8 .8 1.1 .8 .6 1.4 2.87 2. 88 6.8 10 .033 .7 .7 .5 15.5 3. 76 .9 .6 .8 .6 .7 .6 .7 .6 .32 .25 .5 .4 .4 .3 .15 42 1.1 .72 .81 5.8 i 6.7 .018 .31 .23 .20 .21 .18 .13 .23 .18 .17 .19 .18 .18 60 1.3 1.01 2.04 8.82 i 10.2 5.6 .019 .8 .7 .45 44 330 .8 .6 .50 .65 .6 .45 .65 .57 .30 .68 .55 .40 31 2.1 1.18 1.91 7.6 8.9 5.72 .026 .8 .7 .4 82 206 .7 .7 .5 .6 .5 .3 .45 .45 .27 .45 .45 .15 32 1.1 1.1 1.85 5.8 i 6.7 5. 95 .027 .85 .55 .40 107 209 .70 .43 .35 .53 .35 .30 .45 .35 .35 .45 .35 .30 20 1.9 1.07 1.24 7.9 9.2 5. 55 .18 .9 .75 .8 .7 .9 .6 .7 119 .5 .9 .6 .5 .7 .6 .6 26 1.3 1.01 1.07 5.0 8.6 4. 66 .017 None taken 18 109 26 1.0 1.01 1.07 5.9 8.6 4. 66 .017 .6 .5 .4 32 109 Antidunes 0.05 ft. Crest of antidune 12 sec; sample; 0.05 ft screen. Trough of antidune 12 sec; sample; 0.05 ft screen. Channel 8; discharge measurement 4; 0.05 ft screen. Channel 7; discharge measurement 4; 0.05 ft screen. 0.05 ft screen. 0.05 ft screen. 0.175 screen. 0.175 ft screen; bottom of channel smooth and gravelly. Channel 12; 0.05 ft screen. Load taken in main channel after Q measurement. Suspect bank erosion upstream. Channel 76; boulders rolling thru section about sta. 9; shook ground. Sta. 21-23; channel 68; material moving was small. Sta. 23-25; channel 68; 0.04 ft screen. See footnotes at end of table.MORPHOLOGY AND HYDROLOGY, WHITE RIVER, MOUNT RAINIER, WASHINGTON Table 8.—Bed load of the White River—Continued A65 Date, time Channel Depth (feet) Trac- Maximum velocity (fps) Mean veloc- Slope (feet width (feet) Max. Mean tive force Point Float ity (fps) per foot) Load Dimensions* 5 largest boulders (feet) (L) (W) (H) (Lbs per min per ft) Discharge (cfs) Screen-Caught Samples—Continued Remarks 6/20/58, 1630 hrs_. 6/20/58, 1630 hrs. 6/23/58,1500 hrs.. 6/25/58,1900 hrs.. 8/3/58.... 8/5/58,1430 hrs.. 8/6/58, 1030 hrs. 8/6/58,1210 hrs. 8/8/58- 8/9/58- 8/13/58, 1715 hrs.. i 9.3 10.6 0.025 1.7 1.3 1.1 1.2 1.0 .8 1.2 .9 .8 1.3 .8 .8 1.1 .8 .8 » 7.4 8.6 .025 2.0 1.7 1.5 1.5 1.3 1.0 1.4 1.3 1.1 1.2 1.2 1.0 1.4 1.0 1.0 i 11.0 12.6 .022 1.7 1. 5 .9 2.1 1.3 1.0 1.6 1.3 1.1 1.7 1.2 .9 1.3 1.2 1.0 i 7.0 8.0 .176 1.9 1.7 1.7 1.9 1.6 1.5 1.6 1.6 1.6 2.1 1.5 1.4 1.9 1.5 1.5 1.8 21.13 2.04 7.5 7.7 .029 45 .37 . 25 .43 .35 .25 .25 .20 .18 58 3 1.1 3.85 1 7.0 8.0 .056 1 2 . 9 1.1 1.1 1.0 1.6 1.0 1.0 1.1 1.0 1.0 1.1 .9 .8 27 2.4 2.0 5. 36 i 7.8 9.0 .043 1 6 1.3 1.6 1.3 1.2 1.3 1.2 .9 1.6 1.1 1.1 1.5 1.1 1.0 43 2.5 1.3 2.52 19.2 10.5 .031 . 7 1.2 1.1 .9 1.4 1.0 .6 1.2 1.0 .9 1.2 1.0 .6 46 2.5 1.2 2.23 18.0 9.2 1.8 1. 5 2.1 1.8 1.2 1.6 1.2 1.1 1.3 1.2 1.1 2.3 1.0 .9 25 i 8.0 9.2 .050 1 5 1.0 1.5 1.4 1.0 2.4 1.2 1.0 2.0 1.2 1.1 1.6 1.0 .7 60 2 1.3 1.54 i 9.0 10.3 .019 1 3 .7 1.3 1.2 1.0 1.8 1.1 1.0 1.5 1.1 1.0 1.5 1.0 .7 Caught by hand. Not too much moving. Most moved out when pushed and continued to move. Measured on bar head, then moved out. Smooth bottom washes out under foot. Material may be undermined. 1 Velocity read from graph. Float velocity=1.15 point velocity. 2 Mean depth computed using mean depth =0.625 maximum depth. * Tractive force based on calculated mean depth.A66 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS SELECTED BIBLIOGRAPHY Arnborg, Lennart, 1955a, Hydrology of the glacial river Austur-fljot, chap. 7 of The Hoffellssandur—a glacial outwash plain: Geog. Annaler, v. 37, nos. 3-4, p. 185-201. ------- 1955b, Ice marginal lakes at Hoffellsjokull, chap. 8 of The Hoffellssandur—a glacial outwash plain: Geog. Annaler, v. 37, nos. 3-4, p. 202-228. Blench, Thomas, 1956, Scale relations among sand-bed rivers including models: Am. Soc. Civil Engineers Proc., Paper 881, p. 19-25. Brockman, C. F., 1947, Flora of Mount Rainier National Park: National Park Service, 170 p. Brush, L. M., 1961, Drainage basins, channels, and flow characteristics of selected streams in central Pennsylvania: U.S. Geol. Survey Prof. Paper 282-F, p. 145-181. Chow, V. T., 1959, Open channel hydraulics: New York, Mc-Graw Hill, 680 p. Coombs, H. A., 1936, The geology of Mount Rainier National Park: Washington [State] Univ. Pubs, in Geology, v. 3, no. 2, p. 131-212. Crandell, D. R., and Waldron, H. H., 1956, A recent volcanic mudflow of exceptional dimensions from Mount Rainier, Washington: Am. Jour. Sci., v. 254, no. 6, p. 349-363. Davis, W. M., 1902, Base level, grade and peneplain: Jour. Geology, v. 10, no. 1, p. 77-111. Dawdy, D., 1961, Depth-discharge relation of alluvial streams— discontinuous rating curves: U.S. Geol. Survey Water-Supply Paper 1498-C, 16 p. Doeglas, D. J., 1951, Meanderende en verwilderde rivieren (Meandering and braided rivers): Geologie en Mijnbouw, v. 13, no. 9, p. 297-299. Fahnestock, R. K., 1959, Dynamics of stream braiding as shown by means of time lapse photography (abs.): Geol. Soc. America Bull., v. 70, no. 12, pt. 2, p. 1599. Fisk, H. N., 1943, Summary of the geology of the lower alluvial valley of the Mississippi River: Vicksburg, Miss., U.S. Army Corps of Engineers, U.S. Waterways Exper. Sta., 30 October 1943. ------- 1944, Geological investigation of the alluvial valley of the lower Mississippi River: Vicksburg, Miss., U.S. Army, Corps of Engineers, U.S. Waterways Exper. Sta., December 1944. Flint, R. F., 1957, Glacial and Pleistocene geology: New York, John Wiley and Sons, 555 p. Friedkin, J. F., 1945, A laboratory study of meandering of alluvial rivers: Vicksburg, Miss., U.S. Army, Corps of Engineers, U.S. Waterways Exper. Sta., 1 May 1945. Frodin, Gustaf, 1954, The distribution of late glacial subfossil sandurs in northern Sweden: Geog. Annaler, v. 36, no. 1, p. 112-134. Gilbert, G. K., 1877, Report on the geology of the Henry Mountains: U.S. Geog. Geol. Survey Rocky Mtn. Region (Powell), 160 p. ------- 1914, The transportation of debris by running water: U.S. Geol. Survey Prof. Paper 86, 263 p. Harrison, A. E., 1956a, Glacial activity in the western United States: Jour. Glaciology, v. 2, no. 19, p. 666-668. ------- 1956b, Fluctuations of the Nisqually Glacier, Mount Rainier, Washington, since 1750: Jour. Glaciology, v. 2, no. 19, p. 675-683. Hjulstrom, Filip, 1935, Studies of the morphological activity of rivers as illustrated by the river Fyris: Univ. Upsala [Sweden] Geol. Inst. Bull., v. 25, p. 221-527. ------- 1952, The geomorphology of the alluvial outwash plains of Iceland and the mechanics of braided rivers: Internat. Geog. Congr., 17th, Washington 1952, Proc., p. 337-342. ------- 1954a, An account of the expedition and its aims; chap. 1 of The Hoffellssandur—a glacial outwash plain: Geog. 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Survey Prof. Paper 282-A, 37 p. Leopold, L. B., and Wolman, M. G., 1957, River channel patterns—braided, meandering and straight: U.S. Geol. Survey Prof. Paper 282-B, p. 39-85. Mackin, J. H., 1948, Concept of a graded river: Geol. Soc. America Bull., v. 59, no. 5, p. 463-511. ------- 1956, Cause of braiding by a graded river (abs.): Geol. Soc. America Bull., v. 67, no. 12, pt. 2, p. 1717-1718. Matthes, F. E., 1914, Mount Rainier and its glaciers: National Park Service, 48 p. ------- 1930, Geologic history of the Yosemite valley: U.S. Geol. Survey Prof. Paper 160, 137 p. Matthes, G. H., 1941, Basic aspects of stream meanders: Am. Geophys. Union Trans. 22d Ann. Mtg., pt. 3, p. 632-636. Miller, J. P., 1958, High mountain streams—effects of geology on channel characteristics and bed material: Socorro, N. Mex., New Mexico Inst. Mining and Technology, State Bur. Mines and Mineral Resources memo. 4, 53 p. Nevin, C. M., 1946, Competency of moving water to transport debris: Geol. Soc. 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Assoc. Petroleum Geologists Bull., v. 23, no. 8, p. 1199-1227. Schaffernak, F., 1922, Neue Grundlagen fiir die Berechnung der Geschiebefuhrung in Flusslaufen: Deuticke, Leipzig und Wien. Schumm, S. A., 1960, The shape of alluvial channels in relation to sediment type: U.S. Geol. Survey Prof. Paper 352-B, p. 17-30. Sharp, R. P., 1942, Mudflow levees: Jour. Geomorphology, v. 5, no. 3, p. 222-227. Sharp, R. P., and Nobles, L. H., 1953, Mudflow of 1941 at Wrightwood, Southern California: Geol. Soc. America Bull., v. 64, no. 5, p. 547-560. Sigafoos, R. S., and Hendricks, E. L., Botanical evidence of the modern history of Nisqually Glacier, Washington: U.S. Geol. Survey Prof. Paper 387-A, 20 p. Simons, D. B., Richardson, E. V., and Haushild, W. L., 1961, Studies of flow in alluvial channels—Flume studies using medium sand (0.45 min): U.S. Geol. Survey Water-Supply Paper 1498-A, 76 p. Sundborg, Ake, 1954, Map of the Hoffellssandur, chap. 3 of The Hoffellssandur—a glacial outwash plain: Geog. An-naler, v. 36, no. 1, p. 162-168. ------ 1955, Meteorological observations, chap. 6 of The Hoffellssandur—a glacial outwash plain: Geog. Annaler, v. 37, nos. 3-4, p. 176-184. ------ 1956, The river Klaralven: Geog. Annaler, v. 38, nos. 2-3, p. 127-316. Tanner, W. F., 1960, Helicoidal flow, a possible cause of meandering: Jour. Geophys. Research, v. 65, no. 3, p. 993-995. Tiffany, J. B., Jr. and Nelson, G. A., 1939, Studies of meandering of model streams: Am. Geophys. Union Trans., v. 20, pt. 4, p. 644-649. U.S. Waterways Experiment Station, 1935, Studies of river bed materials and their movement, with special reference to the lower Mississippi River: Vicksburg, Miss., Paper 17, 161 p. White, C. M., 1940, Equilibrium of grains on bed of streams: Royal Society (London) Proc., ser. A, v. 174, no. 950, p. 322-334. Wolman, M. G., 1954, A method of sampling coarse river-bed material: Am. Geophys. Union Trans., v. 35, no. 6, p. 951-956. —----- 1955, The natural channel of Brandywine Creek, Penn- sylvania: U.S. Geol. Survey Prof. Paper 271, 56 p. Wolman, M. G., and Brush, L. M., 1961, Experimental study of factors controlling the size and shape of stream channels in coarse noncohesive sands: U.S. Geol. Survey Prof. Paper 282-G, p. 183-210. Wolman, M. G., and Miller, J. P., 1960, Magnitude and frequency of forces in geomorphic processes: Jour. Geology, v. 68, no. 1, p. 54-74.INDEX A Page Ablation deposits, topography and character.._ A19 Abstract........................................ 1 Access to area................................ 2 Acknowledgments...----------------------------- 2 Analyses of particle size...................... 28 pebble counts............--------------- 24 sieve analyses............................. 23 size-distance relation................. 25 size-slope relation........................ 25 Analyses of valley-train materials, size distribution......................................... 24,25 Antidunes___________________________________ 12,23,27 measurement in White River__________________ 28 particle size in crest and troughs______ . 28 Area, relation to discharge___________________ 12 B Bed load, sieve analyses_____ ________ -26,29 transportation of.......................... 26 Blench, Thomas, and Schumm, S. A., cited.. .. 9 Boulder fronts, in mudflow deposits_____________ 20 Braided pattern, abundant load______ _______ . 68 causes____________________________________ 67 erodible banks............................ 67 formation of..................... ...... 53,56 of degrading glacial stream_______________ 66 rapid discharge variation__________________ 68 slope.................................... 68 Braiding..................................... 48 Brandywine Creek_______________________________ 15 channel parameters_______________________ 14,15 Brockman, C. F., cited_______ ______ ..... ... 5 C Carbon River valley train, size distribution in.. 24, 25 Channel parameters, White River and other streams______________________________ 12 Channel pattern, braided__________________________ 53 Channel shape.................................. 9 Climate......................................... 6 Coombs, H. A., cited.............................. 5 Crandell, D. R., and Waldron H . H., cited___ 5 D Debris, materials of______________________________ 19 Depth, relation to discharge______________________ 12 Discharge fluctuations_______________________ 12,55 relation to width, depth, velocity, and area. 12 Doeglas, D. J., quoted__________________________ 58 E East Emmons Creek........................... 40,47 Effective velocity, White River___________________ 31 Emmons Glacier_______ 40 description.._______________________________ 2 erosion by_____________________________________ 5 history................................... 8,6,7 kettles_______________________________ 7,8,9,40 positions____________________________________ 6 profiles of valley train and former glacier surface_______________________________ 8 relation of stream and glacial regimen___ 2 setting and recent history___________________ 2,6 . Emmons Glacier regimen relation to stream regimen............................... 2 relation to valley-train deposits_____________ 59 Page Emmons, S. F., quoted---------------------------- A6 Ephemeral streams, parameters____________________ 14 Equilibrium in regrading glacial stream- 59 F Fieldwork--------------------------------------- 2 Fisk, H. N., quoted__________________________ 57 Flooding_________________________________________ 11 Flume channels, parameters------------- ... 14 Friedkin, J. F., quoted______________ ... 57 Froude numbers------------------------ ... 62 Frying Pan Glacier------------------- ----- 40 flow from----------------------------------- 2 G Geology of area___________________________________ 6 Gilbert, G. K., quoted________________________ 27,30 Glacierized drainage basin, comparison with normal................................. 1 regimen............................— 1,5 Glacier regimen, influence on valley train... .. 59 H Hjulstrom, Filip, cited____________ 2,2 ,30,51,58 Hydrograph, 1958-59______________ - 11 I Iceland, sandurs of.....— ------------- .. 2 Ice tunnels........................ 6,8,38,40,47,48 Imbrication______________________ . . 56 Inter Fork______________________________________ 2 flooding effects--------------------------- 12 Investigations, previous____ .,---- ... .... 2 J Johnson, Arthur, cited__________________________ 6 K Kettles....................................... 46,48 deposition in.............................. 40 formation of............................... 8 Keechelus andesites.... ________ ... 5 L Lane, E. W., quoted_____ ___ --- - --- - 25 Langbein, W. B., quoted. ........ 27 Leopold, L. p., and Maddock, Thomas, cited 2 8,10,14 Leopold, L. B., and Miller, J. P., cited... .... 14 Leopold, L. B., and Wolman, M. G., cited. 51,53, 56 Levees, in mudflow deposits.......... .. 20,21,41 natural.................................... 50 Lodgement till.._____________________________ 19 Longitudinal profile of regrading glacial stream. 59 Longmire, weather records, 1958-59_ -------- 11 M Mackin, J. H„ cited________________________56,58 Matthes, F, E., quoted____ 6 Measurement methods and data analyses... ... 26 Miller, J. P„ cited__ 2 Moraines, dating of_____ . . ______ ... 6 Morainic debris, features of_____________________ 19 Mount Rainier_____________________________________ 5 Page Mount Rainier—Continued active glaciers on...................... A2,3,4 Mount Rainier National Park.......— ----- 2 geology of.................-...........- 5 vegetation of..........................--- 5 Mount Rainier volcanics, rocks of.............. 5 Mudflow---------------------------------------- 40 Mudflow deposits, features of........-.....- 60 particle size______________________________ 21 West Emmons Creek______ 40 N Natural levees________ ________ ______ ______ Nevin, C. M., cited.......................... Nisqually River valley train, size distribution in_____________________ _________- 24, P Particle size of material in suspension------ Particle size-tractive force, measurement in White River...................... Particle size-velocity relation, White River- Pseudomeanders----------------- — 43, R Regrading glacial stream, longitudinal profile and equilibrium__________________ 59 Rigsby, G. P., cited----- ---- ------------- 6 Rouse, H unter, cited----------------------- 28 Rubey, W, W., cited______________ ... ------ 56,59 Russell, R. J., quoted..... — ... 58 S Sarvent and Evans, cited....... ... ..... 6 Sauberer and Dirmhim, cited----- 11 Selected bibliography------------------------- 66 Sigafoos and Hendricks, cited....... ... ---- 5,6 Simons, D. B., and others cited................. 29 Size distribution, valley train materials... 24 Snoqualmie granodiorite...................... 5 Source materials of the load.............. - 19 Source materials, particle size, analyses of. 28 relative stability of------------------ - 21 Source rocks, types______________________________ 5 Southwestern States, channel parameters.. 14 Stampede Pass, weather records, 1958-59— - - H State-discharge relation______________ ... — 12 changes due to scour and fill--------------- 10 adjustments by Savini................... 10,11 hydrographs............................ 11,55 Storm effects...............................- 11.40 Stream-gaging stations, locations ..........3,5,10 Summary......................................... 00 Sundborg, Ake, cited............................. 2 Sunwapta River............................ - 57 Supercritical flow.............................. 27 Suspended load, concentration-discharge relation_________________________________________ 32 particle size.............................. 32 transportation of......................... 82 T Till............................................ 19 A69 S g g g 8 g SA70 INDEX Page Time-lapse photography--------------------- A51.56 Topographip nose, or alluvial fans.......... 47,49,50 Tractive force, White River.................... 31 Transportation, erosion, deposition on valley train............................... 19 Turbulence, White River....................... 31 U U.S. National Park Service, glacier studies. 6 Emmons Glacier_____________________________ 6 V Valley profile, changes during period of study.. 42 Valley train, channel pattern change........... 2, Jfi definition..................................... 1 deposition over............................... 12 materials of.............................. 2,19 morainic debris............................... 19 mudflow deposits_____________________________ 20 particle-size analysis........................ ts present location............................... 6 source materials of the load.................. 19 widening of................................ 4,42 Valley-train deposits.............................. 22 apparent cohesion in......................22,23 as bed-load source.........-..............19,22 particle size________________________________ 22 Valley train formation, equilibrium................. 2 influence of glacier regime............... 2,59 Valley-train materials, size-distance relations... 25,26 size-valley-train-slope relation.......... 25,27 Vegetation........................................ 5,6 Velocity, relation to discharge.................... 12 Velocity measurements..........................10,27,29 Volcanic mudflow, age............................... 5 w Page Waterfall ... A42 47,51,62 47 40,41,42 ... 3,5 12 31 White River Campground White River channel characteristics: 2 adjustment to load and discharge 16,17,18 1958-59 ... 10,62 ... 14.15 cross sections, width-depth ratio, shape factor ... 17,18 11 18 ... 17,18 hydrology and hydraulic characteristics. 2,8 hydraulic roughness.......................... 18 mode of change............................... 16 parameters...............................13, H shape____________________________________15,16,17 shape factor............................. 15,17 slope-discharge relation..................... 53 standing waves............................... 18 velocity measurements........................ 10 water surface slope.......................... 18 width-depth ratio........................ 16,17 White River channel pattern: analysis..................................... 51 braided pattern, causes...................... 57 changes during 1958-59....................... it method of study.......................... 42 description................................ it While River channel pattern—Con. Page features of: deltas.................................... A49 levees...................................49,50 pools........................... 8,40,46,48,49 topographic nose..................... 47,49,50 location on valley train...................... 52 meanders...................................... 43 number and persistence of channels............ 51 relation between number of channels and discharge......................................51,55 relation to channel characteristics........... 63 relation to elevation change................. 66 relation to width of valley train............. 52 White River drainage basin, snow and ice cover___ 11 White River sediment transport, analysis of com- petence.............................. bed load, July-Aug. 1958.................... erosion by.................................. particle size-velocity relations............ tractive force.............................. velocity measurements.......................27, White River study area, physical features........t, 3 White River valley, glacial geology______________ White River valley train elevation and volume change............................ computation of____________________________ cross-section surveys..................... measurement methods....................... on surveyed cross sections................ size distribution in...................... source of deposits, 1957-60............... Width, relation to discharge.................. 12 Width-depth ratio, computation of.............. 56 relation to discharge...................... 16 Winthrop Glacier............................... 5 Wolman, M. G., cited...................2,10,13,14 Wolman, M. G., and Brusfi, L. M., cited.. 10,15,16,43 Wolman, M. G., and Miller, J. P., cited....... 26 Y Yakima Park, climate............................ 5 U. S. GOVERNMENT PRINTING OFFICE : 1963 O - 681-370  Hydraulic Geometry of a Small Tidal Estuary By ROBERT M. MYRICK and LUNA B. LEOPOLD PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-B UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C.CONTENTS Page Abstract_________________________________________________ B1 Introduction________________________________________________ 1 Area of investigation_______________________________________ 2 Description of the marsh____________________________________ 2 Method of study_____________________________________________ 5 Channel shape and drainage network__________________________ 5 Stage-discharge relations_________________________________ 7 Page Bankfull discharge________________________________________ 9 Hydraulic characteristics at a cross section_____________ 11 Theoretical analysis of the hydraulic geometry in the downstream direction, by Walter B. Langbein__________ 15 Measurements of the hydraulic geometry in the downstream direction__________________________________________ 17 References_______________________________________________ 18 ILLUSTRATIONS Page Figure 1. Location map________________________________________________________________________________________________ B2 2. Photographs of typical reaches of Wrecked Recorder Creek______________________________________________________ 3 3. Map of Wrecked Recorder Creek_________________________________________________________________________________ 5 4. Longitudinal profile__________________________________________________________________________________________ 6 5. Detailed map of one upstream tributary________________________________________________________________________ 7 6. Cross sections at principal places of measurement_____________________________________________________________ 8 7. Relation of stream length to stream order_____________________________________________________________________ 9 8. Relation of number of streams to stream order_________________________________________________________________ 9 9. Stage-discharge and stage-velocity relations_________________________________________________________________ 10 10. Sample relations of stage to velocity____________________________________________________________________ 11 11. Stage and velocity as functions of time___ ______________________________________________________________ 12 12. Duration curves for high and low tide, Alexandria, Va., gage______________________________________________ 13 13. At-a-station curves of width, depth, and velocity as functions of discharge------------------------------- 14 14. Downstream curves of width, depth, and velocity as functions of discharge--------------------------------17 TABLES Page Table 1. Particle-size analysis of channel-bank materials_________________________________________________________________________ B4 2. Pollen anlysis of samples of bank material at Section D_____________________________________________________________ 4 3. Characteristics of cross sections of tidal channel and some rivers in nearby area___________________________________ 6 hiPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY By Robert M. Myrick and Luna B. Leopold ABSTRACT A tidal channel in a marsh bordering the Potomac River near Alexandria, Va., was mapped, and current-meter measurements of discharge were made at various locations and at various stages in the tidal cycle. These measurements allowed analysis of the change of width, depth, and velocity with discharge at various cross sections and along the length of the channel. There is also presented a theoretical development of some of these same relations based on hydraulic principles and on the assumption of a uniform distribution of energy and a minimum rate of work in the system as a whole. The change of width, depth, and velocity with discharge downstream developed from the field data checked closely with the theoretically derived values. The estuarine channel differs from a terrestrial one in that discharge at any section in an estuary varies depending on how the flow shaped the entire length of the channel between the point in question and the main body of tidal water. The result is that a tidal channel changes more rapidly in width and less rapidly in depth as discharge changes downstream than does a terrestrial channel. INTRODUCTION The shape and characteristics of a river channel are believed to be determined and maintained by what has been called by irrigation engineers the dominant discharge. Whereas in an irrigation canal the usual or design discharge is ordinarily the modal flow, the designation of a dominant discharge in a river is not as easy. The effective discharge most influential in river morphology and thus equivalent to the dominant discharge of canals is believed to be the flow at bankfull stage. 'Phis view was strengthened by the findings of Wolman and Leopold (1957), and substantiated later by Nixon (1959), that the bankfull discharge has a recurrence interval of 1 to 2 years and is similar among rivers in quite different physiographic settings. Wolman and Miller (1960) extended and generalized this into a concept of effect ive force in geomorphology. The greatest amount of geomorphic work, they postulate, depends not only on the effectiveness of each single event but also on tlie frequency of the events of different magnitudes. The combination of effectiveness and frequency that was most influential in maintaining river channel form and pattern was the bankfull discharge. A tidal estuary is, in a sense, merely one of the ex- tremes in the continuum of river channels. The channel system of an estuary bears some close similarities to that of a river system. It differs, however, in at least one extremely important respect and that is the frequency of the bankfull stage. If, as some investigators believe, the bankfull stage is the one that is most effective in the maintenance of a river channel and is equaled or exceeded once every year or two, one may ask why the channel system of a tidal estuary so much resembles a river network, despite the fact that bankfull stage is attained about twice a day. Observations and measurements of flow characteristics at different sections along a tidal estuary, particularly at bankfull stage, may help to illuminate the position of estuarine channel systems in the continuum of all natural channels. The bankfull stage of a tidal estuary may be defined as the stage at which the water incipiently flows over the adjacent marshlands. Unlike other streams, an estuarine channel having no runoff from the uplands experiences zero velocity at the nadir of ebb tide. Velocity increases progressively as the water deepens but again becomes zero at the crest of flood tide. From this fact arises the question concerning how the hydraulic factors vary along the length of the channel system. There is a collateral question concerning the relation of the bankfull stage to the effective force in channel formation and maintenance. To understand the hydraulic geometry of estuaries better and to compare it with that of an upland channel, this preliminary investigation was undertaken. There are other collateral questions concerning the differences and similarities between upland streams and tidal ones. For example, does the tidal channel exhibit alternations of shallows and deeps which in upland channels we have referred to as pools and riffles? Does the drainage network of the tidal system divide into tributaries of various sizes in such a manner that their lengths and numbers bear a logarithmic relation to stream order as in upland channels? Some evidence on these points was obtained in this investigation, enough to suggest that further investigation would be lucrative. But on all the points studied, B1B2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS the present data are limited, and any generalizations ve suggest on the basis of these data should be viewed as working hypotheses needing further proof. We are grateful to Estella B. Leopold and Anne Davis for their work on the samples obtained for pollen analysis, to H. L. Svenson for notes on the taxonomy of plants in the study area, and to Alfred C. Redfield, for a review of the manuscript and permission to include some of his unpublished data. AREA OF INVESTIGATION Against the west bank of the Potomac River, nearly adjacent to the city of Alexandria, Va., there was until the winter of 1960 a small area of tidal marsh relatively undisturbed by construction or dredging. It can be seen in the location map of figure 1 that the southern of three estuarine channels carries away nearly all the surface runoff originating in the high ground adjacent to the marsh. The central of the three channels, called by us Wrecked Recorder Creek, drains no upland area and is the one studied. The marshland tributary to this channel has an area of about 120 acres. The marsh area and channels draining it receive water only as rain falling directly on the tributary marsh and as flow induced by tidal fluctuation. The water exchanged with the Potomac estuary is here quite fresh (not salty). The size of the channel was such that it could be measured with relative ease. Where it enters the Potomac River, the channel width is about 170 feet. The total length of channel from mouth to the most distant headwater tributary is approximately 5,000 feet. A tide gage operated by the U.S. Coast and Geodetic Survey is slightly less than 2 miles upstream from the mouth of the channel system studied. Cross sections were measured at six locations along the channel system, as will be described. The estuarine channel was 134 feet wide at the most downstream section and 18 feet wide at the uppermost section. Tidal fluctuations affect the whole length of Chesapeake Bay, and tidewater ends at the foot of the steep reach where the Potomac River descends over the Fall Line in the vicinity of Washington, D.C. The tidal estuary studied is thus about 11 river miles downstream from the head of tidewater. Fieldwork was carried on during late spring and summer of 1960. Shortly thereafter, dredging operations for commercial gravel production destroyed the channel system. DESCRIPTION OF THE MARSH The marsh area appears uniformly flat to the eye. In early spring the dried vegetation has been bent over by snow, so that, at low tide one can walk with ease over nearly the full unchanneled area. There are some small areas within the marsh on which timber is the primary cover. Though some of these timbered areas appear to have a slightly higher elevation than average for the marsh, other timber patches appear to be flooded regularly. By midsummer the marsh vegetation is thick and green, standing at least knee high. The photographs in figure 2 show typical channels and vegetation in early summer. MOUNT VERNON 5.3 Ml.1 77°62,30" 0 Vz 1 MILE 1 -------------------1_____________________I Figure 1.—Location map of Wrecked Recorder Creek, a tributary to the Potomac River, near Alexandria, Va. EXPLANATION FOR FIGURE 2 Photographs of typical reaches of Wrecked Recorder Creek, June 10, 1960. A, Small tributary 400 feet below section C, right bank, stage 4.8 feet, falling. B, Section F, viewed from 20 feet downstream, stage 4.6 feet, falling. C, View looking downstream 250 feet upstream from section D, stage 4.6 feet, falling. D, Section D, looking upstream, stage 4.6 feet, falling. E, Left bank anil entrance of small tributary at Iron Hulk Gage near section A, stage 4.92 feet, falling. F, View downstream to mouth of Wrecked Recorder Creek from position 200 feet downstream from section B; Potomac River barely visible in distance ; stage 4.45 feet, falling.HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B3 E FB4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Along the margins of the channel, vegetation is absent below an elevation of 1 foot lower than the bankfull stage. At low tide there is exposed along the walls of nearly all channels a brown silt or clayey silt, containing some organic material. On the bottom along the centerline of the channel, the bed material is highly organic and black and has a texture not immediately obvious in the field. Table 1 includes some typical data on particle-size distribution of samples taken at five of the established cross sections. The median size of these materials tended to be in the range from silt to fine sand. The table also shows the percentage of organic matter in the sample as determined from ignition-loss measurements. Table 1.—Particle-size analysis of channel-bank materials taken 1-2 feet below level of marsh surface, and ignition loss in percent [Data show percentage finer] Particle size (mm.) Section designation (sample from right bank) A C D E F 1.0 100 100 100 100 100 .50 61 85 77 92 91 .25.... 54 73 59 78 75 .125 40 60 43 64 57 .0625 35 59 40 62 48 .0312 30 49 29 48 31 .0156 24 41 22 36 28 .0078 12 18 9 20 21 .0039 5 5 3 12 11 .0019 2 2 1 3 2 Percent ignition loss at 700° F_. 11.3 10.6 12.7 10.9 13.0 The dominant vegetation consists of cattails, Typha latifolia, and probably also T. angustifolia. Both species are to be expected in localities such as this, but T. angustifolia is in general dominant in areas of salty water. The basal part of some stands of Typha is whitened in spring, probably showing the limit of tide. Another plant common to the area is the arrow arum, Peltandra virginica. Some of the plants are unusually luxuriant and might be taken for the yellow water lily, Nuphar advena, especially where partly submerged. The trees growing in some areas slightly higher in elevation than average marsh surface include green ash, Fraxinus lanceolata, which is a common tree in such places in the tidewater region. The willows are probably Salix nigra, common along the Potomac in the Washington area. Underlying most of the marsh area is a thick layer of gravel, presumably Pleistocene in age, which is being dredged for use in highway fills and other construction. The operators of the dredge stated that gravel layers occur 10 to 40 feet below the surface of the marsh. A few samples were taken for pollen analysis at section D at depths ranging from the marsh surface to 14 inches below it. These samples were brown silty clay which in the field appeared to be undisturbed by any recent lateral movement of the channel. A summary of the pollen analysis is presented in table 2. The following interpretation of these samples is quoted from an unpublished report by Dr. Estella B. Leopold (written communication to authors concerning Paleontology and Stratigraphy Branch shipment WR-60-11D). The samples show a decrease of pine and a corresponding increase of hickory from the base to the top of the sampled section. These features, plus the presence of warm-temperate tree types and a small amount of spruce pollen represent an assemblage which is similar to that of the late pine zone or the postglacial pollen Zone B of Deevey (1939). Assuming the samples are postglacial rather than interglacial in age, an interpolation from a marsh section taken near Blackbird, Delaware, would suggest an age of at least 5000 to 7000 years B. P. The small amount of oak, the large amount of pine and the presence of spruce make the samples quite unlike deciduous tree Zones C-l through C-3 as represented at Blackbird, Delaware, as well as in several postglacial sections from Delaware, Virginia, and New Jersey. Table 2.—Pollen analysis of samples taken along channel bank at section D Samples assigned USDS Paleobotanical location Nos. D1599-C D1599-B D1599-A Depth (inches) below surface 0-2 7-9 12-14 Tree pollen (in percent): 0.7 2.6 2.5 35.5 41.8 50.0 .7 .7 1.5 .5 .7 12.8 5.2 8.4 .7 2.5 .7 1.0 1.9 1.9 1.3 6.9 Shrub and herb pollen (in percent): 1.4 1.9 1.9 9.2 9.8 5.4 .7 1.9 2.1 2.6 3.9 .7 17.7 17.6 5.9 1.4 1.0 .7 .5 .7 .5 7.1 7.8 1.0 7.8 5.2 2.5 (141) (153) (202) Ferns and fern allies (actual counts): (1) (13) (1) (13) (10) (1) This preliminary study of pollen from a few samples indicates that the marsh has existed for a considerable time, at least 5,000 years. This knowledge is useful in that it implies sufficient time for establishment of a quasi-equilibrium between forces of erosion and deposition in the construction and maintenance of the channel system.HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B5 METHOD OF STUDY The principal observational data consisted of the gage height and discharge at selected locations along the length of the estuarine channel. The location of the six measuring sections is shown in figure 3. This figure shows in detail only one tributary, the one on which section F is located. Recording gages were installed near the upper and lower sections, and at each there was established a semipermanent staff gage. A level survey established the elevations of the staff gages above an arbitrarily chosen datum. At each of the six cross sections, several complete discharge measurements were made by means of a current meter operated from a canoe. Later a continuously recording current meter and, concurrently, a recording water-stage apparatus were installed and operated, first at one section and then at another, over a period of several weeks at each. A longitudinal profile of the bed extending from section A to the headwater tributary was obtained by sounding and is shown in figure 4. Some samples of bed and bank material for particle-size analysis were collected. CHANNEL SHAPE AND DRAINAGE NETWORK Before embarking on a discussion of the hydraulics of the tidal channels, a description of the drainage network in conjunction with the planimetric maps presented in figures 3 and 5 will provide the reader with soipe picture of the channels studied. Cross sections up to the level of the marsh—that is, to bankfull stage— are presented in figure 6 for each of the six principal measurement locations. In table 3 the principal characteristics of these channel cross sections are compared with river cross sections considered typical for river channels of similar size in the vicinity of Washington, D.C. The table compares width-depth ratios, bankfull discharge, and some other channel parameters. Figure 3.—Map of Wrecked Recorder Creek at Potomac River, south of Alexandria, Va., showing location of cross sections and recording instruments. 675126 0—63-----2B6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 4.—Longitudinal profile of bed of Wrecked Recorder Creek; vertical scale is depth in feet below average level of marsh, which is also equal to 7.2 feet on the arbitrary gage datum of the present study. Table 3 shows that the width of estuarine channels increases with respect to discharge faster than in upland channels. The table does not include enough examples to demonstrate conclusively any generalization about width-depth ratio. Tidal channels are often shallow near the mouth owing to deposition where an estuarine channel joins a large body of water. Table 3.—Characteristics of cross sections of tidal channel and some nearhy rivers for comparison Width (ft) Depth (ft) Width/ depth Bankfull discharge (cfs) Wrecked Recorder Creek near Alexandria. Va.: Section A 134 4.13 32.4 i 566 B-. 82 4. 77 17.2 i 461 C 84 3.83 21.9 i 338 D 45 3.38 13.3 i 216 E- 32 3.00 10.7 i 136 F 18 2. 38 7.6 i 47 Watts Branch near Rockville, Md 15 3.4 4.4 170 Seneca Creek near Dawsonville, Md 86 4.2 21.6 1,320 Brandywine Basin, Pa 100 5.35 18.5 2,700 65 3.4 19.1 1,000 38 2.0 19.0 310 1 For flood tide having a maximum stage of 8.6 ft, and a range in stage of 3.5 ft; this is one combination of range of stage and maximum stage at which maximum velocity during the tidal cycle occurs when the channel is bankfull. The principal parts of the channel network were mapped by planetable, and the entire headwater network of one small tributary was sketched by pace-and-compass methods. The maps of figures 3 and 5 are sufficient to make a preliminary Horton analysis (Horton, 1945), relating stream order to the number and average length of channels. The smallest channel without a tributary is considered to be of first order and is labeled 1 in figure 5. Channels having only first-order tributaries are labeled 2, meaning that they are of second order. The smallest channel where cross-section and discharge measurements were made is a channel of fourth order (fig. 5). The main channel in the network, on which are located sections A to O, is of fifth order. It is recognized that a map of only one example does not represent the variety of conditions that may occur in the field. Yet it is informative to note the relations of stream order to average length of stream shown in figure 7. As can be seen there, first-order channels average about 20 feet in length, and the fifth-order main channel has a length of 5,000 feet. The length-order relations for the estuary are compared with the same relation for ephemeral streams (Leopold and Miller, 1956, p. 18) and for perennial channels in Pennsylvania (Brush, 1961, p. 156). This estuary increases in length with stream order faster than do either ephemeral or perennial river channels. The ratio of the number of streams of a given order to the number of the next higher order (called by Horton the bifurcation ratio) ranges for river channel networks from 3 to 4, often with an average of 3.5. Because only one tributary of the study estuary was mapped in detail, the estimate of the number of streams of various orders is crude. From the limited data for Wrecked Recorder estuary, this ratio averages 3.4. ThisHYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B7 figure may not be typical of other estuaries, and other similar studies would be informative; yet, as can be seen in figure 8, this one sample is very similar to two other types of channels in very different environments. Figure 5.—Detailed map of one tributary of Wrecked Recorder Creek near Alexandria, Va., including all minor channels. Order number of each channel is shown, some of which are estimated (est). STAGE-DISCHARGE RELATIONS The relation of water stage to discharge has been determined for tidal estuaries in many localities, and the characteristic difference between the form of the curve on ebb and flood tide is well known. Nearly all previous observations, however, have been made on relatively large estuaries. With the exception of the studies made by Gilbert (1917, p. 108) in a tidal marsh near San Francisco, no published records are known to us for small tidal channels having no drainage area in the upland. The present investigation adds to Gilbert’s work only the comparison of the relations between different cross sections located along the length of the estuarine channel. Velocity, and thus discharge, at a given stage was dependent both on the maximum stage attained by the particular tidal cycle and on the range of stage in the tidal cycle. Thus, every tidal cycle requires a slightly different stage-discharge curve. The dependence of velocity on range of tide is known from other investigations. Bradley (1957, p. 664), also working in a relatively small estuary, found that the near-bed velocity varied directly with the range of tide. His excellent work does not help solve the particular questions here investigated. All his flow measurements were made close to the estuary bed. Further, he was concerned with a small bay rather than with riverlike estuarine channels. It was impossible for a party of two to make a complete set of discharge measurements at all sections along the stream simultaneously through a single tidal cycle. We chose, therefore, to make as many discharge measurements as possible through a single cycle at each section, and on different days, making similar measurements on different cross sections. Now the cross-sectional area of each section for any given stage is known from the surveys of the section. The problem is to determine the mean velocity for the whole section at selected stages in order to construct a relation of discharge to stage. For two tidal cycles the relation of velocity to stage is shown in figure 9, using section D as an example. Such graphs alone, however, do not give sufficient information to determine the influence of either tidal range or maximum stage. To incorporate these additional variables, two sections were chosen for a more intensive series of velocity measurements covering different tidal cycles. At sections D and F a recording current meter was installed, along with a device that recorded water stage simultaneously. Thus, at a particular distance above the bottom, velocity was measured near the center of the channel continuously for two weeks to sample tidal cycles from spring to neap tides.B8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS VERTICAL SCALE rO -2.0 - 4.0 FEET 10 0 10 20 30 FEET i .... i_____l____I______I HORIZONTAL SCALE Cross sections viewed looking downstream Section F Figure 6.—Cross sections viewed looking downstream of each of the principal measuring places along the channel of Wrecked Recorder Creek. Vertical scale is feet below average level of marsh. The location and depth of the recording current meter were selected on the basis of velocity readings from previous measurements to provide a position where point velocity best correlated with mean velocity. Owing to the variation in water depth throughout the tidal cycle, the depth below the water surface where the meter was installed changed. The position of the recording meter varied from three-tenths to seven-tenths of the full depth below water surface at section />, and from two-tenths to eight-tenths at section F. The velocities recorded at the single point during a given cycle correlated reasonably well with measured mean velocities for the whole section. From this correlation, the relation of mean velocity to stage and to range of stage could be constructed. For each cross section, curves of the type shown in figure 10 were developed to show the stage-velocity relation for each of several values of range in stage. The curves shown by full lines in figure 10 are merely selected examples which comprise three values of maxi- mum stage, each with a different range in stage. For one of these examples, an additional graph is shown (dashed line) representing the condition of another value of range in stage but identical maximum stage. It will be visualized that there may be a very large number of curves making up such a family as that shown in figure 10. The graphs drawn in the interpolation procedure are considered reasonably reliable except for very low velocities, but even there the error in velocity does not exceed 10 percent. Minor variations in individual cycles were neglected in the construction of the graphs. Most data in tidal studies have been presented as graphs of stage or velocity as a function of time in a tidal cycle (for example, Gilbert, 1917, p. 124; Ahnert, 1960, p. 397). Our data plotted in that usual way exhibit the usual features (fig. 11). Minor variations occur in individual tidal cycles; velocity decreases toward zero with time in a somewhat irregular manner in the early part of the ebb tide.HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B9 Figure 7.—Relation of stream length to stream order for Wrecked Recorder Creek and for some other drainage areas. The data from Brush (1961) were extended to account for differences in map scale, and his order 1 was comparable to order 5 in the present study. Dots represent measurement data from this study. BANKFULL DISCHARGE The evaluation of the banks relative to gage datum was determined by visual observations at several sections in the reach during the flood tides. The stage at which the flow started to inundate the marsh ranged from 7.1 to 7.3 feet, and averaged 7.2 feet (relative to our arbitrary datum) for the reach. The difference between the elevation of the marsh at the headwaters and at the mouth is very small. In upland river systems, the higher the discharge the less frequently it is experienced. Except for very low flows, this inverse relation of discharge rate and frequency of occurrence is a general characteristic of rivers. In tidal channels, on the other hand, zero discharge occurs twice in every tidal cycle, and thus there are also two occurrences of high discharge in each cycle. There remain different frequencies of the different values of the peak discharge from cycle to cycle, but the range of values possible is much smaller than the range of possible values of flood flows in an upland river of the same size. Because of this difference, it may be fruitful to examine the frequency of occurrence of the factors which do vary. Figure 12 shows the percentage of time the high and low tides equal or exceed a given stage at the Alexandria, Va., tide gage. The ordinate shows the variation in terms of the datum uesd in Wrecked Recorder Creek. Figure 8.—Relation of number of streams to stream order; data for Wrecked Recorder Creek are compared with average relations for an area of perennial streams and an area of ephemeral streams.BIO PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS DISCHARGE, IN CUBIC FEET PER SECOND Figure 9.—Stage-discharge relation (light line) and st; ge-velocity relation (heavy line) ; a nearly complete tidal cycle having a maximum stage of 6.8 feet is shown and the ebbing limb only is shown for a tidal cycle in which maximum stage was 7.6 feet. Data are for section D. The velocity shown is mean velocity in the cross section. A comparison of our observations with the concurrent values of stage of the Potomac estuary at Alexandria showed that the stage at the two locations varies in a comparable manner, and this indicates that the frequency of stage at the two locations is comparable. To eliminate the effect of the spring and neap tides, a period of 2 lunar months, 58 days, was used in the preparation of the frequency data. The extremes shown by dashed lines were extended on the basis of a straight-line projection. From the graph we find the median high tide for this area is 7.27 feet relative to our local datum. Bank-full stage, 7.2 feet, is, interestingly, approximately the median high tide and is therefore attained on the average every other tidal cycle or once a day. A terrestrial river reaches its bankfull stage but once every 1 or 2 years. This, though true, leaves an incomplete and perhaps false impression in the mind of the student of channel morphology. We are interested, after all, in the frequency of the effective discharge. In an estuarine channel, at some value of stage near high tide, velocity must be zero, for the direction of flow must reverse. In Wrecked Recorder Creek, the velocity is highest at a stage much lower than high tide. The velocity and the stage corresponding to maximum discharge both increase with the maximum stage of of any given tidal cycle. The stage at which the maximum velocity occurs is not well defined by the available data; but as a rough approximation, a tide reaching a maximum stage of 8 to 8i/2 feet is required for the maximum velocity to occur at a stage of 7.2 feet, which is the bankfull stage. Thus, because of the dependence of dis-HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY Bll -1 00 -0.5 0 0.5 1.00 VELOCITY, IN FEET PER SECOND Figure 10.—Sample relations of stage to velocity at section D showing effects of maximum stage and of range in stage. The samples include various maximum stages, 7.0, 7.3, 8.0 feet. Two samples are for the same maximum stage (8.0 feet) but illustrate different ranges of stage, 3.5 feet and 2.0 feet, respectively. charge on velocity at a given stage, the top of banks is also the level of maximum discharge when the high tide occurs at the stage between 8 and 8V2 feet. From an analysis of records of the Alexandria tide gage, a high tide of 8.5 feet is equaled or exceeded about 0.9 percent of the time, and 8.0 feet about 9 percent of the time. Nixon (1959) found that bankfull stage is equaled or exceeded 0.6 percent of the time in English rivers. These results are of the same order of magnitude. Whether this is coincidental or morphologically important cannot be stated in the present state of knowledge. It is a problem worth further work. It is hoped that further investigations on estuarine streams will consider the problem of the dominant discharge. HYDRAULIC CHARACTERISTICS AT A CROSS SECTION In terrestrial rivers, changes in width, depth, and velocity at any cross section accompany a change in discharge. These variables are related to discharge as simple power functions expressed by Leopold and Maddock (1953) in the following form: ic=aQb d=cQf v=kQm where Q is discharge, w, d, and v represent the water-surface width, mean depth, and mean velocity, respectively. The coefficients, a, c, and k, are constants, and b, /, and m are exponents representative of the section.B12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Si Figure 11.—Stage and velocity as functions of time at section D. High tide during the three cycles shown was below, at, and above bankfull stage, 7.2 feet. Hydraulic geometry is a phrase coined to express these and collateral equations at-a-station and in a downstream direction. In the present study, similar curves were constructed for a tidal estuary. It has been previously stated in the discussion of the velocity-stage curve that an unlimited number of velocities are possible at any stage for a given cross section. The effect on at-a-station curves of range in stage and of high tide at one station or section is not large compared with differences between channels in different areas. Figure 13 presents data for three flood tides in the study area. The figure also includes data from Gilbert’s study (1917) and some data collected by the authors in Old Mill Creek, Del. For Wrecked Recorder Creek, figure 13 shows the changes in width, depth, and mean velocity as functions of discharge in flood tide only at section D. Other sections would have similar curves, and the ebbing limb of a tide would differ principally in the direction of the hysteresis loop.HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B13 ____i__i_____i___i ______i_____i_____i_____i_______i____i____i__i_____ 0.1 1 10 30 50 70 90 99 99.9 TIME, IN PERCENT Figure 12.—Duration curves (cumulative frequency) of values of stage at high and low tide, Alexandria, Va., tide gage. Datum is the arbitrary datum used at Wrecked Recorder Creek. Figure 13, thus presents for Wrecked Recorder Creek two sets of curves; each set represents a different tidal cycle and is indicated separately by crosses and triangles. The results of three current-meter measure-, ments, represented by dots, are shown for comparison. Widths for stages above 7.2 feet (bankfull) were determined by a straight-line extension of a width-stage curve, and discharges were based on these widths. The set of curves indicated by a dashed line represent a flow below bankfull stage during the complete tidal cycle. The two tides give such similar results that the slopes for each set of curves are nearly identical. Each tidal cycle will give a separate set of curves, but the slopes are consistent. It can be visualized that in a flood limb of a tidal cycle, width and depth increase progressively from zero discharge through maximum to zero discharge. Velocity, on the other hand, begins with zero, reaches a maximum value at or near the stage of maximum discharge, and decreases to zero at the highest value of depth when flow reverses. Because figure 13 presents a flood tide only, the arrows show only one direction of progression around the hysteresis loop. The respective slopes of the lines representing relations of width, depth, and velocity to discharge were nearly identical on flood and ebb limbs of a tide. Thus, it is possible to discuss the relation of these slopes as exponents in power function equations, and to compare the exponents with those applicable to terrestrial rivers. For Wrecked Recorder Creek, values at given cross sections during flood tide of the exponent constants, i, /, andm, are: Increasing velocity Decreasing velocity Change in width__________________6=0.04 6=—0. 01 Change in depth__________________f— .175 /= — . 04 Change in velocity_______________m= .785 m— 1. 05 For an upland stream from Leopold and Maddock (1953, p. 26) the average values are: 6=0.26 /= .40 m= . 34 In the values for the tidal channel, the left-hand column is undoubtedly the more important from the standpoint of channel morphology. The increasing velocity is related to flow in the channel, whereas when the velocity decreases on a flood tide at least part of the discharge is usually governed by conditions of overflow rather than channel hydraulics. The small value for the exponent b in the estuarine channel was expected because of the relatively vertical banks prominent throughout the reach except at the bends. The median range in tide in the vicinity of the project area is 2.8 feet, although under extreme conditions a range of more than 4 feet may be expected. Increasing velocity occurs principally in the lower segment of the range in stage. As an example, on a high tide of 8.6 feet with a range in stage of 3.6 feet the maximum velocity occurs at a stage of 7.2 feet or 1.4 feet below the maximum stage. It is not known whether the value for the exponent / for Wrecked Recorder Creek is also representative of the numerous areas where a large range in stage occurs during each tidal cycle. Some data are available for comparison as discussed below, but more examples are needed. Old Mill Creek differed from Wrecked Recorder Creek by being longer and deeper. The former also drains some area of upland whereas the latter does not. Gilbert (1917, pi. 33) presented a graph which included the velocity, discharge, and stage for Ravens-wood Slough on the southwest shore of San Francisco Bay. The range in stage for one of the flood tides illustrated was over 7 feet. From his published data, width at any given stage could be measured, and depthB14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS o o o o Ravenswood Slough; southwest shore of San Francisco Bay, Calif. (Gilbert, 1917) ® Flood flow Ebb flow Maximum stage 12.6 feet; range in stage 7.0+ feet Old Mill Creek near Lewes, Del. (Author’s data) 0 Flood flow Wrecked Recorder Creek, Section D Flood flow Maximum stage 8.0 feet; range in stage 3.5 feet Flood flow Maximum stage 7.0 feet, range in stage 2.0 feet Flood flow Actual measurements of maximum sfage 6.85 feet o.oi DISCHARGE, IN CUBIC FEET PER SECOND Figure 13.—Changes at-a-station of width, depth, and velocity as functions of discharge. Data for Wrecked Recorder Creek were collected at section D. Also included are observations at Old Mill Creek estuary, near Lewes, Del., and Ravenswood Slough, San Francisco Bay, Calif.HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B15 versus discharge computed from the width and velocity. The error in reading values of velocity, width, and discharge from his data is less than 10 percent. The three examples presented in this report are only a small sample of the numerous variations in tidal channels. Because of the vast difference in the three channels, it is surprising that the values of the exponents in the equations are as comparable as they are. The following tabulation summarizes the observed values of exponents in the equations for width, depth, and velocity, respectively, relative to discharge at a cross section, and on the rising velocity of a flood tide. Change of- width depth velocity 6 / nn Sec. D, Wrecked Recorder Creek _ — 0. 04 0. 18 0. 78 Old Mill Creek _ _ . 08 . 14 . 78 Ravenswood Slough . 14 . 08 . 78 Average. . 09 . 13 . 78 Average for terrestrial rivers . 26 . 40 . 34 As can be seen in figure 13, the curves for width and depth versus discharge have slightly negative slopes as discharge decreases from its maximum value to zero at high tide. Thus the values of the exponents b and / given above would be smaller rather than larger if averaged with those applicable to decreasing discharge on the floodtide. It must be concluded then that the principal difference in the at-a-station hydraulic characteristics of estuarine and terrestrial streams is that the former have a much more rapidly changing velocity with discharge than do terrestrial rivers. This is compensated by less rapidly changing depth and width with discharge. THEORETICAL ANALYSIS OF THE HYDRAULIC GEOMETRY IN THE DOWNSTREAM DIRECTION By Walter B. Langbein Pillsbury (1939, p. 228-230) defined an ideal estuary as one in which the tidal range, depth, and current are uniform through the length of the channel. By analysis of the motion of water in the channel as induced by the tides at the mouth and as retarded by friction, Pillsbury showed that in such an estuary the width decreases exponentially with the distance up the estuary; thus w'=w0e~nx cot'1’ where w=width at distance x upstream from the mouth where width is w„, e is base for natural logarithms,

where is the angular lag of the primary current behind the hydraulic slope. The value of <£ is itself a function of depth and channel friction, and is evaluated by tables given by Pillsbury (1939, p. 123). Using a Chezy coefficient of 80 and mean depth of cross sections ranging from 2 to 10 feet, the value of the amplitude of the current is calculated to be U=0.35 AD116 Velocities during the tidal cycle will vary sinusoidally in proportion to the amplitude V. Mean velocities at different cross sections vary as the fifth root of the mean depths, therefore, m=// 5. Thus we have the three following hydraulic conditions : m+/+6=1.0 m=f/5 z=-f/2 There are four unknowns; the remaining statement will be supplied by the necessity that the estuary approach a state in which energy is as uniformly distributed as is consistent with the necessity that total work in the estuary as a whole be a minimum. The first condition is satisfied if velocity and depth are uniform; hence, m+f must approach zero. Since m and / are of the same sign, each of the parameters, m and /, also would approach zero. Since m+f+b = 1.0, (1 — 6) approaches zero. The second condition, that of minimum total work, is met as the integral V\°° Qsdx approaches zero. In the upland river, Q, the discharge, is the land drainage and is independent of the channel geometry, and slope is the only variable. The discharge in an estuary, however, is uniquely a function of its size and thus of its width, depth, and length, and so the problem is more complex. The relative work performed during a tidal cycle in an estuary is as follows: p_ «cot

Exponent of width, b Exponent of depth, / Exponent of velocity, to. Exponent of slope. 0. 71 . 24 . 05 -. 12 0. 77 . 23 . 00 0. 74 . 17 . 09 0. 50 . 40 . 10 -. 49 > Preliminary estimates by Alfred C. Redfield and Lincoln Hollister communicated by letter to Leopold, Mar, 30, 1962. 2 Leopold and Maddock (1963) and Leopold (1953). Langbein’s theoretically derived values of these exponents are compared below with the values determined from the present field investigation of Wrecked Recorder tidal stream. Also included are values from field data collected and analyzed by Redfield and Hollister at Barnstable Marsh, Mass. 0.5 m = 0.00 x • • - X 1 1 1 1 1111 „ x X 1 1 1 1 1 1 1 1 DISCHARGE, IN CUBIC FEET PER SECOND Figure 14.—Downstream or along-the-channel changes in width, idepth, and velocity as functions of discharge, at bankfull stage in Wrecked Recorder Creek. The data from Redfield and Hollister are particularly valuable because the channel studied ranged in width from 10 feet near the headwaters to 3,800 feet near the mouth. This is a much greater range in channel size than in our study and yet there is remarkable agreement of the results. The agreement between the field results of this investigation and the theory is very satisfactory. The field .data, however, are few and the scatter is large. More field data are needed. Both theory and field data show that in tidal estuaries, depth tends to be more conservative (a low value of /)B18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS than in upland rivers, so that the width-depth ratio varies rapidly downstream. At the mouth, a tidal estuary is wide and relatively shallow; at its head, it is narrow and relatively deep. The reason for this is that, in contrast to rivers, the discharge at any section is itself a dependent variable depending on how the flow shaped the channel in all the channel length between the point in question and the main bay or body of tidal water. In a terrestrial river, discharge is independent in that it is produced by the watershed and the channel is accommodated to it, rather than by such accommodation modifying the discharge itself. It is this influence of the channel on discharge that makes the rates of change of width, depth, and velocity along the channel different than in terrestrial rivers. REFERENCES Ahnert, Frank, 1960, Estuarine meanders in the Chesapeake Bay area: Geog. Rev., v. 50, no. 3, p. 390-401. Bradley, W. H., 1957, Physical and ecologic features of the Sagadahoc Bay Tidal Flat, Georgetown, Maine: Geol. Soc. America, Mem. 67, p. 641-682. Brush, L. M., Jr., 1961, Drainage basins, channels, and flow characteristics of selected streams in central Pennsylvania: U.S. Geol. Survey Prof. Paper 282-F, p. 145-180. Deevey, E. S., Jr., 1939, Studies on Connecticut lake sediments: Am. Jour. Sci., v. 237, no. 10, p. 691-724. Gilbert, G. K., 1917, Hydraulic mining debris in the Sierra Nevada: U.S. Geol. Survey Prof. Paper 105, 154 p. Horton, R. E., 1045, Erosional development of streams and their drainage basins—hydrophysical approach to quantitative morphology: Geol. Soc. America Bull., v. 56, p. 275-370. Leopold, L. B., 1953, Downstream changes of velocity in rivers: Am. Jour. Sci., v. 251, no. 8, p. 606-624. Leopold, L. B., and Langbein, \V. B., 1962, The concept of entropy in landscape evolution: U.S. Geol. Survey Prof. Pai>er 50O-A, p. 1-20. Leopold, L. B., and Maddock, Thomas, Jr., 1953. Hydraulic geometry of stream1 channels and some physiographic implications : U.S. Geol. Survey Prof. Paper 252, 57 p. Leopold, L. B., and Miller, J. P., 1956, Ephemeral streams— hydraulic factors and their relation to drainage net: U.S. Geol. Survey Prof. Paper 282-A, p. 1-37. Nixon, Marshall, 1959, A study of bankfull discharge of rivers in England and Wales: Inst. Civil Engineers Proc., v. 12, p. 157-174. Pillsbury, G. B., 1939, Tidal hydraulics: Washington, U.S. Army Corps of Engineers, Prof. Paper 34, 281 p. Wolman, M. G., and Leopold, L. B., 1957, River flood plains; some observations on their formation: U.S. Geol. Survey Prof. Paper 282-C, p. 87-109. Wolman, M. G., and Miller, J. P„ 1960, Magnitude and frequency of forces in geomorphic processes: Jour. Geology, v. 68, no. 1, p. 54-74. U.S. GOVERNMENT PRINTING 0FFICE;1963 O—675126ufturm ~ 7 DAY USE 't W WHICH BORROWFD■4* r 2 Drainage Density and Streamflow GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-C documents department] NDV“ 41963 LIBRARY UNIVERSW*-^; CAUWJWWDrainage Density and Streamflow By CHARLES W. CARLSTON PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-C UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402CONTENTS Page Abstract_______________________________________________ Cl Introduction_________________________________________ 1 Previous studies________________________________________ 2 Methods of study__________________________________________ 3 The Jacob water-table model________________________________ 4 Page Application of the Jacob model to landform and stream- flow characteristics___________________________________ C4 The relation of base flow to recharge and drainage density. 6 The relation of the mean annual flood to drainage density. Conclusions______________________________________________ References_______________________________________________ ILLUSTRATIONS Page Figure 1. Map showing location of drainage basins____________________________________________________________________ C2 2. Ground-water table base-flow model after Jacob____________________________________________________________ 4 3. Relation of drainage density to base flow and floods______________________________________________________ 6 hi / CO 00PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS DRAINAGE DENSITY AND STREAMFLOW By Charles W. Carlston ABSTRACT Drainage density, surface-water discharge, and ground-water movement are shown to be parts of a single physical system. A mathematical model for this system, which was developed by C. E. Jacob, can be expressed in the equation T-= WD~2j8h0i in which T is transmissibility, W is recharge, D is drainage density, and ho is the height of the water table at the water table divide. The rate of ground-water discharge into streams, or base flow (Qt), is dependent on and varies directly with transmissibility of the terrane. If W and ho are constant, Qi, oc D~2. Thirteen small, almost monolithologic basins, located in a climatic area in the eastern U.S. where recharge is nearly a constant, were found to have a relation of base-flow discharge to drainage density in the form of Qt per mi2=14D-2. The field relations, therefore, agree with that predicted by the Jacob model. If transmissibility controls the relative amount of precipitation which enters the ground as contrasted with that which flows off the surface, surface-water or overland discharge should vary inversely with transmissibility. It was found that flood runoff as measured by the mean annual flood (Q2.33) varies with drainage density in the form of Q2.33 per mi2 = 1.3D2. The close relation of mean annual flood to drainage density in 15 basins was not affected by large differences among the basins in relief, valley-side slope, stream slope, or amount and intensity of precipitation. It is concluded that drainage density is adjusted to the most efficient removal of flood runoff and that the mean annual flood intensity is due predominantly to terrane transmissibility. INTRODUCTION This report describes the results of an investigation of some of the relations between hydrology and geomorphology. The hydrology of a stream basin involves (1) the overland runoff from the basin, (2) the ground-water recharge rate, which is dependent upon the amount of precipitation in excess of evapotranspiration losses and upon the infiltration capacity of the soil mantle, and (3) the permeability of the bedrock, which affects the rate of yield of ground water to wells, to springs, and to the streams which drain the basin. Stream-flow characteristics were examined in terms of base flow and of height of flood peaks. The geomorphic character of drainage basins, following techniques first suggested by Horton (1945), may be measured or described in a variety of ways. For example, Strahler (1958, p. 282-283) has listed 37 such properties or parameters. In a general way these properties may be divided into four classes: (1) length or geometric properties of the drainage net, (2) shape or area of the drainage basin, (3) relation of the drainage net to the basin area, and (4) relief aspects of the basin. It has been assumed by hydrologists and geomorphologists that certain relations must exist between runoff characteristics and topographic or geomorphic characteristics of stream basins. Much effort has been devoted to statistical correlation of these relations without a clear understanding of how runoff is related to the geomorphic and geologic character of the basins. This paper advances the theory that runoff and drainage density are genetically and predictably related to the transmissibility of the underlying rock terranes. The purpose of this research was not to develop a means of predicting stream-flow characteristics from a landform characteristic. It was to attempt to solve a complex geohydrologic problem by defining a physical system in terms of: (1) the elements of the system, (2) how it operates, and (3) why it operates in the observed fashion. The theory resulting from this type of analysis, if valid, illustrates the basic simplicity and rationality of natural processes. During the progress of research, much effort was devoted to statistical analysis of the data. While very good correlation coefficients (0.96 to 0.98) were obtained the procedure shed no light on the physical cause and effects of the processes. The present theory, formed by inductive and deductive reasoning, is so basically simple that the correlation can be shown adequately in simple graph form and equations can be solved directly by graphical means. The author appreciates Dr. Luna B. Leopold’s penetrating and enlightening critical comments on earlier drafts of this paper. Mr. Frederick Sower, chemist and mathematician, greatly aided the author in the development of the equations. Cl 689426—61 ■2C2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS PREVIOUS STUDIES Early studies of the relation of topographic character of drainage basins to basin-runoff characteristics have been described by Langbein (1947, p. 128-129). The topographic parameters considered important in these studies were: area, channel slope, stream pattern, average basin width, mean length of travel (channel length), and mean relief measured from gaging station. In 1939 and 1940, Langbein (1947) and his coworkers, through assistance from the Works Project Administration of the Federal Works Agency, compiled a large variety of topographic measurements from 340 drainage basins in the northeastern United States. The basins ranged in area from 1.6 to 7,797 square miles. The parameters determined were drainage area, length of streams, drainage density, land slope, channel slope, area-altitude distribution, and area of water bodies. No attempt was made to correlate these topographic measurements with streamflow properties. The measurements were compiled for future studies which would determine their effects on the runoff of streams. Such studies have been carried on in the Water Resources Division of the U.S. Geological Survey by Benson (1959), who participated in the studies reported by Langbein in 1947. As a result of graphical correlation of several topographic parameters with flood flow for 90 New England drainage basins, Benson determined that basin area and channel slope were the most effective determinants of flood flow. These parameters were then statistically computed in correlation with records of 170 gaging stations. Benson’s studies were made entirely of streams in New England where the bedrock lithology, with the exception of the Triassic rocks of the Connecticut Figure 1.—Map showing location of drainage basins studied and discussed in this report. Uncircled numbers and letters correspond to those in table and on graphs.DRAINAGE DENSITY AND STREAMFLOW C3 basin and the Carboniferous rocks of the southeastern New England region, is largely a hydrologically homogeneous complex of igneous and metamorphic rocks. Glacial erosion has modified the land surface of the crystalline rocks, and glacial deposits mantle much of the area. Drainage density among the basins studied ranged from 1.1 to 2.35, as measured on 1: 62,500 scale maps by Langbein. METHODS OF STUDY The area of study comprises the central and eastern United States south of the Wisconsin glacial border. This area was chosen because the landscape is the product of normal denudational processes; landscapes shaped or modified by glacial erosion or deposition, such as those studied by Benson (1959) were thus excluded. The area is generally homogeneous in climate. Average annual precipitation ranges from about 40 to 50 inches and average annual temperature ranges from 50° to 60° F. Four criteria were used in selecting basins having a variety of bedrock geology: (1) homogeneity of bedrock, (2) availability of 1:24,000-scale topographic maps, (3) availability of discharge records, (4) lack of appreciable flow regulation of the streams. These limitations resulted in an unavoidably small sample. The basins, their geology, and their hydrologic and geomorphic parameters are listed in the following table, and their locations are shown in figure 1. Drainage densities were obtained by sampling randomly distributed 1-square-mile plots, whose total area is generally not less than 25 percent of the total basin area. Drainage density is a measurement of the sum of the channel lengths per unit area. It is generally expressed in terms of miles of channel per square mile. Horton (1945, pp. 283-284) and Langbein (1947, p. 133) determined drainage density by measurement of the blue streamlines on topographic maps having a scale of 1:62,500. Langbein (p. 133) pointed out that the number of small headwater streams shown on these maps would vary with the season and wetness of the year in which the survey was made, as well as with the judgment of the topographer and cartographer. The current practice, and the method used in this study, is to show drainage lines wherever a cusp or V-notch of a contour line indicates a channel. By this method, the drainage densities of the basins, on topographic maps having a scale of 1:24,000, range from 3.0 to 9.5. The measurements are believed to be of consistent accuracy because the maps used were all 1:24,000 scale maps and the measurements were made by one operator, the author. Delineation of drainage lines was generally conservative; therefore, drainage-density values, particularly in the higher values, may be lower than those ordinarily measured. An accurate method of distinguishing the ground-water discharge component of streamflow from the surface- or overland-flow component has not yet been developed. Estimates of ground-water discharge of streams, or base flow, are usually made by assuming that all flood peaks are overland-flow discharge and 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12 13 A_ B_ Locality (fig. 1) Basin Gaging station Bedrock lithology Area (sq mi) Drainage density Average minimum monthly flow (CIS per sq mi) Mean annual flood (cfs per sq mi) Mean annual precipita- tion (inches) Sawpit Run near Oldtown, Md__ Shale 5 7. 6 0. 13 62 38 Little Tonoloway Creek near Shale and shale interbedded 16. 9 7. 0 . 29 44 38 Hancock, Md. South Fork Little Barren River with sandstone. Limestone and cherty limestone. 18. 1 9. 5 . 26 77 49 near Edmonton, Ky. Crabtree Creek near Swanton, Sandstone (60-80 percent) and 16. 7 4. 2 . 61 32 42 Md. Totopotomoy Creek near Atlee, some shale. Sand and gravel 6. 0 4. 7 . 35 23 45 Va. Sawmill Creek near Glen Burnie, Sand and clav . . . 5. 1 3. 0 1. 22 19 45 Md. Tuscarora Creek near Martins- Limestone 11. 3 3. 6 . 80 14 40 burg, W. Va. John’s Creek near Meta, Ky __ Sandstone, shale, limestone, and 55. 7 6. 3 . 27 44 44 Fourpole Creek near Hunting- coal. Sandstone, shale, limestone, and 20. 9 8. 0 . 10 86 43 ton, W. Va. East Fork Deep River near High coal. Schist, slate, and greenstone 14. 7 8. 0 . 36 102 46 Point, N.C. Piney Run near Sykesville, Md_ Granite. 11. 4 6. 0 . 84 75 45 Ridley Creek near Moyland, Pa_ Gabbro and schist. ..... 31. 9 5. 0 . 83 38 47 Beetree Creek near Swananoa, Gneiss . — 5. 5 5. 6 . 91 44 49 N.C. Crab Creek near Penrose, N.C.. Granite 10. 9 7. 4 1. 86 72 62 Catheys Creek near Brevard, Gneiss . . .. — . 11. 7 8. 0 2. 45 67 63 N.C. C4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS that the remaining lower flow segments of the hydrograph are base flow. Many surface-water hydrologists believe that the base flow of many streams is largely derived from bank storage, rather than from discharge from the ground water. On the other hand, there are large areas of highly permeable sand formations in the Atlantic Coastal Plain where there is never any discernible overland-flow runoff and where virtually all discharge, including flood peaks, is ground-water discharge. Because of this uncertainty in the definition and computation of base-flow or ground-water discharge, the author used as a measure of base flow average minimum monthly flow, computed by averaging 6 years of monthly minimum discharge measurements. The mean annual flood is determined by plotting annual maximum-flood discharges on modified probability graph paper in which the discharge peaks are plotted against recurrence interval. Gumbel (1958, p. 177) has pointed out that the return or recurrence interval for the mean annual flood is 2.2328 years, the median annual flood is 2 years and the most probable annual flood is 1.582 years. The U.S. Geological Survey has adopted the mean annual flood (Q2.33) as its standard reference value. According to Wolman and Miller (1960, p. 60) most of the work of stream erosion is done by frequent flows of moderate magnitude with a recurrence interval of 1 to 2 years. The use of the Q2.33 flood in the present correlation study is, therefore, generally in scale with the magnitude and frequency of floods which Wolman and Miller have demonstrated to be most important in geomorphic processes. THE JACOB WATER-TABLE MODEL Models depicting the relation of the ground-water table to ground-water drainage have been made by Horton (1936) and Jacob (1943, 1944). Horton (1936, p. 346) considered the shape of the water table as a parabola. He stated that the elevation of the water table (h) at any point X distance from the draining stream is: (1) In this equation, h0 is the elevation of the draining stream, Kt is the transmission capacity of the soil, L0 is the distance from the stream to the ground-water divide, and a is the rate of accretion of recharge to the aquifer, with a being equal to or less than the infiltration capacity. In 1943 and 1944, Jacob developed a parabolic-type equation for ground-water flow in a homogeneous aquifer of large thickness and having uniform accretion from precipitation (1943, p. 566). The model for this equation is shown in figure 2. In a simplified form the equation states: Figure 2.—Ground-water table base-flow model after Jacob (1943). (2) where h0 is the height of the water table above the draining stream, as measured at the ground-water divide, a is the distance from the water-table divide to the stream, W is the rate of accretion to the water table (recharge), and T is the transmissibility (the volume rate of flow through a vertical strip of the aquifer of unit width under a unit hydraulic gradient). Thus, according to equation 2, the height of the water table at the water-table divide is proportional to the square of the distance of the divide from the drainage stream and to the rate of ground-water recharge. The value h0 is also inversely proportional to transmissibility. APPLICATION OF THE JACOB MODEL TO LANDFORM AND STREAMFLOW CHARACTERISTICS Jacob (1944, p. 938-939) computed the transmissibility of the Magothy sand aquifer of Long Island from a water-table contour map of the island and estimated recharge as about 60 percent of the average annual precipitation. He used the following equation for his computation: aW 2 Aii (3) The Jacob ground-water model can be related to a landform model by assuming that the land surface is also a parabola coincident with the water table. Then, the symbol a is equal to L0 or length of overland flow. Thus: (4) Horton (1945, p. 284) pointed out that the average length of overland flow (Z0) is, in most cases, approximately half the average distance between the streamDRAINAGE DENSITY AND STREAMFLOW C5 channels and is therefore approximately equal to half the reciprocal of drainage density (D), or (5) Substituting equation 5 in equation 4 the equation becomes: T- W (6) SD'% or T- WD~2 (7) 00 Jacob’s water-table model as used on long Island, simulates an aquifer having parallel boundaries at a constant head (the sea level on both sides of the island). Jacob pointed out, however, that a serious weakness of the Long Island study was the impracticability of measuring the ground-water discharge along the northern and southern shores of the island. He suggested (1944, p. 939) that the model could be tested more effectively under field conditions where the parallel shores would be replaced by streams draining the water discharged from the water table aquifer. This model is shown in figure 2. The ground-water discharge into such streams would be dependent upon the transmis-sibility of the aquifer, and this discharge could be measured by the gain in flow per unit length of stream. Inasmuch as ground-water discharge into streams would vary directly with the transmissibility, ground-water discharge or base flow (Qb) should also vary according to equation 7 or: Qbcc WD- 8 h0 (8) W and h0 remaining constant, base flow should be related to drainage density in the form: QbccD (9) It may be deduced that as transmissibility decreases, the amount or rate of movement of ground water passing through the system decreases and a proportionately greater percentage of the precipitation is forced to flow directly into the streams in flow over the land surface. Streamflow, therefore, would be derived from ground-water discharge plus overland flow. Both components of discharge should vary inversely in their relative contribution to stream discharge in a regular and predictable system controlled by the transmissibility of the water-table aquifer. As T increases, ground-water discharge into streams increases and surface discharge decreases. As T decreases, there would be a corresponding decrease in base flow and increase in surface discharge. Equation 6, which was derived from Jacob’s basic equation, states that transmissibility varies inversely with drainage density squared. Thus as transmissibility increases, drainage density would decrease, and as transmissibility decreases, drainage density would increase. This relationship may be examined from two different viewpoints. Horton (1945, p. 320) has stated that erosion will not take place on a slope until the available eroding force exceeds the resistance of the surficial materials to erosion. This eroding force increases downslope from the watershed line to the point where the eroding force becomes equal to the resistance to erosion. He named this distance the “critical distance,” and the belt of land surface within the critical distance was termed the “belt of no erosion.” One of the most important factors in determining the width of the belt of no erosion is the infiltration capacity of the soil. Simply stated, the greater the infiltration capacity, the less will be the amount of surface runoff. As infiltration capacity increases, the critical distance also increases because a greater slope length is required to accumulate overland flow of sufficient depth and velocity to start erosion. Thus the infiltration capacity is chiefly responsible for determining the width of the belt of no erosion and the width of the spacing of streams or channels which carry away surface runoff. The infiltration capacity of the soil may be regarded as one part of the general capacity of a terrane to receive infiltering precipitation and to transmit it by unsaturated flow through the vadose zone above the water table and by saturated flow through the aquifer to the streams draining the ground water. This capacity is here termed “terrane transmissibility.” It is recognized that this extends the strict meaning of transmissibility, which is a measure only of saturated flow. Another way of regarding the relation of transmissibility to drainage density is to consider that as T decreases, a concurrent progressive increase in surface flow will occur. Increase in proliferation of drainage channels would provide a more efficient means of transporting the water off the land, and as the water to be so removed increases (with decreasing T) the proliferation and closeness of spacing of the drainage channels would increase. The channel spacing or geometry should operate at peak efficiency during periods of flood runoff. Since flood runoff (Qr) of streams varies in magnitude inversely with the magnitude of their base flow runoff (Qb), and according to equation 9 QbccD~2, then flood runoffC6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS should vary with the positive second power of drainage density, or QJccD2. (10) Chorley and Morgan (1962) have compared the morphometry of two areas of crystalline and meta-morphic rocks: the Unaka Mountains of Tennessee and North Carolina, and Dartmoor, England. Morphometric characteristics which were measured included number of stream segments, mean stream lengths, mean basin areas, mean stream-channel slopes, mean relief of fourth-order basins, mean basin slopes, and mean drainage density. Relief and drainage density were the only parameters which showed significant differences between the two areas. The difference in drainage density between the two areas (Dartmoor, D=3.4; Unakas, D= 11.2) was ascribed to differences in runoff intensity caused by more intense rainfall and greater mean basin slopes. They concluded that the channel system is geared to conditions of maximum runoff. These conclusions are only partially in agreement both with the theoretical and observed relations described in the present paper. THE RELATION OF BASE FLOW TO RECHARGE AND DRAINAGE DENSITY Figure 3 A shows the relation of drainage density to base flow per square mile for 13 basins previously described. As previously stated, average annual precipitation in the 13 basin areas ranges from about 40 to 50 inches and average temperatures range from 50° to 60° F. Recharge (W) is dependent upon the amount of precipitation less the amount of evapotranspiration losses, which is dependent largely on temperature. The variations in rainfall and temperature among the 13 basins are within a sufficiently small range that recharge can be considered to be approximately a constant for the 13 basins. The relation of base flow per unit area to drainage density can be delineated by a regression line which has a slope and intercept such that: Qb per mi2= 14D-2 Equation 8 states that base-flow rate is also directly proportional to recharge. Two basins at the southern tip of the Blue Ridge province were examined to determine qualitatively the effect of significantly higher precipitation or recharge on base flow. In this area annual precipitation is more than 60 inches. The two basins are those of Crab Creek near Penrose, N.C. {A), and Catheys Creek near Brevard, N.C. (B). (See figs. 1, 3.) Their position on the diagram in figure 3A shows a much higher rate of base flow which may be due to the higher recharge rate. The gaging stations for both basins are, however, located in alluvium-filled A B Figure 3.—The relation of drainage density to base flow and floods. A, Base flow (Qfc). B, Mean annual flood (Qj.33). valleys; therefore, the higher base flow discharges may be due in part to higher transmissibility of the alluvium. The relation of drainage density to the 10-day base flow recession coefficient (r10) was also investigated. It was found that this relation could be expressed quantitatively by the equation: Log r10= —0.008D2. THE RELATION OF THE MEAN ANNUAL FLOOD TO DRAINAGE DENSITY In the preceding discussion of the theoretical relation of drainage density to flood runoff it was concluded that flood runoff should vary directly with the second power of drainage density. Flood runoff for the basins studied was computed in terms of the mean annual flood per square mile (Q2.33 per mi2). A regression line having a slope of D2 was plotted on this graph and the best fit gave an intercept with unit drainage density of about 1.3. This gives the equation: Q2.33 per mi2=1.3D2. The departures from this mean trend line were compared graphically with average annual precipitationDRAINAGE DENSITY AND STREAMFLOW C7 and average maximum rainfall intensities as given by the U.S. Weather Bureau. There was no apparent correlation. Graphical correlation was also made between mean annual flood and total relief, local relief, the ruggedness number (the product of relief and drainage density), stream slope, and valley-side slope. If significant correlation of flood peaks with these parameters exists, it is not apparent in the graphical plots. The apparent lack of correlation between rainfall amount or intensity and flood intensity and between relief and flood intensity may be briefly illustrated by reference to points plotted in figure 3B. Basins A and B are located in the region of highest mean annual rainfall and rainfall intensity in the eastern U.S. In addition, the relief (1,500 ft) in these basins places them among those with the highest relief in the total sample, yet they plot on and below the average for the basins. Basin 10 has the highest mean annual flood (102 cfs per mi2), but its relief is only 200 feet. Basin 13 has a relief of 2,600 feet and a high average annual precipitation (49 in.), but departure of its mean flood intensity from the average is not significantly high. Hydrologists have found that the delay time (and hence the attenuation) of flood peaks is composed of two parts; the inlet or overland-flow tune, and the channel-transit time. The present study deals with inlet times of monolithologic terranes. The writer’s analysis of flood runoff per unit area at gaging stations in the Appalachian Plateaus of eastern Kentucky (a basically monolithologic terrane) suggests that inlet time is dominant over channel transit time up to about 75 to 100 square miles in drainage area; however, because the channel transit time increases continuously, for larger basins it tends to become the dominant component of the flood-peak lag. CONCLUSIONS This paper has presented evidence that drainage density, surface-water runoff, and the movement of ground water are parts of a single hydrologic system controlled by the transmissibility of the bedrock and its overlying soil mantle. A mathematical model of such a system, constructed by Jacob (1943; 1944), has been adapted to show that transmissibility (T) is related to ground-water recharge (W), to drainage density (D) and to the height of the water table at the water table divide (ho). The equation for this relation is: rp WD-2 8A0 If W and h0 are constant, the equation may be simplified to T—KD~2. Inasmuch as the rate of base flow (Q„) or ground-water discharge into streams varies with and is controlled by transmissibility, QbocD~2. A total of 13 small, basically monolithologic stream basins were selected in the eastern United States where rainfall and temperature are such that recharge can be considered to be a constant. It was found that the relation of base flow of the 13 streams to drainage density can be expressed by the equation Q„ per mi2= 14D-2. The observed relation therefore is the same as that predicted by the Jacob model. Two stream basins in the southern Blue Ridge province, where rainfall is much higher than that of the 13 basins, have much higher base flows than comparable streams in the lower rainfall region. Flood runoff as measured by mean annual flood (Q2.33) was found to be closely related to drainage density; the equation may be expressed as Q2.33 per mi2= 1.3D2. It is concluded that the terrane transmissibility controls the amount of precipitation which passes through the underground system. The rejected or surface-water component increases with decreasing transmissibility. As surface-water runoff increases, an increase in the proliferation of stream channels is required for efficient removal of the runoff. The close relation of drainage density to mean annual flood per unit area indicates that the drainage network is adjusted to the mean flood runoff. Among the 15 basins in which flood runoff was correlated with drainage density, there are large and significant differences in relief, in valley-side and stream slopes, and in amounts and intensities of precipitation. These factors, however, have no discernible effect on the relation of the magnitude of the floods to drainage density. Transmissibility of the terrane appears to be the dominant factor in controlling the scale of drainage density and the magnitude of the mean annual flood for basins up to 75 to 100 square miles in area. The research reported here should be of interest to geomorphologists m that it provides a quantitative physical model for the origin of one of the most important elements of landform characteristics, drainage density. In addition, it appears that surface-water hydrologists and ground-water hydrologists have far more in common in their hydrologic studies than is generally realized. Transmissibility or permeability is the most important aquifer characteristic in ground-water studies. The results reported in this paper indicate that the transmissibility or permeability of terranes drained by streams is also important in the study of surface-water hydrology. It is hoped that this study provides a theoretical physical basis for interdisciplinary hydrologic studies.C8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS REFERENCES Benson, M. A., 1959, Channel slope factor in flood frequency analysis: Am. Soc. Civil Eng. Proc., Jour. Hydraulics Div., v. 85, pt. 1, p. 1-9. Chorley, R. J., and Morgan, M. A., 1962, Comparison of morphometric features, Unaka Mountains, Tennessee and North Carolina, and Dartmoor, England: Geol. Soc. Am. Bull., v. 73, p. 17-34. Gumbel, E. J., 1958, Statistics of extremes: New York, Columbia Univ. Press, 347 p. Horton, R. E., 1936, Maximum ground-water levels: Am. Geo-phys. Union Trans., Pt. 2, p. 344-357. ------- 1945, Erosional development of streams and their drainage basins; hydrophysical approach to quantitative morphology: Geol. Soc. America Bull., p. 275-370. Jacob, C. E., 1943, Correlation of ground-water levels and precipitation on Long Island, N.Y: Pt. 1, Theory, p. 564-573. 1944, Pt. 2, Correlation of data: Am. Geophys. Union Trans., p. 928-939. Langbein, W. B., 1947, Topographic characteristics of drainage basins: U.S. Geol. Survey Water-Supply Paper 968-C, p. 125-157. Strahler, A. N., 1958, Dimensional analysis applied to fluvially eroded landforms: Geol. Soc. Am. Bull., v. 69, p. 279-300. Wolman, M. G., and Miller, J. P., 1960, Magnitude and frequency of forces in geomorphic processes: Jour. Geology, v. 68, p. 54-74. O 7 DAY USE 7 DAY \£7S jQ/ EARTH y SCIENf ES y* 2^ UBRAWT Channel Patterns and Terraces of the Loup Rivers in Nebraska GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-D Prepared as part of the program of the Department of the Interior for development of the Missouri River basinChannel Patterns and Terraces of the Loup Rivers in Nebraska By JAMES C. BRICE PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL Prepared as part of the program of the Department of the Interior for development of the Missouri River basin UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402CONTENTS Page Abstract..____________________________________________ D1 Introduction_______________________________________________ 1 Pleistocene and Recent deposits________________________ 2 North Loup River valley near Elba______________________ 3 North Loup River valley near Burwell___________________ 5 North Loup River valley downstream from Brewster. 5 North Loup River valley upstream from Brewster__ 6 South Loup River valley near Cumro.................... 8 Summary of Pleistocene and Recent geologic history. 8 Geologic influences on the longitudinal river profiles_ 9 Influence of rock resistance........................ 10 Influence of regional slope and relief_____________ 12 The Middle Loup River at Dunning__________________________ 14 Hydraulic factors in relation to width________________ 14 Hydraulic factors in relation to braiding............. 15 Hydraulic factors in relation to a confluence_________ 21 The Middle Loup River at Dunning—Continued Page Summary of hydraulic factors in relation to channel pattern__________________________________________ D21 Observations on the formation of sandbars, islands, and valley flats__________________________________________ 23 Measurement and description of channel patterns_________ 25 Meandering and sinuosity____________________________ 25 Braiding____________________:___________________ 27 Width............................................... 31 Channel patterns of the Loup Rivers.._______________ 31 Relation of selected hydraulic factors to channel pattern. 32 Bank erodibility________________________________ 32 Discharge____r,_________________________________ 35 Bed material and suspended-sediment concentration. 37 Water temperature________________________________ 38 Longitudinal river slope and valley slope...._______ 39 References cited................................... 40 ILLUSTRATIONS Page Figure 1. Map of Loup River drainage basin___________________________________________......------------------------------- D2 2. General topographic and stratigraphic relations of valley fills in the North, South, and Middle Loup River valleys. 3 3. Generalized geologic section along northwest side of Coopers Canyon near Elba.----------- —--------------- 4 4. Cross section in Coopers Canyon near Elba. _____________________________.........-------------------------- 4 5. Geologic section of north side of North Loup River valley near Burwell____________________________________ 5 6. Stratigraphic diagram of valley fill of the Elba terrace and valley fill of Peorian age exposed in cut bank of North Loup River about 5 miles downstream from Brewster..._____________...---------------------------------------------- 6 7. Cross section of North Loup River valley 6 miles upstream from Brewster----------------------------------- 7 8. Longitudinal river profile, valley profile, and Elba terrace profile, North Loup River.---------— -------- 10 9. Longitudinal river profile and valley profile, Middle Loup River.....------------------------------------- 11 10. Longitudinal river profile and valley profile, South Loup River______________ —---------------------------- 12 11. Location of sections on a reach of the Middle Loup and Dismal Rivers near Dunning_________________________ 14 12. Cross profiles of Middle Loup River at Dunning------------------------------------------------------------ 14 13. Mean depth, mean velocity, local slope, and effective area in relation to effective width at sections B, C2, E, and A, Middle Loup River at Dunning__________________________________________________________________________ 16 14. Cross profiles of section A, Middle Loup River at Dunning___________________________________________________ 18 15. Maximum depth, mean velocity, local slope, and effective area in relation to water width at half mean depth, sec- tion A, Middle Loup River at Dunning_____________________________________________________________________ 19 16. Effective area and local slope in relation to effective width for sections upstream and downstream from con- fluence of Middle Loup and Dismal Rivers_________________________________________________________________ 22 17. Aerial photograph and tracing of photograph, North Loup River, about 1 mile downstream from gaging station near St. Paul____________________________________________________________________________________________ 23 18. Small lobe-shaped dunes on submersed sandbar in North Loup River 8 miles downstream from Brewster----------------- 24 19. Straight reach of North Loup River, same locality as figure 18..------------------------------------------------ 24 20. Delineation of meander belt, axis of meander belt, and channel arcs for measurement of sinuosity and symmetry. . 25 21. Delineation of a reach of the Buyuk Menderes River, Turkey, for measurement of sinuosity and symmetry----- 26 22. Reaches that illustrate different values of the braiding index. T__________.-------------------------------- 29 23. Reaches that illustrate different values of the sinuosity index.________________________________________________ 30 24. Longitudinal river profile and valley profile, Calamus River..____________________________________________ 34 25. Diagrammatic relation of sinuosity index to width and bank erodibility, Calamus River----------------------------- 34 26. Width in relation to mean annual discharge and average maximum daily discharge for different channel pat- terns____________________________________________________________________________________________________ 36 27. Suspended-sediment concentration in relation to temperature and discharge, sections A, E, and B, Middle Loup River at Dunning_____________________________________....._______________________________________________ 38 28. Velocity in relation to temperature and discharge, sections A and E, Middle Loup River at Dunning------— 39 29. Reach slope in relation to the average maximum daily discharges for meandering and braided reaches.------- 40 mIV CONTENTS TABLES Tables 1-2. Values of hydraulic factors at— Pa*« 1. Section A, Middle Loup River at Dunning, Nebr-------------------------------------------------------- D20 2. Gaging stations in the Loup River basin-------------------------------------------------------------- 28PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA By James C. Brice ABSTRACT The North, South, and Middle Loup Rivers drain an area of about 12,000 square miles in central Nebraska. The North and Middle Loup, which rise in the Sandhills, have notably constant discharges because much of their flow is derived from ground water. Moderate sinuosity is characteristic of the South Loup whereas the North and Middle Loup have low sinuosity (except in their upper courses) and tend to be braided. The rivers are bordered by well-defined terraces which are described and interpreted historically as a background for explaining the morphology of the modern rivers. Valley-fill deposits ranging in age from Kansan to Recent are exposed along the rivers, and the fill underlying the youngest major terrace is assigned an age of about 10,000 years on the basis of three carbon-14 age determinations. The rivers are probably slowly degrading, as the lowest terraces are cut on older fill. Along the Middle Loup near Dunning, a decrease in channel width at constant discharge is accompanied by a decrease in effective area and local slope and by an increase in mean depth and mean velocity. In a braided section, bars submersed to a depth equal to or less than half mean depth arf considered equivalent to emersed bars, and the width of the braided section is expressed as the width of flowing water at half mean depth. The effect of braiding at a wide section is to decrease the width of flowing water, and the changes in hydraulic variables that accompany braiding are similar to the changes that accompany flow from a wide single channel to a more narrow single channel. The long river profiles are less steep than the long profile of the Great Plains, owing mainly to the sinuosity of the rivers. The Great Plains profile is nearly straight, and the river profiles are straight for most of their length. The slopes of meandering reaches generally are less steep than the slopes of braided reaches of similar discharge, but local slope correlates more closely with the width of flowing water than with any other aspect of channel pattern. The slope of the long river profiles shows little or no change at the confluence of major tributaries, and it does not show any significant correlation with downstream increase in discharge, which is evidently accommodated mainly by changes in cross section. Slope is regarded as a partially independent variable. Channel patterns are quantitatively described, for geomor-phic purposes, by a braiding index and a sinuosity index. The greater sinuosity of the South Loup, in comparison with the lower courses of the North Loup and Middle Loup, is attributed to the higher concentration of silt and clay in its suspended load. Probably, the silt and clay promotes sinuosity as it becomes incorporated in bank materials and increases the cohesiveness of banks. Within the Sandhills, the North Loup, Middle Loup, and their tributaries tend to be sinuous despite a very low concentration of silt and clay in the suspended load; however, swamp vegetation is common along the stream courses, and it probably promotes sinuosity by increasing bank resistance. In general, where banks are of sand and easily erodible, wide shallow channels tend to develop. Braiding takes place in wide shallow channels by the growth and stabilization of longitudinal sandbars, which probably originate from breaching of transverse bars and dunes. Sinuous reaches are generally narrower than braided or straight reaches of similar discharge. Width, although highly variable locally, increases downstream approximately as the 0.5 power of bank-full discharge. The downstream relation of width to discharge is affected by a downstream increase in variability of discharge. INTRODUCTION The Loup River basin (fig. 1) occupies all of central Nebraska. The part of the basin considered in this report has an area of about 12,000 square miles. Of this area, about 4,500 square miles is mantled with loess and contributes directly to surface runoff, and about 7,500 square miles is mantled with dune sand and does not contribute directly to surface runoff. (See fig. 1.) Annual rainfall averages about 20 inches in the eastern part of the basin and about 15 inches in the western part. The purposes of this report are to describe the channel patterns of the Loup Rivers quantitatively and to determine the factors that have had the greatest influence on the development of these patterns. Among the factors considered were the gradients and deposits of ancestral rivers. Terraces and valley-fill deposits of late Wisconsin and Recent age were studied, and the geologic history of the Loup Rivers was outlined. River widths and channel patterns were measured on aerial photographs, and long profiles wTere prepared from topographic maps. Detailed measurements on the Middle Loup and Dismal Rivers—made by the Geological Survey in connection with investigations of sediment transport—were analyzed and related to channel pattern. Fieldwork, which was done in the summer of 1958, consisted mainly of an investigation of terraces and terrace deposits, and the movement of dunes and bars in the river channels was also observed. DlD2 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA Rivers of the Loup basin are especially suitable for a study of channel patterns and other aspects of river morphology. Different reaches of the North and Middle Loup Rivers represent meandering, braided, straight, or sinuous channel patterns; and the South Loup River is meandering. Gaging stations are well distributed along the rivers (see fig. 1), and the period of record for most of these stations is adequately long. The North and Middle Loup Rivers have in their upper courses notably constant discharges because most of their flow is derived from ground water rather than from surface runoff. Reasonably complete data on suspended-sediment load are available, and investigations of total load have been made in a turbulence flume at Dunning. Topographic maps published between 1951 and 1955 are available for most of the basin; these have a scale of 1:24,000 and a contour interval of 10 feet. Other parts of the basin are covered on recently published topographic maps that have a scale of 1: 62,500 and a contour interval of 20 feet. Aerial photographs, taken in the 1930’s and again in the 1950’s, are available. This report was written under the direction of D. M. Culbertson, district engineer at Lincoln, Nebr. Three carbon-14 age determinations were made by Meyer Rubin. B. R. Colby, M. Gordon Wolman, and S. A. Schumm critically read the manuscript and offered many helpful suggestions. PLEISTOCENE AND RECENT DEPOSITS Pleistocene stratigraphic units that were distinguished in the Loup River basin are shown in figure 2. The basin is underlain by the Ogallala Formation (Pliocene), which crops out at widely scattered localities along the sides of the main valleys. The Ogallala is composed of alluvial gravels, sands, and silts, from which much of the Pleistocene valley fill has been derived. The oldest Pleistocene deposits observed during the present investigation are correlated with the Kansan Glaciation on the basis of their stratigraphic position beneath two buried soils, of which the upper was identified as the Sangamon and the lower as the Yarmouth. The Loveland Formation (Illinoian) on the upland consists of dune sand and loess which commonly are found in alternate layers. Outcrops of the Sangamon soil are common along the valleys of minor tributaries and along roadcuts in the upland. Along the main valleys, the Sangamon soil is developed on silt, sand, or gravel that is presumably of Illinoian age. 101° 400° 99° Figure 1.—Map of Loup River drainage basin.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D3 Peorian Loess of Wisconsin age mantles the southern half of the Loup River basin. The upland loess extends down the valley sides and merges with a contemporaneous valley fill of Peorian age which consists mostly of silt in minor tributary valleys and of silt, sand, and gravel in the main valleys. (See figs. 2, 3, 4.) The physiographic expression of this valley fill of Peorian age, here called the Kilgore terrace, has been mostly obliterated by erosion, but a few distinct remnants of the Kilgore terrace can be seen along the South Loup River near Cumro. The most conspicuous and extensive terrace in the main valleys stands 20-40 feet above river level, and many broad, flat remnants of this terrace occur along the lower 100 miles of the North and Middle Loup Rivers. This terrace is here called the Elba terrace because the town of Elba is built on its surface and because the underlying valley fill is well exposed in the walls of a canyon southeast of Elba. (See figs. 3, 4.) Carbon-14 age determinations indicate that deposition of the valley fill of the Elba terrace began about 10,000 years B.P. Several minor terrace deposits of late Recent age stand along the main streams ait heights ranging from a few feet to 15 feet above river level. NORTH LOUP RIVER VALLEY NEAR ELBA About 0.6 mile southeast of Elba, the Elba terrace is deeply trenched by an intermittent stream that descends from the upland and flows in a canyon northwestward across the terrace and into the North Loup River. (See fig. 3.) Most minor tributaries of the river, both intermittent and permanent, have cut shallow valleys into the terraces that border the river; in the canyon southeast of Elba is one of the few deep cross sections to be found. This canyon, although unnamed on the Elba quadrangle sheet, was called Coopers Canyon by Miller and Scott (1955), and this name is used here. The entrenchment of Coopers Canyon is perhaps explained by the fact that the canyon meets the river on the outside of a meander bend where the meander has made a steep cut into the Elba terrace about 30 feet above river level. Successive shifts of the river against the mouth of the intermittent stream probably formed a succession of headcuts that advanced rapidly upstream during times of heavy runoff and thus formed the canyon. Because the history of Coopers Canyon is not representative of stream history in the Loup River basin, the minor late Recent terrace deposits in the canyon have no regional significance. The valley fill exposed in the nearly vertical walls of Coopers Canyon was described in detail by Miller and Scott (1955). The thickness of exposed valley fill is about 40 feet, of which the lower 12-15 feet is white crossbedded sand and the upper 23-25 feet is silt. Within the silt are several humic zones, which Miller and Scott interpreted as soils, developed during different interglacial episodes of the Pleistocene. The lowest and most conspicuous of these humic zones was named by Miller and Scott the Coopers Canyon gley soil. They attributed this soil to weathering during Figure 2.—General topographic and stratigraphic relations of valley fills in the North, South, and Middle Loup River valleys. Dashed lines are inferred contacts.D4 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA Southwest Figure 3.—Generalized geologic section along northwest side of Coopers Canyon near Elba. Dotted line indicates Elba terrace profile just southeast of canyon. Dashed lines are inferred contacts. Northwest Southeast Figure 4.—Cross section in Coopers Canyon near Elba. Section is 1,100 feet upstream from bridge on State Route 11. the entire interval between the end of the Kansan Glaciation and the beginning of the Cary Stade of the Wisconsin Glaciation. In 1958 the Coopers Canyon gley soil was examined in detail by the writer at several places along the canyon walls, particularly on the northwest wall 100 feet upstream from the railroad bridge that crosses the canyon. Here the soil consists of a band of gray silty clay about 1 foot thick underlain by a band of calcareous silty sand in which snail shells are so abundant that they give the sand the appearance of a coquina. A collection of large and apparently unaltered snail shells from the calcareous silty sand yielded a carbon-14 age of 10,500±250 years (U.S. Geol. Survey lab. No. W-752). As the snail-bearing layer is only about 1 foot above the sand that forms the base of the exposure and as the crossbedding of the sand indicates rather rapid deposition, the sand probably is not much older than 10,000 years. Miller and Scott (1961, p. 1284) published a revision of their previous interpretation (1955) of stratigraphy in Coopers Canyon. They now consider all the alluvium and soils in the high terrace (here named the Elba terrace) as “* * * younger than the Two Creeks interstadial.” Along Coopers Canyon upstream from the point at which the canyon enters the upland, the Elba terrace is absent, probably because it has been removed by lateral erosion. The sides of the canyon are composed of Peorian Loess, of valley fill of Peorian age, and of Loveland Formation overlain by Peorian Loess; set against these sides are minor valley fills younger than the Elba. About 1,100 feet upstream from the bridge where State Route 11 crosses Coopers Canyon, the Sangamon soil is well exposed in the north side of Coopers Canyon; the top of the soil is about 15 feet above the canyon bottom, and at one end of the exposure the soil is truncated by valley fill of Peorian age. (See fig. 4.) The appearance of the Sangamon soil at this locality is typical of its general appearance throughout Nebraska, except that no Cca horizon has developed. The Sangamon soil that has developed in upland areas is commonly a soil complex (some of the horizons are repeated vertically); and thePHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D5 Sangamon soil that has developed on valley fills, like the Sangamon soil in Coopers Canyon, may represent only part of this complex. The Sangamon soil in Coopers Canyon consists of a dark-grayish-brown (10F7? 4/2 moist) humic zone 2 feet thick underlain by a zone of clay enrichment 6 feet thick. The parent Loveland Formation consists of light-yellowish-brown (lOYR 6/4 moist) silt underlain by sand of the same color. NORTH LOUP RIVER VALLEY NEAR BURWELL Pleistocene deposits ranging in age from Kansan to Wisconsin are well exposed in the sharply dissected valley side of the North Loup River 2 miles north of Burwell, Nebr. In the NE14NE14 sec. 11, T. 21 N., R. 16 W., the Sangamon soil forms a conspicuous band on a steep-sided ridge (see fig. 5); and ashy silts crop out at the base of this ridge. Although the Yarmouth soil does not crop out in this ridge, it is well exposed 700 feet west of the ridge; there it is developed on about 4 feet of ashy silt. The ashy silt is underlain by gravel which has been excavated for road building. Although the thickness of the Peorian Loess is only about 20 feet at this locality, it increases rapidly to about 70 feet a few hundred yards to the north. About 3 miles east of Burwell, the uplands north of North Loup River is trenched by Jones Canyon, which is steep sided and deep. In the upper reaches of this canyon, in the NE1^ sec. 9, T. 21 N., R. 15 W., a thickness of 90 feet of Peorian Loess is exposed; the total thickness of the Peorian here is even greater, as the base is not exposed. NORTH LOUP RIVER VALLEY DOWNSTREAM FROM BREWSTER The valley fill of the Elba terrace and the valley fill of Peorian age are well exposed in a deep cut on the outside of a meander bend of the North Loup River near the center of sec. 5, T. 22 N., R. 21 W., about 5 miles downstream from Brewster. (See fig. 6.) A peat bed within the valley fill of the Elba terrace abuts sharply against both sides of a residual ridge of valley fill of Peorian age, and a radiocarbon determination on a sample of the peat (U.S. Geol. Survey lab. No. W-755) yielded an age of 8,400±250 years B.P. The surface of the Elba terrace, which is locally buried by dune sand, stands about 20 feet above the water level of the river. A lower terrace surface, which stands about 10 feet above water level, has been cut on valley fill of the Elba terrace; the peat bed within the Elba extends continuously beneath this lower surface. The valley fill against which the valley fill of the Elba terrace abuts is 45 feet thick and consists of poorly sorted sand and gravel throughout; it is identified as valley fill of Peorian age only on the basis of stratigraphic and topographic position. Rapid deposition of this valley fill is indicated by the absence of buried humic zones, the absence of clay or peat beds, and the presence of large-scale crossbedding in the gravel beds. The following stratigraphic section of dune sand and valley fill of the Elba terrace applies to the cut-bank shown on the left side of figure 6, near the con- North Fir,the 5.—Geologic section of north side of North Loup River valley near Burwell. 719-579 0-64—2D6 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA tact between the Elba valley fill and valley fill of Peorian age: Measured section of Elba Terrace about 5 miles downstream from Brewster Depth (feet) Thickness (feet) Sand, fine-grained, white, aeolian Humic zone in silty, clayey sand; black 0-21 21. 0 (1017? 2/1 moist) __ Sand, medium- to fine-grained, yellowish- 21-22 1. 0 gray - 22-23 1. 0 Clay, sandy, medium-gray. Humic zone in sand; black (10yf? 2/1 23-23. 2 . 2 moist) Sand, light-gray; mottled by filled worm 23. 2-23. 7 . 5 burrows. Sand, medium- to fine-grained, yellowish- 23. 7-25.0 1. 3 gray Sand interbedded with peaty clay and sand. Unit grades horizontally into bed of peat 3 ft thick which contains 25-26 1. 0 abundant upright Equisetum stems Sand, medium-grained, white; contains 26-29 3. 0 laminae of black clay Sand and gravel, poorly sorted, cross- 29-34 5. 0 bedded (Water level of North Loup River 34-41 7.0 at 41 ft) NORTH LOUP RIVER VALLEY UPSTREAM FROM BREWSTER In the vicinity of Pitt Bridge, which crosses the North Loup River 6 miles upstream from Brewster, the Elba terrace forms an extensive flat about 30 feet above water level on the south side of the river, and a succession of lower terraces forms flats on the north side of the river. The most extensive of these lower terraces stands about 12 feet above water level, and other less extensive terraces stand at 4 feet and at 2 feet above water level. About 1,000 feet downstream from the bridge, a section across the 4-foot and the 12-foot terraces is exposed in a meander cut. A peat bed crops out near water level under the 4-foot terrace surface and extends continuously under the 12-foot terrace surface. (See fig. 7.) The 4-foot terrace was evidently cut on the fill that contains the peat bed. The age of 9,000±250 years B.P., obtained on a log from the peat bed (U.S. Geol. Survey lab. No. W-750), indicates that the fill on which both the 4-foot and the 12-foot terraces are cut is the valley fill of the Elba terrace. Where W-750 was collected, underlying the 12-foot terrace is as follows: the fill Measured section of fill underlying 12-foot terrace 6 miles upstream from Brewster Thickness Depth (feet) (feet) Surface soil, very pale brown (10 FT? 8/4 moist); weak humic zone developed on fine-grained sand Sand, fine-grained, white; mottled by numerous filled worm burrows in 0-1. 2 1. 2 upper part.. .. Humic zone, dark-gray (10Ff? 4/1 1. 2-3. 2 2. 0 moist); developed on fine-grained sand. Sand, medium- to fine-grained, white, crossbedded; contains streamers of small pebbles and a few thin layers of 3. 2-3. 5 . 3 dark clay Peat, dark-brown, rubbery; contains abundant Equisetum stems and a few 3. 5-9. 5 6. 0 small logs _ 9. 5-10. 5 1. 0 Sand, medium-gray, silty . 10. 5-11. 0 . 5 (Water level of river at 11 ft) East (downstream) West (upstream) Figure 6.—Stratigraphic diagram of valley fill of the Elba terrace and valley fill of Peorian age exposed in cut bank of North Loup River about 5 miles downstream from Brewster.100 0 100 200 300 400 FEET I i i i I_________I________I I________I Figure 7.—Cross section of North Loup River valley 6 miles upstream from Brewster. 0 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERSD8 CHANNEL PATTERNS AND TERRACES SOUTH LOUP RIVER VALLEY NEAR CUMRO An extensive terrace standing 30-40 feet above river level in many places along the South Loup River is correlated with the Elba terrace of the North Loup River on the basis of height, degree of dissection, and stratigraphic relation with older valley fills. The fill of the Elba terrace is well exposed along the South Loup River at several localities upstream and downstream from Cumro. Silt and fine sand are the main constituents of the fill; however, the proportion of these is not uniform from place to place, and no uniform gradation from coarse sediment at the base of the fill to fine sediment toward the top was observed. Along the South Loup River near Cumro, remnants of a terrace standing 60-80 feet above river level were observed. This terrace is here called the Kilgore terrace because sections of the terrace fill and remnants of the terrace surface can be seen along Kilgore Creek, which is about 3 miles downstream from Cumro. Where sections of the fill underlying the Kilgore terrace are exposed along the South Loup River, as at Satoria, the fill consists mainly of water-laid sand and silt. The fill underlying the Kilgore terrace is considered to be valley fill of Peorian age, contemporaneous with Peorian Loess on the upland, because the Kilgore terrace slopes up and merges with the upland without any break in topography. Moreover, the fill underlying the Kilgore terrace is the only fill observed that is older than the valley fill of the Elba terrace but younger than the Loveland Formation. Minor terraces at various elevations below the Elba terrace surface appear in the vicinity of Cumro and generally along the South Loup River. These minor terraces could not be correlated from place to place along the river, and their variability in height above river level suggests that they are cut terraces rather than fill terraces. Downcutting by the South Loup River is indicated by the presence of abandoned channels that stand at different heights above the present river level and below the level of the Elba terrace. SUMMARY OF PLEISTOCENE AND RECENT GEOLOGIC HISTORY The valleys of the Loup Rivers were originally incised into the Ogallala Formation prior to Kansan time, perhaps as early as the late Pliocene. Exposed Kansan deposits are sparse along the present valleys, and the rivers have probably shifted their courses greatly during the Pleistocene. The valley fill of Kansan age at Burwell consists mainly of gravel and sand, and it filled the bedrock valleys to a height of about 80 feet above present river level. The valley fill of Kansan age was probably incised before depo- OF THE LOUP RIVERS IN NEBRASKA sition of the valley fill of Illinoian age, the top of which stands about 60 feet above present river level; however, no stratigraphic evidence of such incision was found. After the development of the Sangamon soil, the main valleys were incised at least to the present river levels, and the upland was deeply gullied. Incision of the main valleys is indicated by the position of valley fill of Peorian age, which crops out near the level of the present rivers. At upland roadcuts, steep-sided hillocks cut in the Loveland Formation and buried by the Peorian Loess can be distinguished. The hillocks commonly are capped by the Sangamon soil, and on their buried slopes detached slabs of Sangamon soil are commonly incorporated in the base of the Peorian. The origin of these slabs is puzzling, but probably they were detached by erosion from near the top of the slope and then, after sliding some distance down-slope, were incorporated in the accumulating Peorian Loess. The Peorian Loess accumulated to a maximum depth of about 50 feet on the upland near Ord and to a depth of about 90 feet on the upland near Bur-well, and the valley fill of Peorian age accumulated in the main valleys to a height of 60-80 feet above the present river level. Scattered remnants of the surface of the valley fill of Peorian age are preserved, and this surface is here called the Kilgore terrace. Deposition of the Peorian Loess and valley fill was followed by deep incision in the main valleys, inasmuch as the surface of the next younger valley fill (that of the Elba terrace) stands about 30 feet below the surface of the Kilgore terrace and the thickness of the valley fill of the Elba terrace is at least 30 feet. This incision in the main valleys was accompanied by extensive gully and drainage development in the upland. Nearly all the present tributaries, except for gullies of late Recent age, are bordered by remnants of the Elba terrace and, therefore, existed before deposition of the valley fill of the Elba terrace. Deposition of the valley fill of the Elba terrace in the valleys of intermittent tributaries was accompanied by extensive grading of the valley sides, which slope smoothly down to the surface of the fill. Along the largest permanent streams, grading of valley sides is less notable, probably because these streams continued to steepen their valley sides by lateral plana-tion during deposition of the valley fill of the Elba terrace. Minor buried soils are common in the valley fill of the Elba terrace, particularly in the upper part; but the number, degree of development, and position of these soils vary from place to place. The soils, which have poorly developed profiles, probably formed during short intervals, of perhaps no more than a few hundred years each, when deposition of the valley fillPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D9 of the Elba terrace was temporarily halted. The lateral extent of each soil appears to be local rather than general; differences in soil characteristics over short lateral distances probably resulted from differences in drainage that existed during soil development. Close spacing of minor soil and humic zones in the upper part of the valley fill of the Elba terrace suggests slow and intermittent deposition of this upper part. Three carbon-14 age determinations on valley fill of the Elba terrace along the North Loup River indicate that deposition began somewhat earlier than 10,500 years B.P. and was still in progress 8,000 years B.P. Development of buried soils in the upper part of the fill and grading of valley-side slopes must have required a long period of time; hence, incision of the fill and dissection of the associated graded slopes are assigned a more recent date, perhaps 2,000 or 3,000 years B.P. Late Recent terrace deposits, ranging in height from a few feet to 15 feet above present river level, are attributed to lateral cutting rather than to filling because they are mainly underlain by valley fill of the Elba terrace. The Loup Rivers have been degrading their channels continuously during late Recent time, perhaps for the past 2,000 or 3,000 years, and are probably degrading their channels at present. Dune sand covers much of the Loup River basin; however, time did not permit a study of the dunes. H. T. U. Smith (1955) published a brief description of the development of dune forms in the Nebraska Sandhills. The general northwest-southeast trend of the dune ridges shows clearly on aerial photographs, as do the remarkable longitudinal furrows that run the full length of many dune ridges along their crests. The dune ridges are roughly parallel to the prevailing wind direction. The troughs between ridges and the furrows on top of the ridges are probably erosional in that both begin as rounded blowouts and progress downwind, killing the grass in front of them and leaving behind them a furrow, much as a finger leaves behind a furrow as it is pushed across the surface of sand. Mantling of the Peorian Loess by dune sand is conspicuous along the north side of the North Loup River near Burwell; therefore, the Sandhills must have been extended considerably in this vicinity since Peorian time. The Peorian Loess is, however, remarkably free of interbedded dune sand. GEOLOGIC INFLUENCES ON THE LONGITUDINAL RIVER PROFILES The slope and direction of flow of a river are initially determined by the geologic setting in which the river becomes established. Geologic setting includes not only the resistance of the rocks traversed by the river but also the regional slope of the land surface. For modern rivers that flow in valleys made by ancestral rivers, geologic setting includes the gradients and deposits of the ancestral rivers. Rubey (1952, p. 135) pointed out that a river flowing on a predetermined gradient may more nearly or more readily attain equilibrium by the development of a wide, shallow cross section than by the development of a flatter gradient. Development of a flatter gradient might be inhibited by resistant consolidated rock or by gravel beneath the river bed. Development of a wide, shallow cross section would be promoted by easily erodible river banks. Geologic influences on the long profile of a river can be evaluated by relating the river profile to the regional slope, to the valley gradients of ancestral rivers, and to the distribution of different rock units traversed by the river. Inasmuch as the Loup Rivers are flowing mainly on the deposits (valley fill of the Elba terrace), of an ancestral river, the modem rivers have been influenced both by the lithology of the fill of the Elba terrace and by the gradient of the Elba surface. Similarly, the ancestral Loup Rivers were doubtless influenced by the regional slope of the Great Plains and by the lithology of the underlying Ogallala Formation. Longitudinal river profiles and valley profiles were prepared for the North Loup (fig. 8), Middle Loup (fig. 9), and South Loup (fig. 10). Profiles of the North Loup from altitudes of 1,745 to 2,500 feet were measured on Geological Survey topographic maps, scale 1: 24,000, contour interval 10 feet; and from altitudes of 2,500 to 3,200 feet, on Geological Survey topographic maps, scale 1:62,500, contour interval 20 feet. Profiles of the Middle Loup from altitudes of 1,745-2,560 feet were measured on 1:24,000-scale Geological Survey topographic maps having a contour interval of 10 feet; and from altitudes of 2,560-3,300 feet, on 1:62,500-scale Geological Survey topographic maps having a contour interval of 20 feet. Profiles of the South Loup from altitudes of 1,880 to 1,982 feet were measured on 1:24,000-scale Geological Survey topographic maps having a contour interval of 10 feet; and from altitudes of 1,982 to 3,000 feet, on 1: 250,000-scale Army Map Service maps having contour intervals 50 or 100 feet. For the river profiles, horizontal distances along the stream were measured with a map measurer. For the valley profiles, horizontal distances were measured along a line drawn through the inflection points of curves in the stream, and altitudes were taken on the streambank at an inflection point. Between any two points of given elevation, the ratio of valley slope to stream slope is a measure of theDIO CHANNEL PATTERNS AND TERRACES OF THE LOXJP RIVERS IN NEBRASKA sinuosity of the stream; indeed, sinuosity may be defined in this way (Lane, 1957, p. 98). The regional slope of the Great Plains, which is shown for comparison with the profiles, was measured on Army Map Service maps (scale, 1:250,000; contour interval, 100 ft) over the same section of the Great Plains as that crossed by the Loup Rivers. A straightedge was laid perpendicular to the trend of the contour lines that indicate the regional slope of the plains. The prevailing elevation of the upland in the area marked by the east end of the straightedge was subtracted from the prevailing elevation at the west end, and the horizontal distance (about 150 miles) was measured along the straightedge. INFLUENCE OF ROCK RESISTANCE At some places, the channels of the Loup Rivers may be cut into the Ogallala Formation. Certain lithologic units of the Ogallala, particularly the carbonate-cemented sandstone, are moderately resist- ant to channel erosion. The conspicuous bulge in the long profile of the South Loup (fig. 10) between Cumro and Ravenna is attributed to resistant bedrock although no outcrops were observed in the river channel. Information from drilling and from gravel pumping indicates that the channels of the Loup Rivers generally are underlain not by the Ogallala Formation but rather by several tens of feet of alluvial gravelly sand. From place to place along all the Loup Rivers, gravel for road building is pumped from beneath the river channel or from low terraces along the river. However, test drilling has shown no commercial deposits of gravel for long distances along the rivers. For example, no gravel has been found along the South Loup between Arnold and Ravenna. Test drilling at gravel-producing localities has shown that the thickness and distribution of gravel are sporadic even within an area of a few acres. 220 200 180 160 140 120 100 80 60 40 20 0 DISTANCE FROM CONFLUENCE WITH MIDDLE LOUP RIVER, IN MILES Figure 8,—Longitudinal river profile, valley profile, and Elba terrace profile, North Loup River, Nebr. Regional slope of Great Plains Is shown for comparison. Numbers refer to gaging stations in table 2.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Dll Gravel is pumped from depths as great as 40 feet, and nearly all this gravel is less than 1 inch in diameter. Coarser gravel, which is rejected by the Nebraska Highway Department, is separated by screening. The rejected gravel probably represents less than 2 percent of the total sand and gravel pumped, and it consists mostly of pebbles less than 2 inches in diameter; although cobbles as much as 6 inches in diameter were seen, they represent less than 5 percent of the rejected gravel. Competency of the Middle Loup to transport pebbles as much as 1 inch in diameter is indicated by the occasional observance of such a pebble in the artificial turbulence flume at Dunning. Gravel deposits do not seem to be concentrated beneath the bed of the river, but rather they seem to be interbedded with sand and silt down to the maximum depths reached by the pumps. Most of the commercial gravel deposits are probably within the valley fill of the Elba terrace. Gravel lenses in the fill of the Elba terrace were probably derived from older gravel deposits exposed upstream. Some commercial deposits are within older valley fills (of Kansan, Illi-noian, or Peorian age) that have been recently exposed by lateral migration or by downcutting of the rivers. The sporadic lateral and vertical distributions of the gravel as well as its small size indicate that gravel has not prevented the rivers from adjusting their gradients by incision. In general, the valley fills of ancestral rivers are easily erodible and permit the development of wide, shallow cross sections. Nearly everywhere the river banks are formed by terraces cut in late Recent time which are underlain by valley fill of the Elba terrace. This fill consists dominantly of sand along the braided North Loup and Middle Loup and of silty sand along the meandering South Loup. 220 200 180 160 140 120 100 80 60 DISTANCE FROM CONFLUENCE WITH NORTH LOUP RIVER, IN MILES 40 20 Figure 9.—Longitudinal river profile and valley profile, Middle Loup River, Nebr. Regional slope of Great Plains is shown for comparison. Numbers refer to gaging stations in table 2.D12 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA 220 200 180 160 140 120 100 80 60 40 20 0 DISTANCE FROM CONFLUENCE WITH MIDDLE LOUP RIVER, IN MILES Figure 10.—Longitudinal river profile and valley profile, South Loup River, Nebr. Regional slope of Great Plains is shown for comparison. Numbers refer to gaging stations in table 2. INFLUENCE OF REGIONAL SLOPE AND RELIEF The influence of regional slope and relief on the long profiles of rivers can be shown either by comparing the profiles of rivers in mountainous regions with the profiles of rivers on plains or by noting the profile changes along a single river as it flows across different physiographic provinces. For example, the profile of the Arkansas River in the Rocky Mountains upstream from Canon City, Colo., is steep and irregular; it is nearly straight across the Great Plains, but it flattens across the Interior Lowlands and flattens still more as the river traverses the structural trough of the Arkansas Valley and enters the Gulf Coastal Plain. Miller (1958) investigated the characteristics of streams that flow across resistant rocks in mountainous areas onto unconsolidated deposits of the Rio Grande depression. He concluded that channel slope is “* * * a partially independent variable, determined by inherited conditions that the stream can gradually modify, but only within certain limits." In accounting for the shape of long river profiles, many writers give insufficient attention to the influence of regional slope. However, Rubey (1952, p. 135) stressed the significance of regional slope in his discussion of the long profiles of the Missouri and Platte Rivers. He noted that the Missouri River, from about 2,000 miles upstream from Great Falls, Mont., to its mouth, flows at an approximately uniform distance below the older uplands that border it. He remarked that “* * * one is forced to believe either that the discharge and load of the present Missouri River are such that they almost exactly fit the slopes of these old upland surfaces * * * or, as seems more likely, that the profile of the present stream has been greatly influenced by the slope of the earlier surfaces on which it began its work.” Also, he suggested that the Platte River flows on a preexisting slope on which it has attained equilibrium by development of a very shallow cross section rather than a flatter gradient.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D13 The direction of the Loup River courses is closely related to the direction of slope of the Great Plains. In this part of Nebraska the surface of the Great Plains dips about S. 75° E. at about 9.5 feet per mile (0.0018 ft per ft). The upper courses of both the North Loup and the Middle Loup trend about S. 70° E.—that is, almost directly down the dip of the Plains. Beyond the Sandhills, the courses of the two rivers remain generally parallel but swing southward to about S. 40° E. In its upper course the South Loup trends S. 60° E., but between Cumro and Ravenna it swings northward and trends slightly north of east. The trend of the Calamus River is generally uniform at about S. 55° E. The river slopes and the valley slopes of the Loup Rivers are similar although somewhat less steep than the slope of the Great Plains. (See figs. 8, 9, 10). The slopes are less steep owing to the sinuosity of the rivers and their valleys rather than to the incision below the surface of the Great Plains. In fact, the manner of incision of the rivers tends to make their slopes steeper than the slope of the Great Plains because the incision is deeper in the lower courses than in the headwaters. For example, the upper reaches of the North Loup (in the Big Galls quadrangle) are incised to a depth of about 100 feet below the surrounding upland whereas the lower reaches, near the confluence with the South Loup (about 215 river miles downstream), are incised to a depth of about 200 feet below the upland. The long profiles of the Loup Rivers and the profile of the Great Plains are alike in being nearly straight. Although the profile of the North Loup flattens at its lower end and steepens aJt its upper end, the Straightness of most of the profile can be demonstrated by laying a straightedge along it. The profile of the Middle Loup, which is nearly straight from the headwaters to the confluence with the North Loup, has an average slope of about 0.00125 as compared with a slope of 0.0018 for the Great Plains. One explanation for the similarity between the profiles of the Loup Rivers and the profile of the Great Plains is that all are a result of gradation by eastward-flowing streams of similar hydraulic characteristics. There is, however, no convincing evidence that the eastward slope of the Great Plains is fundamentally a result of stream gradation. Instead, the surface of the Great Plains may have been merely flattened originally by subaerial processes generally, then tilted eastward by tectonic movements related to Cenozoic uplift of the Rocky Mountain region. As shown by geologic sections across Nebraska (Condra and others, 1950, fig. 8), the fluvial formations of Cenozoic age (Chadron, Brule, and Ogallala Formations) were deposited generally parallel to an erosional surface on rocks of Cretaceous age. A similar relation between the Ogallala and this erosional surface is shown on geologic sections by Frye and others (1956, p. 19), who also presented convincing stratigraphic evidence that the Ogallala does not represent a series of coalescent alluvial fans sweeping eastward from the Rocky Mountains. Instead, the Ogallala represents a series of alluvial fills in shallow eastwardtrending valleys cut in rocks of Cretaceous age; the fills overlapped laterally onto the gently sloping valley sides, buried the interstream divides, and eventually formed a coalescent alluvial plain. But the thickness of the Ogallala does not change greatly from west to east, and its total thickness (about 400 ft) is insufficient to have much effect on the slope of the Great Plains, which are about 400 miles in breadth. The origin of the eastward slope of the erosional surface on rocks of Cretaceous age remains in doubt, but the overlap of the Ogallala across the Chadron and Brule suggests either a differential tectonic downwarping of the Great Plains or a differential upwarping of the source areas to the west (Frye and others, 1956, p. 50). In addition, the initial eastward dip of the Ogallala probably has been increased by tectonic movements in post-Ogallala time. The profiles of the Loup Rivers have probably been inherited from the Great Plains profile. Many river profiles are characterized by a downstream decrease in slope which generally is attributed to a downstream increase in discharge or to a downstream decrease in particle size of load. Because average discharge remains nearly constant for long distances along the middle courses of the North Loup and Middle Loup, downstream decrease in slope is perhaps not to be expected. Also, there is very little downstream change in particle size of load for any of the Loup Rivers. However, bank-full discharge shows a rather sharp rate of increase downstream along the North Loup and Middle Loup. Both average discharge and bank-full discharge increase downstream along the South Loup, which has a notably straight long profile. Long profiles of the Loup Rivers are not significantly altered by substantial increases in discharge at the confluence of major tributaries. For example, the average discharge of the Middle Loup is increased by a factor of nearly two below the confluence of the Dismal River; yet, this increase in discharge is not accompanied by any significant change in slope. If, as seems apparent for the Loup Rivers, slope is not changed in response to variations in discharge or channel pattern, then the rivers would have little tendency to modify an inherited slope. Hydraulic adjustment to changes 719-579 0-64—3D14 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA in discharge and channel pattern is made mostly by change in cross section. THE MIDDLE LOUP RIVER AT DUNNING HYDRAULIC FACTORS IN RELATION TO WIDTH In their report on sediment transportation, Hnbbell and Matejka (1959) presented detailed tables of basic hydraulic data on different sections of the Middle Loup River at Dunning, Nebr. The location of sections discussed herein is as shown in figure 11. Sections A and E mark the limits of a reach about 7,900 feet in length. Section A has a bank-full width of about 340 feet, and the channel contains emergent bars during ordinary stages. The bank-full width of section C2 is about 85 feet, and that of section E is about 150 feet. Section C2 is free of emergent sandbars, but at section E emergent bars may form near the center of the channel at low flows. The flow at section B is confined by bridge abutments to a width of about 65 feet, but the flow at other sections is not artificially confined, except perhaps by vegetation and by some brush riprap at section C2. Cross profiles of sections A, C2, and E, measured at a discharge about equal to average discharge, are shown in figure 12. The data for each of these sections permit an analysis of the hydraulic adjustments made by the river at constant discharge as it flows through sections of distinctly different widths. The relative variation in river width along the reach at Dunning is typical of the variation along many reaches of the North Loup and Middle Loup. The bed and bank materials are nearly uniform along the reach at Dunning; therefore, variations in width cannot reasonably be assigned to variations in particle size of bank materials. Variations in width might be assigned to variations in vegetal growth on the banks; yet, the present vegetation at section A is similar to that at section E. At both sections, the left bank has a dense growth of willows and other trees whereas the right bank is sodded. Variations in width are perhaps related to past conditions of vegetal growth on the banks. Photographs of the sections are reproduced by Hubbell and Matejka (1959). Basic data for each section include measurements of discharge (Q), effective width (If7,-), effective area (Ac), mean suspended-sediment concentration, local slope, reach slope, and water temperature. Mean velocity was calculated from Q/A,, and mean depth I_J___i_i__1___________I__________I Figure 11.—Location of sections on a reach of the Middle Loup and Dismal Rivers near Dunning, Nebr. Figure 12.—Cross profiles of Middle Loup River at Dunning, Nebr. Section A, C2, and E after Hubbell and Matejka (1959) ; section S after Hubbell (1960).PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D15 from Ae/We. Effective width is defined as the length of a discontinuous line that crosses the channel and that is normal to the direction of flow at every point on the line. Width of sandbars is not included in effective width. Effective area is the sum of the products of effective width and measured depth for each section of a water-discharge measurement. Local slopes were measured with staff gages placed 300 feet upstream and 300 feet downstream from a section. At section A, the slope determined from the readings of these gages is questionable because the slope differed from one individual channel to another and also because transverse flow at the section caused local differences in slope along each bank. The reach slope is based on the difference in water-surface elevation between sections A and E. For each of the hydraulic properties listed above, except slope, 33 measurements were available for section A, 37 for section B, 23 for section C2, and 49 for section E. Of local slope, 20 measurements were available for section A, 11 for section C2, 31 for section E, and none for section B. Some water temperatures were estimated by the writer on the basis of past records of water temperatures for that calendar date, and others were extrapolated from temperature measurements made at another section on the same date. In figure 13, mean value of mean depth, mean velocity, local slope, and effective area are plotted, on rectangular-coordinate paper, against effective width at sections B, C2, E, and A. The mean values were obtained from measurements grouped according to discharge. Mean velocity and mean depth decrease as effective width increases whereas effective area and slope increase as effective width increases. Unfortunately, the scope of the data is insufficient to define adequately the form of the equations that would relate effective width to the other variables. Moreover, the effective widths at section A are not strictly comparable with the effective widths at the other sections because bars, submersed to shallow depth, are commonly present at section A but are rarely present at the other sections. A bar submersed to a depth of 6 inches, for example, would not influence the effective width at section A, but it would influence the hydraulic characteristics of the channel. A measure of width at section A that is about comparable with the effective width at the other sections is the water width at half mean depth. The mean value of the water width at half mean depth at section A, for discharges of 350-399 and 400—449 cfs, is about 215 feet. The points representing these discharges at section A should probably be moved to the left to an effective width of about 215 feet. At the other sections, the effective width as plotted is about equal to the water width at half mean depth. The increase in effective area with increasing width at constant discharge has been noted for many other sandbed streams (B. R. Colby, oral communication, 1961). An explanation of this increase must take into account the necessity for continuity in both bed-mate-rial discharge and water discharge from section to section, regardless of the width of the section. If only water discharge were involved, many different combinations of hydraulic adjustments might be possible; for example, the cross-sectional area might remain constant, and slope might increase enough to increase average velocity (in spite of decreasing depth). Continuity of bed-material discharge, however, requires a decrease in bed-material discharge per foot of stream width as the stream widens; if the width of the stream is doubled, the discharge of bed material per foot of width must be decreased by half. The decrease in bed-material discharge per foot of width requires a decrease in velocity and an increase in effective area. The reach slope remains constant at about 0.0013 foot per foot for the whole range of discharges and widths shown in figure 13. This reach slope corresponds to the local slope expected for a discharge of 450-500 cfs at a width of about 225 feet, which is the average width of the reach. Average width of the reach was determined by dividing the surface area of the reach by its length. Average discharge at Dunning is about 385 cfs and bank-full discharge is about 570 cfs. The reach slope may therefore be adjusted to a discharge between average discharge and bank-full discharge. Data were grouped and graphs were prepared to investigate the relation of local slope to discharge and suspended-sediment concentration at sections A and E. For the range of measurements available, local slope showed no definite relation either to discharge or to suspended-sediment concentration. HYDRAULIC FACTORS IN RELATION TO BRAIDING For a shallow stream of uniform depth that is flowing at bank-to-bank width, the growth of bars or islands in the channel not only divides the flow into braids but also reduces the water width to a value less than bank-to-bank width. The reduction in water width is a measure of the degree of braiding. Section A has well-defined banks, and its bank-to-bank width remained at about 340 feet during the period of measurement (1949-56). The section is in a nearly straight reach of the river; therefore, no tendency for the thalweg to remain near one bank would be expected. AtMEAN OF EFFECTIVE AREAS MEAN OF LOCAL MEAN OF MEAN VELOCITIES MEAN OF MEAN DEPTHS IN SQUARE FEET SLOPES IN FEET PER SECOND IN FEET D16 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA Discharge groups • 300-349 cfs X 400-449 cfs O 350-399 cfs A 450-500 cfs Figure 13.—Mean depth, mean velocity, local slope, and effective area in relation to effective width at sections B, C2, E, and A, Middle Loup River at Dunning.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D17 discharges of 350-500 cfs, the width of flowing water would extend from bank to bank if no bars were present, and the effective width would approximate the bank-to-bank width. To investigate the changes in channel configuration from time to time at section A, 30 cross profiles were plotted (from notes of water-discharge measurements made during the period 1949-56) on a horizontal scale of 40 feet to the inch and a vertical scale of 2 feet to the inch. Seven of these profiles, mostly pertaining to water discharge measurements of 400—449 cfs, are shown in figure 14. The position of greatest depth in the cross profile shifts apparently at random through the width of the stream, although there is some tendency for the deeper channels to be near the banks rather than midway between the banks. Cross profiles of the channel on successive days (such as on Aug. 30 and 31, 1949) show no major changes in configuration, but in a period of 2 weeks the configuration can change completely. For example, compare the cross profile of April 12,1950, with that of April 25. The effect of a bar submersed to a depth of a few tenths of a foot is not very different from the effect of an emersed bar on the flow of water through a given cross section. The water velocity in the shallow depth above the submersed bar is low, and the cross-sectional area of the stream above shallowly submersed bars is only a small part of the total cross-sectional area. Shallowly submersed bars do not confine the flowing water at the surface of the stream, and their width enters into the measurement of effective stream width only if the water above the bar is nearly motionless. To evaluate the effect of all bars and islands on such hydraulic variables as effective area and mean velocity, some means of expressing numerically the configuration of the channel cross profile is needed. Many schemes were tried in an effort to derive a single number that would take into account all the irregularities of the cross profile, but none led to a number that was unique for a particular configuration. However, a measure of water width was devised to take into account the effect of bars submersed to shallow depth. This measure is called the water width at half mean depth. It was measured by laying a straightedge across the plotted cross profile at a distance from the water surface equal to half mean depth; the width of bars that intersected the straightedge were excluded, and the increments of water width at this depth were summed. The choice of half mean depth as a suitable depth at which to measure water width was made by trial-and-error measurement on the 30 plotted cross pro- files at section A. Absolute depth was not chosen because such a choice might lead to the statement of a water width of zero for a very shallow cross section. Of the different fractions of mean depth that were tried, half mean depth seemed most appropriate. Inspection of the velocities and cross-sectional area above bars submersed to a depth equal to or less than half mean depth indicated that only a small part of the water discharge is transmitted above such bars, whereas a significant part of the discharge is transmitted above bars submersed to greater depths. Even where bars submersed to a depth equal to or less than half mean depth occupy much of the width of the cross section, only a small percentage of the total discharge is transmitted above these bars. For example, only about 11 percent of the total discharge was being transmitted above the shallowly submersed bars on the cross section dated Sept. 13,1949 (fig. 14). The water width at half mean depth is only 54 percent of the effective width measured at the water surface For the 30 cross profiles that were plotted, the mean value of half mean depth is 0.36 foot, the standard deviation is 0.04 foot, and the range is 0.28 to 0.47 foot. The graphs of figure 15 show effective area, local slope, mean velocity, and maximum depth at section A in relation to the water width at half mean depth. (See table 1.) In addition, points representing mean values of effective area and mean velocity at sections B, C2, and E (and mean values of local slope at sections C2 and E) are shown for comparison. Maximum depth is not the absolute maximum depth, which is too fortuitous to be significant, but it is the maximum depth above which lies 95 percent of the effective area. It was measured by laying a straightedge across the bottom of the plotted profile, sliding the straightedge upward until it reached the level below which lay 5 percent of the effective area, and then taking the water depth at that level. Mean velocity is computed from discharge divided by effective area. Local slope, which is not available for all the plotted points, is from Hubbell and Matejka (1959, p. 82). Effective area is from the same source and from the original notation of water-discharge measurements. Curves were fitted to the points applying to section A by the method of least squares; correlation coefficients and fitted curves are shown on the graphs. A linear relation is assumed because the range of data is insufficient to warrant the investigation of more complex relations. The correlation coefficients indicate a significant positive correlation with effective area at the 1-percent level and indicate a significant negative correlation with maximum depth and mean velocity, also at the 1-percent level. No significantDEPTH OF BED, IN FEET BELOW WATER SURFACE D18 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA WIDTH, IN FEET Figure 14.—Cross profiles of section A, Middle Loup River at Dunning. Bar widths that are above half mean water depth are indicated by shading.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D19 x h- CL LU Q X < > O K O — LU O C/) o _i cr LU LU > CL Z h- < LU Discharge groups • O 350-399 cfs X X 400-449 cfs ▲ A 450-500 cfs Figure 15.—Maximum depth, mean velocity, local slope, and effective area in relation to water width at half mean depth, section A, Middle Loup River at Dunning. Mean values for sections B, C2, and E as plotted in figure 13 are shown (darker points) for comparison.D20 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA correlation with local slope is indicated at the 5-percent level; therefore, the curve is not shown. Unlike the points representing mean values at sections B, C2, and E, the points representing individual measurements at section A show no alinement according to discharge. This is attributed in part to the fact that significant differences in channel configuration apply to points in the same discharge range; water width at half mean depth is an imperfect measure of channel configuration. In addition, much of the scatter of points is related to differences in water temperature. For example, of the 12 points that lie above the velocity curve, 10 represent measurements made at a water temperature of less than 50° F, whereas only 2 represent measurements made at a water temperature of more than 50°F. Of the 14 points below the velocity curve, only 2 represent measurements made at a water temperature of less than 50°F, and the others represent measurements made at a temperature of more than 50°F. The mechanism by which water temperature affects velocity in the Loup Rivers is complex in that it involves not only a change in water viscosity but also accompanying changes in suspended-sediment concentration and bed form. The relations shown in figure 15 indicate that the forming of emersed and shallowly submersed bars in a wide, shallow channel renders the channel hydraulically similar, at least in effective area and mean Aug. 30 31 Sept. 7-9-13 Mar. 1-21 Apr. 12 25 May 9-23 June 6- July 6-18 Aug. 1-Oct. 4-Nov. 7-29 Feb. 19 Mar. 13 Apr. 25 June 25 July 27 Aug. 15 Sept. 20 Dec. 4. Oct. 22 May 26 Aug. 3. Table 1.— Values of hydraulic factors at section A, Middle Loup River at Dunning, Nebr. Date 1949 1950 1951 1953 1954 1956 Discharge (cfs) Effective width (feet) Width at half mean depth (feet) Effective area (sq ft) Mean velocity (fps) Mean depth (feet) Maximum depth (feet) Local slope (ft per ft) 374 273 270 194 1. 93 0. 71 0. 8 0. 0013 424 272 268 229 1. 85 . 84 1. 1 . 0013 379 279 252 213 1. 78 . 76 1. 2 .00125 399 275 242 221 1. 81 . 80 1. 4 . 0013 419 258 140 187 2. 24 . 72 1. 5 . 00145 426 299 202 168 2. 54 . 56 1. 0 . 00145 430 230 206 190 2. 26 . 83 1. 1 . 00125 432 229 160 172 2. 51 . 75 1. 7 . 0016 426 285 254 194 2. 20 . 68 1. 6 . 0014 477 325 190 197 2. 42 . 61 1. 3 . 0013 391 236 156 171 2. 29 . 72 1. 9 . 00155 356 244 142 178 2. 00 . 73 1. 7 . 0013 366 266 214 196 1. 87 . 74 1. 1 . 0014 402 318 242 221 1. 82 . 70 1. 6 .00125 364 320 216 198 1. 84 . 62 1. 4 380 260 232 190 2. 00 . 73 1. 0 399 330 202 191 2. 09 . 58 1. 4 . 0013 405 256 202 162 2. 50 . 63 1. 2 427 276 228 183 2. 33 . 66 1. 2 . 0015 404 160 118 135 2. 99 . 85 1. 5 452 282 230 190 2. 38 . 67 1. 1 391 288 155 198 1. 97 . 69 1. 8 . 00125 403 323 302 230 1. 75 . 72 . 9 . 00125 350 336 224 210 1. 67 . 62 1. 6 417 256 150 199 2. 10 . 78 2. 0 . 0014 452 330 244 213 2. 12 . 64 1. 0 481 288 168 220 2. 19 . 76 2. 3 386 282 140 186 3. 13 . 66 1. 7 361 343 136 196 1. 84 . 57 1. 5 416 199 142 186 2. 24 . 93 1. 7 Water temper- ature (°F) 64 64 61 59 55 35 35 40 41 SO 70 70 66 64 70 44 48 38 38 32 48 78 81 72 67 35 45 70 70 May 8 50PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D21 velocity, to a more narrow channel. Also, the greater the width of bars in the channel, the greater is its maximum depth. Because the width of section A is very great relative to its depth, the effect of banks is insignificant, and two or three channels transmit a given water discharge at approximately the same effective area and mean velocity as a single channel of the same total width. Although the growth of emersed bars in a channel necessarily causes braiding, a decrease in water width results only if the bank-to-bank width remains constant. If the growth of bars or islands is accompanied by bank erosion, the water width may remain constant or even increase. Also, a reduction in water width can take place without braiding if a single channel is formed. Braiding cannot be equated with a reduction in water width unless a constant bank-to-bank width and the presence of emersed bars or islands are stipulated. The bank-to-bank width at section A, during the period of measurement, possibly remained constant only because the bars were transient. HYDRAULIC FACTORS IN RELATION TO A CONFLUENCE The confluence of the Middle Loup and Dismal Rivers is hydraulically and morphologically similar to the joining of two channels of a braided stream that have been separated by a stabilized alluvial island. In discharge, slope, suspended-sediment concentration, and particle size of load, the Dismal River is similar to the Middle Loup River upstream from the confluence. Therefore, hydraulic adjustments made at the confluence are pertinent to a consideration of adjustments made in a braided stream where channels join or divide. Measurements made on the same day and at about the same discharge have been given by Hubbell (1960) for six sections on the Middle Loup and Dismal Rivers (see fig. 11); the measurements of water temperature, water discharge, effective area, effective width, and local slope are for all six sections and for 7 different days during 1956. Local slope was measured for a reach 300 feet upstream and 300 feet downstream from each section. In addition, Hubbell reported similar measurements made on 4 days at four additional sections (here designated Si, S2, T1? and T2), which are 300 feet upstream and 300 feet downstream from sections S and T, respectively. The cross profiles of sections E and S are shown in figure 12. Figure 16 shows the relation of effective area and local slope to effective width for sections upstream and downstream from the confluence. Measurements of effective width and area at each section have been placed into two groups according to discharge and water temperature, and the means of the values within each group are plotted. The upstream sections have been paired and summed as E plus R and F plus P, so that the combined discharge of each pair approximately equals the discharge downstream from the confluence. Hubbell (1960, p. 29) used dimension-analysis and multiple-regression techniques to demonstrate the relations of hydraulic variables at individual sections upstream and downstream from the confluence. The purpose of figure 16 is to show graphically the relations that are considered to be of greatest geo-morphic significance. Points in the same discharge group show a linear distribution when plotted on double log paper, and the variables are probably related by a power function. (A power function is also suggested by the distribution of points in fig. 13.) The curves are visually approximated rather than statistically constructed because the scatter of points is small and because no particular significance is attached to the slope value of the curve. In general, the sum of the widths of the Middle Loup and the Dismal upstream from the confluence is less than the width of the Middle Loup downstream from the confluence. Also, the sum of effective areas for two sections upstream from the confluence is less than the effective area downstream from the confluence. Increase in width at the confluence is probably related to a fortuitous change in bank erodibility, inasmuch as vegetal growth along the banks is generally dense except along the left bank downstream from the confluence (Hubbell, 1960, p. 42). Despite some scatter of points, figure 16 indicates a regular increase in effective area with increase in effective width, regardless of whether effective width applies to the sum of two channel widths or to the width of a single channel. A similar increase in effective area with increasing width has been shown for a single channel (fig. 13). SUMMARY OF HYDRAULIC FACTORS IN RELATION TO CHANNEL PATTERN Among the major aspects of channel pattern are the total width of flowing water and the manner in which this width is distributed in the cross section—whether in a single channel or (as in the braided pattern) in multiple channels. The Middle Loup River at Dunning afforded a good opportunity for investigation of the relation of width, at constant discharge, to other hydraulic variables. The river flows through sections of greatly differing width along a short reach that has no significant downstream increase in discharge. Also, there was available a large number of measurementsD22 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA at sections having greatly different widths and a narrow range of discharges. In effect, the variable of discharge was held constant, so that the relation of width to effective area, mean velocity, mean slope, and mean depth could be distinguished. Hydraulic changes that occur when water flows from a wide section to more narrow sections were then compared with hydraulic changes that occur when channel width decreases, owing to braiding, at section A. Also avail- able were detailed measurements of width and other hydraulic variables upstream and downstream from the confluence of the Middle Loup and Dismal Rivers near Dunning. The confluence of these two rivers is considered analogous to the confluence of two channels in a braided stream; therefore, sections upstream and sections downstream from the confluence were compared as to the relation of total width to hydraulic variables. MEAN OF EFFECTIVE WIDTHS, IN FEET O Discharge < 750 cfs, temperature > 50°F • Discharge >750 cfs, temperature< 50°F Letters refer to sections Figure 16.—Effective area and local slope In relation to effective width for sections upstream and downstream from confluence of Middle Loup and Dismal Rivers.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D23 The conclusions drawn from these investigations are that the effect of braiding at section A is to decrease the width of flowing water and that the changes in effective area and mean velocity accompanying braiding are similar to the changes accompanying flow from a wide single channel to a narrower single channel. In other words, the effect of braiding on a wide channel is to render it hydraulically similar, at least in effective area and mean velocity, to a narrower single channel. Similarly, if effective width downstream from a confluence is greater than the sum of effective widths upstream from the confluence, effective area is increased and mean velocity is decreased. No definite relation between width and local slope was discerned, partly because the range of data for local slope was less than that for the other variables. Local slope perhaps increases as width increases from section to section; however, no significant correlation was discerned between local slope and braiding at section A or between local slope and width upstream and downstream from the confluence of the Middle Loup and Dismal Rivers. OBSERVATIONS ON THE FORMATION OF SANDBARS, ISLANDS, AND VALLEY FLATS In laboratory experiments by Leopold and Wolman (1957), braiding began with the deposition of an initial central bar which consisted of coarse particles that could not be transported under local conditions. In natural streams, such a bar may become vegetated and grow into an island, which divides the flow of the stream so that the channel pattern is braided. Bars and islands of the Loup Rivers are built of fine- and medium-grained sand, which is the ordinary bed material of the rivers; their origin cannot be explained as the selective deposition of coarser particles. As seen on aerial photographs, most submersed bars in the Loup Rivers are either lobe shaped or wedge shaped, and the point of the lobe or wedge is generally oriented downstream. (See fig. 17). A typical bar is about 300 feet long and about 200 feet wide at its widest part. Small lobe-shaped dunes having heights as much as 6 inches and lengths as much as several feet were seen in shallow water on the surface of bars (fig. 18) and elsewhere. Vs _____ - * X'-v-s • .r 1000 1 2000 3000 Feet __1 .—------J Figure 17.—Aerial photograph and tracing 0f On tracing, submersed bars are shown in River flows toward right. Photograph, North Loup River, about l mile downstream from gaging station near St. Paul, lightly stippled pattern emersed bars in heavily stippled pattern, and islands are blank.D24 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA Figure 18.—Small lobe-shaped dunes on submersed sandbar in North Loup River 8 miles downstream from Brewster. River flows toward right. Sandbar is being rapidly eroded as salient at lower right migrates upstream. The large bars and the small dunes probably acquired their similar shapes in the same general way. The forming of small dunes was observed in shallow water at the river edge. First, nearly straight dune ridges several feet long and oriented transversely to the current were formed. The ridges advanced downcurrent and grew rather uniformly in height until the current broke through them in several places, forming scallops from which lobes of sand were built ahead of the original dune ridge. Some lobes increased in height as they were built forward. Increase in height was augmented if a lobe advanced up the backslope of another lobe ahead. The bar of figure 18 is in a straight, narrow reach of the North Loup River; the reach is shown in the background of figure 19. The bar is not typical of those previously described, but it was studied because the high adjoining bank (Elba terrace surface) afforded a vantage point not found at many places along the Loup Rivers. The channels on either side of the bar are kept clear of sand deposits by high water velocity and by sand boils, which are nearly vertical clockwise-moving eddies that rise from the stream bottom and carry sand in suspension. Erosion of the downstream end of the bar was observed to take place along a deep salient, shown in figure 18, that moved rapidly upstream. Many Loup River bars probably originate from sand ridges built across the river bed at right angles to the Figure 19.—Straight reach of North Loup River, same locality as figure 18. Elba terrace, at right, stands 34 feet above river. Note small willow-covered island in foreground. prevailing current. Channels are formed where the current breaks through the ridges, and lobe-shaped bars are built in front of these channels. If a lobeshaped bar increases sufficiently in height, the current is diverted around it and forms channels on either side; the bar may continue to grow in the quieter water between these channels by the advance of small dunes on its surface. The downstream end of the bar may become pointed by the convergence of channels on either side, and the bar as a whole may become wedge shaped. Diversion of current around submersed bars creates braided patterns, observable on aerial photographs (fig. 17), that are the same as the patterns made by diversion around emersed bars and islands. Bars are built during a high river stage to heights that are above the water level during an average river stage. Some of these bars emerge when the river recedes, and they then become islands if vegetation grows on them (fig. 19) and prevents their destruction. The valley flats (flood plains) of the North and Middle Loup Rivers are built by a modification of the lateral accretion process: islands are successively added to the valley flat as the course of the river shifts laterally. Many islands are separated from the valley flat by a single narrow channel that is dry during an average river stage. (See island at lower center, fig. 17.) Aerial photographs show many former islands that have become part of the valley flat by the disappearance of narrow channels which separated them from the flat. A succession of vegetation begins with weeds on recently emersed bars; weeds are replaced by willows as the bar becomes an island; and finally, after the island has become part of the valley flat, thePHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D25 willows are replaced by grass. Grassed areas that are still recognizable as former islands are higher than the younger willow-covered islands. This greater height cannot be attributed to a thicker accretion of overbank deposits that accumulated over a longer period of time because overbanks flows are rare in the upper courses of the North Loup and Middle Loup. Moreover, grassed former islands show a succession of heights as much as about 8 feet above present river level. This succession of heights is an indication of degradation by the river. MEASUREMENT AND DESCRIPTION OF CHANNEL PATTERNS According to the usage of Leopold and Wolman (1957), the term “channel pattern” applies to the plan view of a river reach and includes meandering, braided, and straight channels. The channel pattern of a river usually varies from place to place. MEANDERING AND SINUOSITY Leopold and Wolman (1957) computed the sinuosity of a reach as the ratio of thalweg length to valley length. Lane (1957) computed a “tortuosity ratio” as the ratio of the length of the stream channel to the length of the stream measured along the axis of the valley. Many rivers on broad valley floors have meander belts that are curved. For such rivers, the sinuosity that is a function of the size, shape, and repetition of individual channel arcs should be considered separately from the sinuosity that is a function of the curvature of the meander belt. Otherwise, a different measure of sinuosity would be obtained for two reaches whose channel arcs are identical but whose meander belts are differently curved. If the axis of a series of symmetrical sine curves is bent, the sinuosity as measured in relation to a straight line joining the ends of the axis is changed; but the radius of curvature of individual arcs remains the same. Figure 20 illustrates the delineation of the meander-belt axis, which passes through the inflection points of arcs. Individual channel arcs are thus delineated above and below the meander-belt axis. The channel sinuosity of a reach, here called the sinuosity index (S.I.), is the ratio of length of channel to length of meander-belt axis. Where no meander belt can be distinguished, the sinuosity index is the ratio of length of channel to length of valley axis. Sinuosity of a meander belt, if desired, may be determined from the ratio of length of meander-belt axis to length of valley axis. Figure 20.—Delineation of meander belt, axis of meander belt, and channel arcs for measurement of sinuosity and symmetry. Sinuosity is easy to define and measure, but meandering is difficult to define and measure. A discussion and a definition of meandering are given by Lane (1957), and a summary of the geometry and hydraulics of river meandering is provided by Leopold and Wolman (1960). According to general usage, a meandering stream has a moderate to high degree of sinuosity, and the sinuosity is somewhat symmetrical. In defining meandering, some writers specify a migration of meander bends. Leopold and Wolman (1957) stipulated that meandering reaches have a greater sinuosity than irregularly sinuous reaches and that a sinuosity of 1.5 or greater indicates meandering. However, the use of sinuosity alone is not satisfactory for identification of a meander; on the basis of other criteria, these authors recorded a meander that has a sinuosity of 1.12 (Leopold and Wolman, 1960, p. 792). They reported consistent correlations between meander length and channel width; large rivers have large bends and small rivers have small bends, so that the general aspect of both large and small meandering rivers is similar in plan view. The symmetry of a reach consisting of several channel arcs depends both on the properties of individual arcs and on the group properties of the sequence of arcs. An individual arc may be described by its mean radius; its form may be described by the ratio of length to height, which is here called the form ratio. The arc length is the length of the meander-belt axis that is intercepted by the arc, and the radius of an arc is approximately equal to half the arc length. Group properties of a sequence of arcs include the statistical variation in arc length, height, and form ratio, the regularity of repetition for arcs of average size and form, and the sinuosity of the meander-belt axis. The average arc height, length, and form ratio may be determined for a reach. The symmetry of the reach may then be compared with an ideal reach that consists of this average arc regularly repeated along a straight line of the same length as the actual meander-belt axis. If the consistency in size, form, and repetition of arcs in this ideal reach is con-D26 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA sidered to be 100-percent symmetry, then the symmetry of the actual reach may be expressed as a percentage of 100 according to the following scheme: Percentage symmetry of arc length = 100-arc height = 100- 100 (mean deviation of length) mean length 100 (mean deviation of height) mean height form ratio = 100- 100 (mean deviation of form ratio) mean form ratio The aspect of symmetry that depends on the regular repetition of arcs can be expressed roughly by dividing the number of arcs of approximately average size in the actual reach by the number of arcs in the ideal reach. Arcs in the actual reach whose length does not deviate more than 25 percent from average length may be considered to be of approximately average size. Finally, the sinuosity of the actual meander-belt axis relative to the straight axis of the ideal reach may be expressed as a percentage by dividing the sinuosity into one, which is the sinuosity of a- straight line. In order to evaluate the significance of symmetry in the definition of meandering, measures of symmetry were applied to a reach of the river from whose name the term “meander” is derived—the Maiandros River in Turkey (now named the Buyuk Menderes). This reach, shown in figure 21, is redrawn from a longer Figure 21.—Delineation of a reach of the Buyuk Menderes River, Turkey, for measurement of sinuosity and symmetry. reach reproduced by Lane (1957) on a scale of 1 inch to 10,000 feet. Symmetry of the reach is perhaps adequately indicated by the following measures: Arc length.—mean, 2,600 ft; mean deviation, 800 ft; percent symmetry, 70. Arc height.—mean, 1,700 ft; mean deviation, 570 ft; percent symmetry, 66. Form ratio.—mean, 1.8; mean deviation, 0.6; percent symmetry, 67. Repetition of arcs.—number of arcs of approximately average size, 10; number of arcs in ideal reach, 38; percent symmetry, 30. Straightness of meander-belt axis.—sinuosity, 1.3; percent straightness, 77. Sinuosity index of channel.—1.56. If these measures of symmetry are valid and if this reach is representative of the Buyuk Menderes, then the type example for meandering rivers has a rather low degree of symmetry. If symmetry is to be stipulated in applying the term “meandering” to randomly selected reaches, some arbitrary limit of symmetry will have to be established. No simple and rapid means of expressing symmetry quantitatively is apparent although a great, many different ways of expressing symmetry could be devised. An experiment carried out by Friedkin (1945, p. 15) indicated that a minor lack of homogeneity in bank materials has a pronounced effect on both symmetry and sinuosity. When a small percentage of cement was distributed unevenly through the bank materials, a laboratory river developed low symmetry and a sinuosity index of 1.17, as compared with a high symmetry and a sinuosity of 1.4 for a similar laboratory river in homogeneous bank materials. In this report, sinuous reaches are described by means of the sinuosity index, and no attempt is made to describe the symmetry of the sinuosity. If the sinuosity index of a reach is 1.3 or greater, the reach is further described as meandering, whether the sinuosity is symmetrical or not. Ideally, a straight reach has a sinuosity index of 1, but reaches that have sinuosity indices of less than 1.05 are described as straight. Reaches that have sinuosity indices between 1.05 and 1.3 are described as sinuous. The term “meandering” is more satisfactory for the description of a stream course in general than for the description of short randomly selected reaches. The term “meandering” thus is applied to a sinuous stream that has, from place to place along its course, one or a series of symmetrical arcs, the length of which is related to the width of the stream. The sinuosity index is greatly influenced by the choice of reaches to be measured. Few streams meander continuously along their courses, and very sinuous reaches are usually separated by less sinuous reaches. If the reach to be measured is restricted to the more sinuous parts, then a higher sinuosity index will obviously be obtained than if the reach is extended beyond the more sinuous parts. In this study, reaches were not selected on the basis of channel pattern but on the basis of location at a gaging station. EachPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D27 reach is about 2 miles in length along the stream and extends about 1 mile upstream and about 1 mile downstream from a gaging station. If the sinuosity index of a reach chosen in this way is about 1.3 or greater, the reach is meandering in the sense that seems to be generally accepted. For example, the small-scale rivers considered to be meandering by Friedkin (1945, pis. 33,41) have sinuosity indices of 1.25-1.3. BRAIDING Leopold and Wolman (1957) defined a braided river as “* * * one which flows in two or more anastomosing channels around alluvial island.” Friedkin (1945) stipulated that the braided channel is extremely wide and shallow and that the flow passes through a number of small interlaced channels separated by bars. According to Lane (1957), “* * * a braided stream is characterized by having a number of alluvial channels with bars or islands between meeting and dividing again, and presenting from the air the intertwining effect of a braid.” Also, Lane proposed a genetic classification of braided streams in which the major categories are “braiding due to steep slope” and “braiding due to aggradation.” Within each major category, streams are further classified according to whether they are degrading, at equilibrium, or aggrading. The term “multiple channel streams” was proposed by Lane to include not only streams having a distinctly braided appearance but also streams having scattered islands (such as the lower Illinois River). Also included are distributaries on deltas and anastomosing distributaries on deltas and on alluvial fans. The term “multiple channel” has much merit, but the classification of multiple-channel streams on a purely genetic basis is impractical. The causes of braiding are poorly understood, and a sound descriptive classification is a first step toward understanding braiding. Several descriptive features of probable genetic significance and of use in descriptive classification may be suggested. One such feature is the size of the islands relative to the width of the channel. Small islands are probably formed by the growth of bars in the channel whereas very large islands are more likely formed by diversion and splitting of the channel during overbank discharge, as in the crevass-ing of the natural levees of delta distributaries. The location of islands near the entrance of minor tributaries probably has genetic significance, as Rubey (1952) suggested. High sinuosity of individual channels of braided streams, together with the presence of lakes and undrained depressions on islands, may be characteristic of low-slope braided streams such as the Upper Mississippi River near Waukon, Iowa. No scheme for describing quantitatively the braided pattern of a reach or for distinguishing between braided and not braided reaches was found in the literature. Leopold and Wolman (1957) specified that islands develop from bars that emerge and become stabilized by vegetation. The presence or absence of vegetal cover seems to be the only practical criterion for distinguishing an island from a bar in the field and on aerial photographs. As defined here, an island has vegetal cover and is ordinarily emersed during a bank-full stage; a bar has no vegetal cover and is ordinarily submersed during a bank-full stage, but it may be emersed during lower stages. Braiding that changes in pattern with river stage is characterized by bars and is termed “transient.” Braiding that remains nearly constant in pattern is characterized by islands and is termed “stabilized.” The following ratio, called the braiding index (B.I.), has been found satisfactory for description of channel braiding, either transient or stabilized: ^ j _2[sum of lengths of islands and (or) bars in reach] length of reach measured midway between banks The braiding index is a measure of the sum of island or bar perimeters in a reach and hence of the increase in bank length that results from braiding. Most islands or bars are narrow, and their perimeters are approximately twice their lengths. Lengths of islands or bars in a reach can be measured and summed in the same operation by use of a chartometer (map measure). The stabilized-braiding index is determined by measuring islands only; the transient-braiding index, by measuring bars only; and the total-braiding index, by measuring both islands and bars or by adding the other two indices. Submersed bars are not measured. Aerial photographs are much more satisfactory than topographic maps for measurement of the braiding index because the braided channel pattern is conventionalized on many topographic maps. Also, the representation of a particular feature in the channel as an island or as a bar depends on the judgment of the cartographer. Bars are not shown on many maps. The stabilized-braiding index for a particular reach should remain nearly constant for different river stages whereas the transient-braiding index will change greatly with changing river stage. A decision must be made as to which index of braiding—transient, stabilized, or total—most satisfactorily represents the channel morphology of a particular river. For the Loup Rivers, the total-braiding index was used because both bars and islands are characteristic featuresD28 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA of the river channels. A reach that is both meandering and braided may be described by using both the sinuosity index and the braiding index. Although the braiding index as defined herein is useful for morphologic description, it has little quantitative hydraulic significance. The braiding index has no consistent relation to water width, which is, as previously shown for the Middle Loup at Dunning, the dimension to which hydraulic variables can be related. The writer was not able to devise a braiding measure that has both morphologic and hydraulic significance. Distinction between braided reaches and reaches having few islands or bars is arbitrary, but a consistent distinction may be made by means of an appropriate value of the braiding index. For the Loup Rivers, a value of 1.5 for the total-braiding index was selected to distinguish braided from not braided reaches, because reaches having an index of less than 1.5 did not have a braided appearance. The braiding index, like the sinuosity index, is influenced by the choice of reaches: short reaches that are restricted to the vicinity of islands will have a higher braiding index than long reaches sampled at random. Most of the bias that results from the selection of short reaches can be avoided by specifying an adequate reach length in terms of bank-full widths; for example, a reach having a length equal to 10 bank-full widths would probably be satisfactory. The braiding indices shown in table 2 apply to reaches about 2 miles long, extending about 1 mile upstream and 1 mile downstream from each gaging station. Channel patterns of reaches having different indices of braiding and sinuosity are illustrated in figures 22 and 23. These reaches were selected to illustrate the range of channel patterns of the Loup Rivers and do not correspond exactly to the reaches referred to in table 2. Table 2.—Values of hydraulic factors at gaging stations in the Loup River hasin Gaging station Average discharge for period of record to 1957 (cfs) Number of years of record to 1957 Average of maximum daily discharges (cfs) Coefficient of deviation, discharge Mean width (feet) Stand- ard devia- tion, width (feet) Coefficient of variation, width (percent) Chan- nel slope 1 * (ft per ft) Index of braiding Total Stabi- lized Tran- sient Index of sinuosity Suspended sediment Concentration (ppm) * <0.062 mm (percent) 3 Bed material <0.062 mm (per- cent) <0.50 mm (per- cent) <2 mm (per- cent) Middle Loup River: 1. Near Seneca*. 2. At Dunning*. Dismal River: 3. At Dunning*. 201 385 308 570 460 100 160 80 0.00137 .0013 <0.1 <.l <.l <0.1 <.l <.l <0.1 <.l <.l 1.3 1.06 1.3 4 770 «800 4 940 Middle Loup River: 4. Near Milburn. 5. At Walworth. 6. At Sargent__ 7. At Arcadia__ 8. At Loup City* 9. At Rockville* * 786 799 813 799 823 804 5 1,250 0.16 12 1,550 .38 5 1,470 .21 12 2, 580 .81 7 2,000 .40 2 4,340 345 530 595 320 587 581 98 229 175 185 155 113 28.4 43.2 29.4 57.8 26.4 19.4 .00156 .00159 .00135 .00120 .00131 .00127 1.0 .2 1.8 .9 2.2 .5 2.5 1.2 2.5 <.l 2.3 .4 .8 .9 1.7 1.3 2.5 1.9 1.05 1.06 1.05 1.07 1.15 1.06 4 820 4 900 4 1,100 4 1,250 4 1,200 4 1,250 21 73 92 17 0 23 0 91 99 91 99 South Loup River: 10. Near Cumro...* 11. At Ravenna____ 12. At St. Michael_ 165 198 248 7 1,535 12 3,645 12 4,830 .91 .81 1.06 119 205 2(J9 26 63 52 23.2 30.7 24.9 .00083 .00108 .00096 <.l .5 .5 <.l <.l <.l .1 .5 .5 2.4 4 2,100 1.6 4 3,000 1.4 8 3,170 68 81 1.0 North Loup River: 13. At Brewster. 14. At Taylor.*. 15. At Burwell*. 378 448 485 830 1,120 1,215 .29 .41 .46 267 270 320 98 93 90 36.7 34.4 28.1 .00119 .00133 .00147 .8 1.5 1.6 .3 .7 .3 .5 .8 1.3 1.04 1.03 1.15 4 510 4 600 4 680 19 23 .1 69 92 Calamus River: 16. Near Burwell. 17. Near Harrop* 288 12 250 («) 550 450 .19 170 67 43 19 25.3 .00109 28.4 .00102 <.l <.l 1.05 1.6 4 360 29 .1 73 91 North Loup River: 18. At Ord____________ 19. At Scotia........ 20. Near Cotesfield... 21. Near St. Paul_____ 826 828 903 888 5 2,250 12 4,250 .54 6 4,300 12 5,850 .58 453 515 417 501 140 125 113 108 30.9 24.3 27.1 21.6 .00110 .00121 .00120 .00107 1.7 .4 1.5 .7 1.2 <.l 1.5 <.l 1.3 .8 1.2 1.5 1.06 1.09 1.04 1.2 4 800 4 1,000 4 1,100 8 1,430 0 92 28 62 2.0 75 95 0 75 97 i Measured from topographic map. 3 Discharge weighted. 3 Average of samples. 4 Based on periodic measurements. * Based on daily measurements. • Partial record.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D29 A-CALAMUS RIVER NEAR BURWELL, OCT 22,1949 (TOTAL B, I., 0.2;S. i., 1.05) 5.-NORTH LOUP RIVER UPSTREAM FROM TAYLOR OCT 22, 1949 (TOTAL B. I., 1.6;STABI LIZED B. I., 1,3; S. I., 1.1) C.-MIDDLE LOUP RIVER UPSTREAM FROM ARCADIA, OCT 24, 1949 (TOTAL B. I., 2.7; STABILIZED B. I., 1.1; S. I., 1.05) * 1000 o 1000 2000 3000 4000 5000 FEET I___I____l_______1_1_________________I__________________I___________________I__________________I__________________I Figure 22.—Reaches that illustrate different values of the braiding index. Sandbars are shown in stippled pattern, islands are blank, and scarps of minor cut terraces are hachured. Traced from aerial photographs.D30 CHANNEL PATTERNS AND TERRACES OP THE LOXJP RIVERS IN NEBRASKA A.-CALAMUS RIVER NEAR HARRUP. OCT 28,1949 (S. I., 1 .7; TOTAL B. I., 0.08) 5.-SOUTH LOUP RIVER IN VICINITY OF CUMRO, OCT 11, 1949 (S. I., 2.1) C.-SOUTH LOUP RIVER AT ST. MICHAEL, NOV 8, 1951 (S. I.. 1.6; TOTAL B. I., 0.5) Figure 23.—Reaches that Illustrate different values of the sinuosity index. Sandbars are shown in stippled pattern, islands are blank, and scarps of minor cut terraces are hachured. Traced from aerial photographs.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D31 WIDTH For each of the gaging stations listed in table 2, except Middle Loup River near Seneca, river width at the gaging station, at six sections upstream, and at six sections downstream was measured on aerial photographs. Width of the Middle Loup near Seneca was measured on a Geological Survey topographic map. Sections were spaced about 800 feet apart; therefore, the reach between the end sections was about 9,600 feet in length. Mean width for each 9,600-foot reach was determined from the 13 individual measurements of width. At each section, width was measured between river banks, which are stabilized by vegetation and are clearly shown on the aerial photographs. For braided reaches, the widths of vegetated islands were not included in river width; therefore, the river widths represent the total widths of the stabilized channels. The widths of unvegetated sandbars in the channel, whether emersed or submersed, were included in the measurement of channel width. The widths as measured thus differ from the effective widths reported by Hubbell and Matejka (1959) for the Middle Loup at Dunning; there emersed sandbars were excluded from measurements of width. Coefficients of width variation, as determined by dividing standard deviation by mean width, range from about 19 percent to about 58 percent. CHANNEL PATTERNS OF THE LOUP RIVERS A general impression of the sinuosity of the Loup Rivers may be obtained from figures 8, 9, and 10 by comparing each river profile with the corresponding valley profile. For a given reach, the ratio of valley slope to river-channel slope equals the sinuosity index; and where the two slopes are nearly parallel, the reach is nearly straight. At most places along the North Loup and Middle Loup, no meander belt can be distinguished; and the valley width, which includes the width of the Elba terrace surface, is no more than 20 times the width of the river. The maximum width of the South Loup valley is nowhere more than twice the width of the meander belt. For the Loup Rivers, therefore, slope of meander belt and slope of valley axis are nearly equal. The North Loup has a sinuosity index of about 1.6 in its upper reaches. This value gradually decreases downstream, and below Brewster the sinuosity is so low that the valley slope and stream slope are nearly parallel. Symmetrical, well-defined meanders are present only in a short reach near Brewster and in a reach near the confluence with the Middle Loup. Most of the lower 130 miles of the North Loup trends in broad arcs (S.I.=1.1-1.05) or in nearly straight reaches (S.I.= 1.01-1.05) that commonly meet to form a right-angle bend. Scattered islands divide the channel from place to place, but in general the North Loup is only slightly braided. The reach at Ord (total B.I., 1.7; stabilized B.I., 0.4) is typical of the more braided reaches of the river. For the reaches measured at gaging stations (table 2), the average total B.I. is 1.4 and the average stabilized B.I. is 0.4. According to Leopold and Wolman (1957), extremely short reaches of a channel may be straight, but reaches that are straight for distances exceeding 10 times the channel width are rare. They did not specifically define what is meant by a straight channel, but the straight channels in their illustrations show a slight degree of sinuosity. Long reaches that have a sinuosity index of less than 1.01 are fairly common along the North Loup and Middle Loup. A reach of the North Loup downstream from Scotia is 2% miles long (a distance of about 30 times river width) and has a sinuosity index of 1.01. The straightness of Loup River reaches may be exceptional. In general, the Middle Loup is somewhat less sinuous than the North Loup and somewhat more braided. Both of these rivers have a low degree of braiding as compared with a river such as the Platte. In a 2-mile reach near the village of Silver Creek, Nebr., the Platte had a total braiding index of about 8 on May 2, 1951. For the reaches measured at Middle Loup gaging stations, the average total B.I. is 1.6 and the average stabilized B.I. is 0.4. Although a comparison of the North Loup and Middle Loup as represented on Geological Survey topographic maps (scale 1:24,000 issued 1951-55) would indicate that the Middle Loup is much more highly braided than the North Loup, no great difference between the rivers could be seen on the aerial photographs from which the maps were made. The maps were evidently drawn by different cartographers who had different opinions about the representation of a braided stream. The Middle Loup is moderately sinuous in its upper reaches, but the parallelism of its stream slope and valley slope (fig. 9) shows that its sinuosity is generally very low. A fewr symmetrical meander arcs may be seen in the river course near Dannebrog and at Boelus, but otherwise the sinuosity of the Middle Loup shows little symmetry. The sinuosity of the South Loup is apparent from a comparison of its river slope with its valley slope (fig. 10) . Except for the reach between elevations 2,000 and 2,150 feet, which has a steep slope and low sinuosity, the average sinuosity index of the river is about 1.4. The degree of symmetry shown by most reaches is represented by figure 23B; but a higher symmetry,D32 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA represented by figure 236', is possessed by a few reaches. Channel arcs gradually increase in size downstream as water discharge and river width increase. (See figs. 23/?, C.) Small vegetated islands and emersed bars occur from place to place in the channel of the South Loup, particularly in the lower 20 miles of its course; but the degree of braiding is generally very slight. Most of the smaller flowing tributaries in the Sandhills (such as Goose Creek, which enters the North Loup upstream from Brewster) are meandering. The Calamus River meanders for most of its course, but the last 7 miles upstream from its confluence with the North Loup is characterized by wide, nearly straight reaches that contain abundant sandbars. In the loess-mantled part of the Loup River basin, most tributaries (such as Clear Creek, Oak Creek, and Mud Creek) are meandering, and the meanders seem to he incised. Channel stability of the North Loup River was studied by comparing aerial photographs made in August 1938 with photographs made in May 1957. Measurements were made on photographs of a reach near Ord, Nebr. Islands in this reach showed no measurable change in size, number, or position during the 19-year interval. The pattern of sandbars was of smaller scale and greater complexity on the 1938 photograph. The outside of a bend had migrated laterally by about 80 feet in 19 years, but the inside of the same bend showed little change. A straight reach showed a maximum of about 50 feet of lateral migration at one place, but at most places the river banks showed little change. RELATION OF SELECTED HYDRAULIC FACTORS TO CHANNEL PATTERN Lane (1957) discussed the factors affecting stream-channel form or shape. He did not specifically define the terms “channel form” and “channel shape,” but he used these terms interchangeably to mean the aspects of a stream channel in plan view and in cross profile. Since his paper mainly is on the aspect of a stream channel in plan view, his conclusions about channel form readily apply to channel pattern. According to Lane, the major factors that affect channel form are water discharge, longitudinal slope, sediment load, resistance of bed and banks to scour by flowing water, vegetation, temperature, geology, and work of man. Some of these factors influence channel form directly; some, indirectly. In selecting the fewest factors for adequate differentiation of channel forms, Lane considered not only the relative magnitude of the effect of each factor but also both the feasibility of expressing each quantitatively and the availability of data for each. A major objective of his paper is to show that differences in channel form can be satisfactorily explained by differences in slope, bed-bank material, and average discharge. The factors described by Lane are undoubtedly the major variables in the determination of channel pattern; and his selection of slope, bed-bank material, and average discharge as adequate variables for explaining channel pattern is reasonable. However, the usefulness of average discharge may depend on its correlation with other measures of discharge (such as bank-full discharge), inasmuch as an average discharge will not move the bed or bank material of many streams (M. G. Wolman, written communication, 1961). The writer regards bank erodibility as the most significant single variable, and bank erodibility depends mainly on the particle size of bank material and on vegetal growth along the banks. The influence of diversion dams on channel patterns of the Loup Rivers has not been evaluated in this report. Diversion dams for irrigation have been built on the North Loup near Taylor, Burwell, and Ord and on the Middle Loup near Sargent and Arcadia. In addition, the Boelus Power Canal diverts water from the Middle Loup into the South Loup near the confluence of the two rivers. Most of the irrigation works have been built since 1945, although the Boelus Power Canal has been in operation since 1936. Comparison of river reaches immediately upstream from diversion dams with reaches immediately downstream indicates that the diversion has had little effect on previously established bank-full widths or on vegetated islands. BANK ERODIBILITY Loup River banks are typically low and overgrown by grass or other vegetation, but cut banks were found and examined at many localities. The banks of the North and Middle Loup Rivers are dominantly of sand. The sand generally contains streamers of fine gravel. Most cut banks contain a few layers of silty sand; and in several places, layers of dark clayey sand, probably deposited under swampy conditions, were seen. The low banks and point-bar deposits of the South Loup are of silty sand but in many places the Elba terrace deposits, against which the river has cut laterally, are mainly of silt and clay. The generally meandering pattern of the South Loup is attributed to the silt and clay content of its banks; such a content gives these banks a moderate degree of resistance to erosion. In comparison with the North Loup and Middle Loup, the much higher content of particles less than 0.062 millimeter in diameter in the suspended load of the South Loup, as well as the higher suspended-sediment con-PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D33 centration, is considered to be a satisfactory indication of the higher silt and clay content of the South Loup’s bank materials. Schumm (1960) reported a relation between the width/depth ratio of a channel and the silt-clay content of its bed and banks. As the percentage of silt and clay in the bed and banks increases, the channel becomes deeper and narrower. In a later paper (1963), Schumm showed, for some alluvial rivers on the Great Plains, that sinuous channels are characterized by a high percentage of silt and clay in their perimeters. The writer was unable to devise a sampling procedure that would give a significant representation of the silt-clay content of banks along the Loup Rivers. The cohesive effect of silt and clay depends not only on its amount but also on its distribution in the bank materials. For example, silt and clay concentrated in layers gives a very different cohesive effect than would the same amount of silt and clay distributed uniformly through the bank materials. No attempt has been made to describe quantitatively the influence of vegetation on bank erodibility, but this influence is significant. In the Sandhills, where the water table is high, swamp vegetation grows along the river courses and confines the channels. The meandering of the courses of the Calamus River and minor flowing tributaries in the Sandhills is attributed to the influence of swamp vegetation, which decreases bank erodibility. The ability of bank materials to confine a stream depends partly on the discharge of the stream. As the discharge of a stream increases, the depth and mean velocity increase, and the banks not only are higher but also are subjected to greater erosive action. For bank materials of a given cohesiveness, a small stream would be expected to have a channel of lower width/depth ratio than a large stream. The Calamus River is described in detail because it exhibits variations in channel pattern that are attributed to the influence of vegetation on bank erodibility. The major factors that influence channel pattern, except the factor of vegetation, seem to be constant. For a distance of about 7 river miles upstream from its confluence with the North Loup, the Calamus has low sinuosity and is slightly braided (fig. 22A). Farther upstream, the Calamus becomes distinctly meandering (fig. 23A). Discharge measurements for a section near Harrop, which is about 25 miles upstream from the confluence, and for another section near the confluence at Burwell show that discharge over the lower 25 miles of the river course is nearly constant. (See table 2.) Bank materials are derived from dune sand and are uniform from place to place. Differences in bank erodibility are attributed to differences in vegetal growth along the banks; swamp vegetation is denser along the upper course of the river, probably because of the higher water table. The water table is higher along the upper course of the river because the sand dunes that border the river valley are higher than those along the lower course, which is near the edge of the Sandhills. At some places within the Sandhills, such as near Ashby, Nebr., the water table stands sufficiently high that flat areas between the dunes are marshy or are occupied by lakes. The long profile of the lower 25 miles of the Calamus is shown in figure 24. The profile was drawn from Geological Survey topographic maps (scale, 1: 24, 000; contour interval, 10 ft), and plotted points represent locations at which successive 10-foot contours cross the river. The slope is about 0.0011 foot per foot for the entire 25-mile river profile except for two minor bulges in the profile. The valley slope is slightly steeper in the upper part of the profile. Sinuosity indices and mean widths were measured on the topographic maps for 15 reaches, each of which is bounded by successive 10-foot contour lines and represents a 10-foot drop in altitude and a horizontal distance of about 10,000 feet along the river. The reaches range in sinuosity index from 1.01 to 2.1, and they range in mean width from 50 to 300 feet. The wider reaches, which are within 5 miles of the confluence, are slightly braided (braiding index, about 0.1) but contain numerous submersed sandbars. River slope of the Calamus, as measured on the topographic maps, shows no consistent relation to width nor to sinuosity index. A reach having a mean width of about 50 feet has the steepest slope (0.0016), and another reach having the same mean width has the gentlest slope (0.0008). Along many of the 10,000-foot reaches, the river shows little variation in width, and any significant difference in slope between uniformly narrow reaches and uniformly wide reaches would have been recorded by the 10-foot contours. Like the Middle Loup at Dunning, the Calamus River evidently adjusts from wide sections to narrow sections not by change in slope but by change in effective cross-sectional area. Valley slope, as measured along a line that passes through the inflection points of river bends, also shows no consistent relation to width or to sinuosity index. Figure 25 shows the relation between width and sinuosity index. Inasmuch as slope and discharge are nearly uniform, width may be regarded as the practical measure of bank erodibility—the wider a section, the more erodible its banks. Low sinuosities are asso-D34 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 DISTANCE FROM CONFLUENCE WITH NORTH LOUP RIVER, IN THOUSANDS OF FEET Figure 24.—Longitudinal river profile and valley profile, Calamus River, Nebr. Profile extends 25 miles upstream from confluence with North Loup River. n BANK ERODIE ULITY — —> LOW — M( DDE r RAT F IGH / / ✓ \ / / / / / s \ / ^ 1 ✓ / ' S • • \ \ X \ \ \ ' \ X > X \ > .01----lx _____________________ 50 60 70 80 90100 200 300 WIDTH, IN FEET Figure 25.—Diagrammatic relation of sinuosity index to width and bank erodibility, Calamus River. Width is regarded as a practical measure of bank erodibility. ciated with high and low bank erodibilities, and high sinuosities are associated with intermediate bank erodibilities. The relations among channel pattern, bank erodibility, and slope along the Calamus River, as interpreted by the writer, are summarized as follows: For a given discharge, channel pattern is determined mainly by bank erodibility and valley slope. Differ- ences in bank erodibility are mainly determined by vegetal growth along the banks. Valley slope is mainly predetermined by regional slope and by the slope of the Elba terrace. A moderate degree of bank erodibility, controlled by swamp vegetation in the upper part of the river segment under consideration, led to the development of a meandering pattern and a moderately narrow width. Along reaches where vegetal resistance to bank erosion was high, the river course is nearly straight and tends to be narrow. Along reaches having low vegetal resistance to bank erosion, as in the lower part of the segment, the river developed a broad, shallow, and somewhat braided course. The general slope of the long river profile is rather uniform, and hydraulic adjustment to local variations in width is mostly made by changes in cross-sectional area rather than by changes in slope. Friedkin (1945) attributed changes in channel pattern along the length of the Mississippi River to differences in bank erodibility. His experiments with small-scale rivers showed that entirely resistant banks led to the development of a deep, narrow channel having a flat channel slope and that less resistant banksPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D35 led to a slowly meandering channel having a moderately flat slope. A braided pattern having steep slopes was developed where banks were very erodible. Friedkin’s conclusions apply satisfactorily to the Loup Rivers except that a braided reach may have an only slightly greater channel slope than a meandering reach of nearly the same discharge. Also, a lush growth of vegetation may serve to confine a stream between otherwise erodible banks. DISCHARGE Leopold and Wolman (1957) tentatively concluded that channel width is mainly a function of discharge and that the quasi-equilibrium width of a channel is determined by bank-full or barely overbank discharges. Width also is related to average discharge. Leopold and Maddock (1953) showed that, for many rivers, width increases downstream as the 0.5 power of average discharge. The Loup River system is exceptional among the eight rivers plotted by Leopold and Maddock, inasmuch as width increases downstream as the 0.75 power of average discharge. The concept of bank-full discharge is difficult to apply to the upper reaches of the Loup Rivers, which have such constancy of discharge that they rarely attain the bank-full stage. In addition, the Loup Rivers are bordered by a series of low terraces of different heights. The height of the river banks varies from place to place and depends on which terrace level the river has cut against laterally. For an approximation of bank-full discharge, an average of the maximum daily mean discharges for the years of available record during the period 1946-57 was computed for each gaging station. For some stations, only 2 years of record was available for the period. (See table 2.) For those reaches attaining a bank-full or overbank stage, the average of the maximum daily mean discharges agrees reasonably well with the bank-full discharge as estimated from gage heights. For the Loup Rivers, mean width was plotted against average discharge and also against bank-full discharge. Very different results were obtained from the two plots. Width increases for each river according to the following powers of average discharge: South Loup, 1.8; Middle Loup, 1.3; and North Loup, 0.7. Width increases for each river according to the following powers of bank-full discharge: South Loup 0.6; Middle Loup, 0.8; and North Loup, 0.45. In general, therefore, bank-full width increases at a greater rate relative to average discharge than to bank-full discharge. Figure 26 shows the relation of width to mean annual discharge and bank-full discharge for Loup River reaches at the gagin stations listed in table 2. When all points on the plot are considered, the visually approximated curve for meandering reaches has a slope of about 0.5. The point representing the Middle Loup near Seneca does not fall on this curve. However, the measurements for the Middle Loup near Seneca were made on a topographic map of scale 1:62, 500 whereas measurements for the other reaches were made on aerial photographs. Points representing braided or sinuous reaches and straight reaches are widely scattered; but the widths represented are generally greater than the widths of meandering reaches. Much of the scatter of points is attributed to the pronounced local variation in width that is characteristic of the North Loup and Middle Loup. This variation in width is, in turn, attributed to banks whose credibility is high insofar as particle size is concerned but whose erodibility is much decreased locally by vegetation. Variability of discharge is a possible factor in the development of channel pattern and river width. A rought measure of variability of discharge was obtained by relating the discharge for the month having the highest flow during a water year to the discharge for the month having the lowest flow. For the period 1946-57, averages were computed for the maximum monthly discharges and for the minimum monthly discharges. The mean deviation of the average monthly maximums and minimums (relative to the average discharge for the period of record) was divided by the average discharge to give a coefficient of deviation of discharge. (See table 2.) An accurate variability index, such as that proposed by Mitchell (1957), requires the use of flow-duration curves, which are not available for the Loup Rivers. For the Loup Rivers, no direct relation between variability of discharge and channel pattern is apparent, but variability of discharge may have a significant effect on increase in width with downstream increase in discharge. The channel pattern of the Calamus River changes greatly along the last 25 miles of its course despite the fact that variability of discharge is low and constant. The reach having the highest coefficient of deviation of discharge (1.06 for the South Loup River at St. Michael) is meandering, and many reaches in the upper Middle Loup that have low coefficients of devaition (probably less than 0.2) are also meandering. Downstream increase in width with increasing average discharge—higher for the Loup Rivers than for other rivers plotted by Leopold and Maddock (1953)—may perhaps be attributed to the downstream increase in variability of discharge shown by the Loup Rivers (table 2). River width increases not only with downstream increase in all measures ofMEAN WIDTH, IN FEET MEAN WIDTH, IN FEET D36 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA a 6 -8 19 E# J!1 4 15 4 « 20 12 S n 13. + 14 10 o 16. J_ 2 • ( J -3 17 L) o 100 1000 MEAN ANNUAL DISCHARGE, IN CUBIC FEET PER SECOND + Braided reach, total B. I. 1.5 • Reach having B. I. <1.5, S. I. <1.3 O Meandering reach, S. I. S'1.3 , Standard deviation of width 100 1000 10,000 100,000 AVERAGE MAXIMUM DAILY DISCHARGE, IN CUBIC FEET PER SECOND Figure 26.—Width in relation to mean annual discharge and average maximum daily discharge (approximately bank-full discharge) for different channel patterns. Numbers refer to gaging stations listed in table 2.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D37 discharge but also with increase in bank-full discharge relative to average discharge. BED MATERIAL AND SUSPENDED-SEDIMENT CONCENTRATION In the northern part of the Loup River basin, where the flow of the Loup Rivers is derived mainly from ground water rather than from surface runoff, the rivers derive most of their total sediment load from bank erosion. Any material supplied by surface runoff is fine- or meduim-grained sand because the ground is almost entirely mantled by dune sand. In the loess-mantled southern part of the basin, surface runoff makes substantial contributions to streamflow, and the sediment supplied is mainly silt. J. C. Busby (written communication, Nebraska Univ. 1950) investigated the bed material, particularly the sands, of the Middle Loup River. The bed material was sampled at depths ranging from 1 inch to 10 feet by driving an iron casing into the river bed at seven locations. Penetration by the casing into the bed at some of the sampling points was halted by coarse gravel or clay at a depth of about 5 feet, but the river bed at most points was free of gravel to a depth of at least 9 feet. In general, the median diameter of the sand increases somewhat as the depth below the surface of the stream bed increases. A typical increase is shown by the sampling of the Middle Loup River near Loup City in that the median diameter of bed material is about 200 microns in the first foot and about 400 microns in the ninth foot. Busby computed the average median diameter for the first 3 feet in each cross section and plotted the results to show downstream changes. The averages were 0.27 millimeter at Thedford, 0.38 millimeter at Dunning, 0.25 millimeter at Walworth, 0.26 millimeter at Loup City, and 0.29 millimeter at Rockville. Very little sand coarser than 0.8 millimeter was found in any of the samples. In general, the bed material is well sorted, and the size distribution is only slightly skewed. In their report on sediment transportation in the Middle Loup River at Dunning, Hubbel and Matejka (1959) discussed size analyses of bed material taken with a piston-type sampler from cores about 2 inches in diameter and about 6 inches in length. For the coarest of 187 samples analyzed, 80 percent of the material was finer than 4 millimeters and 27 percent was finer than 0.5 millimeter. The samples were collected at irregular intervals between September 1949 and September 1952. Bed material has been sampled by the Geological Survey at four stations along the North Loup over periods ranging from 1 year at Cotesfield to 4 years at Burwell. For the South Loup River, bed material has been sampled only at St. Michael. Analyses of 24 samples collected by the Geological Survey during 1952 and 1953 indicate that the bed material of the South Loup is mainly fine- to medium-grained sand. Bed material of the South Loup is somewhat finer than that of the North Loup or Middle Loup, but all three rivers have sandy beds. Averages of particle-size analyses of bed material from the Loup Rivers are shown in table 2. According to these analyses, most of the bed material of the Loup Rivers is medium- or fine-grained sand; there is little difference in size of bed material from one river to the next or from place to place along the same river. Even where the Loup Rivers appear to be only slightly turbid, the suspended-sediment discharge is obviously a significant part of the total sediment discharge. The measured suspended-sediment discharge of the Middle Loup River at Dunning, where the river appears to be clear rather than muddy, averages about one-half of the total sediment discharge as measured at the turbulence flume (Hubbell and Matejka, 1959). Average suspended-sediment concentrations, in parts per million, were computed for each station by J. C. Mundorff of the Quality of Water Branch in Lincoln, Nebr. The South Loup River, which exhibits the most consistently meandering pattern, also has the highest suspended-sediment concentration. The average amount of clay (particles less than 0.004 mm in diameter) in the suspended sediment of the South Loup River at Cumro is about 24 percent, whereas the average amount for stations on the North Loup and Middle Loup is about 9 percent. Schumm (1963, p. 1094) suggested that the silt-clay content of the bed and banks may be representative of the type of sediment load transported by a stream. The higher the percentage of total sediment load transported as wash load (silt and clay), the higher is the silt-clay content of bed and banks and the more sinuous is the stream likely to be. This suggestion seems valid and it is supported by comparison of the South Loup with the North Loup and Middle Loup. Unfortunately, information on total sediment load is not available for most rivers. The braided pattern is sometimes attributed to an excess of total load supplied to the stream—that is, to a supply of sediment that exceeds the capacity of the Stream. The capacity of a stream is difficult to determine either by theory or by measurement. An excess of total load might be indicated by aggradation, and no excess of total load might be indicated by degradation. Geological evidence indicates that the Loup Rivers are slowly degrading.D38 CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA WATER TEMPERATURE Hubbell (1960, p. 22) discussed the significant influence of water temperature on the hydraulics of the Middle Loup River at Dunning. Dune heights on the river bed become distinctly lower as water temperature decreases, and the bed is smoothest at water temperatures near 32° F. Small sediment discharges and large flow resistances are associated with high temperatures whereas large sediment discharges and small flow resistances are associated with low temperatures. Hubbell also noted that the shallowest depths, narrowest widths, and highest velocities are associated with the lowest temperatures. Figures 27 shows the relation of suspended-sediment concentration to water temperature and discharge at sections A, E, and B on the Middle Loup at Dunning. Data from Hubbell and Matejka (1959) were grouped according to temperature and discharge, and means of the groups were plotted. Means of water temperature applied to measurements grouped according to successive 10°F temperature intervals, beginning with the interval 30°-40° F. Within each temperature interval, means of mean suspended-sediment concentration were based on measurements grouped according to discharge. For example, a mean was computed for all suspended-sediment concentrations listed for discharges of 350-399 cfs at water temperatures of 40°-49° F. For a given discharge range, suspended-sediment concentration decreases sharply as temperature increases. Figure 28 shows the relation of velocity to water temperature and discharge at sections A and E. Means of the variables were grouped as described for figure 27. The decrease in velocity with increase in temperature accounts for part of the scatter of points in figure 15, for which temperature was not taken into account. For the Middle Loup at Dunning, temperature has a significant effect on suspended-sediment concentration and velocity as well as an effect (probably indirect) on bed form; however, these effects do not necessarily apply to streams in general. Because much of the suspended sediment in the Middle Loup is fine sand, increase in viscosity with decrease in temperature has a significant effect on the vertical distribution of suspended sediment. For streams whose suspended sediment is mostly silt and clay, increase in viscosity has less effect on vertical distribution of suspended sediment. The change in bed form of the Middle Loup is probably related to a slight change in velocity through a critical value, above which dunes are built and below which dunes are flattened. For other streams, change in temperature might not cause velocity to change through this critical value. A measurable effect of temperature on channel pattern and slope might be expected because of the significant influence of temperature on Loup River hydraulics. The measurements at sections A and E on the Middle Loup at Dunning were analyzed to detect any effect of temperature on effective width, effective area, or slope. The number of measurements made at similar discharges for the whole range of water temperatures was insufficient for any definite conclusions to be drawn. However, both effective 2000 O 1000 cn < < cn h- 200 2000 □ LLl o Z LlI gj 1000 z> z < X \ \ \ 'x, \ \ A K \ o \ 1 \ o \ oN • o \ • SEC' ION A .' r-r~l 200 30 A A 0 A A X 8 X • 9 Ox • A o O • SEC! 'ION B Discharge groups • 300-349 cfs o 350-399 cfs x 400-449 cfs a 450-500 cfs O A \ \ V X A \ \ \ \ . \ • ► • •°J X SEC riON E — O 50 90 30 50 90 MEAN OF WATER TEMPERATURES, IN DFOREES FAHRENHEIT Figcre 27.—Suspended-sediment concentration in relation to temperature and discharge, sections A, E, and B, Middle Loup River at Dunning, Nebr.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D39 o o tz UJ CL o o > z < o z < 3.0 2.5 2.0 1.5 3.5 3.0 2.5 + A A + O A SECTION A O + © o 2.0 A + O + 0 + A SECTION E • o i 8 4 tt 30 40 50 60 70 80 90 100 MEAN OF WATER TEMPERATURES, IN DEGREES FAHRENHEIT Discharge groups o 300-349 cfs + 400-449 cfs o 350-399 cfs a 450-500 cfs Figure 28.—Velocity In relation to temperature and discharge, sections A and E, Middle Loup River at Dunning, Nebr. width and effective area show a general tendency to increase as temperature increases. At section A, for discharges in the range of 400-449 cfs, mean effective width increases by 20 percent as water temperature increases from 35° to 65 °F, and mean effective area increases by 30 percent. At section E, for discharges in the range of 350-399 cfs, mean effective width does not change as temperature increases from 34° to 74°F, but effective area increases by 21 percent. Changes in local slope could not be related to changes in water temperature. In spite of the influence of temperature on width at section A. there is nevertheless a significant correlation of width to effective area, mean velocity, and maximum depth. (See fig. 15.) LONGITUDINAL RIVER SLOPE AND VALLEY SLOPE Inspection of figures 8, 9, and 10 shows that the longitudinal river profiles of the North, South, and Middle Loup Rivers are nearly straight except for some steepening in the upper reaches and some flattening in the lower reaches. River profiles of the North Loup and Middle Loup are similar in slope, but in valley profile the Middle Loup is slightly steeper than the North Loup. The general river slope of the South Loup (0.0011 ft per ft) is somewhat gentler than that of the Middle Loup (0.00125 ft per ft). The general valley slope of the South Loup (0.0018 ft per ft) is steeper than that of either the Middle Loup or North Loup. General straightness of the river profiles is attributed mainly to the straightness of the predetermined slope over which the rivers flow—the slope of the Great Plains. Because of the near constancy of slope that accompanies a significant variation in discharge and channel pattern, no good correlation of slope with discharge and channel pattern would be expected. However, river slopes of the 9,600-foot reaches at gaging stations were measured on topographic maps and plotted against bank-full discharge (fig. 29). The plot shows that the slopes of most meandering reaches are less than the slopes of most braided reaches of similar discharge. The points for both meandering and braided reaches represent an insufficient range in slope and discharge for any general relations to be established. Also, channel width should be taken into account simultaneously with the other variables. Indeed, analysis of channel characteristics of the Middle Loup at Dunning indicates that local slope shows little or no correlation with any variable except width. For a given discharge and size of bed material, braided reaches of rivers in general probably have steeper channel slopes and greater effective widths than meandering reaches. Lane (1957) plotted slope-discharge relations for meandering sandbed streams, for braided sandbed streams, and for streams in gravel. He proposed that, for streams in each of these categories, slope varies inversely as the fourth root of average discharge. Also, meandering sandbed streams have lower slopes than braided sandbed streams of the same average discharge. Leopold and Wolman (1957) plotted channel slope against bank-full discharge for rivers that represent a wide range of discharges and bed-bank materials. A line, whose slope varies inversely as the 0.44 power of discharge, separates meandering streams from braided streams on their plot; and meandering streams have lower slopes than braided streams of similar bank-full discharge. Lane (1957, p. 36) discussed the influence of valley slope on channel pattern. According to one viewpoint, a river meanders on a steep slope to reduce channel slope to an equilibrium value; therefore, the steeper the valley slope, the more sinuous the channel. Lane argued against this viewpoint; he maintained that steep valley slopes promote the braided pattern and that low valley slopes promote the meandering pattern. The lower the valley slope, the more sinuous the stream is likely to be, except that the sinuosity may be again reduced on very low slopes.D4Q CHANNEL PATTERNS AND TERRACES OF THE LOUP RIVERS IN NEBRASKA AVERAGE MAXIMUM DAILY DISCHARGE, IN CUBIC FEET PER SECOND + Braided reach, total B. I. -•? 1.5 • Reach having B. I. <1.5, S. I. <1.3 ® Meandering reach, S. I. ^ 1.3 Figure 29.—Reach slope In relation to the average maximum daily discharges for meandering and braided reaches. Numbers refer to gaging stations listed in table 2. Experiments were done by Friedkin (1945, p. 10), to investigate the effect of valley slope on sinuosity. Within the range of slopes used in the experiments (0.006-0.009), sinuosity (and also the length and width of individual bends) increased as valley slope increased. However, the rate of increase in sinuosity was much reduced as the slopes became higher, and further increase in valley slope would perhaps have resulted in a decrease in sinuosity. No generalities about the relation between valley slope and channel pattern are valid without the specification of other variables, of which bank erodibility, bed material, and discharge are probably the most significant. Under suitable conditions, streams will meander on steep slopes. For example, Leopold and Wolman (1960, p. 774) described a meandering channel having a slope of 0.023 and a discharge of about 2.4 cfs that was carved in glacier ice by streams of melt water flowing on the glacier surface. Dury (1959) described meandering gullies on a steep-sided “spoil-heap.” Relations between discharge and bank erodibility that are favorable for development of the meandering pattern probably cause development of maximum sinuosity at some optimum value of valley slope and cause a decrease in sinuosity at slopes both greater and smaller than this optimum value. The reach of the South Loup that is on the steepest segment of the longitudinal valley profile (fig. 10) has the lowest sinuosity index. Elsewhere, the valley slope is nearly constant, but different reaches have considerably different sinuosities. The differences in sinuosity are attributed to local differences in bank erodibility. For a given valley slope, river slope necessarily decreases as sinuosity increases. Because the valley slope of the South Loup is nearly constant upstream from the steep segment of its profile, the river slope upstream from this segment decreases as sinuosity index increases. REFERENCES CITED Condra, G. E., Reed, E. C., and Gordon, E. D., 1950, Correlation of the Pleistocene deposits of Nebraska: Nebraska Geol. Survey Bull. 15A, p. 1-2. Dury, G. H., 1959, The face of the earth: Baltimore, Md., Penguin Books, 223 p. Friedkin, J. F., 1945, A laboratory study of the meandering of alluvial rivers: Vicksburg, Miss., U.S. Waterways Expt. Sta., 40 p. Frye, J. C., Leonard, A. B., and Swineford, Ada, 1956. Stratigraphy of the Ogallala formation (Neogene) of northern Kansas: Kansas Geol. Survey Bull. 118, p. 1-92. Hubbell, D. W., 1960, Investigations of some sedimentation characteristics of sand-bed streams. Progress reports: U.S. Geol. Survey open-file rept., 78 p., 15 figs. Hubbell, D. W., and Matejka, D. Q., 1959, Investigations of sediment transportation, Middle Loup River at Dunning, Nebraska: U.S. Geol. Survey Water-Supply Paper 1476, 123 p.PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS D41 Lane, E. W., 1957, A study of the shape of channels formed by natural streams flowing in erodible material: U.S. Army Corps of Engineers, Missouri River Div., Omaha, Nebr., Sediment Ser. 9,106 p., 15 figs. Leopold, L. B., and Maddock, Thomas, Jr., 1953, The hydraulic geometry of stream channels and some physiographic implications: U.S. Geol. Survey Prof. Paper 252, 57 p. Leopold, L. B., and Wolman, M. G., 1957, River channel patterns—braided, meandering, and straight: U.S. Geol. Survey Prof. Paper 282-B, 85 p. ------1960, River meanders: Geol. Soc. America Bull., v. 71, p. 769-794. Miller, J. P„ 1958, High mountain streams—effect of geology on channel characteristics and bed material: New Mexico Inst. Mining and Technology Mem. 4, 53 p. Miller, R. D., and Scott, G. R., 1955, Sequence of alluviation along the Loup Rivers, Valley County area, Nebraska : Geol. Soc. America Bull., v. 66, p. 1431-1448. Miller, R. D., and Scott, G. R., 1961, Late Wisconsin age of terrace alluvium along the North Loup River, central Nebraska—a revision: Geol. Soc. America Bull., v. 72, p. 1283-1284. Mitchell, W. D., 1957, Flow duration of Illinois streams: Springfield, Illinois Dept. Public Works and Buildings, Div. Waterways, 189 p. Rubey, W. W., 1952, Geology and mineral resources of the Hardin and Brussels quadrangles (in Illinois) : U.S. Geol. Survey Prof. Paper 218, 179 p. Schumm, S. A., 1960, The effect of sediment type on the shape and stratification of some modern fluvial deposits: Am. Jour. Sci., v. 258, p. 177-184. ------1963, Sinuosity of alluvial rivers of the Great Plains: Geol. Soc. America Bull., v. 74, p. 1089-1100. Smith, H. T. U., 1955, Use of aerial photography for interpretation of dune history in Nebraska, U.S.A.: Intemat. Congr., 4th, Rome-Pisa 1953, Assoc, for Study of the Quaternary, 7 p. U.S. GOVERNMENT PRINTING OFFICE : 1964 O—719-579-i '] /V-< I 4IV earth .Sciences sc . i :£**** UBRARY a^2. ' /h^o, V x >* • ' Sediment Yield of the Castaic Watershed, Western Los Angeles County California—A Quantitative Geomorphic Approach GEOLOGICAL SURVEY PROFESSIONAL Prepared in cooperation with State of California Department of IVater Resources DOCUMENTS department I APR 9 1958 1 LIBRARY \ uhwcrsity of cm mm\ PAPER 422- FSediment Yield of the Castaic Watershed, Western Los Angeles County California—A Quantitative Geomorphic Approach By LAWRENCE K. LUSTIG PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-F Prepared in cooperation with State of California Department of Water Resources UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1965UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 65 centsCONTENTS Page Abstract_______________________________________________ FI Introduction__________________________________________ 1 Statement of the problem and the approach employed______________________________________________ 2 Acknowledgments and personnel______________________ 2 The Castaic watershed___________________________________ 2 Physical description of the area___________________ 2 Location and extent____________________________ 2 Topography and drainage________________________ 2 Climate________________________________________ 3 Vegetation and soils___________________________ 4 Summary of geology_____________________________ 5 Sources of sediment________________________________ 6 Long-term channel erosion______________________ 8 The San Gabriel watersheds_____________________________ 10 Physical description of the area__________________ 10 Location and extent___________________________ 10 Topography and drainage_______________________ 10 Climate_______________________________________ 11 Vegetation and soils__________________________ 11 Summary of geology____________________________ 11 Sediment-yield data_______________________________ 11 Page Quantitative geomorphology_____________________________ F12 General discussion___________________________________ 12 Basic-data collection________________________________ 12 Geomorphic parameters________________________________ 13 Relief ratio_____________________________________ 14 Sediment-area factor_____________________________ 15 Sediment-movement factor_________________________ 17 Total stream length______________________________ 17 Transport-efficiency factors_____________________ 18 Discussion of results________________________________ 19 Significance of correlations_____________________ 19 Similarity between the Castaic and the San Gabriel Mountains watersheds___________________ 20 Estimated sediment yield of the Castaic watershed_____________________________________________ 20 Range of the sediment-yield values for the Castaic watershed___________________________________ 21 Conclusions_______________________________________________ 22 References________________________________________________ 22 ILLUSTRATIONS [Plates are in pocket] Plate 1. Index map showing the location of selected watersheds, Los Angeles County, Calif. 2. Map of the Castaic watershed, Los Angeles County, Calif., showing the drainage net, approximate lithologic dis- tribution, traces of major faults, and sediment-sample locations and particle-size distributions. Figures 1-7. Photographs showing— Pa8e 1. View downstream at the single-stage sampling site in the lower reach of Castaic Creek------------- F3 2. The Liebre Mountain area from the south__________________________________________________________ 4 3. The metamorphic-igneous complex in Elizabeth Lake Canyon___________________■--------------------- 5 4. The relatively undeformed Tertiary sedimentary rocks in the southern part of the Castaic watershed.. 6 5. A typical outcrop of the Tertiary sedimentary rocks in the Castaic watershed_____________________ 7 6. Rockfall of conglomerate blocks in subbasin 2 that has produced a natural debris dam------------- 9 7. Exposure of the roots of a tree in Ruby Canyon by channel erosion________________________________ 10 8. Graph showing relation of sediment yield and relief ratio for watersheds in the San Gabriel Mountains- 16 9. Diagrammatic longitudinal section of a watershed that contains three main groups of hills, showing the geometry of the relief ratio and sediment-area factor__________________________________________________ 16 10-15. Graphs showing relation, in watersheds in the San Gabriel Mountains, of sediment yield and— 10. Sediment-area factor____________________________________________________________________________ 17 11. Sediment-movement factor________________________________________________________________________ 17 12. Total stream length_____________________________________________________________________________ 18 13. Transport-efficiency factor Ti__________________________________________________________________ 18 14. Transport-efficiency factor T2__________________________________________________________________ 18 15. Transport-efficiency factor 7j__________________________________________________________________ 19 mIV CONTENTS TABLES Pago Table 1. Weight percentage of granules, sand, and silt-clay in the granule-to-clay size fraction of samples from the Castaic watershed__________________________________________________________________________________________________ F7 2. Data on reservoirs and sediment yield of watersheds in the San Gabriel Mountains, Calif_____________________ 11 3. Morphometric data and geomorphic parameters of the Castaic drainage basin and of watersheds in the San Gabriel Mountains, Calif___________________________________________________________________________________________ 14 SYMBOLS u Order of a stream, where unbranched tributaries are designated as first order and the confluence of two streams of a given order is designated by the next higher order number Au Area of a basin of order u, where u is the highest order number of the streams contained Lu Length of stream or stream channel of order u LL Total stream length Nu Number of streams of order u LN Total number of streams 6U Gradient of stream channel of order u dg Ground slope, measured orthogonal to contours E Mean basin elevation D Drainage density, where D=LL/Au F Stream frequency, where F=NJAU Rb Bifurcation ratio, where Rb=NJNu+i Ra Basin-area ratio, where Ra=AJAu_1 Rl Stream-length ratio, where Rl=Lu/Lu_i Rc Stream-channel-slope ratio, where RC=0U/0U+1 R{ Ruggedness index, where R{=DXE Rn Relief ratio; Rb=H/Lb, where H is basin relief and Lb is basin length Q Water discharge Sv Sediment yield SA Sediment-area factor, where Sa — Av/cos 6g and Ap is planimetric area SM Sediment-movement factor, where SM=SAX sin 6g Te Transport-efficiency factors, where Te= Tu T2, Tn S L _ Tx Transport-efficiency factor, where I\=Rby.LL T2 Transport-efficiency factor, where T2=LNXRC T3 Transport-efficiency factor, where T3=(Ni + N2)(Rc„) + (N2+N3)(R'h)+. ■ . + (Nb_1+Ab) Ic Intercept value giving the sediment yield of Castaic drainage basin r Simple-correlation coefficient rT Rank-correlation coefficient H„ Null hypothesis for one-sided tests a Confidence interval X Linear-scale ratio , is the mean bifurcation ratio and 2L is total stream length, (5) a transport-efficiency factor, T2 = XN X f?c, where ZV is the total number of streams and TZC is the mean stream-channel-slope ratio, and (6) a transport-efficiency factor, T,3=(iVi-|-Ar2)(i?Cl/2)-|-(Arj+Nj)(RC2/^)-\- (iV„_i-f Nn) (Rcn_1/n), where the subscripts designate stream order. These parameters are plotted against the known long-term sediment yield as simple regressions for six watersheds in the San Gabriel Mountains. Both parametric and nonparametric tests of the computed correlation coefficients show that the relationships are significant at the 95-percent confidence level. For these watersheds, relief ratio correlates poorly with sediment yield and cannot be used. Variation in basin shape is thought to be the primary cause of this failure. The value of each parameter computed for the Castaic watershed is substituted into the appropriate regression-line equation obtained for watersheds in the San Gabriel Mountains; thus, a series of sediment-yield values is provided. The data suggest that an estimate of 250 acre-feet per year will approximate the long-term sediment yield of the Castaic watershed. This estimate depends, to some extent, on an assumption of balance between the dynamic factors and the geometric properties of the Castaic watershed and the watersheds in the San Gabriel Mountains. Additional support for the estimated annual sediment yield is provided by an assessment of the yield that would be expected (1) from consideration of the contrast in effective precipitation between the Castaic and San Gabriel watersheds, and (2) from consideration of the difference in drainage area between the Castaic watershed and the neighboring Piru watershed. These methods provide sediment-yield values of about 280 and 220 acre-feet per year, respectively, which suggest the possible range in average annual sediment yield and lend added credence to the previous estimate of 250 acre-feet per year. Data on the size distribution of the granule-to-clay size fraction of sediments in the Castaic watershed are presented. It is shown that the sedimentary rocks and the metamorphic-igneous complex that occur in the watershed contribute approximately equal quantities of sediment in the silt-clay size range of about 8 percent, whereas the granitic rocks contribute one-half of this amount. The contribution of the granitic rocks is less because these rocks crop out over a smaller area. Sediment in the sand-sized range is abundant everywhere in the watershed, and certain suggestions for future debris-dam locations based upon the yield of sand from subbasins, are given. The net long-term channel erosion in the Castaic watershed is discussed on the basis of data on the cores of trees whose roots have been exposed by channel erosion. The data suggest that the channels do not contribute a large percentage of the total sediment yield and, hence, that sheet erosion of hillslopes is responsible for most of the sediment production. INTRODUCTION The life expectancy of reservoirs has been of practical importance to man since he first began his interference with the natural location of water supplies. The ancient civilizations often rose or fell in accord with the success or failure of their aqueduct and storage systems in arid regions (Glueck, 1959), as attested to by the many abandoned cities discovered by the modern archaeologist. Because we can no longer afford such disruption, considerable study is devoted to the several problems pertinent to the location of any proposed reservoir site. A segment of the California State Water Project requires the construction of three terminal reservoirs for water storage in southern California. One of these reservoirs is to be created by the construction of a dam below the junction of Castaic and Elizabeth Lake Canyons in the Castaic watershed in southern California. One factor that will influence the life expectancy of this proposed reservoir is the anticipated long-term sediment yield, which is discussed in this report. FlF2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS STATEMENT OF THE PROBLEM AND THE APPROACH EMPLOYED That the magnitude of the problem of sediment yield above reservoir sites is a function of the planned storage capacity and of the operation of such reservoirs rather than of the absolute yield should be recognized at the outset. A sedimentation rate of 200 acre-feet per year in a given watershed, for example, will result in a life expectancy of 50 years if the planned capacity is but 10,000 acre-feet and of 500 years if the capacity is to be ten times as great. This comparison is, of course, axiomatic but it is the reason why equal sedimentation rates in different watersheds may be described as either of critical or negligible import. This report, however, is concerned solely with the absolute long-term sediment yield of tbe Castaic watershed; the problem treated is how best to determine this yield. There are three possible methods for determining the sediment yield of a given watershed. As outlined by Gottschalk (1957) these methods are (1) obtaining sediment-load data directly, (2) estimating rates of erosion within the watershed, and (3) comparing the watershed with neighboring basins for which sediment-yield data are available. Because most of the streams in the Castaic watershed are intermittent and few data on water and sediment discharge are extant, the first of these methods is not applicable to the problem. The second method would require that pins be driven into valley walls and that careful surveys of stream channels be conducted over a period of years. When combined with some data on dendrochronology, such an approach might provide an estimate of the rate of erosion within the watershed and the long-term yield to be expected. Because of one of the boundary conditions of the problem, namely that an estimate of the sediment yield be provided within a year, this approach could not be employed either. The last method (3) was therefore chosen by process of elimination. Although the time limitation just mentioned did not allow as complete an exploration of this method as might be deemed desirable, particularly in regard to comparison by field studies, it did not hinder comparison by morphometric analysis. This tool of quantitative geomorphology provides the main basis for the estimate of sediment yield given in this report. Supported by field observations, sediment analyses, and other data, this approach offered the sole possibility of success under the given conditions. The neighboring watersheds that provide the basis for comparison are in the San Gabriel Mountains. Sediment-yield data on the basins selected for study encompass a period ranging from approximately 30 to 40 years, and the basins meet the requirements for a comparative approach. ACKNOWLEDGMENTS AND PERSONNEL The study reported on here was conducted under the supervision of George Porterfield, who accompanied the writer in the field on several occasions, aided in selection of locations for the installation of single-stage sampling devices, and lent much encouragement during various phases of the investigation. Mr. H. V. Peterson accompanied the writer during an initial reconnaissance of the Castaic watershed and neighboring watersheds in the San Gabriel Mountains and provided several useful suggestions regarding the conduct of the study, as well as stimulating discussions of the problems involved. The writer received the benefit of critical review from S. A. Schumm, L. B. Leopold, and H. V. Peterson and gratefully acknowledges their several suggestions, which did much to improve this report. Sediment samples from the canyons of the Castaic watershed were collected with the assistance of A. D. Lovelace whose capable service in the field is appreciated. The size distribution of these samples was determined by Y. L. Gamble by means of the visual-analysis method. Single-stage sampling devices were planned and assembled by J. M. Knott and were installed in the Castaic watershed by Mr. Knott, in company with the writer. Sediment-yield data on the watersheds in the San Gabriel Mountains were kindly provided by M. F. Burke, Division Engineer, Los Angeles County Flood Control District. This project was conducted in cooperation with the California Department of Water Resources. THE CASTAIC WATERSHED PHYSICAL DESCRIPTION OF THE AREA LOCATION AND EXTENT The Castaic watershed is in western Los Angeles County, to the northwest of the San Gabriel Mountains (pi. 1). Its areal extent of approximately 137 square miles is bounded by Antelope Valley in the Mohave Desert to the north, San Francisquito Canyon to the east, U.S. 99 to the south and southwest, and by the Piru watershed to the west. The area of approximately 18 square miles that is contiguous with the Castaic watershed and that borders it on the northeast is not considered in this report. Although this small area contains lakes, it is topographically separated from the Castaic watershed, and surface flow between the two areas does not normally occur. TOPOGRAPHY AND DRAINAGE Elevations within the Castaic watershed range from about 1,200 to 5,700 feet; the mean basin elevation,SEDIMENT YIELD OF THE CASTAIC WATERSHED, CALIFORNIA F3 computed by methods to be later described, is 3,240 feet. Canyons are both deep and steep sided. The mean ground-slope angle is about 40°, indicating that level ground is not prevalent save along stream courses. Many of the steepest slopes occur within that part of the basin in which sedimentary rocks crop out (pi. 2). Erosion along joint planes and the tendency toward block fracturing of nearly horizontal strata have produced many canyons whose lower walls approach the vertical. The drainage is moderate to good, its density being 2.23 miles per square mile. Most streams within this fifth-order watershed are intermittent, but the flow persists during all but the hottest months of the year. Surface flow in Castaic Canyon normally does not reach the gaging station (pi. 2) because of seepage into the permeable alluvium of the stream-channel floor and because of water use from wells near and below the junction of Castaic and Elizabeth Lake Canyons. Surface flow ceases near the junction of Castaic and Fish Canyons (pi. 2), except when storm runoff occurs. Surface flow in Elizabeth Lake Canyon generally continues to a point slightly nearer the basin mouth but then becomes subsurface flow for the same reasons. Personal observation of surface flow in nearly all canyons tributary to Castaic and Elizabeth Lake Canyons and of the presence of water snakes in a few of the streams indicates that the general impression of aridity conveyed by the lower reach of the basin (fig. 1) is not representative of the entire watershed. CLIMATE The climate of the Castaic watershed is similar to that existing elsewhere in southern California. Summers are hot and dry and temperatures often reach or exceed 100°F.; winters are generally mild and wet. Temperatures are below freezing during the winters at higher elevations, but the number of freeze-thaw cycles per year is not known. Cyclonic storms that move eastward and northeastward from the Pacific Ocean Figure 1.—View downstream at the single-stage sampling site in the lower reach of Castaic Creek. Note the sparseness of vegetation in this area which characterizes the lower part of the watershed. The boulders and cobbles visible in the right foreground are abundant throughout the area to be occupied by the proposed reservoir. The single-stage sampler is 5 feet high, it is bolted to reinforcing rods that are set in concrete.F4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS during the winter months provide most of the annual precipitation. Precipitation data have been obtained at only one station within the watershed; this station is in Elizabeth Lake Canyon (pi. 2) at an elevation of 2,075 feet. Total annual precipitation may fluctuate considerably from year to year, and periods of drouth recur intermittently. The total precipitation recorded in 1961, for example, was 8.26 inches, whereas the following year it was 24.72 inches at this station. Even a long-term mean value at a single station cannot accurately reflect the orographic and areal variations in precipitation that would be expected within the watershed, however, and more data are sorely needed. A regional isohyetal map (California Water Resources Board, 1953), based on the data of the Elizabeth Lake station and on long-term records elsewhere, shows that annual precipitation in the watershed ranges from about 14 to 22 inches per year. This is the best estimate that can be made at present. Maximum precipitation occurs, of course, at the higher elevations; the percentage attributable to snowfall is not known. VEGETATION AND SOILS The lower reach of the Castaic watershed appears to be semiarid, as previously mentioned. The vegetation in this area consists of scattered sage and some phreato-phytes, but riparian species of trees and shrubs and grasses grow along most of the stream channels elsewhere in the watershed. An assemblage of chaparral and sage provides moderate to good cover on the slopes and ridges of most of the watershed and is more characteristic of the basin than is the assemblage of the lower reach. Woodland communities are prevalent only at higher elevations and along the basin divide; oaks are common in these areas. Immature alluvial soils border the stream-channel flats; these soils generally lack profiles and range from Figure 2.—View of the Liebre Mountain area from the south. The granitic rocks of Liebre Mountain are separated from the sedimentary rocks in the foreground by a valley that coincides with a fault zone at the mountain front. (See dashed line.) Valley-wall slopes are moderate to steep in the area and vegetative cover is more abundant than in the lower part of the watershed.SEDIMENT YIELD OF THE CASTAIC WATERSHED, CALIFORNIA F5 a few inches to about 2 feet in depth. Residual rock fragments mixed with finer material mantles the steep slopes and the parts of the watershed at higher elevations; neither soil profiles nor zonation was noted. Soils tend to be thickest in the northeastern part of the watershed, where foliation planes of the metamorphic rocks are exposed to weathering. SUMMARY OF GEOLOGY The geology of the area has not been mapped in detail; and because of the time limitations of this study, only the more general aspects could be recorded. Lithologies are therefore grouped into three major types in the following discussion: igneous, metamorphic, and sedimentary. The lithologic boundaries as well as the surface trace of major faults shown on plate 2 should be regarded as both approximate in location and inferred over much of their extent. The Castaic watershed is bounded by two strike-slip fault zones, which are the San Andreas to-the north and the San Gabriel to the south. The watershed proper contains several major and many minor faults. In the absence of stratigraphic and other data, the nature of these faults cannot be determined but the Liebre Mountain and Clearwater faults (pi. 2) that trend approximately east to west may, in part, also represent strike-slip movement. The general aspect of the Liebre Mountain area as viewed from the south is shown in figure 2. The Liebre Mountain area consists wholly of granitic rocks that contain intermediate to mafic inclusions. This granitic mass is bordered on the southeast and east by a metamorphic complex that consists predominantly of gneiss, schist, and metasedimentary rocks, all of which are intruded in many areas by granite and by aplitic dikes. A typical exposure of this complex is shown in figure 3. Sedimentary rocks of Tertiary age crop out elsewhere in the watershed (pi. 2). South of the Liebre Mountain and Ruby Canyon faults, these Tertiary strata dip Figure 3—View of the metamorphic-igneous complex in Elizabeth Lake Canyon. A dike has intruded the metasedimentary rocks that arc ground. Note the steep slope of this valley wall, which bounds the channel in Elizabeth La e anyon. visible in the right fore- 776-695 0—65-----2F6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS steeply toward the south and in some places approach the vertical. Beyond these zones of deformation, to the south, dips become more gentle; and in large areas the strata are nearly horizontal (fig. 4). The entire sedimentary sequence in the basin forms the east limb of a syncline that plunges northwest (Eaton, 1939). The sedimentary rocks comprise a coarse facies, consisting predominantly of alternating conglomerate and sandstone. The conglomerate ranges from pebble to boulder size; individual clasts consist of a wide variety of igneous and volcanic rock types. The sandstone ranges from fine to very coarse grained and from well indurated to highly friable. This wide range in both grain size and in degree of cementation combines with alternating thicknesses of beds to produce outcrops that resemble shale-sandstone sequences (fig. 5). True shales are extremely scarce, however, and the sediment yield from the sedimentary rocks of the watershed is derived from only a few siltstones and silty sandstones, in addition to the rock types already cited. Jointing and fracturing of the sedimentary beds are widespread and are a major factor in the production of sediment in the watershed. SOURCES OF SEDIMENT To ascertain whether a large percentage of the total sediment yield of the Castaic watershed might be attributable to a single lithologic source or to an individual subbasin, 61 samples were collected within the basin. These samples were obtained from talus slopes below outcrops as well as from the beds and banks of stream channels. All are composite samples; in stream channels, for example, as many as 18 individual samples were taken at a given cross section to provide the data shown for a given location on plate 2. The average weight percentages of granules, sand, and silt-clay particles in samples are grouped by lithologic source and by subbasin in table 1. These data provide the basis for much of the following discussion. Figure 4.—View of the relatively undeformed Tertiary sedimentary rocks in the southern part of the Castaic watershed. The strata crop out as a series of ledges in the canyon and are visible in the center of the photograph. The ledges probably reflect differential resistance to weathering of the sedimentary strata.SEDIMENT YIELD OF THE CASTAIC WATERSHED, CALIFORNIA F7 Table 1.— Weight percentage of granules, sand, and silt-clay in the granule-to-clay size fraction of samples from the Castaic watershed Source Number of samples Granules (2-4 mm) Sand (0.062-2 mm) Silt-clay « 0.062 mm) Sedimentary rocks 33 8.6 82.6 8.8 Metamorphic-igneous complex.. - 21 14.3 77.7 8.0 Granitic rocks 7 20.6 75.5 3.9 Castaic Creek drainage basin Elizabeth Lake Canyon drain- 47 10.8 81.7 7.5 age basin 14 15.1 75.3 9.6 Subbasin 2 19 9.7 82.9 7.4 Subbasin 3 13 13.8 80. 1 6.1 Subbasin 5 3 13.0 76.5 10.5 Consideration of the data listed in table 1 shows that the silt-clay contribution from the area in which sedimentary rocks crop out (pi. 2) is slightly greater than, but nearly equal to, the silt-clay fraction that is contributed by the metamorphic-igneous complex. The difference in the silt-clay yield from these two lithologic groups may result from inadequacies of sampling; if so, then the difference is apparent rather than real. The granitic rocks of the Liebre Mountain area contribute only one-half the silt-clay yield of either the sedimentary or metamorphic-igneous source areas. The granitic rocks may contribute more fine sediment than would be suspected, however. They are generally weathered to a depth of 1 or more feet in the Liebre Mountain area, and the weathering of ferromagnesian minerals, that precedes the breakup of granites, must produce some clay. Moreover, the high average percentage of granules (20.6) in the granitic sediment and the decrease in abundance of this size class to the south (pi. 2) is significant. Granules consist of polymineralic aggregates of sand-sized particles that are rapidly reduced to these particles through weathering, regardless of the frequency and duration of transport (Lustig, 1963). The breakup of granules produces a change in the size distribution of sediments mainly because sand will be added, but some silt and clay will also be produced. Because the streams that drain the Liebre Figuke 5 —View of a typical outcrop of the Tertiary sedimentary rocks in the Castaic watershed. The alternation of thick to massive beds and thinner strata is typical of many sequences in the watershed. The rocks in this section are all highly friable. The thin beds consist of siltstone and silty sandstone. Note the jointing and tendency toward block-fracturing in this exposure. The massive unit visible in the center is approximately 4 feet thick.F8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Mountain granitic area (pi. 2) flow through the sedimentary outcrop area of the sedimentary rocks to the south, this secondary contribution of fine sediment from the granitic rocks is somewhat masked. The lithology of the source area does not appear to exert a pronounced influence upon the production of fine sediment within the watershed. Most of the sediments are sands and coarser elastics; the rather low percentages of silt-clay listed in table 1 would be still smaller if total sediment had been considered rather than the granule-to-clay size fraction. The sediment that occurs in the proposed reservoir site today reflects this dominance of coarse elastics throughout the watershed. Composite samples from cross sections in Castaic Creek and the Elizabeth Lake Canyon channels contain approximately 85 percent sand in the granule-to-clay size fraction, and no marked skewness toward the smaller sizes occurs. Numerous boulders are, in fact, present at the proposed damsite. Because many of these large particles seem to have been transported fairly recently, their occurrence in these wide, shallow channels of gentle slope must result from either high velocity or density flows. Therefore, the production of sediment in the area in which sedimentary rocks crop out may be greater than is suggested by their silt-clay content. As previously mentioned, jointing and fracturing of the sedimentary rocks are widespread in. the watershed. Slopes are unstable and rock fragments are frequently heard tumbling down canyon walls to the channels below. Much of the sandstone is so friable that it disintegrates to its constituent particles when it falls to the ground from a height of about 4 feet. Sediments so produced occur mainly within the Castaic Creek drainage basin and they may compose much of the total sediment yield because the water discharge from this basin may be greater than the discharge from the Elizabeth Lake Canyon drainage basin. The conglomeratic facies, however, is well cemented, and jointing and fracturing of these rocks produce large blocks that also fall to canyon bottoms. One such rockfall is shown in figure 6. It forms a natural debris dam above the single-stage-sampler location in subbasin 2 (pi. 2). The author suggests that the present effectiveness of this natural barrier be increased in order to further reduce the sediment yield from the contributing drainage systems in that area. The Fish Canyon drainage basin (subbasin 3, pi. 2) may also yield much sediment that is transported by Castaic Creek because it is relatively large in area and because the major tributaries head near the basin divide where precipitation is greatest. For these reasons, a sediment sampling station was located near the mouth of this subbasin but data have not yet been obtained. Subbasin 5 (pi. 2) is thought to be a source of much sediment in the Elizabeth Lake Canyon drainage basin. The small number of samples taken in this area (table 1) may in part account for the greater silt-clay abundance listed; however, field observation shows that fracturing and shearing of the rocks has occurred along a major fault zone that coincides with Ruby and Tule Canyons. If the precipitation and discharge in these canyons is sufficient, the sediment yields will be high. For these reasons a single-stage sampler was also installed at the mouth of Ruby Canyon. In summary, the entire watershed provides an excellent sediment source despite the fact that silt and clay are not abundant. The sediment yield of the Castaic Creek drainage system may prove to be greater than the yield of the remainder of the watershed owing to the occurrence of friable sedimentary rocks and to a probable higher water discharge. Subbasins 2, 3, and 5 (pi. 2) are worthy of consideration as good locations for construction of debris dams. LONG-TERM CHANNEL EROSION A discussion of the sources of sediment is incomplete unless the possible sediment yield from channel erosion is considered. The roots of many of the riparian species of trees that grow along stream channels in the Castaic watershed have been exposed by channel erosion. In one such exposure (fig. 7) approximately 4 feet of net channel erosion has apparently occurred along the stream reach within the lifespan of the tree. If a sufficient number of such exposures are present in a watershed, determination of the net channel erosion is possible; and, hence, some estimate of the sediment yield from this source can be obtained simply by coring and dating the trees. Although an extensive investigation of the inherent possibilities of the method was not undertaken for this study, some pertinent data were obtained. The total stream length of the Castaic watershed is approximately 307 miles. If an average bankfull channel width of 20 feet (which appears reasonable from field observations) is assumed, the total channel area is 32,420,000 square feet. The examples of root exposure that were noted ip the field suggest that approximately 3 feet of net channel erosion has occurred within the lifespan of the trees in many parts of the watershed. If a rectangular channel cross section is assumed, the volume of sediment that has been removed is 97,260,000 cubic feet. The trees that were cored proved to be less than 100 years in age, and none observed in the field are thought to be much older. If the channel erosion has occurred within the lastFigure 6.—Rockfall of conglomerate blocks in subbasin 2 that has produced a natural debris dam. The block in the right foregound is 5 feet high, measured from the channel floor; but much larger conglomerate blocks also occur. Note the coarse nature of the well-indurated conglomeratic facies and the trace of joint planes that are visible on the nearly vertical canyon wall in the background. The effectiveness of this natural dam could be easily increased by blasting out additional material from the canyon walls above the site. 100 years, then an average of approximately 22 acre-feet of sediment per year can be attributed to this process in the watershed. Two major qualifications of the foregoing calculation must be considered. First, the calculation is based upon the arbitrary assumption that channel erosion occurs everywhere within the watershed. There is little doubt that aggradation occurs today along certain parts of the drainage system; reaches that are aggrading cannot be detected, however, unless detailed surveys are made over a period of years. If this assumption alone is considered, the calculation provides too great an estimate. A second qualification that is perhaps more significant is the fluctuation between aggradation and degradation during the 100-year period considered. This fluctuationF10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 7.—Exposure of the roots of a tree in Ruby Canyon by channel erosion. Approximately 4 feet of net channel erosion is indicated at this point. Note the cobbles and boulders that are visible within the root system, suggesting that much coarse material has been removed in this reach of the stream. depends upon the kinds and sequence of runoff events. More sediment is probably attributable to channel erosion during any long time interval than is suggested by the method of calculation that is outlined here. Despite these qualifications, the estimated long-term channel erosion of 22 acre-feet per year is useful. It will be shown in this report that a reasonable estimate of the total long-term sediment yield of the Castaic watershed is 250 acre-feet per year. If the estimated yield from channel erosion had proved to be a large percentage of this total yield, then the validity of the total yield would be subject to considerable question. The results of many studies indicate that in most watersheds the sediment contribution by sheet erosion of valley slopes far outweighs the contribution by channel erosion. The estimate of channel erosion provided here suggests, therefore, that the Castaic watershed does not differ in behavior from most watersheds in regard to sources of sediment. The data suggest that perhaps 20 percent of the total sediment yield is derived from channel erosion. THE SAN GABRIEL WATERSHEDS The watersheds in the San Gabriel Mountains that provided a basis for the comparative approach to the problem of sediment yield will be described as a group rather than individually. Emphasis will be placed upon differences or similarities in physical characteristics among the watersheds studied. PHYSICAL DESCRIPTION OF THE AREA LOCATION AND EXTENT The watersheds are in the San Gabriel Mountains, a range approximately 70 miles long and 25 miles wide that trends in an east-west direction (pi. 1). As shown on plate 1, the San Dimas, Big Dalton, Sawpit, and Big Santa Anita watersheds are in the central and eastern parts of the range, along its southern front. The Big Tujunga watershed occupies a more central position with respect to the interior of the range, and the Pacoima watershed lies along the west margin of the mountains. These watersheds range in area from about 3 to 82 square miles; all are therefore smaller in drainage area than the Castaic watershed (137 sq mi). TOPOGRAPHY AND DRAINAGE Although elevations in the San Gabriel Mountains exceed 10,000 feet at the east end of the range, elevations in the watersheds studied are somewhat comparable to those in the Castaic drainage basin (1,200 to 5,700 feet). Elevations in the Sawpit, Big Santa Anita, and San Dimas watersheds range from about 1,300 to 5,700 feet and are therefore nearly identical with the range of elevations in the Castaic watershed. Elevations are lower in the Big Dalton watershed, ranging from 1,600 to 3,500 feet, and are higher in the Pacoima and Big Tujunga watersheds, ranging from 1,700 to 6,500 feet and from 2,300 to 7,100 feet, respectively. Thus, only the Pacoima and Big Tujunga watersheds exhibit greater maximum elevations than the Castaic watershed. Canyons are narrow, deeply dissected, and steep walled throughout the San Gabriel watersheds. Valley-wall slopes in the San Gabriel Mountains seem to be much steeper than those in the Castaic watershed when viewed in the field, owing in part to a greater difference in the relief of the canyons. Valley-wall slopes are in fact steeper in many watersheds in the San Gabriel Mountains, but the mean ground-slope angle in these watersheds, computed as later described, is generally about 5° less than in the Castaic watershed. Streamflow tends to be intermittent near the mountain front and perennial in interior parts of the range. Except for Sawpit watershed (pi. 1), drainage densities are comparable to that of the Castaic watershed (D=2.23 miles per sq. mi.). The drainage density of the Sawpit watershed is 3.64 miles per square mile; this value is 1.33 to 1.77 miles per square mile greater than the value for the drainage densities of the other basins studied.SEDIMENT YIELD OF THE CASTAIC WATERSHED, CALIFORNIA Fll CLIMATE The climate in the watersheds of the San Gabriel Mountains is similar to that in the Castaic watershed: summers are hot and dry, and winters are mild and wet. Most of the total precipitation is produced by the same cyclonic storms that move eastward and northeastward during the winter months. Because the large mass of the San Gabriel Mountains intercepts and forces upward the incoming moist air, more rainfall is induced along the south half of the mountains than in the Castaic watershed. Total annual precipitation ranges from about 20 to 40 inches along an east-west belt between the foothills and the summit of the range. In the watersheds of the San Gabriel Mountains that were studied, precipitation is probably 5 to 10 inches greater than that in the Castaic watershed; it also occurs more frequently. VEGETATION AND SOILS The greater precipitation in the watersheds of the San Gabriel Mountains generally supports more vegetation; hence, the ground-cover density is greater than that of the Castaic watershed. Vegetation, like rainfall, is also orographically controlled; and, in addition to a greater ground-cover density, a greater abundance of woodland communities that include spruce, pine, and oak distinguishes these watersheds from the Castaic drainage basin. Soils of the watersheds in the San Gabriel Mountains are similar in occurrence to those in the Castaic drainage basin but generally possess a higher clay and organic content and tend to be thicker. Anderson and Trobitz (1949) have described the soils as rocky, sandy loams that are generally less than 3 feet thick and lack profiles. Soils are as thick as 6 feet (Maxwell, 1960) in a few places, however. SUMMARY OF GEOLOGY The San Gabriel Mountains are a structurally complex range that contains many major and minor faults. Although the geology of the area is much better known than is that of the Castaic watershed, for purposes of this report and for reasons of consistency, only the litho- logic contrast between the two areas need be noted here. The watersheds studied occur within areas that consist of several igneous and metamorphic rock types, the metamorphic types including schist, gneiss, and meta-sedimentary rocks. Clastic sedimentary rocks are generally lacking in the watersheds of the San Gabriel Mountains whereas such rocks crop out over approximately one-third of the Castaic drainage area; therein lies the chief geologic difference between the two areas. Joints and fractures are very common in the rocks of the San Gabriel Mountains. It has not been possible to quantitatively assess the abundance of these features relative to similar zones in the Castaic watershed, however. SEDIMENT-YIELD DATA Because any comparative study must depend upon the reliability of the information that is used for standard or known values, some discussion of the sediment-yield data from the watersheds of the San Gabriel Mountains is warranted. These data (table 2) are derived from repeated surveys of the respective reservoirs during the past 30 to 40 years. Although surveys of each of the reservoirs have not generally been made during the same year, if the number of years of sediment accumulation (table 2) is divided by the total number of surveys, the average number of years between surveys is seen to range from about 2 to 5. These data probably are internally consistent and reliable for the following reasons: (1) All data are derived from reservoirs, (2) the period of record is sufficient to include the years of major storms as well as relatively dry years, and (3) the methods used to compute the sediment accumulation in each reservoir is identical. In summary, the sediment yield of these watersheds, expressed in acre-feet per year, is probably satisfactory for the purposes of this report. Sediment yield is also expressed in acre-feet per year per square mile, and the yield from the major storm of 1938 is shown in the same units (table 2). These data will be cited elsewhere in this report, where appropriate. Table 2.—Data on reservoirs and sediment yield of watersheds in the San Gabriel Mountains, California Reservoir Dam completion date Initial capacity (acre-feet) Latest survey date Loss of storage capacity (percent) Number of years of sediment accumulation Total number of surveys Sediment yield (acre-ft per yr) Sediment yield (acre-ft per yr per sq mi) Sedmient yield of the flood of 1938 i (acre-ft per sq mi) 476 May 1962 42.9 2 35.58 9 8.43 2.52 20.65 18.70 30.19 31053 January 1962 April 1962 . 17.5 32.25 6 5.71 1.27 Big Santa Anita.. March 1927 1376 54.2 2 35.50 16 43.18 4.00 September 1922. February 1929... July 1931 1496 April 1962 51.3 40. 50 9 21.70 52. 62 113.23 1.34 13. 46 20.85 6060 May 1962 24.4 2 35.75 8 1.87 Big Tujunga.. 6240 July 1962 34.9 31.75 11 1.38 18.29 1 Surveys made prior to 1938 do not have a common date. All reservoirs were surveyed between 1934 and 1936, except for Big Tujunga which was not surveyed between the initial debris year of 1930-31 and 1938. The data given are therefore based upon the assumption that the total sediment yield between 1938 and the previous survey date accrued solely during the flood of 1938. 2 Dam construction was sufficiently advanced to trap debris from a storm in February 1927, and the first debris-year is assumed to be 1926-27. 3 The initial topographic survey was made in November 1934. The capacity determined at that time is given as the initial capacity; observation suggested that little sediment accumulated prior to 1935.FI 2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS QUANTITATIVE GEOMORPHOLOGY GENERAL DISCUSSION The methods of quantitative geomorphology are widely applicable to problems involving erosion and sedimentation. The major impetus for studies in this field was provided by Horton (1945), who set forth many of the principles and parameters that are today applied to drainage-basin studies. Subsequent investigators, notably Strahler (1950, 1952, 1954, 1957, 1958) and several of his students (Miller, 1953; Schumm, 1956; Melton, 1957; Coates, 1958; Broscoe, 1959; Morisawa, 1959; Maxwell, 1960), have enlarged these concepts, introduced new parameters, and extended the data to include a wide variety of geographic regions. In addition to providing a sound basis for quantitative landform description and comparison, these and other studies have led to a better understanding of geologic processes and of the interrelationship between geomor-phic characteristics and the hydrology of drainage basins. Sherman (1932) initially demonstrated that the unit hydrograph varied with basin shape and slope, but Langbein and others (1947) provided one of the first mathematical treatments, relating discharge and drainage area. Anderson (1949; Anderson and Trobitz, 1949) was among the first to apply multiple-regression methods to hydrologic problems in watersheds, and to relate forest-cover density to discharge and sedimentation. Potter (1953) showed that peak flow was correlative with the length and slope of the principal channel, and Morisawa (1959) extended Anderson’s multiple regression approach to relate peak intensity of runoff with several additional geomorphic variables. In a highly sophisticated statistical treatment, Maxwell (1960) computed the correlations among peak discharge and storm rainfall, cover density, antecedent rainfall, and nine geomorphic parameters taken five at a time. In similar fashion, Benson (1962) related the T-year annual peak discharge to climatic and geomorphic factors by multiple-regression analysis. From the foregoing abbreviated survey of the literature, it can be concluded that many geomorphic parameters can exert an effect upon the discharge from a given watershed. This conclusion is true if, as shown for example by Hack (1957), Q=MJ, (i) because Au=j(Nu, Lu, 2A, 0, ...) (2) and discharge is therefore a function of these same variables. It is reasonable to suppose then that certain geomorphic parameters will also affect sediment yield if S,=m, (3) that is, if sediment yield is some function of water discharge, and the methods of quantitative geomorphology should, therefore, be applicable to the problem of determination of the anticipated sediment yield of a given watershed. In addition to stream discharge, however, sediment yield is undoubtedly a complex function of a veritable host of climatic, geologic, edaphic, and other characteristics of a watershed. If sufficient data on these characteristics are available, then the methods of multiple regression will enable one to derive an equation expressing sediment yield in terms of all the variables. The applicability of such an equation, however, will still depend upon qualitative factors such as the arbitrary assignation of numerical values for the erodibility of rocks of various lithologic character. Such procedure is, in any event, precluded in the problem considered in this report because available hydrologic data for the Castaic drainage basin (pi. 2) are scarce. Knowledge of the long-term sediment yield from watersheds in the San Gabriel Mountains, however, is equivalent to knowledge of the cumulative effect of all the variables operative in that area. Determining the role of geomorphic factors in the production of sediment from the watersheds of the San Gabriel Mountains should therefore be possible by simple linear regression of each parameter with the known sediment yield. Determination of the same geomorphic parameters for the Castaic watershed should then provide a means by which some reasonable estimate of anticipated sediment yield can be made. With this approach in mind, the methods of quantitative geomorphology were applied; a morphometric study of the Castaic and San Gabriel Mountains watersheds was undertaken in a search for significant parameters. BASIC-DATA COLLECTION The morphometric data listed in table 3 were obtained from U.S. Geological Survey topographic maps having a scale of 1:24,000. These maps depict the mountainous terrain of the areas studied in considerable detail. Some errors, both of location and of omission of first-order stream channels, do occur, however. Several examples of such errors have been cited by Coates (1958) and Maxwell (1960), among others. Maxwell proved their occurrence in the San Dimas watershed of the San Gabriel Mountains by a careful field check and revised the maps involved accordingly. Time limitations prevented a similar correction of the maps employed in the present study, and for this reason the effect of typical map discrepancies upon the data should be noted.SEDIMENT YIELD OF THE CASTAIC WATERSHED, CALIFORNIA F13 The morphometric properties most readily altered by omission of first-order streams are: stream-channel order, number of streams, stream-channel length, basin order, and drainage density and other derived parameters. Such parameters are a function of map scale (Giusti and Schneider, 1963, for example), and any first-order stream channel can generally be shown to be of much higher order number if detailed mapping on a larger scale is undertaken (Leopold and Miller, 1956), however. The values obtained for drainage density (table 3), for example, may be less than the true values because drainage density is proportional to total stream-channel length. Basin-order designations are also less, for similar reasons. The values obtained can still be used for comparative purposes, however, because they have been affected to the same degree; the fact that the maps including the basins studied are of equal scale obviates the difficulties imposed by the discrepancies of these maps. Stream order and basin order were designated in accord with Strahler’s (1952) modification of the usage of Gravelius (1914) and Horton (1945). Stream-channel lengths were measured with dividers set at 0.01 mile. The use of dividers provided more consistent replicate results than could be obtained with a map measurer. This was particularly true for measurement of the lengths of highly sinuous streams. The mean stream-channel length of streams of each order (Lu) and the cumulated lengths of streams of each order (2Lu) were computed for each watershed; the values are listed in table 3. Drainage areas were measured with a compensating polar planimeter, and the mean values of replicate sets of measurements were recorded. Areas of subbasins of each order (Au) within a given watershed were cumulated in order to verify total drainage area. The results accorded to the nearest 0.05 square mile. The mean area of each basin order (Au) and the cumulated area of each basin order (2Au) were computed for each of the seven watersheds (table 3). Random-number overlays, of a size sufficient to cover the area of each watershed, were prepared to determine the mean ground-slope angle of each basin. The procedure, as described by Strahler (1954) is simple. At each of 100 random locations within each basin, the slope was measured over a 200-foot reach orthogonal to contour lines. The mean ground-slope angle (dg) was calculated from these sets of 100 measurements to the nearest one-half of a degree. Replicate measurements of slope at alternative sets of 100 random locations were made within the largest of the watersheds that were studied to test sufficiency of sample size. The expected reduction of the standard deviation of the means of these replicate sets, compared to that 776-J095 O—65——2 of the original set of values, occurred, and the variance from the mean of the means was within the limits of error of the measurements. It was concluded, therefore, that a random sample of 100 ground-slope angles was sufficiently large to be representative of the population of ground-slope angles of each watershed. The value of the sine of each of the 100 ground-slope angles within each watershed was recorded, and the mean value (sin dg) was computed. The reasons for computing the sine rather than the tangent of the angles will be discussed later. The cosine of the mean ground-slope angle was also computed to calculate SA, sediment area (Sa=AJcos dg). These data are listed in table 3. Elevations at each random location in a basin were read from the maps, and the means of each set of 100 values (E) were computed (table 3). These values are thought to be representative of the mean basin elevations. Proportional dividers were used to interpolate between adjacent contours to the nearest 5 feet. This same procedure was followed in measuring elevation differences along stream channels. These values were used to compute the mean stream-channel gradients of streams of each order (0U) within each basin (table 3). The number of streams of each order (Nu) within a basin was determined by inspection, and relief ratio (Rh=H/Lb) was computed in accord with Schumm’s (1956) procedure. All other parameters that are listed in table 3 are derived properties that require stream length, basin area, stream-channel gradient, or the number of streams of a given order for their computation. These derived parameters are: drainage density (D=~EL/Au), stream frequency (F=NJAU), bifurcation ratio (Rb=NJNu+l), stream-length ratio (Rl=LJLu-i), basin-area ratio (Ra=AJAu_i), stream-channel-slope ratio (Rc=6u/du+i), and the ruggedness index {Rt = DXE). GEOMORPHIC PARAMETERS Each of the parameters listed in table 3, obtained as described in the preceding section, was considered both individually and in various combinations for possible correlation with the sediment yield of the watersheds of the San Gabriel Mountains. Although a trial-and-error approach will suffice in seeking the correlation of individual parameters with sediment yield, the choice of combinations of these parameters must be governed by consideration of both the relationship between two parameters of a paired set and the expected influence of the paired set upon sediment yield. One would not, for example, expect a product of mean stream length of a given stream order and mean basin elevation (LUXE) to produce an explicable correlation withF14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Table 3.—Morphometric data and geomorphic parameters of the Castaic Watershed Basin order Area of basin of order u (sqmi) Mean area of basin of order u (sq mi) Number of streams of order u Total length of streams of order u (miles) Mean length of streams of order u (miles) Mean channel slope of streams of order u (feet per foot) Mean basin elevation (ft) Mean ground-slope angle (degrees) Stream frequency (number per sq mi) Castaic 5 ,4.1=83.28 Ai=0A5 N, =187 22/1=200.21 Xi—1.07 01=0.189 2i=2.25 Aa=79.60 ,43=1.85 Ni=43 21/3=245.54 Zi=5.71 03 =.078 Fi=.54 Aj=82.12 ,43=6.32 N3=13 22,3=269.42 Zs=20.72 03 =.043 3240 40.5 F3=. 16 A«=111.72 ,44=27.93 Nt=4 2^4=294.62 Z4= 73.66 Si=.Q27 *4=.04 As=137.63 An=137.63 N3=l 22,6=307.49 Zs=307.49 05 =.014 Xs=,01 Sawpit 3 Ai=2.29 ,4i = .18 N,=13 22/1=8.13 Zi = .63 0i = .292 F,=5.6S ,4.3=1.94 Ai=.39 Ni=6 22,3=9.89 Zi=1.98 02=.201 3010 33.5 F2=2.5S As=3.34 As=3.34 Ns=l 2X3=12.17 Zi=12.17 03=.083 X3=.30 Big Dalton 3 ^1=4.30 Ai=.72 N,=6 22^i=6.31 Zi=1.05 0i = .207 Fi=1.40 At=AAl ,43=2.21 Nj=2 2LS=8.60 Zi=4.30 03 =.060 2620 30.5 Ft=A5 A3=l. 60 As=4.50 N,=l ZXt=8.80 Zj=8.80 03 =.038 F3=. 22 Big Santa Anita . 3 Ai=6.69 ,4i=.39 N,=17 2ii = 15.50 Zi = .91 0i = .279 Fi=2.54 Ai=8.06 ,43=2.69 N,=3 22/3=21.47 Zi=7.16 03 =.142 3485 34.0 F3= .37 ^3=10.79 ,43=10.79 N3=l 22.3=23.72 Zs=23.72 03 = .055 F3=.09 San Dimas 4 A, = 10.64 A, = .88 N,= 12 22/i=19.02 Zi = 1.59 Si = .155 Fi=1.14 Aj=7.54 Ai=1.89 Ni=4 2X3=23.50 Zi=5.88 Si=.109 Fi=.53 As=9.95 Aj=4.98 N3=2 22/3=27.21 Zs=13.61 S3=.059 Fj=,20 Ai=16.17 Ai=16.17 Ni =1 2X4=30.27 Zi=30.27 Si=.020 Ft=. 06 Pacoima 3 Ai=16.81 A, = .34 N,= 49 22/i=46.26 Zi = .94 Si = .228 Fi=2.9l ,4.3=12.43 Aj=1.78 N,=7 2x1=55.66 Zi=7.95 Si=.109 4050 33.5 Fi=.56 As=28.10 ,43=28.10 Ns=l 22/3=61.24 Z3=61.24 Ss=.073 P’s =.04 Big Tujunga 5 Ai=43.52 Ai=.32 Ni = 136 2X1=116.01 Zi=.85 Si = .216 Fi=3.13 Ai=53.61 Ai=1.73 N,=31 2X2=156.98 Zi=5.06 Si=.091 Fi=.58 Aa=54.71 Ai=6.84 N,=8 2Xa=176.32 Zi=22.04 S3=.056 4540 30.5 Fs=.15 Ai=64.35 ,44=32.18 Nt=2 22,4=185.11 Z4=92.56 Si=.046 Ft=. 03 Aj=82.00 ,45=82.00 Nt=l 2L5=189.07 Zs=189.07 Ss=,018 F6=.01 sediment yield. Although each of the variables in this example may indeed bear some relation to sediment yield, their relation to each other is not obvious, and the results of a correlation between the pair and sediment yield would defy explanation. In this sense, all possible combinations of parameters have not been tested for correlation with sediment yield. The parameters discussed below are, except for relief ratio, those which are interpreted as meaningful and which correlated well with sediment yield. Certain of the parameters to be described appear in this report for the first time, and comparison with previous results is therefore not possible. Among the more common parameters, such as drainage density or ruggedness index, only total drainage area provided a good correlation with sediment yield. As previously indicated (equations 1, 2, 3), if this were not true, then the relationship between sediment yield and various parameters that are a function of drainage area could not be investigated. The relationship between sediment yield and drainage area will, however, be held in abeyance for discussion in a subsequent section of this report. The linear-regression lines that relate sediment yield with each of the following geomorphic parameters were fitted by the least-squares method. Correlation coefficients computed for each pair of variates are cited below where appropriate; the statistical significance of these coefficients will be explained in a subsequent subsection entitled “Discussion of results.” RELIEF RATIO The relief ratio of a watershed (Schumm, 1956) is defined as Rh=H/Lb, where H is the difference in elevation between the basin divide and mouth and Lb is the maximum basin length measured along a line as nearly parallel to the principal channel as possible. The relief ratio is therefore a dimensionless number that approximates overall watershed slope. Because of lack of adequate maps, or for ease of computation, relief ratio has often been correlated with sediment yield in preference to treating hypsometric data as suggested by Langbein and others (1947) and Strahler (1952). In addition, it is logical to assume that the energy input of such hydrologic factors as runoff and discharge should be, in part, a function of this parameter. Morisawa (1959) found that relief ratios of watersheds in the Appalachian Mountains correlated well with peaks of discharge and rainfall-runoff intensities, and Hadley and Schumm (1961) obtained a good correlation with sediment yield in the Cheyenne River drainage basin. Accordingly, relief ratio for each of the watersheds in the San Gabriel Mountains was among the first parameters determined during the morphometric study for this report. The linear regression of sediment yield with relief ratio for the basins studied is shown in figure 8. The rather poor correlation (r=0.658) between these variates is apparent, and the data are included here for illustrative purposes only.SEDIMENT YIELD OF THE CASTAIC WATERSHED, CALIFORNIA F15 drainage basin and of watersheds in the San Gabriel Mountains, California Drainage density (mile per sq mi) Ruggedness index Bifurcation ratio Stream-length ratio Basin-area ratio Stream- channel-slope Relief ratio Sediment-area factor Sediment- Transport efficiency factors ratio factor Ti T, Tz Ni/N,=4.35 Zs/Z,=4.17 Aj/At=1.23 0i/02=2.42 Ns/Ns=3.3I Z4/Z* =5=3.56 Ai/A3=1.36 02/03 = 1.81 2.23 7,225 Ns/Ni=3.25 Zs/Z!=2.63 Az/A2=1M 03/04=1.59 0.058 180.97 116.16 1146.94 481.12 694.64 Ni/N6=4.00 Za/Zi=5.34 Ai/Az = M fc/«i=1.93 Rb=3.73 Wl=4.18 R.=1.15 iee=i.94 Ni/N,=2.60 L3[L2=6.15 A3/A,=1.72 0i/02=1.45 3.64 10,956 IV,/Nj=5.00 Zi/Zi=3.17 A2/Ai=.85 02/03 = 2.42 .340 4. 01 2.18 46.25 36.86 40.62 -Rb=3.80 Rl.=4.66 i?o=1.29 Rc=1.94 Ni/IVj=3.00 Z3/Z,=2.05 AzlAi=1.02 0i/02=3.45 1.95 5,109 |n^N3=2.00 Z2jLi=4.09 A2Ai = 1.03 02/03=1.58 .061 5.23 2.59 22.00 22.68 32.34 5Tb=2.50 Si=3.07 Ra=1.03 Re=2.52 N,/Ni=5.67 Z3/Zs=3.31 Xa/A,=1.34 0i/02=1.96 2.20 7,667 Nj/N3=3.00 Zj/Zi=7.86 AilA\=1.20 02/03 = 2.58 .230 13.01 7.18 102.95 47.67 49. 52 ^=4.34 Rl=5.59 Ra=1.27 Re=2.27 N,/N,=3.00 Zi/Z3=2.22 Ail 43=1.63 0i/02 = 1.42 1.87 6,180 N2IN3=2.00 N3/Ni=2.00 Z3/Z!=2.31 Li/Zi=3.70 43/^2=1.32 A2/Ai = .72 02/03=1.85 03/04=2.95 .139 19.28 10.34 70.53 39.33 42.67 Rl=2.33 Rl=2.74 Ra=1.22 Rc=2.07 Ni/Arj=7.00 A3/Z,=7.70 A3/A2=2.20 01/02=2.09 2.18 8,829 N2/N3=7,00 Z^/Zi=8.46 A2/Ai = .74 02/03 = 1.49 .070 33.70 18.34 428.68 102.03 128.96 Rb=7.00 Rl=8.08 TTa = 1.50 Re=1.79 Ni/N2*=4.39 A5/Z(=2.04 As//L=1.27 ft/9j=2.37 n2/n3=sm Z1PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS THE DISTRIBUTION OF BRANCHES IN RIVER NETWORKS By Ennio V. Giusti and William J. Schneider ABSTRACT Bifurcation ratios derived from streams ordered according to the Strahler system are not wholly independent of the stream orders from which they are computed and, within a basin, tend to decrease in a downstream direction. The bifurcation ratios computed from two successive constant orders of streams within equal-order basins increase with the area of the basin but tend to become constant where the basin reaches a certain size. The distribution of the number of major tributaries (streams of one order-lag) is exponential with a maximum frequency of two and can be expressed as the probability function p=e-HM-.. The distribution of the number of smaller tributaries is both unimodal and skewed and there appears to be an orderly succession of distributions from exponential to normal from the major tributaries to the smallest fingertip branches. INTRODUCTION Many geomorphic studies make use of an ordering system of stream branches proposed by Horton (1945). A fundamental parameter of this ordering system is the bifurcation ratio, which is defined as the ratio of the number of stream branches of a given order to the number of stream branches of the next higher order. This ratio can be expressed by where Rt=bifurcation ratio, •/Vu=number of streams of given order, and iV„+I = number of streams of next higher order. According to Horton (1945, p. 290), the bifurcation ratio varies from a minimum of 2 in “flat or rolling drainage basins” to 3 or 4 in “mountainous or highly dissected drainage basins”; it is a parameter used in equations giving the number of streams in a basin. As expressed by Horton, the equation is Nu=Rt('~u) (2) where s=order of main stream and «=given order. Strahler (1957) expresses the equation as log Nu=a—bu (3) where the antilog of b is the bifurcation ratio. He further states that the bifurcation ratio is “highly stable and shows a small range of variation from region to region.” The average (mean) is about 3.5. This paper discusses the distribution of bifurcation ratios and further analyzes the distribution of the number of any order tributary in a basin. SOURCES OF DATA Several sources of data were used for this study. Special photogrammetrically prepared drainage maps of 130 square miles of the Yellow River basin in the Piedmont province of Georgia were analyzed to determine the distribution of streams in the drainage network of the basin. The maps, compiled at the scale of 1:24,000 delineated all drainage courses visible on aerial photographs taken at a flight height of 7,200 feet. Additional data were obtained from 108 standard topographic maps, ranging in scale from 1:24,000 to 1:250,000, of the Piedmont province. Data published by Melton (1957), Coates (1958), and Leopold and Langbein (1962) were used also for analyses. Streams were classified according to a system of ordering proposed by Horton (1945) and modified by Strahler (1957). A summary of the distribution of number of streams of a given order within the Yellow River drainage system is shown in table 1. The 130-square-mile area of the Yellow River basin consisted of two seventh-order streams. The table lists separately the distribution of streams in each seventh-order basin. VARIABILITY OF BIFURCATION RATIOS The number of streams of any order in each of the two seventh-order basins that comprise the part of the Yellow River drainage system analysed here are plotted against their order in figure 1. The figure is typical in that some curvature exists in the range of higher order subbasins. This effect is also shown in GlG2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 1.—Relation between number of streams and stream order for the Yellow River. figure 2 where the bifurcation ratios computed as Ay Ay N2/N3 * * * Nt/N, from the values of table 1 are plotted against the basin order. The position of the point suggests that the bifurcation ratio is highly variable. In order to clarify this variability, mean bifurcation ratios and their standard deviation were computed from 26 subbasins of fifth, sixth, and seventh order < ex < o CX r> 2 3 4 5 6 7 BASIN ORDER (u + l) Figure 2.—Relation between bifurcation ratios and order for two seventh-order basins. within the Yellow River drainage system. These data are shown in table 2. Both figure 2 and table 2 suggest (1) that bifurcation ratios expressed as the number of next lower order tributaries or AI,_i are smaller than bifurcation ratios computed from lower order tributaries and (2) that bifurcation ratios tend to become constant and less variable (smaller standard deviation) when expressed as ratios between the numbers of lower order branches. Figure 3 suggests a relation between the bifurcation ratio and the drainage area. This relation, however, depends entirely upon the relation between order and area and can be expressed as follows: Basins of equal order but variable areas tend to have the smallest bifurcation ratios in the smallest areas; the ratios increase with increasing areas up to a certain size, beyond which the bifurcation ratios tend to become constant. Table 1.—Distribution of streams, by order, within the Yellow River drainage system Number of streams (iV«) of order— 1 2 3 4 5 6 7 A 4,026 812 200 33 8 2 1 B 4,823 1,060 233 42 11 4 1 Table 2.—Means and standard deviations of bifurcation ratios for Yellow River drainage N-1 N, N,-t N,-i N.-3 N,-i Nr* Nr- 3 N.-1 Nr-, Nr* N,-S 3.2 4.5 4.7 4.8 i 4.6 2 4. 75 1.3 1.5 .75 .54 1 Computed from 6 basins of 6th order and 2 basins of 7th order. 2 Computed from 2 basins of 7th order.DISTRIBUTION OF BRANCHES IN RIVER NETWORKS G3 20 r o o o • a S —4----- 0 S =5 ll— 0.01 0.05 0.1 0.5 1.0 DRAINAGE AREA (A), IN SQUARE MILES Figure 3—Relation between drainage area and bifurcation ratios Ni'.N* for subbasins within the Yellow River basin. The fact that bifurcation ratios become smaller as they are computed from higher order subbasins is basically due to the branching process. Consider figure 4 where a fourth-order basin derived by a random-walk process is shown. For clarity, the same basin is shown successively in figure 4 with the number of streams of increasing order. It is apparent that as the stream order increases the percentage of streams that coalesce into a higher order tributary also increases and that this increase is due to the diminishing amount of area available. In figure 5, all the percentages of coalescing streams for the Yellow River and for the random-walk model developed by Leopold and Lang-bein (1962, p. A18) have been plotted against their order. The increase in a downstream (with higher order) direction, particularly for the Yellow River, is evident. Thus, bifurcation ratios AT.-, N, tend to become constant for values of u > s — 2, where s is the order of the main stem. DISTRIBUTION OF BIFURCATION RATIOS Frequency distributions of bifurcation ratios are shown in figures 6 and 7. The distributions are derived from data on all fourth- and higher order subbasins and random sampling of first-, second-, and third-order subbasins within the Yellow River drainage system. Data obtained by Melton (1957) fro n arid-to-humid mountainous basins in Arizona, New Mexico, Colorado, and Utah, and data obtained by Coates (1958) for small basins in the humid Interior Low Plateaus province are included also. According to Horton (1945, p. 296), equation 2 becomes Rn=Ng-1 (4) if u=s— 1. This relation indicates that the bifurcation ratio, R„, is equal to the number of streams of the next to the highest order for a given drainage basin. Thus, the bifurcation ratio as defined by equation 4 can be equated to the number of major tributaries to a given stream. The major tributaries can also be defined as streams of one order-lag. Thus, it follows that a drainage system will have streams of one order-lag (major tributaries), two order-lags (smaller tributaries), and so on, and the number of tributaries may be defined as N, N,_2, * * * Y,_n, the subscripts indicating the relative position within the drainage system. Figures 4 and 5 show distributions of bifurcation ratios which are grouped according to their order-lag. Thus ratios ~ in basins of order 4 can be written as N3 ratios ^s 2 and can be compared to ratios ^ from third-order basins which are also expressed as Ns_2 Nt-i The distributions of number of major tributaries (streams of one order-lag) are shown in figure 8. These distributions differ considerably from those of figures 6 and 7. Statistical parameters of the distributions of figures 6, 7, and 8 are tabulated in table 3.G4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 4.—Random-walk model of a fourth order basin (from Leopold and Langbein, 1962, fig. 8, p. A18).DISTRIBUTION OF BRANCHES IN RIVER NETWORKS G5 Table 3—Selected, statistical data for distributions shown in figures 6, 7, and 8 Source of data Statistic Yellow River Melton (1958) Coates (1959) Leopold and Lang- bein (1962) Topo- graphic maps (blue lines) Sample size 270 84 84 84 132 45 45 60 60 92 269 Basin order, s 2-7 3 4 4 3,4 4 4 3 3 2,3 2-6 N. i Nt-i Nr-1 NrJ Nr-1 Nr-, N.-i N,-i N,-, N. i Nr-1 Mean bifurcation ratios, Rb 3. 48 Nt-i 4.36 Nr-1 3.88 Nr- J 4. 72 3.04 Nr-1 4.04 N,-t 6.02 3. 40 N.-i 4.38 3.23 3.27 Standard deviation of bifurcation ratios, S(Rj) 1.79 1.97 1.09 1. 05 1.43 1.17 3.24 1.62 1.17 1.83 1.66 Skewness1 of bifurcation ratios, Sk(Rb). .83 .18 -.11 .21 .73 .034 .31 .86 .32 .67 .77 1 Skewness calculated by the approximate formula, skewness= mean—mode standard deviation Figure 5.—Relation between percentage of coalescing tributaries and order. DISTRIBUTION OF NUMBER OF MAJOR TRIBUTARIES Because the distributions for the various data shown in figure 8 did not differ significantly from each other, the variations among the moments of these distributions were assumed to be due to sampling errors, and all data of figure 8 were combined. The resulting distribution is shown in figure 9 as a semilogarithmic function of the form (5) where /=the frequency of occurrence, in percent, jV(s-i)=number of streams of one order-lag, and e=base of Naperian logarithms. The computed regression equation is which can be simplified to /=100e~*Vi- (6a) Because equation 6a represents a distribution, it may become a probability function as P=«-*V 1 (7) from which the probability, p, can be computed for any given number of major tributaries. Probabilities of occurrences for values of Ns~i up to 10 are shown in table 4. This table shows that more than one out of three streams similar to those studied here will have two major tributaries, and about one out of five three major tributaries. A second-order stream will most frequently have two first-order streams as branches; a third-order stream two second-order branches, and so on. Table 4—Probability of occurrence of number of major tributaries to a stream Number of major tributaries (N 8—0 Probability of occurrence (p) 2.____________________________________ 0.368 3..____________________________________ .233 4_ ................................... . 135 5................................... .082 6 ..................................... 050 7 ..........-.......................... .030 8 ........................................018 9 ..................................... .011 10 .................................... . 007 Melton (1958) defines a “conservative drainage system” as “one having the minimum number of channel segments necessary for the highest order of the system.” His measure of conservancy for any order u is 1 (8) A /=99.48e-°-51w«-i (6) and maximum conservancy corresponds to Su—0.G6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS COATES (1958), THIRD ORDER BASINS (1957), THIRD ORDER BASINS 40 ~ 20 \ 10 UJ 3 O uj n cr 0 BIFURCATION RATIOS (N^Ng) FROM YELLOW RIVER, THIRD ORDER BASINS BIFURCATION RATIOS (N2:N3) FROM YELLOW RIVER, FOURTH ORDER BASINS BIFURCATION RATIOS (N2:N3) FROM MELTON (1957), FOURTH ORDER BASINS 40 r- O < 5 30 - o cr ~ 20 - > o O' Figure 0.—Distributions of bifurcation ratios defined as 40 i- \ \ DATA FROM \ YELLOW RIVER \ \ \ \ \ \ \ 30 - 20 - 10 / . DATA FROM \ MELTON (1957) 'f \ \ \ \ \ \ \ \ \ _L ! 1 2 3 BIFURCATION RATIOS (Nj:N2) Figure 7—Distribution of bifurcation ratios AT.-3-5-lV.-2. This concept applied to the major tributaries only gives Su-Pfi-1 (8a) where maximum conservancy refers to streams having only two major tributaries; this relation, according to the probabilities computed, has a one-in-three occurrence. The manner and the number of joining tribu- taries no doubt greatly affects the hydrology of the drainage system, and the number of major tributaries probably bears some relation to the shape of the basin. DISTRIBUTION OF NUMBER OF SMALLER TRIBUTARIES The subbasins of the Yellow River were further investigated in terms of the smaller tributaries presentDISTRIBUTION OF BRANCHES IN RIVER NETWORKS G7 50 r- o < o cr 40 E 30 >- o z LU 3 O 20 10 \ \ DATA FROM COATES (1958) \- J___L V. —• I 11 50 r- 40 30 20 10 \ \ \ \ • \» N 50 40 \ DATA FROM MELTON (1957) \ 30 20 10 ~t----~t 8 10 \ \ \ DATA FROM LEOPOLD (1962) \ \ -------------• 10 NUMBER OF STREAMS OF ORDER ONE LESS THAN BASIN ORDER (Ns-i) 50 40 o < 30 - \ - 20 - >- o 10 - O' DATA FROM YELLOW RIVER \ \ J____L l -4—*--♦ 50 I- 40 - 30 20 _ \ 10 - DATA FROM STANDARD TOPOGRAPHIC MAPS AT VARIOUS SCALES V \ J___L 579 11 2357 NUMBER OF STREAMS OF ORDER ONE LESS THAN BASIN ORDER (Ns_i) ~1---«— » 9 11 Figure 8.—Distributions of number of major tributaries or bifurcation ratios defined as ;V,-1 (data from various sources as shown). therein by combining the number of branches of two order-lags (such as first-order branches in third-order basins, and second-order branches in fourth-order basins) into one distribution. Similarly, branches of three order-lags (such as first-order branches in fourth-order basin and second in fifth) and four order-lags (such as first in fifth and second in sixth) were combined into one distribution for each. Figure 10 shows the distributions for various order-lags; the progression from exponential to unimodal skewed distribution with increasing order-lag is apparent. As a measure of the symmetry of the distributions, skewness was calculated for each distribution in figure 10. Results are shown in table 5. Skewness appears to decrease with increasing order-lag, and only the number of the smallest fingertip branches in higher order basins may be normally distributed. Table 5.—Skewness of distributions for given order lags Order lag Skewness s — 1.................. 0.83 s —2..................... 93 s —3...................... .88 s —4______________________ . 67 NUMBER OF BRANCHES IN RIVER NETWORK The minimum number of branchings of a stream is given by Melton (1957) in the equation Nu=(9) This equation states that a second order stream will have at least two first-order branches, a third-order stream at least two second-order branches and four first-order branches, and so on. A graph of this equation is shown in figure 11 together with a straight fine fitted through the modal values of the distributions of figure 10. The stream branches of the YellowPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS G8 NUMBER OF STREAMS OF ORDER ONE LESS THAN BASIN ORDER (Ns-l) Figure 9.—Combined distribution of number of major tributaries, from figure 6. River obviously multiply much more rapidly than the minimum growth. In a previous study, Giusti (1963) showed that the number of subbasins, Na, of any given area, a, within a region or basin of a given area, A, could be expressed by the equation #,=0.3 -• (10) CL Similarly, from equation 2 or 3, the number of stream branches can be expressed in terms of the order of the subbasins with respect to the order of the main stem. The fact that the relation based on area is hyperbolic indicates a geometric progression. However, by definition, the number of branches follow a geometric progression and the order (or order-lags) an arithmetic progression. Consequently the relation between number of branches and their order is exponential. A further equation of Melton (1958) gives the number of channel segments in a basin as T 1.75 #=0.8147— (11) where L is the total length of channels and R is the relative relief. There are then several equations from which the number of channel segments in a basin can be computed, and usage of one or another will depend on the preference of the user. However, because of the variability of the bifurcation ratios, the time required for arranging the stream segments of a drainage system into orders, and the lack of true portrayal of the drainage systems on topographic maps, equations 10 and 11 will possibly be more applicable in practice. SUMMARY AND CONCLUSIONS River networks have been analyzed in terms of the number of branches ordered according to a system proposed by Horton (1945) and modified by Strahler (1957). All networks were dendritic and in different climatic and geologic environments. The bifurcation ratios computed from two successive constant orders in any stream network vary according to the two stream orders from which they are computed, and decrease in a downstream direction. Bifurcation ratios for equal-order basins increase somewhat with the area of the basins but tend to become constant for basins beyond a certain size. These two factors—the order used for computation and the size of the area—also affect the parameters of the distributions of bifurcation ratios. Bifurcation ratios tend to become constant for ratios made between number of branches which are two or more orders removed from the order of the main stem (two order-lags). The distribution of the number of one order-lag branches (considered the major tributaries to a given stream) was found to be exponential, with a maximum frequency at two. On an average, one out of three rivers of any given order will have two major tributaries, and one out of five rivers will have three. The distribution of number of higher order-lag branches or number of smaller tributaries was found to be both unimodal and skewed, and only the number of smallest fingertip branches may approach a normal distribution.FREQUENCY (f), IN PERCENTAGE FREQUENCY (f), IN PERCENTAGE DISTRIBUTION OF BRANCHES IN RIVER NETWORKS G9 30 |— A \« \ 20 - I - I 10 - \* \ \ \ \ \ • X 48 88 128 168 208 248 NUMBER OF THREE ORDER LAG TRIBUTARIES (Ns- 3) NUMBER OF FOUR ORDER LAG TRIBUTARIES (Ns_4) Figure 10— Distributions of number of tributaries of varying order lag in Yellow River basin.NUMBER OF TRIBUTARIES OF GIVEN ORDER LAG (Ns.u) G10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 11.—Growth curves of stream branching. Lower curve is minimum growth, upper curve from model values of distributions of figure 10. The variability of bifurcation ratios found in this study indicates that bifurcation ratios must be carefully defined in terms of the two successive orders from which they are computed and the order of the main stem. Comparisons of undefined bifurcation ratios may lead to erroneous conclusions. REFERENCES CITED Coates, D. R., 1958, Quantitative geomorphology of small drainage basins of Southern Indiana: New York, Columbia Univ. Dept. Geology Tech. Rept. 10. Giusti, E. V., 1963, Distribution of river basin areas in the conterminous United States: Bull. Internat. Assoc. Scientific Hydrology, v. 8, no. 3. Horton, R. E., 1945, Erosional development of streams and their drainage basins—hydrophysical approach to quantitative morphology: Geol. Soc. America Bull., v. 56, no. 3, p. 275-370. Leopold, L. B., and Langbein, W. B., 1962, The concept of entropy in landscape evolution: U.S. Geol. Survey Prof. Paper 500-A, 20 p. Melton, M. A., 1957, An analysis of the relations among elements of climate, surface properties, and geomorphology: New York, Columbia Univ. Dept. Geology, Tech. Rept. 11. Melton, M. A., 1958, Geometric properties of mature drainage systems and their representation in an E, phase space: Journ. Geology v. 66, no. 1. Strahler, A. N., 1957, Quantitative analysis of watershed geomorphology: Am. Geophys. Union Trans., v. 38, no. 6, p. 913-920. O 4JRiver Meanders— Theory of Minimum Variance GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-H- , 'River Meanders— Theory of Minimum Variance By WALTER B. LANGBEIN and LUNA B. LEOPOLD PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL The geometry of a meander is that of a random walk whose most frequent form minimizes the sum of the squares of the changes in direction in each unit length. Changes in direction closely approximately a sine function of channel distance. Depth, velocity, and slope are adjusted so as to decrease the variance of shear and the friction factor in a meander over that in an otherwise comparable straight reach of the same river PAPER 422-H UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 20 cents i4 CONTENTS Page Abstract___________________________________________________ HI Introduction________________________________________________ 1 Meander geometry____________________________________________ 2 Path of greatest probability between two points____ 2 The sine-generated curve------------------------------- 3 Analysis of some field examples________________________ 4 Comparison of variances of different meander curves. 4 Meander length, sinuosity, and bend radius_____________ 5 Page Meanders compared with straight reaches---------------- H8 Field measurements___________________________________ 9 Reduction of data____________________________________ 9 Comparison of results_____________-_____________ 11 Interpretive discussion-------------------------------- 13 References_____________________________________________ 15 ILLUSTRATIONS Page Figure 1. Most frequent random walks..._______________________________________________________________________________________ H3 2. Plan view of channel and graph of direction angle <#> as a function of distance along a meander__________________ 3 3. Maps of meander reaches of several rivers and graphs showing relation between channel direction and channel distance________________________________________________________________________________________________________ 6 4. Map of laboratory meander and plot of thalweg direction against distance______________________________________________ 8 5. Comparison of four symmetrical sinuous curves having equal wavelength and sinuosity___________________________________ 9 6. Plan and profiles of straight and curved reaches, Baldwin Creek near Lander, Wyo_____________________________________ 10 7. Plan and profile of curved and straight reaches of Popo Agie River, % mile below Hudson, Wyo______________________ 11 8. Plan and profile of straight and curved reaches on Pole Creek at Clark’s Ranch near Pinedale, Wyo__________________ 12 9. Relation of sinuosity to variances of shear and friction factor__________________________________________________ 14 10. Relations between velocity, depth, and slope in straight and curved reaches, Pole Creek, near Pinedale, Wyo______ 14 TABLES Page Table 1. Field data for Pole Creek at Clark’s Ranch near Pinedale, Wyo_______________________________________________________ H13 2. Comparison between straight and curved reaches in five rivers___________________________________________________ 14 3. Comparison of coefficients of correlation_______________________________________________________________________ 14 in^ ^ 2q (3 % SYMBOLS a ratio of variances (table 3) c a coefficient (equation 1) k sinuosity, ratio of path distance to downvalley distance R radius of curvature of a bend r coefficient of correlation (table 3) s unit distance along path x a factor or variable p probability of a particular direction (equation 1) a2 variance of a particular factor, x water depth energy slope total path distance in a single wavelength number of measurements v velocity a a constant of integration (equation 2) y unit weight of water X wavelength tv pi, ratio of circumference to radius p radius of curvature of path a standard deviation 4> angle that path at a given point makes with mean downpath direction co maximum angle a path makes with mean downpath direction rvPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS RIVER MEANDERS—THEORY OF MINIMUM VARIANCE By Walter B. Langbein and Luna B. Leopold ABSTRACT Meanders are the result of erosion-deposition processes tending toward the most stable form in which the variability of certain essential properties is minimized. This minimization involves the adjustment of the planimetric geometry and the hydraulic factors of depth, velocity, and local slope. The planimetric geometry of a meander is that of a random walk whose most frequent form minimizes the sum of the squares of the changes in direction in each successive unit length. The direction angles are then sine functions of channel distance. This yields a meander shape typically present in meandering rivers and has the characteristic that the ratio of meander length to average radius of curvature in the bend is 4.7. Depth, velocity, and slope are shown by field observations to be adjusted so as to decrease the variance of shear and the friction factor in a meander curve over that in an otherwise comparable straight reach of the same river. Since theory and observation indicate meanders achieve the minimum variance postulated, it follows that for channels in which alternating pools and riffles occur, meandering is the most probable form of channel geometry and thus is more stable geometry than a straight or nonmeandering alinement. INTRODUCTION So ubiquitous are curves in rivers and so common are smooth and regular meander forms, they have attracted the interest of investigators from many disciplines. The widespread geographic distribution of such forms and their occurrence in various settings had hardly been appreciated before air travel became common. Moreover, air travel has emphasized how commonly the form of valleys—not merely the river channels within them— assumes a regular sinuosity which is comparable to meanders in river channels. Also, investigations of the physical characteristics of glaciers and oceans as well as landforms led to the recognition that analogous forms occur in melt-water channels developed on glaciers and even in the currents of the Gulf Stream. The striking similarity in physical form of curves in these various settings is the result of certain geometric proportions apparently common to all, that is, a nearly constant ratio of radius of curvature to meander length and of radius of curvature to channel width (Leopold and Wolman, 1960, p. 774). This leads to visual similarity regardless of scale. When one looks at a stream on a planimetric map without first glancing at map scale, it is not immediately obvious whether it is a large river or a small stream. Explanation of river meanders have been varied. It has been suggested that meanders are caused by such processes as—- regular erosion and deposition; secondary circulation in the cross-sectional plane; seiche effect analogous to lake seiches. Attempts, however, to utilize these theories to calculate the forms of meanders have failed. Although various phenomena—including some of those mentioned above, such as cross circulation—are intimately involved in the deviation of rivers from a straight course, the development of meanders is patently related to the superposition of many diverse effects. Although each of these individual effects is in itself completely deterministic, so many of them occur that they cannot be followed in detail. As postulated in this paper, such effects can be treated as if they were stochastic—that is, as if they occurred in a random fashion. This paper examines the consequences of this postulate in relation to (1) the planimetric geometry of meanders, and (2) the variations in such hydraulic properties as depth, velocity, and slope in meanders as contrasted with straight reaches. The second problem required new data. These were obtained during the snowmelt season of high discharge in the years 1959-64 by the junior author with the invaluable and untiring assistance of William W. Emmett and Robert M. Myrick. Leon W. Wiard was with us for some of the work. To each of them, the authors are very grateful for their important contribution, measured in part by the internal consistency of the various field data, and the satisfactory closures of surveys made under difficult conditions. HIH2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS MEANDER GEOMETRY Meandering in rivers can be considered in two contexts. the first involves the whole profile from any headwater tributary downstream through the main trunk—that is, the longitudinal profile of the river system. The second context includes a meandering reach of a river in which the channel in its lateral migrations may take various planimetric forms or paths between two points in the valley. The river network as a whole is an open system tending toward a steady state and within which several hydraulically related factors are mutually interacting and adjusting—specifically, velocity, depth, width, hydraulic resistance, and slope. These adjust to accommodate the discharge and load contributed by the drainage basin. The adjustment takes place not only by erosion and deposition but also by variation in bed forms which affect hydraulic resistance and thus local competence to carry debris. Previous application of the theory of minimum variance to the whole river system shows that downstream change of slope (long profile) is intimately related to downstream changes in other hydraulic variables. Because discharge increases in the downstream direction, minimum variance of power expenditure per unit length leads toward greater concavity in the longitudinal profile. However, greater concavity is opposed by the tendency for uniform power expenditures on each unit of bed area. In the context of the whole river system, a meandering segment, often but not always concentrated in downstream rather than upstream portions of the system, tends to provide greater concavity by lengthening the downstream portion of the profile. By increasing the concavity of the profile, the product of discharge and slope, or power per unit length becomes more uniform along a stream that increases in flow downstream. Thus the meander decreases the variance of power per unit length. Though the occurrence of meanders affects the stream length and thus the river profile, channel slope in the context of the whole river is one of the adjustable and dependent parameters, being determined by mutual interaction with the other dependent factors. In the second context of any reach or segment of river length, then, average slope is given, and local changes of channel plan must maintain this average slope. Between any two points on the valley floor, however, a variety of paths are possible, any of which would maintain the same slope and thus the same length. A thesis of the present paper is that the meander form occupies a most probable among all possible paths. This path is defined by a random-walk model. PATH OF GREATEST PROBABILITY BETWEEN TWO POINTS » The model is a simple one; a river has a finite probability p to deviate by an angle d from its previous direction in progressing an elemental distance ds along its path. The probability distribution as a function of deviation angle' may be assumed to be normal (Gaussian) which would be described by dp=c exp g (—-— ) (1) where a is the standard deviation and where c is defined by the condition that f dp= 1.0. The actual meander path then corresponds to the most probable river path proceeding between two points A and B, if the direction of flow at the point A and the length of the path between A and B is fixed and the probability of a change in direction per unit river length is given by the probability distribution above. This formulation of the problem of river meanders is identical to that of a class of random-walk problems which have been studied by Von Schelling (1951, 1964). The Von Schelling solution demonstrated that the arc length s is defined by the following elliptic integral: *=- f - d* ....... (2) a J V2(a —cos ) where is the direction angle measured from the line AB and a is a constant of integration. It is convenient to set of—COS a (3) in which co becomes the maximum angle the path makes from the origin with the mean direction. Curves for co=40°, 90°, 110°, which all show patterns characteristically seen in river meanders, are given in figure 1 (after Von Schelling, 1964, p. 8). Von Schelling (1951) showed that a general condition for the most frequent path for a continuous curve of given length between the two points, A and B, is VI ^f=a minimum, (4) pz where As is a unit distance along the path and p is the radius of curvature of the path in that unit distance. But since "-(!£> (5> where A<£ is the angle by which direction is changed in distance As, therefore, S(A$)1 2 . . /cv = a minimum. (6) As 1 Acknowledgments arc due A. E. Scheldegger for his assistance in clarifying the mathematical relationships of this section.RIVER MEANDERS—THEORY OF MINIMUM VARIANCE H3 Since the sum of all the directional changes is zero, or 2>*=0. (7) equation 6 is the most probable condition in which the variance (mean square of deviations in direction) is minimum. Figure 1.—Examples of most frequent random-walk paths of given lengths between two specified points on a plane. Adapted from H. von Schelling. THE SINE-GENERATED CURVE For graphing meanders it is easier to make use of the approximation that 2(l-cos „) [l-(£)2] (8a) is a close approximation for d=u sin ——1 s CO the central zero. With (9) or as a function of distance along the channel path (B). Inasmuch as the ratio co/^fl —cosco) is nearly constant (=1.05), over the range of possible values of co, meander path lengths, M, are inversely proportional to the standard deviation a; thus, M=6.6/ has its largest value co, as can be seen in figure 2. The graph on figure 2A has been constructed for a value of co=110°, and corresponds almost exactly to the 110° curve of figure 1 calculated by the exact equations of Von Schelling. It will be noted that the plan view of the channel (fig. 2A) is not sinusoidal; only the channel direction changes as sinusoidal function of distance (fig. 2B). The meander itself is more rounded and has a relatively uniform radius of curvature in the bend portion. This can be noted in the fact that a sine curve has quite a straight segment as it crosses the x -axis.H4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS The tangent to the sine function in figure 2B at any point is A/As which is the reciprocal of the local radius of curvature of the meander. The sine curve is nearly straight as it crosses the zero axis. Therefore, the radius of curvature is nearly constant in a meander bend over two portions covering fully a third of the length of each loop. There are, of course, plenty of irregular meanders but those which are fairly uniform display a close accordance to a true sine curve when channel direction is plotted against distance along the channel, as will be shown. ANALYSIS OF SOME FIELD EXAMPLES Some field examples presented in figure 3 will now be discussed briefly with a view to demonstrating that among a variety of meander shapes the sine-generated curve fits the actual shape quite well and better than alternatives. Figure 3A, upper part, shows the famous Greenville bends at Greenville, Miss., before the artificial cutoffs changed the pattern. The crosses in the lower diagram represent values of channel direction, —relative to a chosen zero azimuth, which for convenience is the mean downstream direction—plotted against channel distance. The full curve in the lower diagram is a sine curve chosen in wavelength and amplitude to approximate the river data. A sine curve that was fitted to the crosses was used to generate a plot of channel direction against distance which has been superimposed on the map of the river as the dashed line in the upper diagram. It can be seen that there is reasonable agreement. The same technique has been used in the other examples in figure 3 chosen to cover a range in shape from oxbows to flat sinuous form, and including a variety of sizes. Figure 2>B shows a stream that not only is not as oxbowed as the Greenville bends but also represents a river of much smaller size, the Mississippi River being about three-quarters of a mile wide and Blackrock Creek about 50 feet wide. Figure 3(7 shows the famous Paw Paw, W. Va., bends of the Potomac River, similar to and in the same area as the great meanders of the tributary Shenandoah River. The curves in these rivers are characterized by being exceptionally elongated in that the amplitude is unusually large for the wavelength and both are large for the river width, characteristics believed to be influenced by elongation along the direction of a fracture system in the rock (Hack and Young, 1959). Despite these peculiarities the sine-generated curve fits the river well. Figures 3D and 3E are among the best known incised meanders in the western United States, where the Colorado River and some of its tributaries are a thousand or more feet below the lowest of the surrounding benches, the cliffs rising at very steep angles up from the river’s very edge. Even these provide a plot of direction versus length which is closely approximated by a sine curve. By far the most symmetrical and uniform meander reach we have ever seen in the field is the Popo Agie River near Hudson, Wyo., for which flow data will be discussed later. The relation of channel direction to distance is compared with a sine curve in the lower part of figure 3F. The laboratory experiments conducted by Friedkin (1945) provide an example of near-ideal meanders. The meander shown in figure 4 is one of those that developed greatest sinuosity. The channel shown on figure 4A was developed in Mississippi River sand on a “valley” slope of 0.006, by rates of flow that varied from 0.05 to 0.24 c-fs over an 18-hour period on a schedule that simulated the fluctuations of discharge on the Mississippi River. The sand bed was homogeneous and the meander mapped by Friedkin is more regular than those usually observed in natural streams. The direction angles of the thalweg were measured at intervals of 2.5 feet. These direction angles are plotted on the graph, figure 4B. Since each meander had a thalweg length of 50 feet, the data for the two meanders are shown on the same graph. Where the two bends had the same direction only one point is plotted. The data correspond closely to the sine curve shown. COMPARISON OF VARIANCES OF DIFFERENT MEANDER CURVES As a close approximation to the theoretical minimum, when a meander is such that the direction, , in a given unit length As is a sine function of the distance along the curve, then the sum of squares of changes in direction from the mean direction is less than for any other common curve. In figure 5, four curves are presented. One consists of two portions of circles which have been joined. Another is a pure sine curve. The third is a parabola. A fourth is sine-generated in that, as has been explained, the change of direction bears a sinusoidal relation to distance along the curve. All four have the same wavelength and sinuosity— the ratio of curve length to straight line distance. However, the sums of squares of the changes in direction as measured in degrees over 10 equally spaced lengths along the curves differ greatly as follows:RIVER MEANDERS—THEORY OF MINIMUM VARIANCE H5 Curve 2 (A )2 Parabola____________________________________ 5210 Sine curve__________________________________ 5200 Circular curve______________________________ 4840 Sine-generated curve________________________ 3940 The “sine-generated” curve has the least sum of squares. The theoretical minimum curve is identical within practical limits of drawing. MEANDER LENGTH, SINUOSITY, AND BEND RADIUS The planimetric geometry of river meanders has been defined as follows =a sin ■— 2x (10) has the dimension of L~l or the reciprocal of length. As previously shown, a is inversely proportional to meander length. Bend radius is related to wave length and is virtually independent of sinuosity. Defined as before, as the average over the % of channel length for which is nearly linearly related to channel distance, bend radius R is Since 0 ranges from -(-0.5w to —0.5to over this near-linear range, A<£=«. Substituting for to its algebraic equivalent 3 in terms of sinuosity, where equals the direction at location s, to is the maximum angle the meander takes relative to the general direction of the river, and M is the channel length of a meander. The values of to and M can be defined further as follows: The angle to is a unique function of the sinuosity and is independent of the meander length. It ranges from zero for a straight line path of zero sinuosity to a maximum of 125° for “gooseneck” meanders at point of incipient crossing. The sinuosity, k, equals the average of the values of cos over the range from 0=0 to 0=co. Thus a relationship can be defined between k and co. An approximate algebraic expression 2 is V"fc—T —£— or —°V¥- Sinuosity as measured by parameter k, as explained, is thought to be a consequence of the profile development which is only secondarily influenced by reaction from the meander development. In the random-walk model, the standard deviation of changes in direction per unit of distance is a which 1 Relation of« to sinuosity, k. where k= M 2 cos <£As 0=« sin '^2ir. With an assumed value of w, values of at 24 equally spaced intervals of s/M were computed. The reciprocal of the average value of cos equals the sinuosity, k. « Values of a> and k so computed were as follows: u (radians) 0.5______ 1.0______ k . 1.06 . 1.30 (0 (radians) 1.5....... 1.75...... k 1.96 2. 67 The values of k approximate the following or k=4.84/(4.84-a>2) “W? and since M=k\, bend radius equals 13 Yfc —1 Some typical values for bend radius in terms of sinuosity are k Bend radius (») „ wavelength lialto—i--:---f-r— bend radius 1.25_________________ 0. 215X 4.6 1.5................... . 20X 5.0 2. 0................... . 22X 4. 5 2. 5._______________— . 24X 4. 2 These values agree very closely with those found by measurement of actual meanders. Leopold and Wolman (1960) measured the meander characteristics of 50 samples, including rivers from various parts of the United States, that range in width from a few feet to a mile. For this group, in which sinuosities were dominantly between 1.1 and 2.0, the average ratio of meander length to radius of curvature was 4.7, which is equivalent to .213X. The agreement with values listed above from the theory is satisfactory. The curve defined by 4>=w sin jj 360° is believed to underlie the stable form of meanders. That actual meanders are often irregular is well known, but as observed above, those meanders that are regular in geometry conform to this equation. Deviations (or “noise”), it is surmised, are due to two principal causes: (1) shifts from unstable to a stable form caused by random actions and varying flow, and (2) nonhomogeneities such as rock outcrops, differences in alluvium, or even trees. The irregularities are more to be observed in “free” (that is, “living”) meanders than in incised meanders. During incision, irregularities apparently tend to be > See footnote 2, this page. 789-282—6< 2,H6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 3—MAPS OF RIVER CHANNELS AND A, Upper, Map of channel compared to sine-generated curve (dashed); lower, plot of actual channel direction against distance (crosses) and a sine curve (full line). B, Upper, Map of channel compared to sine-generated curve (dashed); lower, plot of actual channel direction against distance (crosses) and a sine curve (full line). C, Upper, Map of channel compared to sine-generated curve (dashed); lower, plot of actual channel direction against distance (crosses) and a sine curve (full line). D, Upper, Planimetric map of river; lower, plot of actual channel direction against distance (crosses) and a sine curve (full line).RIVEK MEANDERS—THEORY OF MINIMUM VARIANCE H7 o 10 DISTANCE ALONG CHANNEL 20 30 . IN THOUSANDS OF FEET PLOTS OF CHANNEL DIRECTION AGAINST DISTANCE E, Right, Planimetric map of river; left, plot of actual channel direction against distance (crosses) and a sine curve (full line). F, Upper, Planimetric map of river; lower, plot of actual channel direction against distance (crosses) and a sine curve (full line).H8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 4.—Map of laboratory meander and plot of thalweg direction against distance. A, Plan view; B, Comparison of direction angles with sine curve. averaged out and only the regular form is preserved, and where these intrench homogeneous rocks, the result is often a beautiful series of meanders as in the San Juan River or the Colorado River. MEANDERS COMPARED WITH STRAIGHT REACHES Meanders are common features of rivers, whereas straight reaches of any length are rare. It may be inferred, therefore, that straightness is a temporary state. As the usual and more stable form, according to the thesis of this report, the meander should be characterized by lower variances of the hydraulic factors, a property shown by Maddock (Written commun., 1965) to prevail in self-formed channels. The review of the characteristics of meanders and the theories which had been advanced to explain them showed that quantitative data were nearly nonexistent on many aspects of the hydraulics of flow during those high stages or discharges at which channel adjustment actively takes place. Because there are certain similarities and certain differences between straight segments or reaches of river and meandering reaches, it also seemed logical to devote special attention to comparison of straight and curved segments during channel-altering flow. An initial attempt to obtain such data was made during 1954-58 on a small river in Maryland, near enough to home and office to be reached on short notice during the storm flows of winter. Though something was learned, it became clear that rivers which rise and fall in stage quickly do not allow a two-man team with modest equipment to make the desired observations. Attention was then devoted to the rivers which drain into the Missouri and Colorado River systems from the Wind River Range in central Wyoming, where good weather for field work generally exists at the base of the mountains during the week or two in late spring or early summer when snowmelt runoff reaches its peak. Some of the smaller streams presented the difficulty that the area of study lay 15 to 20 miles from the farthest watershed divide—a distance sufficient to delay the peak discharge by 9 to 12 hours. Maximum snowmelt in midafternoon caused a diurnal peak flow which reached the study area shortly after midnight and meant surveying and stream gaging by flashlight and lantern illumination of level rod and notebook. The plan of field work involved the selection of one or more meanders of relatively uniform shape which occurred in the same vicinity with a straight reach of river at least long enough to include two riffles separated by a pool section. No tributaries would enter between the curved and straight reaches so that at any time discharge in the two would be identical. It was assumed and checked visually, that the bed and bank materials were identical in the two reaches as well. Later bed sampling showed that there were some differences in bed-material size which had not been apparent to the eye. A curve introduces an additional form of energy dissipation not present in a comparable straight reach, an energy loss due to change of flow direction (Leopold and others, 1960). This additional loss is concentrated in the zone of greatest curvature which occurs midway between the riffle bars.- The additional loss is at a location where, in a straight reach, energy dissipation is smaller than the average for the whole reach. The introduction of river bends, then, tends to equalize the energy dissipation through each unit length, but does so at the expense of greater contrasts in bed elevation. Moreover, curvature introduces a certain additional organization into the distribution of channel-bed features. The field measurements were devised to measure and examine the variations in these quantities. Though more will be said about individual study reaches when the data are discussed, a salient feature was apparent which had not previously been noticed when the first set of such field data was assembled; this feature can be seen in figure 6. Where the profiles of bed and water surface are plotted on the same graph for comparison, it is apparent that at moderately high stage—from three-fourths bankfull to bankfull—the larger-scale undulation of the riverbed caused by pool and shallow has been drowned out, as the streamgager would say, or no longer causes an undulation in the water-surface profile of the meander. In contrast, at the same stage in the straight reach the flat and steep alternations caused by pool and riffle are still discernible.RIVER MEANDERS—THEORY OF MINIMUM VARIANCE H9 On the other hand, the profile revealed that the undulation of the bed along the stream length is of larger magnitude in the curved than in the straight reach. Considering the features exemplified in figure 6, one could reason as follows. In a straight reach of channel the dunes, bars, pools, and riffles form more or less independently of the channel pattern owing to grain interaction. To the occurrence of pool and riffle the channel must adjust. The riffle causes a zone of greater-than-average steepness which is also a zone of greater-than-average energy expenditure. The pool, on the other hand, being relatively free of large gravel on the bed and of larger depth relative to bed roughness elements, offers less resistance to flow and there the energy expenditure is less than average. The result is a stepped water-surface profile—flat over pool and steep over riffle. Once pools and riffles form with their consequent variations in depth, width remaining relatively uniform, then slopes must vary. If slopes vary so that bed shear remains constant, then slope would vary inversely with depth. However, this result would require that the friction factor vary directly as the square of the depth. If, on the other hand, the friction factor remained uniform and the slope varied accordingly, then bed shear would vary inversely as the square of the depth. Uniformity in bed shear and the friction factor is incompatible, and the slope adjusts so as to accommodate both about equably and minimize their total variance. FIELD MEASUREMENTS The general program of field measurements has been described. The main difficulty experienced was that the combination of conditions desired occurs rarely. A straight reach in proximity to a regular meander Figure 5— Comparison of four symmetrical sinuous curves having equal wavelength and sinuosity. curve does not have a high probability of occurrence, and to find such a combination, airplane photographic traverses at about 1,000 feet above the ground were flown for some distance along the east front of the Wind River Range.4 Even after reaches were chosen from the photographs, ground inspection resulted in discard of significant proportions of possible sites. When a satisfactory site was located, a continuous water-stage recorder was established; benchmarks and staff gage were installed and the curved and straight reaches mapped by plane table. Cross sections were staked at such a spacing that about seven would be included in a length equal to one pool-and-riffle sequence. Water-surface profiles were run by leveling at one or more stages of flow. Distances between water-surface shots were equal for a given stream; 10-foot distances were used on small streams having widths of 10 to 20 feet and 25-foot spacing used for streams 50 to 100 feet wide. Usually two rods, one on each streambank, accompanied the instrument. Shots were taken opposite one another along the two banks. Water-surface elevations were read to 0.01 foot on larger rivers, and to 0.001 foot on small ones. For the small rivers an attachment was used on a surveying rod which in construction resembled a standard point gage used in laboratory hydraulic practice for measurement of water-surface elevation. Velocity measurements by current meter were usually made at various points across the stream at each cross section or at alternate ones. In the small streams these current-meter measurements were made by wading, those in the larger rivers from a canoe. For canoe measurements a tagline had to be stretched across the river at the staked cross sections, an operation fraught with some difficulty where brush lined the banks and when the velocity was high at near-bankfull stage. Figures 6, 7, and 8 show examples of the planimetric maps and bed and water-surface profiles. The sinusoidal change of channel direction along the stream length for figure 7 was presented in figure 3. REDUCTION OF DATA Mean depths and mean velocities were calculated for each cross section. The average slopes of the water surface between the cross sections were also computed from the longitudinal profile. To reduce these quantities to nondimensional form, they were expressed in ratio to the respective means over each reach. The variances of these ratios were computed by the usual formula 4 The authors acknowledge the assistance of Herbert E. Skibltcke and David E. Jones in the photo reconnaissance.H10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS where t— < O 1 1 1 a 1 1 1 "■ 1 1 1 //Straight ■—1—1—1— reach Bed profiles/^ «o» _ ^— Meander Water-surface profiles /A Meander, p— k ‘ St 1 1 1 ■aight reach ^ \ ^ 1 1 I \ - __L^ 1 100 200 B DISTANCE ALONG CHANNEL, IN EEET 300 400 Figure 6.—Plan and profiles of straight and curved reaches, Baldwin Creek near Lander, \yy0 4 planimetric map showing location of reaches; B, Profiles of centerline of bed and average of water-surface elevat ions on the two sides of the stream'; curved reach is dashed line and straight reach is full line.RIVER MEANDERS—THEORY OF MINIMUM VARIANCE Hll Width is not included in the analysis because it is relatively uniform, and has no characteristic that distinguishes it between curved and straight reaches. Table 1 shows a sample computation for Pole Creek, and table 2 lists a summary for the five streams surveyed. COMPARISON OF RESULTS The results are listed in table 2. It will be noticed that the variance of slope in each reach is larger than that of depth or velocity. The contrast, however, is less in the meandering than in the straight reach; this reduction may be noted by the figures in the column headed a, which is the ratio of the variance of slope to the sum of those of depth and velocity. Among the and of the friction factor—again considered as ratios to their respective means—were also computed. Bed shear is equal to the product yDS, where y is the unit weight of water, D is the depth, and S is the slope of the energy profile. The unit weight of water is a constant that may be neglected in this analysis of variances. In this study water-surface slopes will be used in lieu of slopes of the energy profile. This involves an assumption that the velocity head is small relative to the accuracy of measurement of water surface. In any case, where the velocity-head corrections were applied, they were either small or illogical, and so were neglected. Similarly, the variance of the quantity DS/v3, which is proportional to the Darcy-Weisbach friction factor, was computed.H12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS A uj 0 + 00 5 + 00 10+00 Stationing along curved reach 15 + 00 ^ DISTANCE ALONG CHANNEL. IN FEET Figure 8—Plan and profile of straight and curved reaches on Pole Creek at Clark’s Ranch near Pinedale, Wyo. A, Planimetric map showing location of reaches; B, Profiles of centerline of bed and average of water-surface elevations on the two sides of the stream; curved reach is dashed line and straight reach is a full line. five rivers studied, this ratio is uniformly less in the meandering reaches than in the straight reaches. The data listed in table 2 also show that the variance in bed shear and in the friction factor are uniformly lower in the curved than in the straight reach. The data on table 2 suggest that the decrease in the variance of bed shear and the friction factor is related to the sinuosity. As shown on figure 9, the greater the sinuosity the less the average of these two variances. When one considers, for example, that the variances of depth and velocity are greater in the curved reach, and even that the variance of slope may be greater (as for example Pole Creek), the fact that the product DS and the ratio DS/F2 have consistently lower variances must reflect a higher correlation between these variables in the meandered than in the straight reach. This improved correlation is shown in the following tabulation (table 3) of the respective coefficients of determination (= squareof the coefficient of correlation.) Figure 10 shows graphical plots for the Pole Creek data listed in table 2. The graph shows that in the meanders, depths, velocities, and slopes are more systematically organized than in straight reaches. To summarize, the detailed comparisons of meandering and straight reaches confirm the hypothesis that in a meander curve the hydraulic parameters are so adjusted that greater uniformity (less variability) is established among them. If a river channel is considered in a steady state, then the form achieved should be such as to avoid concentrating variability in one aspect at the expense of another. Particularly a steady-state form minimizes the variations in forces on the boundary—RIVER MEANDERS—THEORY OF MINIMUM VARIANCE H13 Table 1.—Field data for Pole Creek at Clark’s Ranch near Pinedale, Wyo., June 1964 [Dlscharge=350 cfs, % bankfull] Depth Velocity (v) Mean depth (D) in reach Mean velocity (v) in reach Slope (S) in reach DS (ratio DS/»i(ratio Station (feet) Feet Ratio to mean Feet per sec. Ratio to mean (ratio to mean) (ratio to mean) Feet per 1,000 ft. Ratio to mean to mean) to mean) Straight reach o 1. 52 0. 79 3. 67 1. 30 150 _ 2. 26 1. 17 3. 26 1. 12 0. 98 1. 21 1. 67 1. 20 1. 18 0. 81 300 1. 92 1. 00 2. 44 . 89 1. 08 1. 01 2. 14 1. 55 1. 67 1. 64 450 2. 01 1. 05 2. 44 . 87 1. 02 . 88 1. 14 . 83 . 85 1. 10 600 1. 79 . 93 3. 52 1. 25 . 99 1. 06 . 80 . 58 . 58 . 52 750 1. 69 . 88 2. 12 . 76 . 90 1. 01 2. 24 1. 62 1. 46 1. 43 900 2. 29 1. 19 2. 26 . 80 1. 04 . 78 . 30 . 22 . 23 . 38 Mean 1. 92 2. 81 1. 38 Variance . 019 . 042 . 26 . 25 . 21 Curved reach o 2. 68 1. 23 2. 24 0. 82 150 1. 96 . 90 2. 45 . 90 1. 06 ' 0.88 1. 03 0. 50 0. 53 0. 69 300 1. 44 . 66 3. 46 1. 27 . 78 1. 08 2. 20 1. 07 . 84 . 72 450.. 2. 37 1. 09 2. 45 . 90 . 88 1. 08 2. 00 . 98 . 86 . 74 600 3. 77 1. 74 1. 63 . 60 1. 42 . 75 . 70 . 34 .48 . 85 750 1. 95 . 90 2. 62 . 96 1. 32 . 89 1. 14 . 56 . 74 1. 22 900 . 1. 25 . 58 4. 10 1. 50 . 74 1. 23 3. 94 1. 92 1. 42 . 94 1050. . 1. 25 . 58 4. 02 1. 48 . 58 1. 49 3. 97 1. 94 1. 13 . 51 1200 2. 34 1. 08 2. 20 . 81 . 83 1. 15 3. 17 1. 55 1. 29 . 98 1350 2. 67 1. 23 1. 98 . 73 1. 16 . 77 .27 . 13 . 15 . 25 Mean. 2. 17 2. 72 2. 05 Variance . 067 . 088 . 41 . 15 . 07 that is, the steady-state form tends toward more uniform distribution of both bed stress and the friction factor. INTERPRETIVE DISCUSSION After a review of data and theories concerning river meanders Leopold and Wolman (1960) concluded that meander geometry is “related in some unknown manner to a more general mechanical principle” (p. 774). The status of knowledge suggested that the basic reason for meandering is related to energy expenditure, and they concluded (p. 788) as follows. “Perhaps abrupt discontinuities in the rate of energy expenditure in a reach of channel are less compatible with conditions of equilibrium than is a more or less continuous or uniform rate of energy loss.” Subsequent work resulted in the postulate that the behavior of rivers is such as to minimize the variations in their several properties (Leopold and Langbein, 1963), and the present work shows how this postulate applies to river meanders as had been prognosticated. Total variability cannot be zero. A reduction in the variability in one factor is usually accompanied by an increase in that of another. For example, in the meander, the sine-generated curve has greater variability in changes in direction than the circle. This greater variability in changes in direction is, however, such as to decrease the total angular change. The sine-generated curve, as an approximation to the theoretical curve, minimizes the total effect. The meandering river has greater changes in bed contours than a straight reach of a river. However,H14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Table 2.—Sinuosity and variances for straight and curved reaches on five streams Sinuosity: ratio of path distance to downvailey distance. Variances: a'o} depth; a], velocity; as, slope; a, bed shear; a}, Darcy-Weisbach friction facton as °IW, 2 2 Stream Sinuosity (fc) °D 5, coefficient of linear determination between depth and slope; rjs, coefficient of linear determination between velocity and slope] Straight reach Curved reach tds r% _2 TdS Pole Creek 0.044 0.27 0.72 0.90 Wind River .072 .062 .55 .42 Baldwin Creek .56 .28 .42 .73 Popo Agie River .39 .49 .58 .62 Mill Creek 0 o >"0 1 2 0 1 2 3 4 WATER-SURFACE SLOPE, IN FEET PER HUNDRED FEET X Curvec reach X X X X * r =0.95 O o cr UJ CL J— - 2 >- o u o o r =C oo .52 i Straight reach 1 H LlJ UJ X £ UJ o 3 2 1 0 o ° o r = —0 O 0 21 o Straig re It ach X X < r=- c 9 -0.85 Curved reach X X > * 0 1 2 0 1 2 3 4 WATER-SURFACE SLOPE, IN FEET PER HUNDRED FEET Figure 10.—Relations between velocity, depth, and slope, in straight and curved reaches, Pole Creek, near Pinedale, Wyo. r=correlation coefficient. these changes in depth produce changes in the slope of the water surface and so reduce the variability in bed shear and in the friction factor, as well as to lessen the contrast between the variances of depth and slope.RIVER MEANDERS—THEORY OF MINIMUM VARIANCE H15 These considerations lead to the inference that the meandering pattern is more stable than a straight reach in streams otherwise comparable. The meanders themselves shift continuously; the meandering behavior is stable through time. This discussion concerns the ideal case of uniform lithology. Nature is never so uniform and there are changes in rock hardness, structural controls, and other heterogeneities related to earth history. Yet, despite these natural inhomogeneities, the theoretical forms show clearly. This does not imply that a change from meandering to straight course does not occur in nature. The inference is that such reversals are likely to be less common than the maintenance of a meandering pattern. The adjustment toward this stable pattern is, as in other geomorphic processes, made by the mechanical effects of erosion and deposition. The theory of minimum variance adjustment describes the net river behavior, not the processes. REFERENCES Hack, J. T., and Young, R. S., 1959, Intrenched meanders of the North Fork of the Shenandoah River, Va.: U.S. Geol. Survey Prof. Paper 354-A, p. 1-10. Langbein, W. B., 1964, Geometry of river channels: Jour. Hydro. Div., Am. Soc. Civil Engineers, p. 301-312. Leopold, L. B., Bagnold, R. A., Wolman, M. G., and Brush, L. M., 1960, Flow resistance in sinuous or irregular channels: U.S. Geol. Survey Prof. Paper 282-D, p. 111-134. Leopold, L. B., and Langbein, W. B., 1962, The concept of entropy in landscape evolution: U.S. Geol. Survey Prof. Paper 500-A, 20 p. Leopold, L. B., and Wolman, M. G., 1957, River channel patterns—braided, meandering, and straight: U.S. Geol. Survey Prof. Paper 282-B, 84 p. ------ 1960, River meanders: Bull. Geol. Soc. Am., v. 71, p. 769-794. Von Schelling, Hermann, 1951, Most frequent particle paths in a plane: Am. Geophys. Union Trans., v. 32, p. 222-226. ------ 1964, Most frequent random walks: Gen. Elec. Co. Rept. 64GL92, Schenectady, N.Y. U.S. GOVERNMENT PRINTING OFFICE:t966IAn Approach to the Sediment Transport Problem From General Physics By R. A. BAGNOLD PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-1 From considerations oj energy balance and oj mechanical equilibrium, a mathematical expression is derived relating the rates of sediment transport as bedload and as suspended load to the expenditure of power by a statistically steady flow of water UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 35 cents (paper cover)CONTENTS Page Introduction________________________________________________ II General considerations_______________________________________ 2 Essential features of granular flow_____________________ 2 Implication: the existence of an upward supporting stress________________________________________________ 2 Restrictions of conditions to be considered_____________ 3 Bedload and suspended load and the work rates of their transport_____________________________________________ 3 Definitions_____________________________________________ 3 Transport work rates__________________________________ 4 Principle of solid friction and bedload work rate__ 4 Suspended-load work rate________________________________ 5 Transport work rates and available power; the general sediment-transport relationship_______________________ 5 The general power equation______________________________ 5 Available stream power__________________________________ 5 The general transport relationship in outline form_ 6 Bedload transport efficiency eb at high transport stages.. 6 Concept of a moving flow boundary_______________________ 6 Critical stage beyond which the bedload efficiency eb should be predictable______________________________ 8 Effect of inadequate flow depth on the values of eb-- 9 Variation of the dynamic bedload friction coefficient tan a________________________________________________ 10 The suspended transport efficiency es of shear turbulence. 12 Critical flow stages for suspension____________________ 15 The effective fall velocity V for suspension of heterogeneous sediments______________________________ 17 The effective mean grain size D for heterogeneous bedloads_____________________________________________ 18 Washload.......................................... 18 Page The final transport rate relationship___________________ 118 Existing flume data_______________________________________ 19 Limitations and uncertainties________________________ 19 Wall drag and flow depth_____________________________ 20 Estimation of appropriate fall velocity V____________ 21 Comparison of the theory with the experimental flume data_______________________________________________ 21 Gilbert 5.0-mm pea gravel____________________________ 21 Gilbert 0.787-mm sand______________________________ 21 Gilbert 0.507-mm sand______________________________ 21 Simons, Richardson, and Albertson 0.45-mm sand__ 26 Gilbeit 0.376-mm sand______________________________ 26 Gilbert 0.307-mm sand_____________________________ 26 Barton and Lin 0.18-mm sand_________________________ 26 Laursen 0.11-mm silt_________________________________ 26 General comments_____________________________________ 26 Experiments by Vanoni, Brooks, and Nomicos______ 27 Relation between i and co over the lower, transitional stages____________________________________________ 28 Transport of fine silts in experimental flumes_______ 28 The available river data, nature and uncertainties___ 29 Eneigy-slope estimation_____________________________ 29 Bedload and suspended load__________________________ 29 Deficiency of sediment supply_______________________ 29 Values of the stage criterion 0_____________________ 30 Comparison of predicted with measured river transport rates___________________________________________________ 30 Comparison of river transport data with data for wind- transported sand________________________________________ 35 Conclusion________________________________________________ 35 References________________________________________________ 37 ILLUSTRATIONS Pago Figure 1. Diagram of the relation of normal force or stress to tangential force or stress between solids in moving contact- - 14 2-5. Graphs: 2. Flow relative to moving boundaries__________________________________________________________________ 7 3. Values of theoretical bedload efficiency factors, in terms of mean flow velocity, for quartz-density solids in an adequate depth of fully turbulent water______________________________________________________ 8 4. Values of the solid-friction coefficient tan a in terms of the bed-stress criterion 0=t/(ite=m'bgUb tan a=ib tan a (3) Here the angle a is associated with the average angle of encounter between individual grains, and tan a is the ratio of the tangential to the normal components of grain momentum resulting from the encounters. SUSPENDED-LOAD WORK RATE The suspended-load work rate can be inferred simply and indisputably. The suspended solids are falling relative to the fluid at their mean fall velocity V. But the center of gravity of the suspension as a whole does not fall relative to the bed. Hence the fluid must be lifting the solids at the velocity V. The rate of lifting work done by the shear turbulence of the fluid must therefore be . V suspended-load work rate=m^V=t,= (4) U a The factor As= V/Us may be regarded as the counterpart of tan a. The fluid shear turbulence is in effect pushing the suspended load up a notional frictionless incline V/U,. TRANSPORT WORK RATES AND AVAILABLE POWER; THE GENERAL SEDIMENT-TRANSPORT RELATIONSHIP THE GENERAL POWER EQUATION When any kind of continuing work is being steadily done the principle of energy conservation can be expressed in terms of the time rates of energy input to, and output from, a specified system by the equation rate of doing work=available power—unutilized power or in an equivalent alternative form rate of doing work=available power X efficiency (5) This basic equation has long been familiar to engineers in other fields—such as mechanical, hydromechanical, and electrical—concerned with working machines and with power transmission. It has attracted but little attention, however, in the field of channel hydraulics, and no conventional symbol has been allotted to the quantity power. The power equation appears first to have been applied to sediment transport by Rubey (1933) and later by Velikanov (1955). It was again suggested by Knapp (1941), and was later introduced by me in a paper (Bagnold, 1956) wherein the flowing fluid was regarded as a transporting machine. But none of these ideas have been followed up by taking the logical step of plotting experimental transport rates against stream power. AVAILABLE STREAM POWER The available power supply, or time rate of energy supply, to unit length of a stream is clearly the time rate of liberation in kinetic form of the liquid’s potential energy as it descends the gravity slope S. Denoting this power by 12, Sl=pgQS where Q is the whole discharge of the stream. The mean available power supply to the column of fluid over unit bed area, to be denoted by co, is therefore “=fIow width=flow^width = MdSu= ™ The kinetic energy co liberated in unit time and subsequently dissipated as heat must not be confused with the energy stored within the fluid at any given instant. The liberated energy is as little related to the stored energy as is the inflow and outflow of water through a reservoir related to the quantity of water stored in the reservoir. Confusion is sometimes caused by the seeming paradox that at a given tractive or motive force an increase of resistance has the effect of reducing the power dissipation even though the extra resistance element itself introduces an added element of power dissipation. The reason is immediately clear from the electrical analogy. The electric power dissipated at a given voltage E is PR—E2/B. Hence if we increase the resistance R to R-\-R', the power dissipation is decreased in the ratio R/(R-\-R') even though the extra series resistance R' itself dissipates a new power element PR'. Uncertainties as to the precise energy state existing at any point or cross section have led many to prefer momentum to energy considerations. However, in statistically steady channel flow no such uncertainties arise as to the time rate of energy supply and dissipa-IG PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS tion, provided the system considered is defined as representative of the flow along a length of channel sufficient to include all repetitive irregularities of slope, cross section, and boundary, and the consequent periodic energy transformations. The available power and the transport work rates, regarded as so defined, have precise average values along a channel although their values at any specific cross section may be imprecise. The available power has the same dimensions and quality as the rate of transporting work done. And since the dynamic transport rate i of the solids also has these dimensions and quality, there can be no reasonable doubt that the transport rate is related primarily to the available power, as in all other modes of transport. THE GENERAL TRANSPORT RELATIONSHIP IN OUTLINE FORM The available power u constitutes the single common supply of energy to both the transport mechanisms. Applying equations 3 and 5 for bedload transport, we have tan a=et£<) or i(,=;- (/) tan a where eb is the appropriate efficiency, necessarily less than unity. The work power ebw is dissipated directly into heat, in the process of solid friction. It has gone. Hence, for turbulent fluid flow, the power available to do the work of supporting a suspended load is co(l — et). Applying equations 4 and 5 for suspended-load transport, we have therefore is =f=esa)(l—eb) or i,=u (1—et) (8) Us v Adding equations 7 and 8 the expression for the whole transport rate i is i=i>+i-=“(sk+fw<1-e*)) (9) The derivation of this general outline relationship seems to me straightforward and logical. It involves neither assumption nor approximation, and it is applicable both to turbulent and to laminar fluid flow. In laminar flow the second term disappears, for suspension as here defined cannot- occur, all the solid material being transported as bedload. We have now to infer the values of the four parameters eb) tan «, e„ and Us. By an approximation which should be reasonably close for most purposes, we can at once reduce the number of parameters to three. For since the travel of the suspended solids is unopposed, they can be assumed to travel at the same velocity as the fluid surrounding them. The error introduced by substituting the mean velocity u of the fluid for the mean velocity U, of the suspended solids lies only in the differing distributions of fluid and solid discharges with distance from the bed. The possible small error introduced into what follows by making this substitution should be borne in mind. BEDLOAD TRANSPORT EFFICIENCY eb AT HIGH TRANSPORT STAGES CONCEPT OF A MOVING FLOW BOUNDARY The conventional kinematic definition of a boundary to shear flow is a surface at which the flow velocity is zero relative to the boundary. In channel hydraulics the boundary is assumed axiomatically to be fixed relative to the ground. An equally valid, and more general, dynamic definition of a boundary to shear flow is a surface, or a zone of finite thickness, at or within which the fluid shear stress is reduced to zero by transfer to another medium. The boundary medium may or may not be fixed relative to the ground. At the threshold of motion of the solids both eb and es are evidently zero. The solids are all stationary on the bed. So the flow of the fluid is relative to a boundary which is stationary relative to the ground. As the available power is increased, however, more and more solids move over the bed as bedload, and consequently more and more of the boundary shear stress is applied to the stationary bed indirectly in the form of the solid-transmitted frictional stress T via the moving bedload solids. In this transitional range of flow stage, therefore, the fluid flow is relative to a boundary which is partly moving and partly stationary. Both the internal structure of the flow and the velocity distribution are likely to be complex. As the bedload increases, however, a critical stage must be reached at which so great a number of bedload solids cover the stationary surface as a moving layer that in effect this layer occludes the stationary surface from the fluid flow above it. At and beyond this critical stage, therefore, the fluid flow is wholly relative to a boundary which is itself moving relative to the ground. The fluid shear stress t at this moving boundary may be regarded as disappearing, the fluid-transmitted shear stress being converted progressively through the thickness of the layer into the solid-transmitted shear stress T.* < Experiment (Bagnold, 1956, p. 242) has indicated that at granular concentrations prevailing immediately over a stationary grain bed at the higher flow stages less than 1 percent of the shear stress is maintained by the intergranular fluid, the remainder being maintained by the solid-transmitted stress T.APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 17 As a first step toward arriving at the efficiency es under these conditions, let the moving granular flow boundary be replaced by a continuous carpet. The carpet is supposed in contact with the stationary bed, and has a varying weight m'g proportional to the applied shear stress. The motion of the carpet is opposed by solid friction at its underface. The fluid flow above the carpet applies a fluid shear stress t to it. Provided the carpet is thin compared to the depth of the flow, the resisting shear stress T at its underface is equal to r above it. The conditions are as sketched in figure 2A. The problem is now as follows: If the flow is of given depth and has a given mean flow velocity u relative to the ground, what is the maximum limiting rate at which the flow can do work in transporting its boundary along the ground? The mean velocity of the flow relative to its own boundary is u— Uc, where Uc is the transport velocity of the bouudary. For generality, let the flow law be Hence the maximum transport efficiency is as follows: TUC 1 e‘ TU (1+71.) (10) which is one-third for fully turbulent flow and one-half for laminar flow. These results are unaffected by the replacement of the single carpet layer by a number of superimposed carpet layers in solid contact with one another, [provided the total thickness is negligible in comparison to the flow depth. Let the superimposed layers each have some unspecified thickness t proportional to the number of real granular solids each represents. The shear force on any layer t„, moving at a velocity Uh, is the work rate of moving the layer is work rate is therefore thUh and The whole r S (bi IA) , r ~,u r=a(u— Uc)n where n is 2 for fully turbulent flow and 1 for laminar flow. If the thickness of the carpet boundaiy is negligible compared to the flow depth, T—t and the rate of transporting work done is TUC= TUe—aUe(U— Ue)n This function has two zero values, when Uc equals zero and u, and one maximum value, when Uc=u/(l+n). As a second step consider the essential difference of condition between that of a continuous-carpet flow boundary and that of a boundary consisting of dispersed solids. This difference lies in the fact that whereas the fluid shear stress t is applied directly to a continuous boundary, its application to a dispersed granular boundary necessarily involves relative motion between the constituent grains and the fluid in their immediate neighborhood. Hence the transport velocity Ut, of the boundary material is less than the velocity uc of the boundary it constitutes (fig. 2B). The transfer of stress from fluid to solids involves a local dissipation of energy. This introduces a further efficiency factor eg=Utluc, so that eb=ec- eg, where, as we have seen, ec is one-third for fully turbulent flow. The limiting value of ee follows from the same line of reasoning as before, applied to the local flow in the neighborhood of the bedload grains rather than to the whole flow. Consider this time a single representative bedload grain. Under steady average conditions the fluid force F urging it along is in equilibrium with an equal mean frictional force applied to it intermittently by the bed. The force F varies as (uc—Ut)n’ where n' varies between 1 and 2 according to the local Reynolds number R = (uc—Ut,)D/p, being 1 in the ultimate Stokes law region and 2 for large grains. The work rate U^Uc—Ut,)”' has a limiting value when UtJuc=-eg= l/(ri/ + l). The exponent n' for a given grain size D and a given R, that is, for a given slip velocity uc— Z7», can be obtained from the slope of the experimental log curve of Figure 2.—Flow relative to moving boundaries. A, Flow relative to a moving carpet. B, Flow relative to a moving granular boundary of negligible thickness. C, Effect of inadequate flow depth and appreciable boundary thickness.18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS drag coefficient versus R. If the slope at any point on the curve is a, then n'—2—a. Since uc= (uc— Ut) ~~y~ and uc=~ u 71 O for fully turbulent flow, values of eg and can be found corresponding to a range of arbitrary values of the mean flow velocity u. These efficiency values are shown in figure 3 for various grain sizes D (quartz-density grains in water). The bedload efficiency eb should range from 0.11 for large grains and large flow velocities to 0.15 for very fine grains and low flow velocities. The whole rate of energy expenditure in transporting the bedload consists of the work rate ebo^ec- ego> plus the ineffective power dissipation ecco(l —eg) involved in the local transfer of stress from fluid to solids. It might at first sight be thought that in the transport of large grains since this latter dissipation takes place through the creation of wake eddies some of it may be available to maintain the turbulent suspension of smaller grains. But, as later becomes apparent, this kind of turbulent eddy is unlikely to possess the essen- tial quality of boundary shear turbulence necessary to maintain a suspension. Therefore, the overall power loss attributable to the bedload transport is ecw=3w, so that only fco remains available to maintain a suspended load. CRITICAL STAGE BEYOND WHICH THE BEDLOAD EFFICIENCY eh SHOULD BE PREDICTABLE The foregoing argument is restricted to conditions in which the moving bedload solids are sufficiently numerous to interpose an effective flow boundary between the free fluid flow above and the stationary bed below. By the foregoing dynamic definition of a flow boundary the above condition is fulfilled when virtually the whole applied fluid stress is transferred to the moving bedload solids. The critical bedload stage appears therefore to be definable by a critical value of the bed stress r. A rough estimate of this critical value can be obtained as follows. The topmost stationary grain layer of the bed effectively occludes the fluid flow above it from the stationary layer immediately beneath it. Hence Fiqube 3.—Values ol theoretical bodload efficiency factors, in terms of mean flow velocity, for quartz-density solids in an adequate depth of fully turbulent water.APPKOACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 19 this occlusion can be expected to persist if the whole topmost layer is set in motion as bedload over the layer beneath, which has now become the topmost stationaiy layer. The immersed weight of the original topmost layer, which is now in motion, is (cr-p)gD-Ca where D is the mean grain size, and C0 is the static volume concentration, that is, (1—porosity), whose average value may be put at 0.6 to 0.7. The bed shear stress r required to maintain the motion of this load is r = (a—p)gDC0 tan a Introducing 9 as the dimensionless bed shear stress Tj{a—p)gl), the critical stage should occur when, approximately, 6X=C0 tan a (11) From experiment, tan a appears to vary according to the conditions of shear from 0.375 to 0.75 owing to the variation of fluid-viscosity effects with variation of grain size and mass. The estimated range of 6X from about 0.5 for grains smaller than 0.3 mm to about 0.25 for grains larger than 2.0 mm is sketched in figure 4. This range is derived from a more basic relationship, figure 5, introduced under the later heading “Variation of the dynamic bedload friction coefficient tan a.” It should be realized that any direct manifestation of this critical stage concerning the bedload by a change in the trend of experimental plots of total transport rate against the power may, for fine grains, be masked by a more pronounced change resulting from the attainment of a corresponding critical stage at which the suspended load becomes fully developed. However, as suggested previously (Bagnold, 1956, p. 256), the critical bedload stage 6X is likely to be broadly associated with the value of 6 at which bed features disappear or at any rate cease to create appreciable form drag. The experimental evidence (figs. 9-13) shows the correspondence to be moderate. This evidence is somewhat indefinite. The change from large-scale dunes to flat beds occurs gradually over a considerable range of stage, and different observers tend to discriminate differently. The experimental scatter adds to the uncertainty. As can be seen, some reported dunes widely overlap with reported flat beds. More precise experimental work is needed before a satisfactory criterion is found by which to predict for a given sediment the critical stage at which the change of ud occurs in the transport-rate plots. Figure 4.—Values of the solid-friction coefficient tan a in terms of the bed-stress criterion 9=Tl( •a T> u O + P_ 2 P V Down Fluid masses partaking isotropic turbulence Up Fluid masses partaking shear turbulence Figure 7.—Schematic diagram showing postulated asymmetry in the normal eddy velocity components of shear turbulence- In both isotropic and shear turbulence, continuity requires that the total discharge, as given by the positive and negative areas, must be zero. In isotropic turbulence the symmetry makes the upward and downward elements of momentum flux equal, but the asymmetry in shear turbulence gives a residual upward momentum flux.PHYSIOGRAPHIC ANI) HYDRAULIC STUDIES OF RIVERS I 14 Consider a representative unit volume of fluid within the boundary region where the normal fluctuations are decreasing toward the boundary. Inside this unit volume let a minor mass p(% — a) be moving upward at root-mean-square velocity v'nv, and the remaining major mass p(K+a) be moving downward at a velocity v'An, the asymmetry a being positive. The total normal momentum must be zero, so v'iu 1—2 a v’m 1+2a but the normal momentum fluxes, per unit area of a shear plane, are unequal. There results a residual unidirectional momentum flux /, upward into the body of the flow, of magnitude /=pC 1—2a 2 PV dn l + 2a 2 P»up • 2 a 1—2a l+2a (14) balanced below by an equivalent excess of mean static pressure at the boundary. If the velocity r'p is determined by the shear stress r, then for a given r the flux j has a limiting maximum value when the asymmetry a=^ (V2—1)=0.207. Shear turbulence results, broadly, from a general dynamic flow instability. And again broadly, within the region of its generation, the intensity of any dynamic instability disturbance proceeds spontaneously to the limit set by the approach of the energy losses involved to the energy supply. It is to be expected therefore that this particular energy loss, associated with j, would be maintained at its maximum value. So there is a reasonable probability that the asymmetry a has the critical value 0.207 within the boundary region. It must, however, decrease toward zero through the body of the flow as the turbulence approaches isotropy with increasing distance from the boundary. The relation in equation 14 can be expressed hi terms of the overall mean measured velocity v' by addition y.n_j/2 1 2a . ,2 l+2q ,2 1 2a o 2 +Vdn 2 W“p l+2a whence (15) f—2apv'2—0A\4pv'2 on the above assumption The measured quantity V has been found to increase from zero at the boundary to a sharp maximum, and then to decrease progressively with distance from the boundary (Laufer, 1954; Townsend, 1955). The ratio i’max/'W* appears to be in the neighborhood of unity. Thus the normal momentum flux entering the body of the flow above the plane of v'm&x is y=0.414pr:2tl«0.414r This value would be well below the limit of sensitivity of Laufer’s manometer. So the excess reaction pressure at the boundary could not have been detected. Significantly however an energy loss toward the boundary by pressure transport was inferred from the energy balance drawn up. (See also Townsend, 1955, p. 217 and fig. 9.12.) The propagation velocity of the flux j being V7/p, the supply of lifting power to the body of the flow is /V7/p=(2a)3/2p«/3=0.266p^3.J (16) It remains to express this relation in terms of the whole power supply co=riZ, from the rather inadequate experimental data. According to Laufer’s work the ratio v'm&x/u* varies with the Reynolds number of the flow, and is approximately 1.0 at R=3X104 and approximately 1.1 at R=3X105. This ratio being denoted by b and the flow coefficient u\u* by c, the suspension efficiency e„ should be given by e<=./Vyyp=0 266&3/c (17) At flow velocities u of the same order, b would increase with increasing flow depth. And since c increases likewise, it is possible that b3/c may remain constant over a wide range of conditions. Experimental flume conditions being taken as a standard, since they cover approximately the same experimental range of R, b may be put at say 1.03, and c appears to range only between 16 and 20 for high-stage flows and for the sand range of grain sizes. Putting c=18, 0.266X1.1 18 =0.016 (17a) Laufer’s measurements refer to flows past smooth boundaries, and the effect of boundary roughness on the ratio b appears to be uncertain, as is also the effect of the presence of transported solids. However, for test purposes I decided to ignore these uncertainties and to assume that the suspension efficiency e, has the universally constant value 0.015 for fully developed suspension by turbulent shear flow. This assumption gives the numerical coefficient in the second term of equation 9 the round figure of f X0.015=0.01. It may of course be fortuitous that this value makes the general transport relation in equation 9 accord surprisingly well with the comprehensive range of transport data to be presented later. From the river evidence the figure might be 25 percent larger but no more. The postulated upward lifting power supplied to the flow body from below would also account for theAPPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 115 hitherto unexplained persistence of secondary circulations in long straight channels, by creating large-scale instabilities of the same nature as density instabilities due to thermal gradients. Incidentally, the theory may shed some light on the truth of the legend that the surface of a stream is slightly raised in the center. If the stream cross section is dish shaped or trapezoidal, so that the local r is negligible at the sides, and if the stream is unladen with suspended sediment, the maximum predicted excess height in midstream would be j/pg «OAdS. In natural rivers this maximum would rarely amount to 10~3 feet. It would be still less if the excess head were relieved by secondary circulations. Since writing this section I have seen Prof. S. Irmay’s highly relevant paper (Irmay, 1960). In the latter, my postulated normal stress j is predicted mathematically in terms of acceleration, by an entirely different approach along the lines of Reynolds’ (1895) treatment of the Navier Stokes equations. Irmay has shown that the Reynolds approach leads to a mean acceleration A, the normal component of which, in the present notation, is d{vn) * dy (Irmay’s equation 113) Hence if Y is the flow depth from the boundary up to the no-shear plane, the upward momentum flux j \ is given by Aydy—pv,2, since v' —0 at the boundary This flux is shown to be generated in the region of maximum v' close to the boundary (see Irmay’s fig.4). My own approach appears to explain the mechanism of its generation there. As can be seen from Irmay’s figure 2, based on Laufer’s data for R = 3X104, j—pv'y= approx. 0.375r which compares well with my value of approximately 0. 41r. Thus my conclusions receive strong independent support. In the light of these new ideas it would seem that the basis of the conventional theory of suspension may need serious reassessment. For the existence, inherent in shear turbulence, of an upward fluid stress appears incompatible with theories purporting to explain suspension in the absence of such a stress. Two immediate implications of Irmay’s more detailed analysis may be noted here. 1. The normal concentration profile of suspended solids is related to the rate of decrease with distance from the boundary of the normal momentum flux 208-578—66--4 jy=pv'v2 as modified by the transfer of upward momentum to the solids. If Cy is the weight of suspended solids per unit volume at distance y from the boundary, the work rate needed to maintain the suspension of the solids in a layer dy is CvVdy. The fluid lifting power is —d(/s/*)Vp- Whence r_______1 d{f'2)^_ P d(J)'y3) ‘ VpF dy V dy The required modification of the v' profile may be deducible from the measurable modification of the Karman constant by a suspension of solids. 2. The no-shear region in Laufer’s experiments was that of the central axis of a closed duct. So the normal fluctuations remained finite there. At the free surface of an open flow the normal fluctuations are inhibited. The dynamic Reynolds stress pvn must therefore fall sharply to zero as it does at the shear boundary below, being converted into an excess of static head. Thus the whole integral needs to be taken to an upper limit just below the free surface. In the thin intervening zone there must exist a large negative acceleration, downward away from the surface, possibly of the same order as the large positive acceleration, found to exceed 2g, at the shear boundary. This seems likely to account for the hitherto unexplained phenomenon observable in flume experiments on sediment transport, namely the inability of transported solids ever to touch the actual surface film, together with the existence of a thin layer of relatively sediment-free fluid immediately beneath the surface. CRITICAL FLOW STAGES FOR SUSPENSION It is reasonable to suppose that no solid can remain suspended unless at least some of the turbulent eddies have upward velocity components v'„v exceeding the fall velocity V of the solid. The turbulence has a spectrum of such velocity components, of which v'w is the mean. Some eddies have greater upward velocities and some less. Hence the stages marking the threshold and full development of suspension should be definable by critical values of the ratio v'aJV, the threshold occurring at some value less than unity and full development at some higher value. From equation 15, r'p=1.56i' when a=0.207. And v' varies as the shear velocity sfr/p. The Laufer (1954)PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS I 16 data being used as before, v' appears to have an average value of about 0.8m* over the flow depth. Whence as a tentative approximation we can write v'—1.25 Vr/p. So unit value of the ratio v'UB/V can be defined by l:2o2r=pV2. Substituting r=0s(a=0.245 mm F=3.0 cm per sec 1 D 50 percent =0.088 mm, and Vd so peroent would be 0.6 cm per sec. Whereas, V=3 cm per sec is five times bigger. Dv=Da=0.245 mm, corresponding to approximately D so percent. The three biggest grades together constitute only 6 percent of the suspended material, but contribute 66 percent to the value of V. 2 Fines neglected. Elkhorn River at Waterloo, Nebr. 1 {8-26-52) Size limit Percent P Dp pDp (mm) Vp (cm pVp (cm per sec) (mm) finer (mm) per sec) 0.5 100 } 0.04 0.35 0.014 5 0.2 .25 96 } .19 .176 .0335 1.9 .36 .125 77 } .15 .088 .0132 .6 .09 . 0625 62 } .13 .044 .0057 .16 .0208 .0312 49 } -21 .022 .0046 .04 .0084 . 0156 28 } .04 .011 .00044 .01 .0004 .0078.... 24 } .06 .0055 .00033 .0025 .00015 . 0039 18 2.18 1.00 Da=0.0718 mm V=0.68 cm per sec 1 D 50 peroent=0.033 mm, and Vd so peroent would be 0.09 cm per sec. Vis 7.6 times bigger. 2>v=0.092 mm, which is 1.3 times Da. Dv corresponds to approximately D 70 percent. > Fines neglected. Rio Puerco near Bernardo, N. Mex.1 (8-10-59) Size limit Percent P Dp pDp (mm) Vp (cm p Vp (cm per sec) (mm) finer (mm) per sec) 1.0— 100 } 0.002 0.71 0.00142 10 0.02 .5 99.8 ) .006 .35 .00210 5 .03 .25. 99.2 } .041 .176 .00720 1.9 .078 . 125 95.1 } .083 .088 .00555 .6 .049 .0625 86.8 } .153 .044 .00670 .16 .0244 .0312 71.5 } .101 .022 .00220 .04 .0040 .0156 61.4 1 .011 .0011 .01 .001 .0078--.. 51.4 \ .014 .0055 .00007 .0025 .000035 . 0039 50.0 2.50 Total... 1.00 Da=0.026 mm F=0.206 cm per sec 1 Dsn p.roont=0.0039mm, and Vd» percent would be 0.0013 cm per sec. Vis ICO times bigger. i>t'=0.016mm, which is smaller than Dp. Zip corresponds to approximately Du percent. The four biggest grades together constitute only 5 percent of the suspended material, but contribute 60 percent to the value of V. ‘ Fines neglected.118 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS The comparatively large discrepancies between V and VDSo percent in the above examples refer to wide size distributions, and to those which extend well into the Stokes law range of sizes. Naturally, the discrepancy diminishes with decrease of distribution width (see following example). Rio Grande at Cochiti, N. Mex.1 (6-17-58) Size limit Percent P Dr pD (mm) Vp (cm pVp (cm per sec) (mm) finer (mm) per sec) 2.0 100 } .01 1.5 .015 22 22 1.0 99 j .249 .71 .177 10 2.49 .5 74.1 } .455 .35 .158 5 2.27 .25 28.6 } .238 } .012 .176 .042 1.9 .35 .125 .0625 4.8 .088 .0011 .6 .01 2.036 Total 1.00 D.=0.363 mm F=5.34 cm per sec 1 Dh o percent — 0. 34 mm, and Vdso percent would be 4.7 cm per sec. So V is only 1.3 times larger. JDr=0.37 mm corresponds to approximately Deo percent. 2 Fines neglected. The effective mean fall velocity V thus depends both on the distribution of p versus I) and on the position of this relative to the curve of fall velocity versus D. It therefore bears no general relationship to Tz>50 percent* It should perhaps be emphasized here that the effective mean fall velocity V = T? v V-„ is involved in the transport of suspended load only. It is not involved in the transport of bedload. Moreover, the summation of pVv refers to the actual size distribution of the suspended material. V may be considerably smaller than the value computed from the overall size distribution of the whole material. THE EFFECTIVE MEAN GRAIN SIZE D FOR HETEROGENEOUS BEDLOADS The bedload friction coefficient tan a and the dimensionless bed-stress parameter 6=t/( where i=-—- j—0.62 times the conventional transport <7 rate by dry weight for quartz-density grains in water; u—stream power pdSu in units consistent with i; eb is given in figure 3; tan a is given in figure 4; V— effective fall velocity 2pVv for the suspended material; andAPPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS the coefficient 0.01 against the second suspension term is the theoretical suspension efficiency 0.015 reduced by the factor 2/3 on account of the stream power already dissipated in bedload transport. For mean bed grain sizes Da less than 0.5 mm, e6/tan a can as a first approximation be taken as 0.17 for 6 less than unity. Equation 20 can of course be written as I=a (s^+0-011) <20“> in terms of the whole width of the flow. If C"=0.62<7 is used for the transport concentration by immersed weight, equation 20 may be put in the alternative form or C'=s(-^-+ \tan a i C ( e„ co S \tan a 0.M |) -0.01 |) The ratio (206) suspended transport rate bedload transport rate =0.01 u tan a (21) This equation is approximately equal to 0.06 y for grains smaller than 0.5 mm. This ratio should increase with decreasing grain size. For a given overall size distribution it should also increase with increasing stage, but this increase may be masked by a progressive increase in V as larger grains are brought into suspension. Inadequate flow depth, however, should cause the bedload rate ib to increase disproportionately with increasing stage, by a factor of 3 or less. The load ratio susPen(|e(^Joa^, that, js the ratio of bedload the quantities of suspended and unsuspended solids present over unit bed area, appears to be considerably smaller. The suspended load is iju and the bedload is i6/I76, where U„lu=et, which is of the order of 0.13 (fig. 3). The load ratio is therefore (22) Clearly, if the bedload travels slower than the suspended load the quantity of bedload present must be correspondingly greater. This emphasizes the need to distinguish between transport concentrations and spatial concentrations. The present looseness of definition tends to a confusion of thought. I 19 Equations 20, 20a, and 206 can be written in the alternative dimensionless forms i V=I VJL V e»V co u 0 u S u tan au (20 c) The equivalent terms on the left are now the proportion of stream power expended in sediment transport if the whole of the load were suspended. This approximates to reality when the bedload term on the right is small compared with 0.01, that is, when V/u is small. i V If we neglect the bedload work, the proportion — -=- of stream power expended in sediment transport is constant, as was suggested by Rubey (1933, p. 503). Rubey’s values for this proportion, inferred empirically from river data, were rather larger, around 0.025 instead of 0.01. This is understandable because the bedload work element was not taken into consideration and because the effective fall velocity V was estimated in a different way. EXISTING FLUME DATA LIMITATIONS AND UNCERTAINTIES Very few of the many laboratory transport-rate measurements extend to the higher stages with which we are here concerned and which seem to be prevalent in many rivers. Although now half a century old, Gilbert’s (1914) work still provides the majority of the available data. This data deficiency may be attributed in part to a lack of appreciation of the real conditions the experiments were aimed at reproducing; but it seems to be due mainly to two outstanding experimental weaknesses: (a) underestimation of the pumping capacities required, and consequent inability to maintain adequate flow depths at the higher stages; and (6), resulting from a, increasingly serious disturbances of the flow associated with unnaturally high Froude numbers at inadequate flow depths. Several uncertainties arise in the interpretation of the few relevant sets of data available: 1. A large scatter exists within most sets of data. No serious investigation has been made into the cause. So it has remained uncertain how far this scatter may be systematic. A correlation with variations of flow depth has long been known qualitatively to exist, but unfortunately— 2. We have no reliable way of estimating the proper reduction factor to be applied in the evaluation of the tractive stress r and of the available power co to make allowance for ineffective wall drag in rectangular flumes.120 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 3. Although in some sets of data the Aopercent grain size of the total transported solids is given, together with that of the whole stock of experimental material, the size distributions of the transported solids are omitted in all sets of data. Further, even were the transported distributions to be given, it would be impossible in such shallow flow depths to make any reliable estimate of the size distributions of the suspended loads as distinct from the bedloads. Hence the effective fall velocity V of the suspended loads cannot be determined with any accuracy. In this respect the data obtained from natural rivers is likely to be more relevant. For over the greater prevailing flow depths the material collected by a sampler and its size distribution have reasonable likelihood of representing the suspended load alone. Similarly, no information is available as to the mean size of the bed-surface grains as an indication of the mean bedload size in flume experiments. Hence any estimate of the effective value of the bedload criterion 8= is open to some {=pdSu. The plots are arranged in descending order of grain size. Both the quantities i and u have the dimensions of energy rates per unit of boundary area. The units used for both are pounds-feet per second-foot2, equivalent to pounds per second-foot. Superimposed are comparative plots of the theoretical values of i predicted according to equation 20. This equation, it should be remembered, was derived without recourse to information obtained from any flume experiment. In plotting the factual data I have attempted to distinguish between reportedly “flat” or “smooth” beds and “dune” or “sand-wave” beds, and between these and “antidunes” occurring at the unnaturally high Froude numbers obtained in small-scale experiments. The difficulty here is that one condition merges gradually into the next, which makes the reported distinctions a matter of personal judgment. Reported “transition” conditions have been referred to the next lower condition. Thus “transition” from dune to flat is plotted as dune, and that from flat to antidune as flat. The antidune condition would not have intervened had the experimental flow depths been larger, and it may be assumed, I think, that the beds would have remained flat. I have distinguished the antidune condition merely as an indication that the violent disturbances of the flow are likely to have rendered the measurements less reliable. To complete the information, the corresponding values of u and 0 are plotted in the following illustrations. In the absence of any information as to the actual mean sizes of the grains in transit close to the bed surface, the bedload grain size D in the denominator of 9 has been given the value corresponding to the overall mean size of the material stock. The actual size D in 9 may well be larger by a factor approaching 2. GILBERT (1914) 5.0-mm PEA GRAVEL Figure 9 A In terms of 9 the highest stage reached is only 0.21, whereas according to figure 4 the critical stage should not have been reached until 0=0.26. Hence this whole plot would seem to lie within the lower, transitional stages. According to figure 8 it seems very likely that suspension would be negligible, the entire transport taking place as bedload. In consequence, the predicted values of i have been calculated using the first term only of equation 20. The flow depth being around 14 grain diameters only, it would certainly have been inadequate at the higher stages. One would therefore expect the measured transport rates to have been too large by a progressively increasing factor, consistent with the plot. GILBERT (1914) 0.787-mm SAND Figure 9B Depth range 75 to 110 diameters Approaches adequacy? Max 0=0.8. Suspension possibly approaching ing full development (fig. 8). H=0.18 ft per sec. Average flow velocity u—3.5 ft per sec over the higher stages. Ratio of suspended to bedload transport rates approx 0.86 according to equation 21. That is, bedload transport rate still exceeds that of suspen- T , suspended load i, . , _ sion. Load ratio —=—n—3---------=«&—=0.15. bedload ib Critical stage: predicted from figure 4, 0=0.4. From plot, 0 between 0.3 and 0.5, as indicated by onset of flat beds, and between 0.4 and 0.5, as indicated by change of trend in data plot. GILBERT (1914) 0.507-mm SAND Figure 104 Depth range 120 to 200 diameters. Probably adequate. Max 0=1.2. Suspension probably fully developed (fig. 8). (F=0.13 ft per sec. Ratio approx 1.6 over the higher stages. Suspension beginning to dominate. Load ratio 0.29. Critical stage: predicted from figure 4, 0=0.5. From plot, flat beds, 0 between 0.4 and 0.7; change of trend, 0 between 0.5 and 0.6.TRANSPORT RATE (i ), IN POUNDS PER SECOND FOOT (IMMERSED WEIGHT) 122 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 9.—Comparative plots of theoretical and experimental transport rates (flume experiments). A, Gilbert 5.0-mm gravel. B, Gilbert 0.787-mm sand.TRANSPORT RATE ( i ), IN POUNDS PER SECOND FOOT (IMMERSED WEIGHT) APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 123 1.0 10~ Figure 10—Comparative plots of theoretical and experimental transport rates (flame experiments). A, Gilbert 0.507-mm sand. B, Simons, Richardson, and Albertson 0.45-mm sand. x o B |Q°I III II 2 3 4 6 8 10" 234 68 2 34 68 10" 1.0 10"2 POWER (w), IN POUNDS PER SECOND FOOT 10" 1.0 0.8 0.6 2 3 4 6 8 1.0 Figure 12—Comparative plots of theoretical and experimental transport rates (flume experiments). A, Barton and Lin 0.18-mm sand. B. Laursen 0.11-mm sand.126 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS SIMONS, RICHARDSON, AND ALBERTSON (1961) 0.45- mm SAND Figure 10B Depth range 140 to 700 diameters. Adequate? No evident correlation with flow depth, in spite of the wide depth range. However, there is an unfortunately large uncorrelatable scatter. This is probably due to too infrequent measurements of the transport rate during the runs. The scatter is noticeably greater over the lower, transitional stages, when large long-period fluctuations in the discharge of solids from the flume are to be expected as alternate dunes and troughs pass the end of the flume. Significantly, the transport-rate measurements were made by periodic sampling at the immediate exit from the flume. Max 0=1.2. F=0.1 ft per sec. Average ratio i,lib=2.2 over the higher stages. Load ratio 0.4. Critical stage: predicted from figure 4, 0=0.5. From plot, flat beds, 0=approx 0.5; change of trend, 0=approx 0.5. GILBERT (1914) 0.376-mm SAND Figure 114 Depth range 160 to 240 diameters. Max 0=1.5. F=0.086 ft per sec. Average ratio ijib=2.3. Load ratio 0.42. Critical stage: predicted 0=0.5. From plot, flat beds, 0= approx 0.6; change of trend, 0 between 0.5 and 0.6. GILBERT (1914) 0.307-mm SAND Figure 11B Depth range 200 to 300 diameters. Max 0 = 1.1. F=0.066 ft per sec. Average ratio iJib—2.Q. Load ratio 0.47. Critical stage: predicted 0 = 0.5. From plot, flat beds, 0 between 0.5 and 0.6; change of trend, 0 between 0.6 and 0.7. BARTON AND LIN (1955) 0.18-mm SAND Figure 12/1 Depth range 500 to 2700 diameters. Max 0 = 1.4. F = 0.035 ft per sec. Average ratio i,/ib= 4.8. Load ratio 0.86. Critical stage: predicted 0=0.5. From plot, flat beds, 0 between 0.5 and 0.6; change of trend, 0 between 0.5 and 0.7. Inexplicable scatter. LAURSEN (1957) O.ll-mm SILT Figure 12B Max 0 = 1.6. F = 0.0175 ft per sec. Average ratio is/ib = 6.1. Load ratio 1.1. Critical stage: predicted 0 = 0.5. From plot, flat beds, value of 0 uncertain; beginning of trend coincidence, 0 between 1.1 and 1.4. GENERAL COMMENTS The mutual consistence between the experimental results is remarkable in view of the differences in the experimental conditions—different experimenters and methods of measurement, different apparatus from nonrecycling to recycling flumes, and different gram-size distributions. There is also a surprising general agreement, all differences being within the limits of experimental error, between these results and those predicted by the present theory, over the 50-fold range of grain sizes covering nearly the whole range of transport modes from transport as bedload alone to transport in which suspension greatly predominates. The scatter tends to obscure evidence of the critical stage at which the theory becomes operative. This stage appears to be predictable to a fair approximation by a critical value of the bedload criterion 0 between 0.5 and 0.6 for Gilbert’s data. But these data refer to narrow grain-size distribution. Other plots suggest that the critical stage occurs at considerably larger values of 0. However, this discrepancy may well be apparent rather than real. For with a wider size distribution the finer grades tend to be removed into suspension, leaving the bedload consisting of grades appreciably coarser than the mean stock size. Thus, were 0 to be based, as it theoretically should be, on the mean bedload size, its value would be appreciably smaller. It would however be unwise to speculate further about this until experiment is improved toward repeatability by the avoidance of the scatter, and until methods are devised for measuring the size distribution of the grains in transit close over the bed. Obviously a comprehensive investigation is needed to determine the causes of this unfortunate scatter, including the relative effects on the transport rate of (a) wall drag at various flow stages and for various grain sizes, and (b) the real effective height to which the bedload rises at high-flow stages. Further, for more accurate application of the theory, it is evident that more attention should be paid to the size distributions of the suspended load, so that the effective value of F can be better estimated. With regard to the scatter, the plots show that it is far narrower in the predicted transport-rate values than in the experimentally measured values. Since the predicted values are based on the same experimental values of the flow quantities to and u and on a systematically estimated value of F, it follows that either there were large variations in the real effective value of F from run to run, which seems unlikely, or alternatively the experimental scatter originated in the measurement of the transport rates.APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 127 It is significant that the experimental scatter is worse, on the whole, for modem experiments done in recycling flumes, in which this measurement has to be by sampling, than for Gilbert’s experiments done in a nonrecycling flume, in which the transport-rate measurement was direct, by integrated weight. However accurately the sampling method may measure the instantaneous discharge of solids, it cannot make accurate measurement of the mean value of a spontaneously fluctuating discharge unless samples are taken systematically over a period covering several fluctuations. EXPERIMENTS BY VANONI, BROOKS, AND NOMICOS That the scatter which mars most of the experimental results can be greatly reduced by proper experimental design is shown in figures 13 A and B by the results of experiments carried out at California Institute of Technology (Brooks, 1957). X o $ Q 2 2 O o Ll_ Q z o o Ld (/) (£ C/) O z ID O Q. 2 34 6 8 x 2 34 68 2 3 4 6 8 8 i i r i i i i i i i i i i - Vanoni and Brooks 0.14 mm x Nomicos 0.17 mm I 1 I I \ j Laursen 0.04 mm o Depth 0.24-0.3 ft B Depth 0.24 ft • Depth 0.38-0.67 ft a Depth 0.53 ft n Width 0.875 ft o* x Width 3 ft Width 2.7 ft ° o o < 4 ox o o o o o 2 oD o ° o o 10"2 o - 8 — o 6 “ □ o _ > 4 o 3 XX 2 _ o X — □ - icr3 - - 8 o 6 EXPLANATION □ o Dunes 3 — ■ • _ Flat bed X X 2 Predicted by equation 20 A B c 1 1 1 I J ^ 1 1 1 1 J 1 1 1 1 1 1 1 1 h cr o a- f. I— z < 10 1.0 8 10" 8 8 1.0 23468 234 10“l 10'2 10“l 10"1 POWER (w), IN POUNDS PER SECOND FOOT Figure 13.—Comparative plots of theoretical and experimental transport rates (flume experiments). A, Vanoni and Brooks 0.14-mm sand, B, Nomicos 0.172-mm sand, C, Laursen 0.04-mm silt.128 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS These experiments differ from all others in that efforts were made to maintain constant flow depth, that is, to vary the tractive conditions alone, without varying the boundary conditions simultaneously. Figure 13.4 shows the measured transport rates of the same sand, Z?50 percent—0.14 mm, at two different flow depths of approximately 0.24 and 0.54 feet in the same recycling and tilting flume 2.8 feet wide. Though each of the two superimposed plots is highly consistent within itself, there is a systematic discrepancy between them. This discrepancy emphasizes the need to find out by well-designed and critical experiment the answers to the following questions: Is the discrepancy due to differences in relative wall drag; or is it due to excessive transport of bedload at the smaller depth, this depth supposedly being inadequate in the sense already discussed; or is it inherent in the arbitrary method of estimating the mean flow depth, as a factor of to, over a rippled and highly irregular flow boundary; or is it perhaps a combination of all three? In figure 13B are plotted the transport rates of another sand of considerably wider size distribution, Z)a=0.172 mm, measured by Nomicos in a narrower flume but 0.875 feet wide. The flow depth was kept at 0.24 feet. The results are again consistent, and this consistency discloses a clearer picture than before of the abrupt change of trend which occurs at the critical stage. Superimposed are the theoretical transport rates calculated from equation 20 on the same arbitrary assumption as before that the effective suspension fall velocity V—xA'Z'pVv where the summation is taken over the given size distribution for the whole stock of material. RELATION BETWEEN i AND w OVER THE LOWER TRANSITIONAL STAGES It is not the purpose of this paper to consider con-conditions over the lower, transitional stages between the threshold of motion and the critical stage. The very marked linearity of the logarithmic plots in this region is nevertheless noteworthy. It appears in all the relevant plots in which transport by suspension predominates: figures 10B, 124, 1225, 134, and 13B. It is particularly striking in the last two figures. The transport rate i increases as w”, where the mean value of n approximates 3. I offer no explanation. TRANSPORT OF FINE SILTS IN EXPERIMENTAL FLUMES Laursen repeated his own experiments on the 0.11-mm sand, using a still finer material having a mean size of 0.038 mm. The measured transport rates are plotted in figure 13C together with the theoretical rates given by equation 20 on the same standard assumption that V=jiSpVp over the size distribution of the stock material. On this basis 17=0.0047 ft per sec, corresponding to a uniform D\ =0.04 mm. As can be seen, the measured rates are too large by an order of 10, and this might at first be taken to indicate the breakdown of the theory. A study of the report, however, suggests that such an anomaly is rather to be expected in view of the particular experimental conditions. The solids enter a recycling flume more or less uniformly dispersed throughout the flow depth. The coarser grades fall to the neighborhood of the bed, where under normal experimental conditions they are transported along the bed as bedload. At the exit they are remixed with the circulating load. However, at the high transport concentrations prevailing for fine materials having a small fall velocity V—concentrations approaching 10 percent in Laursen’s experiments—the bed boundary is invisible. So the flow has to be stopped to allow the suspension to settle out before depth measurements can be made. This deposition had the reported effect of blanketing the bed to the extent that the ripple features were partly obliterated. As a result, it seems reasonable to suppose, the subsequent transport of the coarser bed grains being prevented, the circulating load would become progressively finer as the coarser constituents were progressively trapped. (It is significant that deposition over the first 70 ft of the flume was reported to be continuing while transport measurements were made over the final 20 ft.) No analysis, unfortunately, appears to have been made of the size constitution of the circulating load. It may well have consisted mainly of the 15 percent of the bed stock which was less than 0.02 mm in size, for which the effective fall velocity V would be of the order of 0.0007 ft per sec. The flow velocities being around 2 ft per sec, this value of V inserted into equation 20 would give predicted transport rates of the same order as those measured. The theory does indeed become inapplicable when the effective grain size is reduced below, say, 0.015 mm, as is evident from considerations of spatial concentration. Two-phase flow at very high concentrations is beyond the scope of the present paper. It has already been discussed in previous papers (Bagnold, 1954, 1955, 1956).APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 129 THE AVAILABLE RIVER DATA, NATURE, AND UNCERTAINTIES The considered river data consist of 146 sets of records taken on various dates at various stations on certain rivers in western conterminous United States by the U.S. Geological Survey. These sets were selected some years ago by L. B. Leopold, before the present theory had been conceived, as being most likely to be reasonably reliable. Each set comprises the flow data, the transport rate measured by sampling the flow, and integrated size analyses both of the sampled solids transported and of samples taken simultaneously from the bed surface. The factual information is in this respect more complete and more readily comparable with the theory than is the experimental evidence. Further, both the width to depth ratios and the flow depths are much larger; so the uncertainties arising from the influence of side boundaries, from the likelihood of inadequate flow depth, and from Froude number disturbances are absent or greatly reduced. There are, on the other hand, a number of other uncertainties. ENERGY-SLOPE ESTIMATION In many sets of data all reference to slopes was omitted from the original records. So the necessary information had to be obtained subsequently from map contours. Provided no discontinuities of profile at local rapids have been overlooked, and provided the river flow is not artificially constricted locally—for example, by bridge works—this map slope may be assumed to coincide with the true-energy slope at bankfull or flood stage. In other data sets the recorded slope is the directly measured slope of the local water surface. Comparison with the map slope shows that in general the slope of the local water surface may be smaller at low river stages by a factor of 2 or more, owing to local irregularities of bed profile. Only at one station have the true energy slopes been computed from measurements of both water surface and bed profiles. Naturally such costly and time-consuming measurements are unlikely to be made unless a need for them is clearly established. BEDLOAD AND SUSPENDED LOAD Unlike flume experiments in which it is possible to measure the total transport rates directly, the transport rates obtained for rivers are derived from samples taken from the body of the flow, integrated over the cross section. So there is an indeterminate deficiency in respect to the unsampled transport passing close over the bed. This unmeasured transport is often referred to loosely as that of “bedload.” Since, however, the magnitude of it depends entirely on the inadequacy of the humanly devised method of measurement, it is unrelated to the magnitude of the real bedload transport as defined by its dynamic mechanism. As already pointed out, the maximum height above the bed to which the real bedload may rise, at high flow stages, is not known. So how much of it may have been included in the sampling and how much excluded cannot be determined. It is known for instance that by placing large-scale obstacles on a riverbed the local turbulence is increased to the extent that a large proportion of the normal bedload is thrown up into temporary suspension and can thus be included in the sampling. Since at high river stages the bed is frequently invisible, there may be undetected obstacles upstream doing the same thing. In face of this uncertainty I have had to assume, as an arbitrary systematic assumption, that the real and fictitious bedloads are the same. Consequently, the recorded transport rates are assumed to refer to the suspended load alone. Consistently, the comparable predicted rates have been computed from equation 20 using only the second, suspension term 0.01 u/V. Since the first, bedload term e»/tan a is nearly constant at around 0.18, the effect of ignoring the bedload becomes serious only when the suspension term is of the same small order. The error may then mount to a factor of, say, 1.5 either way. The suspension term in many data sets is, however, much larger. The above assumption also introduced possibly more serious error. The effective fall velocity V=hpVv is computed directly from the recorded size distribution on the assumption that this refers to the suspended load. But the undetected inclusion of even a small proportion, say 5 percent only, of the coarser bedload may so alter the pattern of the size distribtuion as to have a profound effect on the computed value of V. The value of \ would be too large, and the predicted transport rate may in consequence be several times too small. As can be seen from the sample computations of V given earlier, the summation is very sensitive to small changes in the proportions of the few largest grades. DEFICIENCY OF SEDIMENT SUPPLY It may well happen, on the other hand—for example, after a flood stage has removed much of the transportable material from the riverbed—that the river transports less sediment than it could if more transportable130 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS sediment were available. The predicted rates might then he considerably larger than the actual measured rates. VALUES OF THE STAGE CRITERION 0 The size analysis of the bed surface being given, 0 should be capable of direct determination from it by putting D = Da = 'Zj)Dll. However, some uncertainty arises from the fact that, again unlike laboratory conditions, the largest bed grains according to the analyses are often greater by a factor of 30 or more than the largest transported grains. So it is open to some doubt whether or not the whole bed surface can be assumed to be mobile. Again, though the value of 0 is an approximate guide in default of a better one, it is not, as pointed out earlier, a precise criterion for either the disappearance of dune features on the bed or the change of trend in the transport rate versus power curves. As is apparent from figures 9 to 11, dune features often persist at the higher flow stages where the theory agrees with the data. Accordingly agreement with the river data is found even when dunes are known to be present. In view of these various uncertainties, many discrepancies, in both directions, must be expected when comparing the predicted with the measured transport rate. COMPARISON OF PREDICTED WITH MEASURED RIVER TRANSPORT RATES In contrast to laboratory conditions, in which the same stock of sediment material taken from some natural deposit of limited size distribution is transported at progressively increasing flow stages, the material transported by a river, together with that of the local bed surface, changes from season to season, from day to day, and often from hour to hour, according to the ever-changing tributary influxes upstream. A group of records taken at the same gaging station on different dates would therefore show little if any correlation. So it would be unprofitable to plot the transport rates. Instead, each record should be treated as though it were the result of a separate and independent experiment done on the transport of a different sediment stock. Accordingly I have tabulated the river records (table 1) so as to compare the transport rates measured on each separate occasion with the rate predicted for that occasion from the flow and sediment conditions then prevailing. Table 1.—Data from 1^6 individual river measurements comparing measured with predicted transport rates [Slope: M, map slope; L, measured slope of local water surface; ES, estimated true-energy slope. Basic data compiled by L. B. Leopold from various sources, mostly measure -ments by U.S. Geol. Survey. The compilation included arbitrarily chosen individual measurements to illustrate conditions over a range of discharges. Measuremen t stations in the tabulation were limited to those for which complete discharge and suspended-load data were available. Data are available in the files of the U.S. Geol. Survey, Washington, D.C.] Bed Suspended load Transport rate Date USGS serial Discharge (cfs) Depth (feet) Slope Mean velocity (ft per sec) a Grain size (mm) Bed Fall Measured C' i. Predicted Discrepancy predicted No. Q d Maxi- mum D shear stress 0 velocity V=XpVP (cm per sec) Concentra- tion C f. 0.01 U Mean 2>. 10 0.4 0.72 2.24X10-3 8. 75 2.9 0.33 4- 3-57 2 3,460 9.5 1.59X10-* (M) 2.83 16 .24 1.1 .083 3.88X10-* 15.2 10.0 .66 5-14-57 3 59,000 26.5 1.59X10-4 (M) 4.7 8 .9 .91 .159 3.0 XKH 11.7 8.9 .76 5- 9-58 4 31,000 18.8 1.59X10-4 (M) 3.96 16 1.9 .2 .24 3.19X10-» 12.5 5.0 .42 4-24-57 5 31,760 14.9 1.59X10-4 (M) 5.47 4 .36 1.4 .122 6.76X10-* 26.4 13,5 .51 10-18-57 6 83,600 27.5 1.59X10-4 (M) 6.3 .215 7. 07X10-3 24.8 9.0 .36 10-24-57 7 51i300 23.3 1.59XKH (M) 5.0 16 .7 .175 4.24X10-* 16.5 8.9 .54 COLORADO RIVER (OF THE WEST) i Taylors Ferry, Ariz. 5- 2-56 248 7,900 8.57 1. 73X10“* (L) 2.63 2 0.4 0.7 1.75 7.6 X10-8 0.27 0.45 1.67 9- 6-56 218 9,880 9.41 1.47X10-4 (L) 2.99 8 .64 .4 1.75 1.18X10-* .5 .515 1.03 9-20-56 219 6,625 7.45 1.73X10-4 (L) 2. 55 4 .8 .3 2.83 2.07X10-* .72 .44 .61 10- 5-56 220 6,400 7.38 1.78X10-4 (L) 2.49 2 .45 .55 1.17 2.69X10-* .935 .675 .68 9-15-55 208 10,845 9.9 2.16X10-* (L) 3.08 2 .35 1.15 1.73 1.59X10-* .45 .53 1.16 12-15-55 209 4,200 4.92 3.3 X10-* CL) 2.49 1 .31 1.0 2.0 9.4 XIO-s 1.75 .37 .21 12-29-55 210 4,760 5.4 2.6 X10-* (L) 2.53 1 .17 1.5 1.96 1.4 XKH .33 .39 1.17 3- 5-56 211 7,762 7.02 1.9 XHH(L) 3.17 2 .2 1.25 1.78 1.16X10-* .38 .53 1.4 3-21-56 212 7,767 7.7 2.16X10-* (L) 2. 89 2 .19 1.65 1.93 1.51X10-* .43 .45 1.05 4- 3-56 213 10,733 9.81 2.07X10-* (L) 3.12 2 .37 1.03 1.44 2.44X10-* .73 .64 .88 214 9,533 9.25 2.24X10-* (L) 2.93 1 .19 2.04 2.1 9.39X10-5 .26 .42 1.6 5-17-56 215 7,409 7.67 2.27X10-* (L) 2.76 4 .36 .9 2.26 1. 51X10-* .38 .37 .96 5-31-56 216 8,480 7.61 1.87X10-* (L) 3.17 16 1.2 .215 1.77 7.7X10-5 . 255 .88 2 4.1 8-21-56 217 10,480 9.7 2.33X10-* (L) 3.08 2 .5 .85 1.66 5.0 X10-8 .13 .57 4.4 See footnotes at end of table.APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS I 31 Table 1.—Data from 146 individual river measurements comparing measured with predicted transport rates—Continued Bed Suspended load Transport rate Date USGS serial Discharge (cfs) Depth (feet) Slope S Mean velocity (ft per sec) u Grain size (mm) Bed Fall Measured V COLORADO RIVER (OF THE WEST)—Continued Below Needles, Ariz. 10-17-56 239 8,600 8.61 1.77X10-* (L) 3.02 2 0.34 0. 94 0. 893 3. 79X10-* 1.32 0.67 0.51 7-31-56 240 10,365 10. 77 1. 53X10-* (L) 2.86 16 .58 .53 1.89 1.13X10-* .455 .455 1.0 8-10-55 250 12,316 11.69 1.96X10-* (L) 3. 07 0.5 .3 1.46 2.94 1.1 xio-* .35 .31 .89 9-30-55 251 12,670 11.86 2.2 X10-< (L) 3.17 1 .3 1. 63 3.02 2.44X10-* .69 .5 .725 12- 8-55 252 3,855 5. 24 2. 77X10-* (L) 2.44 1.5 1.0 .27 3.34 2.6 X10-* .58 .22 .38 1- 5-56 253 7, 024 7.73 2. 77X10-* (L) 2.81 2 .25 1.6 2. 48 1.92X10-* .43 .34 .79 3- 7-56 254 7, 780 10.0 2.13X10-* (L) 2.3 3 .3 1.3 1.8 1. 52X10-* .44 .38 .86 4-13-56 255 10,357 10.88 1.93X10-* (L) 2. 75 8 .35 1.15 3.71 4.7 X10-* 1.55 .31 .2 Lees Ferry, Ariz. 10-16-55 3,560 4.3 2.23X10-3 (M) 2.47 1 0.20 9 0.265 1.01X10-3 0.27 2.8 10 6- 3-56 62,000 18.1 2.23X10~3(M) 8.95 1 .32 23 1.37 6.25X10-3 1.74 1.97 1.13 6-25-58 33,100 16.6 2.23X10"3(M) 5.21 1 .21 30 .48 1.8 X10-3 .5 3. 24 6.48 4-14-56 10,000 7. 73 2.23X10-»(M) 3.55 6 .33 11 .52 3.0 X10-3 .835 2.04 2.45 Grand Canyon, Ariz. 4-12-56 c 11,300 9.73 1.89X10-»(M) 4.08 0.9 0.34 10 0.25 4.57X10-3 1.5 4.9 3.25 5-31-56 K 54,300 20.1 1. 89X1(H(M) 9. 02 4.0 .45 16 1.19 9.0 X10-3 2.95 2.27 .77 5-17-55 F 27,110 14.6 1. 89X10-HM) 6.29 .5 .2 27 .7 8. 88X10-3 2.9 2.7 1.06 Palo Verde Weir, Ariz. (1.4 miles below) 5-31-56 203 12,180 9.2 1.6 X10-*(L) 2.87 16 1.2 0.23 2.86 9.27X10-5 0.35 0.30 0.85 5- 2-56 201 12,170 8.9 1.13X10-HL) 2.98 16 1.3 .14 2.11 1.09X10-* .64 .43 .66 5-17-56 202 10,660 8.67 1.47X10~*(L) 2.78 >16 2.98 2.14X10'* .9 .29 .32 204 11,860 9.74 6.0 X10-5(L) 2.64 >16 3.6 2. 06X10-* .97 .70 .73 9- 6-56 205 9,880 8.95 1.63X10-*(L) 2.41 16 1.6 .17 1.55 1.42X10-* .65 .48 .74 9-26-56 206 10,200 8.8 6.6 X10-*(L) 2.58 3 0.28 .57 1.43 1.46X10-* 1.38 .54 .39 10- 5-56 207 8,440 8. 25 1.27X10-*(L) 2.31 4 .22 .9 1.03 2. 69X10-* 1.3 .69 .53 9-16-55 241 9,461 9. 93 6.3 X10-«(L) 2.93 8 2.2 .05? 1.6 7.1 X10-» .83 .55 .66 12-14-55 242 4,412 6. 87 ].06X10-*(L) 2.04 8 .63 .2 2.0 5.2 XIO-s .3 .3 1 12-28-55 243 4,174 6.9 1.13X10-*(L) 1.93 8 .86 .17 2.75 1.12X10-5 .062 .2 3.2 3- 6-56 244 6, 778 7. 74 1.4 X10-*(L) 2.66 >16 1.5 .14 2.1 1.29X10-* .58 .38 .65 245 9,633 9.25 1.03X10-*(L) 3.08 >16 6.47 1.68X10-* 1.0 .14 .14 246 12,819 10.87 1.27X10~*(L) 3.27 >16 2.85 1.69X10-* .84 .35 .42 4-19-56 247 10,900 9.00 1.2 X10-*(L) 2.75 >16 1.76 6.3 X10-5 .32 .47 1.46 COLORADO RIVER (OF TEXAS) Columbus, Tex. 4- 2-57 8 2,250 5.06 2.3 XKH(M) 2.04 2 0.36 0.62 0.030 1.57X10-* 4.25 20 4.7 5- 9-57 9 13,000 11.34 2.3 X10-'(M) 2.54 16 1.17 .4 .23 4.48X10-* 1.2 3.3 2.7 9-26-57 10 16,100 21.7 2.3 X10-'(M) 5.34 8 .5 1.9 .92 1.94X10-3 5.2 1.7 .33 4- 9-59 11 8,500 6.8 2.3 X10-‘(M) 2. 93 1 .3 .98 .89 1.68X10-3 4.55 1.0 .22 9-25-57 12 4,600 4.5 2.3 X10-*(M) 2.48 2 .35 .55 .08 1.97X10-* 5.3 9.3 1.75 5- 6-57 13 31,800 16.18 2.3 X10-*(M) 3. 93 12 .55 1.25 .59 6. 87X10-* 1.85 2.0 1.08 9-22-58 14 24,600 11.2 2.3 X10-*(M) 4.57 1 .34 1.4 .2 2.62X10-3 7.0 6.8 .98 10-17-57 15 36,700 18.2 2.3 X10-*(M) 3.95 16 1.5 .52 .15 2. 57X10-3 6.9 5.9 .86 9-25-57 16 18,500 10.1 2.3 X10-*(M) 4.12 .5 .24 1.8 .83 5.25X10-3 14.2 1.5 .104 5- 4-58 17 2,460 13.0 2.3 X10-*(M) 3.37 8 .36 1.6 .41 2.34X10-3 6.2 2.4 .39 4-27-57 18 31,550 12.86 2.3 X10-*(M) 5.0 1 .3 1.85 1.0 4.38X10-3 12.8 1.5 .12 ELKHORN RIVER Near Waterloo, Nebr. 3-26-52 257 2,830 3.07 4. 03 X 10-* (ES) 3. 79 0.5 0.075 0.68 3.43 X 10-J> 5.25 1.7 0.32 3.7 X10~*(ES) 5. 51 .48 6 X 10-3 10.0 3.4 .34 5.73 4.33 X 10“* (ES) 5.23 .48 5. 01 X 10-3 7.2 3.3 .45 6,520 5.46 4. 75 X 10-* (ES) 4. 55 .46 4.14 X 10-3 5.4 3.0 .55 2.9 3. 72 X 10-* (ES) 3.68 .59 1.52 X 10-3 2.5 1.80 .75 262 3. 02 4.07 X 10“* (ES) 2.78 .65 1 X 10-3 .95 1.27 1.34 2. 84 4.38X 10-* (ES) 2.08 .62 6 ix io-* .85 1.0 1.18 1, 500 2. 84 4.24 X 10~* (ES) 2.14 .43 6.98 X 10"* 1.02 1.53 1.5 1,820 3.44 3. 64 X 10-* (ES) 2.05 .08 3.41 X 10-3 5.8 7.5 1.3 6-26-52 266 6i 960 4.38 4.67 X 10~* (ES) 6.00 2 .29 1.33 .18 2.1 X 10-2 28.0 10.0 .36 7- 2-59 267 2,722 3. 92 2. 68 X 10-* (ES) 3. 02 1 .21 .93 .12 3. 7 X 10-3 8.6 7.6 .88 7- 2-59 268 2,272 4. 0 2.68X 10-* (ES) 2.43 .087 3.42 X 10-3 7.9 8.4 1.06 7- 2-59 269 2,387 3. 74 2.68X10-*(ES) 2.77 8 1.2 .15 .086 3. 3 X 1CH 7.2 9.6 1.33 270 2,387 3. 8 2. 68 X 10~* (ES) 2.26 .086 3.28 X 10-3 4.75 7.9 1.65 7- 2-59 271 2,240 3.7 2.52 X 10~* (ES) 2.64 1 .21 .83 .0855 3.41 X 10-3 8.4 9.4 1.12 7- 2-59 272 2,240 3. 78 2.52 X 10-* (ES) 2.13 .079 3.08 X IO"3 7.6 8.1 1.07 See footnotes at end of table.132 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Table 1.—Data from 146 individual river measurements corn-paring measured with predicted transport rales—Continued USGS serial No. Discharge (cfs) Q Depth (feet) d Slope 5 Mean velocity (ft per sec) u Bed Suspended load Transport rate Grain size (mm) Bed shear stress 0 Fall velocity F=SpFP (cm per sec) Concentra- tion C Measured (L) 6.91 16 .86 1.02 1.16 4.42 X 10-3 2.16 1.78 .83 6- 4-58 82 8,160 4.34 1.15 X 10-3 (L) 6.92 1 .32 2.8 1.64 2. 58 X 10-3 1.4 1.26 .9 5-13-58 83 8,320 4.46 1. 27 X 10-3 (L) 6. 88 12 .48 2.2 1.21 4. 74 X 10-3 2.3 1.73 .75 5-27-58 84 10,100 4.8 1.2 X10-S(L) 7. 71 12 .41 2.6 1.39 3.04 X 10-3 1.57 1.66 1.06 San Antonio, N. Mex. 6-26-58 85 378 1.21 3 4.5 X 10- (L) 1.58 8 0.30 0.34 .97 9.04 X io-* 1.24 0.48 0.39 0 68) 6-19-58 86 3,080 2.59 3 3.5 X 1 3.99 .5 0.115 2.0 .344 3. 73 X 10-3 5.1 3.5 0.68 («. 95) 6-20-58 94 3,810 3.62 * 3.5 X 10-4 5.6 .5 .157 1.51 .36 4.21 X 10-3 7.5 4.7 .62 Cl.l) 6-10-58 95 4,230 4.07 *6.0 X 10-4 16 2.13 1.05 2.37 7.44 X 10-3 2.0 .87 .43 6- 3-58 278 8,590 8.96 2.4 X 10-3 (I.) 7.1 >16 3.51 1.15 1.8 1.72 X 10-3 .44 1.15 2.65 6-12-58 279 5,000 7.19 2.31 X 10-3 (1,1 5.56 >16 2.3 1.35 2.7 1.11 X 10-3 .3 .62 2.0 6-17-58 280 2,240 4.69 1.63 X 10-3 (1,) 4.23 16 1.1 1.3 7.2 1.86 X 10-3 .73 .17 .24 6-24-58 281 1,130 1.44 1.31 X 10-3 (I,) 3.35 >>16 ? ? 2. 79 3.16 X 10-< .15 .36 2.4 5-26-58 282 10,100 10.21 2.35 X 10-3 (L) 7. 08 >16 1.54 1.0 1.62 2. 99 X 10-3 .79 1.3 1.6 Cocliita, N. Mex. 6-24-58 283 1,000 1.71 1.18X1CH (L) 2.23 8 2.1 0.18 1.18 2.47X10-* 0.13 0. 56 4.3 6-17-58 284 2,040 2.24 1.18X10-3 (L) 3.2 8 1.0 .5 5.4 5.24X10-3 2.75 .18 .065 6-17-58 285 5,060 4.03 1.13X10-3 (L) 4.22 >16 2.3 .37 8.1 8. 84X10-3 4.9 .16 .032 0- 3-58 286 8,680 4.85 1.27X10-3 (L) 6. 07 >16 3.5 .33 3.5 3.11X10-3 1.5 .51 .34 5-12-58 287 8,900 4.09 1. 2 X10-3 (L) 6.64 >16 2.23 .42 1.34 4.55X10-3 2.35 1.48 .63 5-20-58 288 8,920 4.34 1. 2 X10-3 (L) 6. 51 >16 5.0 .2 1.33 3.11X10-3 1.6 1.47 .92 5-26-58 289 9,810 4.39 1.27X10-3 (L) 6.68 >>16 2.0 3.11X10-3 1.5 1.0 .66 See footnotes at end of table.APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 133 Table 1.—Data from 146 individual river measurements comparing measured with predicted transport rales—Continued USGS serial No. Discharge (cfs) Q Depth (feet) d Slope 5 Mean velocity (ft per sec) u Bed Suspended load Transport rate Grain size (mm) Bed shear stress d Fall velocity V=XpVp (cm per sec) Concentra- tion C Measured £_ij_ S a Predicted i. 0.01O V Discrepancy predicted measured Maxi- mum D Mean Da RIO GRANDE—Continued San Fiiipe, N. Mex. 6-24-58 290 1,020 2.17 1.0 X10-» (L) 2.6 >>16 1.33 5.77 Xlfr-4 0.36 0.59 1.6 6-17-58 291 2,200 2.73 9.1 X10-< (L) 4.44 16 1.27 0.36 2.66 1.1 X10-3 .75 .5 .66 6-17-58 292 5,010 4.47 1.51X10-3 (L) 5.99 >>16 5.3 2.74 X10-* * 1.12 .29 .26 6- 9-58 293 5,120 4.49 1.51X10-3 (L) 6.06 16 1.0 1.3 6.5 6.3 XIO-s 2.6 .28 .11 6- 7-58 294 7,520 5. 57 1.52XHH (L) 6.7 (•) (•) m 1.4 4.62 X10-3 2.5 1.44 .58 5-12-58 295 8, 200 5. 56 1.76X10-* (L) 7.19 >>16 2.9 4.41 X10-3 1. 55 .75 .48 6- 3-58 296 8,590 6.0 1. 68X10-S (L) 7.16 16 2.1 0.9 2.3 2.430X10-* .9 .94 1.05 5-21-58 297 9,140 5. 86 1.8 X10-* (L) 7.43 >>16 1.56 2.79 X10-3 .90 1.4 1.4 5-26-58 298 9,720 6.17 1.93X10-S (L) 7.53 >>16 1.53 2.58 X10-3 1.15 1.46 1.3 RIO PUERCO Bernado, N. Mex. 8-10-59 8-26-59 11- 3-59 1 2 3 1,600 2,010 31.8 2.62 2.64 .97 1.05X10-»(M) 1.05X10“3(M) 1.05X10-3(M) 6.58 7.31 1.37 1 2 2 0.29 .33 .24 1.8 1.6 .8 0.20 .22 .0074 1.41X10-1 1.65X10-1 4.8 X10-3 83 97 28.4 10 10 56 0.12 .10 2 SAN JUAN RIVER Shiprock, N. Mex. 5-31-51 275 6,120 5.49 4.1 X1(H (L) 6.3 o (•> (•) 1.9 2.07XKH 3.12 1 0.32 SCHUYLKILL. RIVER Philadelphia, Pa. 9- 3-59 256 13,000 8.08 5.3 X10-« (L) 4.9 >>16 Stony 0.27 4.88XKH 0.57 5.5 9.6 1 Data from Colorado River (of the West) unpublished, from U.S. Bur. of Reclamation, on file in U.S. Qeol. Survey, Washington, D.C.; locations shown on map published by the U.S. Inter-Agency Committee on Water Resources (1961). 2 Below critical stage. * Map slope is 7.9X10-4. Local pool? * Discrepancy using map slope. » Map slope is 6.3X10-4. Local pool. «No bed. The comparison is given in terms of the dimensionless ratio C'/S=is/w, where C is the transport concentration by immersed weight and is assumed to refer to suspended load only. The effective fall velocity V—’Z,pVv is given in centimeters per second for easy reference to standard values such as those plotted in Report 12 of the U.S. Inter-Agency Committee on Water Resources (1958). The predicted ratios is/w=0.01u/V have, however, been calculated in consistent units of u and V. The prevailing values of the stage criterion 9 have been added, together with the maximum and mean grain sizes of the bed. In summary of the comparative transport rates (table 1), the predicted rates are within a factor of 1.5, either way, of the measured rates in 49 percent of the data sets and within a factor of 2 in 65 percent of the data sets. Moreover the overall geometric mean of all the 146 discrepancy ratios is as near parity as 0.73. When all the uncertainties are considered this measure of agreement is very promising. The discrepancies given in table 1 are analysed in figure 14 according to river and gaging station. There is, as figure 14 shows, a marked correlation between the discrepancies and their relevant stations. Although for most of the data the points are significantly concentrated close to the parity line, the results from some stations, for example, Colorado River of the West at Lees Ferry, Ariz., and Colorado River of Texas at Columbus, Tex., are very erratic. It should be remembered that energy effects calculated from data obtained from a single cross section of a river are likely to be unrepresentative. In addition most of the gaging stations involved were permanently sited for the original and simpler purpose of gaging water discharge only; for that purpose the effects of local energy interchanges are immaterial. The Lees Ferry station, for instance, is sited just downstream of an abrupt widening of the river channel. The discrepancies shown above the parity line in figure 14 indicate too large a predicted value; these discrepancies, however large, can readily be explained by concurrent infilling of a local scour, by a general dearth of suspendable material, or by the flow stage being subcritical. In many plots where the predicted rates are several times too large, the stage as indicated by 6134 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS COLORADO (OF THE WEST) RIO GRANDE _____A_____ Figure 14.—Transport-rate discrepancies (predicted/measured), from table 1, grouped according to river and station. either is below the critical value or may well have been had the actual bed-surface conditions been better known. Big discrepancies in this direction do not occur when 6 is large. The many large discrepancies in the other direction indicate too small a predicted value and are of more interest. A search for their probable cause led me to inspect the recorded size distributions of the sampled material transported, for some common abnormality. While the sampler is lowered through the river to final contact with the bed and is then withdrawn, it seems inevitable that some of the coarser bedload grains become included in the integrated sample. The proportion in most samples may be small, but in the presence of large-scale bed features such as boulders, gravel bars, and old bridge debris it may be very appreciable. The inclusion of bedload transported by another mechanism, and therefore likely to have a differentAPPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 135 size distribution, should, it seem reasonable to suppose be manifested in the combined size distribution. The conventional integrated percentage-less-than method of presentation of size distributions discloses but little useful detail. Indeed it is designed to smooth out irregularities. When, however, the distribution is converted back into its original grade proportions p, and these are plotted as log p against log D, distinguishing features can readily be spotted. Exceptionally low predicted transport rates appear to be associated with size distributions having an actual or inferable second hump toward the larger end of the size scale. Conspicuous examples are shown in figures 15.4 and B. When these curves are adjusted by the removal of the second hump and the appropriate increase of the remaining p values, the summation ~2pVp is reduced to the extent that the predicted value of 0.01 u/V is increased to the right order of magnitude. In some plots the second hump is merged into the main one to the extent that its presence is difficult to distinguish. Figure loC gives an example. Here again, a reasonable adjustment based on the rather slight indications is found to remove the discrepancy in the transport rates. It should be a simple matter to test this hypothesis by duplicating the sampling. If a sample taken over the full depth, to contact with the bed, were to show the second hump as in figures 154. and B, owing to the inclusion of bedload material, another sample taken on the same occasion down to say only two-thirds of the depth should show the second hump to be absent or considerably reduced. In this connection, although the majority of the discrepancies in table 1 are comparatively small, those in the direction of too low a predicted value are markedly preponderant. In view of the close agreement of the laboratory data, it would seem advisable, before attempting to apply some empirical correction to the coefficients of equation 20, to investigate the extent to which responsibility for this general tendency can be attributed to the inevitable inclusion of coarser bedload material in the samples of supposedly suspended loads. Considerably clearer inferences could be drawn from distribution graphs such as those of figure 15 had the size analyses been made in grade intervals of -J2, instead of the wider intervals of 2 and sometimes 4. Some further points are noteworthy in connection with table 1. Although the 6 values shown cannot be regarded as precise, they do appear to indicate that in the types of river sampled, at any rate, conditions usually are those of the high transport stages to which the present theory is applicable. Since the present theory takes both modes of sediment transport into consideration, it should be applicable to rivers which transport relatively large materials—pebbles, gravels, and boulders—mainly as bedload. None of these types of river, however, are included in the table, for the reason that no data are available owing to lack of means of measurement. Hence the bedload part of the theory remains untestable except by Gilbert’s laboratory experiments done at clearly inadequate flow depths and over a too limited range of transport stages. Lastly, the need is evident for accurate determinations of the true-energy slopes of rivers. COMPARISON OF RIVER TRANSPORT DATA WITH DATA FOR WIND-TRANSPORTED SAND From the viewpoint of general physics, a broad theory of the present kind would be expected to be consistent with the facts over a still wider field. For it is contrary to experience that Nature restricts the operation of her basic principles to particular phenomena, such as the transport of solids by a particular fluid. If the theory is soundly based, therefore, it should be consistent with evidence on the transport of windblown sand. This evidence indicates in the first place that the transport of sand—as opposed to fine dust—over the ground is by the bedload mechanism alone, suspension by air turbulence being negligible over the range of wind speeds commonly experienced. Hence the first term of equation 20 should alone be operative. Owing to the very low dynamic viscosity of air, the conditions of the bedload motion are wholly inertial; so tan a should be constant at its lower value. The theory would therefore predict that the transport rate of a given windblown sand should, to a close approximation, be proportional to the available power. The quantitative data from wind-tunnel experiments is definite on this point (Bagnold, 1941; Zingg, 1950). The transport rate is indeed proportional to the power. So to this extent the theory is entirely consistent. CONCLUSION The foregoing theory constitutes an attempt to explain the natural process of sediment transport along open channels quantitatively, by reasoning from the general principles of physics and from the results of certain critical experiments. General relationships have been derived independently of any quantitative data drawn from experiments on channel flow. To this extent the theory is rational rather than empirical. Consideration has been confined to transport conditions at the higher stages of flow. For here the process appears to be much simpler than over the lower, transitional stages. An understanding of this simpler process is directly relevant to much of riverflow, and should be relevant indirectly to the more difficult problems presented by the lower, transitional stages.136 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Fiqure 15—Examples of anomalous size distributions of assumedly suspended river loads, associated with anomalous transport rates predicted from them (suggesting that the sampling had included a proportion of coarser bedload material). Within the flow region considered, the derived relationships are, I believe, fairly and evenly consistent with the available facts over a range of conditions far wider than for any previous theory. Wide consistency of this kind tests the general form of a theoretical relationship in a way that consistency over a narrow range of conditions connot do. If the general form appears sound, the relationship is worth additional study by others who may modify the parameters in the light of further factual knowledge to bring the relationship into closer approximation everywhere without detriment to its generality. No theoretical results can, however, be properly tested unless the facts against which they are tested areAPPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 137 themselves both adequate and certain. If the facts, are in doubt, then a fair approximation is all that can be expected. Many serious factual inadequacies and uncertainties in the existing knowledge about sediment transport have already been pointed out. The majority could undoubtedly be removed by experimental researches of a critical kind, if these were imaginatively and scientifically designed and carried out for this specific purpose only, regardless of either convention or immediate practical utility. Given adequate facilities, each of the researches I have in mind should be capable of completion within a short period of say 2 years. Continued tolerance of longstanding factual uncertainties cannot but have an adverse effect on the status of research in this field. The most serious factual inadequacy in the field of sediment transport is, I suggest, our lack of data on the unsuspended transport of bedload by a turbulent fluid. The reason is of course our inability to separate this transport experimentally from a concurrent transport in turbulent suspension. Consequently we cannot check theoretically predicted transport rates by either mechanism separately. So any theory must extend to cover the prediction of both transport rates before any verification is possible. This difficulty remains insuperable so long as the belief prevails that bedload, although unsuspended, is yet somehow activated and transported by the agency of turbulence. The present theory however denies this belief, on the direct visual evidence that the saltating motion characteristic of bedload transport persists also under laminar flow in the entire absence of turbulence. The bedload transport relationship it derives is applicable both to turbulent and laminar conditions. This immediately opens the possibility that the transport of bedload can be studied quantitatively under conditions of laminar flow in a way that is impossible under practical turbulent conditions except within the narrow range of large grains at very low flow stages. Unfortunately no quantitative experiments on sediment transport by laminar flow have ever been done; for on the above belief, such experiments would be irrelevent and unpractical. For the same reason even simple qualitative experiments have been so rare that few if any present-day workers have had opportunity to observe the reality of transport by laminar flow and the closeness of its similarity to transport by turbulent flow. It is hard to avoid the conclusion that progress in the field of sediment transport has been retarded by a refusal to appreciate the significance of experimental results which may appear superfically to be unpractical. REFERENCES Bagnold, R. A., 1941, Physics of blown sand: New York, William Morrow. -------1954, Experiments on the gravity-free dispersion of large spheres in a Newtonian fluid under shear: Royal Soc. [London] Proc. A 225, 49. ------- 1955, Some flume experiments on large grains but little denser than the transporting fluid, and their implications: Inst. Civil Engineers Proc., Pt. 3. -------1956, Flow of cohesionless grains in fluids: Royal Soc. [London] Philos. Trans., v. 249, p. 235-297. Barton, J. R., and Lin, P. N., 1955, A study of sediment transport in alluvial channels: Fort Collins, Colorado State Univ., Agr. and Mech. Coll. Rept. 55, JRB 2. Brooks, N. H., 1957, Mechanics of streams with movable beds of fine sand [Includes experiments by V. A. Vanoni and N. H. Brooks, and by G. Nomicos]: Am. Soc. Civil Engineers, v. 83, no. HY 2. Durand, R., 1952, Proceedings of colloquium on hydraulic transport of coal: London Natl. Coal Board. Gilbert, G. K., 1914, The transportation of debris by running water, based on experiments made with the assistance of E. C.Murphy: U.S.Geol. Survey Prof. Paper 86, 263 p. Irmay, S., 1960, Accelerations and mean trajectories in turbulent channel flow: Am. Soe. Mech. Engineers Trans., December. Knapp, R. T., 1938, Energy balance in stream flows carrying suspended load: Am. Geophys. Union Trans., p. 501-505. Laufer, J., 1954, The structure of turbulence in fully developed pipe flow: Natl. Advisory Comm, for Aeronautics Rept. 1174. Laursen, E. M., 1957, An investigation of the total sediment load: Iowa State Univ. Inst. Hydrol. Research. Prandtl, L., 1952, Essentials of fluid dynamics: London, Blackie & Son. Reynolds, Osborne, 1885, On the dilatancy of media composed of rigid particles in contact: Philos. Mag., 5th ser., v. 20, p. 469-489. ------- 1895, On the dynamic theory of viscous incompressible fluids and the determination of the criterion: Roy. Soc. [London] Philos. Trans., v. 186 A., p. 123. Rubey, W. W., 1933, Equilibrium conditions in d<5bris-laden streams: Am. Geophys. Union Trans., 14th Ann. Mtg., p. 497-505. Shields, A., 1936, Anwendung der Anhlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung: Berlin Preuss, Versuchsanstalt fur Wasser, Erdund Schiffbau, no. 26. Simons, D. B., Richardson, E. V., and Albertson, M. L., 1961, Flume studies using medium sand (0.45 mm): U.S. Geol. Survey Water-Supply Paper 1498-A, 76 p. Townsend, A. A., 1956, The structure of turbulent shear flow: Cambridge Univ. Press. U.S. Inter-Agency Committee on Water Resources, 1958, Some fundamentals of particle size analysis: U.S. Inter-Agency Comm. Water Resources, Sub-Comm. Sedimentation Rept. 12, 55 p. ------- 1961, River basin maps showing hydrologic stations: U.S. Inter-Agency Comm. Water Resources, Sub-Comm. Hydrology, Map 59. Velikanov, M. A., 1955, Dynamics of channel flow—v. 2, Sediments and the channel [3d ed.]: Moscow, State Publishing House for Tech.-Theoretical Lit., p. 107-120 [In Russian], Zingg, A. W., 1950, Annual report on mechanics of wind erosion: U.S. Dept. Agriculture Soil Conserv. Service. U.S. GOVERNMENT PRINTING 0FFICE:I966 . ' D/~ ' TT^F. >-*•&-........yt 7r V i ^ ” 5 H I ✓ UAT Resistance to Flow in Alluvial Channels H 2* ► GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-JResistance to Flow in Alluvial Channels By D. B. SIMONS and E. V. RICHARDSON PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 42 2-J UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 — Price 50 cents (paper cover)CONTENTS Page Abstract------------------------------------------------ J1 Introduction -------------------------------------------- 1 Experimental equipment and data collection--------------- 2 Forms of bed roughness and flow phenomena---------------- 4 Bed configuration without sediment movement----- 4 Ripples--------------------------------------------- 5 Dunes_______________________________________________ 6 Plane bed with sediment movement-------------------- 7 Antidunes------------------------------------------- 8 Chutes and pools____________________________________ 9 Regimes of flow in alluvial channels-------------------- 10 Lower flow regime__________________________________ 11 Upper flow regime__________________________________ 11 Transition_________________________________________ 11 Comparison of flume and field conditions________________ 12 Variables_______________________________________________ 12 Concentration of bed-material discharge------------ 14 Fine-sediment concentration________________________ 14 Dependent variable_________________________________ 15 Change of variables-------------------------------- 15 Shape factor for the reach and cross section---- 15 Seepage force-------------------------------------- 16 Resistance coefficient_____________________________ 16 Slope _________________________________________ 16 Depth_________________________________________ J18 Page Variables—Continued Resistance coefficient—Continued Size of bed material___________________________ 19 Fall velocity---------------------------------- 19 Apparent viscosity and density----------------- 22 Gradation of bed material______________________ 23 Prediction of form of bed roughness--------------------- 24 Velocity distribution----------------------------------- 26 Plane bed with sediment movement------------------- 26 Roughness coefficients_________________________ 28 Evaluation of resistance to flow------------------------ 29 Evaluating resistance to flow by adjusting slope — 30 Evaluating resistance to flow by adjusting depth of flow__________________________________________ 38 Adjusting depth to average alluvial grain roughness------------------------------------ 41 Alluvial sand-bed roughness---------------- 41 Baffle and cube roughness__________________ 45 Adjusting depth to smooth boundary roughness ------------------------------------------ 46 Baffle and cube roughness---------------. 48 Gravel and cobble roughness---------------- 48 Alluvial sand-bed roughness________________ 49 Determination of average velocity-------------- 52 Summary and conclusions_________________________________ 56 Literature Cited________________________________________ 60 ILLUSTRATIONS Page Figure 1. Size-distribution curves for the sands used in the 8-foot-wide flume-------------- J3 2. Size-distribution curves for the sands used in the 2-foot-wide flume------------- 3 3. Idealized diagram of the forms of bed roughness in an alluvial channel----------- 5 4. Photograph showing upstream view of ripple bed----------------------------------- 6 5. Photograph showing side view of ripple bed--------------------------------------- 6 6. Graph showing variation of V with V, and D for flow over a ripple bed------------ 6 7. Photograph showing upstream view of dune bed----------------------------------- 6 8. Photograph showing side view of the dune bed shown in figure 7------------------- 7 9. Photograph showing upstream view of plane bed------------------------------------ 8 10. Graph showing comparison of resistance to flow for flow over a plane bed with and without movement of the bed material----------------------------------- 8 11-15. Photographs showing: 11. Downstream view of antidune flow---------------------------------------- 8 12. Side view of the antidune flow shown in figure 11_______________________ 9 13. Downstream view of chute-and-pool flow------------------------------ 10 14. Downstream view of the chute and breaking antidune shown in figure 13 10 15. Side view of the breaking antidune shown in figure 14------------------ 10 16. Graph showing relation of average velocity and stream power for runs using sand I 10 17. Plan view of alternate bar development in the large flume at large width-depth ratio ___________________________________________________________________ 13 18. Photograph showing large alternate bar in Marala Ravi Canal, Pakistan-------- 14 111IV CONTENTS Page Figure 19-45. Graphs showing: 19. Change in Darcy-Weisbach f with slope, depth, and fall velocity of bed material __________________________________________________________ J17 20. Relation of depth to discharge, Elkhorn River near Waterloo, Nebr._____ 19 21. Approximate relation of length of dunes to median fall velocity of bed material, at constant depth-------------------------------------------- 20 22. Change in resistance to flow with temperature______________________________ 21 23. Variation in resistance to flow with concentration of fine sediment____ 21 24. Apparent kinematic viscosity of water-bentonite dispersions________________ 22 25. Variation of fall velocity with percentage of bentonite in water___________ 22 26. Variation of fall velocity with temperature________________________________ 23 27. Variation of resistance to flow with slope as a function of size distribu- tion of bed material___________________________________________________ 23 28. Relation of form of bed roughness to stream power and median fall diameter of bed material______________________________________________ 24 29. Relation of r02 V/R'A, V*, and regime of flow for laboratory and field conditions_____________________________________________________________ 25 30. Typical vertical-velocity-distribution curves in alluvial channel__________ 26 31. Variation of slope of vertical-velocity distribution and shear velocity for plane bed with sediment movement------------------------------------- 27 32. Relation between intercept of vertical-velocity distribution and shear velocity ______________________________________________________________ 28 33. Comparison of roughness height with (0.27 mm) resulted from screening the coarse sand fraction from sand IIX. The difference between sands IL. and 1I3 (0.32 mm) resulted from natural changes during the experiments. Sands III (median diameter about 0.45 mm), consisting of a mixture of feldspar and quartz with some mica, were obtained Figure 2. -Size-distribution curves for the sands used in the 2-foot-wide flume. by screening the gravel from a sand deposit of the Cache la Poudre River at Fort Collins, Colo. The differences in the size distribution among sands HE (0.45 mm), IIP. (0.47 mm), and III* (0.54 mm) were caused by natural changes of the bed material as they were used in the experiments. Sand IV (0.93 mm) was a natural river sand, mostly quartz, obtained by screening the material coarser than 5 mm from the bed material of the North Platte River near Scottsbluff, Nebr. Bed material V, for which both the sieve size (0.68 mm) and the fall-diameter size (0.36 mm) distributions are given, was a lightweight aggregate (expanded illite) marketed by the Ideal Cement Co. under the trade name Idealite. Sands VI (0.33 mm) were almost pure silica obtained from the Black Hills Silica Corp., Hill City, S. Dak. Sand VI] passed through a U.S. standard No. 40 sieve and was retained on a No. 60 sieve, whereas sand VL> was prepared by mixing six sands in predetermined proportions to obtain a graded sand having the same median diameter as sand VIi. The sands were placed in the flumes to a depth of about 0.7 foot. The general procedure for each run was to recirculate a given discharge of the water-sediment mixture until equilibrium flow conditions were established. Equilibrium flow is defined as flow which has established a bed configuration and slope consistent with the fluid, flow, and bed-material characteristics over the entire length of the flume, neglecting entrance- and exit-affected reaches—that is, the time-average water-surface slope of the flow is essentially constant and parallel to the time-average bed surface, and the time-average concentration of the bed-material discharge is constant. Equilibrium flow should not be confused with steady uniform flow; in equilibrium flow, velocity may vary at a point or from point to point. Steady uniform flow, as classically defined (dV/dt = 0, dV/dx = 0), does not occur in an alluvial channel unless the bed is plane and the flow is steady. After equilibrium flow was established, average water-surface slope (S), discharge of the water-sediment mixture (Q), water temperature (T), depth (D), velocity distribution in the vertical (v„), total sediment concentration (CT), and the geometry of bed configuration (length, L; height, h), and shape) were determined. Water-surface slope was measured with a Lory point gage and a precision level by determining water-surface elevations at definite intervals along the flume (every 5 ft in the 8-ft-wide flume; every 2 ft in the 2-ft-wide flume). Average water-surfaceJ4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS slope was obtained by averaging several individual slope determinations made during a run. Discharges were measured with calibrated orifice meters located in the return-flow pipes. Water temperature was measured to the nearest 0.1 °C with a mercury thermometer. Depth was determined by subtracting mean bed elevation from mean water-surface elevation. Velocity distribution in a vertical was measured with a calibrated Prandtl pitot tube; however, the mean velocity of the cross section was computed from the discharge and cross-sectional-area data, V = Q/A. Total sediment concentration was measured by traversing the outflow nappe at the end of the flume with a width-depth integrating sampler. Suspended-sediment concentration was measured with a depth-integrating sampler. Bed configuration was measured with the Lory point gage and, in later runs, with a newly developed sonic depth sounder (Richardson and others, 1961). Photographs of the bed and water surfaces for all runs were taken with a still camera; normal and time-lapse sequences of the flow of water and sediment were photographed with a 16-mm movie camera. Many of the flow phenomena observed are recorded in the Geological Survey film “Flow in Alluvial Channels.” The filming was done by the Survey staff, assisted by personnel from Bandolier Films, Albuquerque, N. Mex., in the laboratory at Colorado State University and in the field near Albuquerque, on the Rio Grande. A complete documentation and description of all basic data collected from 1956 to 1961 by the Geological survey at Colorado State University is included in a data report, Professional Paper 462-L. FORMS OF BED ROUGHNESS AND FLOW PHENOMENA The bed configurations (roughness elements) that may form in an alluvial channel are plane bed without sediment movement, ripples, ripples on dunes, dunes, plane bed with sediment movement, antidunes, and chutes and pools. These bed configurations are listed in their order of occurrence with increasing values of stream power (VyDS) for bed materials having d-M less than 0.6 mm. For bed materials coarser than 0.6 mm, dunes form instead of ripples after beginning of motion at small values of stream power. The typical form of each bed configuration is shown in figure 3. These bed-roughness elements are not mutually exclusive occurrences in time and space in a flume or a natural river. They may form side by side in a cross section or reach of a natural stream, giving a multiple roughness; or they may form in sequence, in time, producing variable roughness. Multiple roughness is spacially related to variation in shear stress (yDS), stream power (VyDS), or alluvial bed material. The greater the width-depth ratio of a stream or flume, the greater is the probability of a spacial variation in shear stress, stream power, or bed material. Thus, the occurrence of multiple roughness is closely related to the width-depth ratio of the stream or flume. Multiple roughness in these experiments occurred when the width-depth ratio was greater than 20. Variable roughness is related to changes in shear stress, stream power, or reaction of bed material to a given stream power. An example of the effect of changing shear stress or stream power is the change in bed form which occurs with changes in depth during a runoff event; this action has frequently been observed in natural rivers. An example of the effect of the bed-material-stream-power relation is the change in bed form that occurs with change in the viscosity of the fluid as the temperature or concentration of fine sediment varies. It should be noted that a transition occurs between the dune-bed and the plane bed; either bed configuration may occur for the same value of stream power. In the following sections, bed configurations and their associated flow phenomena are described in the order of their occurrence with increasing stream power. BED CONFIGURATION WITHOUT SEDIMENT MOVEMENT If the bed material of a stream moves at one discharge but not at a smaller discharge, the bed configuration at the smaller discharge will be a remnant of the bed configuration formed when sediment was moving. The problems are in knowing when beginning of motion of the bed material occurs (Kramer, 1935; Shields, 1936; and White, 1940) and which bed form may develop. The bed configuration after the beginning of motion may be any of the preceding ones, depending on flow and bed material. Prior to the beginning of motion, the problem of resistance to flow is one of rigid-boundary hydraulics. After beginning of motion the channel is alluvial, and the problem related to defining resistance to flow under the alluvial condition is the main subject of this report. Plane bed without movement was studied to determine the shear stress for the beginning of motion and the bed configuration that would form after beginning of motion. A plane bed for this study wasRESISTANCE TO FLOW IN ALLUVIAL CHANNELS J5 obtained artifically by screeding the bed. Within the accuracy of visual observation, Shields’ relation for the beginning of motion was adequate. After beginning of motion, the plane bed changed to ripples for sand smaller than 0.5 mm, and to dunes for 0.93-mm sand. This was contrary to Liu’s (1957) findings; he reported a plane bed for a range of shear stresses after beginning of motion. Knoroz (1959) reported that a plane bed did not persist after the beginning of motion and that ripples do not form for coarser sands. Resistance to flow is small for a plane bed without sentiment movement. In the flume runs, values of C/'/g ranged from 15 to 20. RIPPLES Ripples are small triangle-shaped elements having gentle upstream slopes and steep downstream slopes. In this study they ranged from 0.4 foot to 2 feet in length and from 0.02 foot to 0.2 foot in height and were narrow normal to the direction of flow. (See figs. 4 and 5.) Resistance to flow is large, and the resulting discharge coefficient (C/^Sg) ranged from 7 to 12. As depth increased, resistance to flow due to roughness decreased. (See fig. 6.) Thus, there is a relative roughness effect produced by the ripple bed. Figure 6 shows that resistance to flow is independent of sand size when the bed configuration is one of ripples. This is because the ripple shape is independent of sand size and the effect of grain roughness is small relative to the form roughness. Knoroz observed that the length of the separation zone when the bed form was either ripples or dunes was about 10 times the height of the ripple or the dune. The separation zone downstream from a ripple causes very little, if any, disturbance on the water surface, and the flow contains little suspended bed material. The water is clear enough that the bed configuration can be photographed through the run-J6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 4.—Upstream view of ripple bed. Bed material is sand II; slope = 0.00108, depth = 0.57 foot, discharge = 7.2 cfs, C/y/~g = 11.2. Figure 5.—Side view of ripple bed. Bed material is sand_II; slope = 0.00023, depth = 1.01 feet, discharge = 7.7 cfs, C/y/g = 11.1. ning water. The bed-material discharge is small, ranging from 10 to 200 ppm. DUNES When the shear stress or the stream power is gradually increased for flow over a bed having ripples or, if the bed material is coarser than 0.6 mm, over a plane bed, a rate of bed-material transport, a magnitude of velocity, and a degree of turbulence will soon be achieved that cause sand waves called dunes to form. At smaller shear-stress values, the dunes will have ripples superposed on their backs. (See fig. 7.) These ripples will disappear at larger shear values, particularly if coarse sands (d50 > 0.4 mm) are involved. Dunes are large triangle-shaped elements similar to ripples. (See figs. 7 and 8.) Their lengths range from 2 feet to many feet, depending on the scale of 0.06 0.08 0.10 0.12 0.14 0.16 SHEAR VELOCITY , IN FEET PER SECOND Figure 6.—Variation of V, V*, and D for flow over a ripple bed. the flow system. Dunes formed in the large flume used in this study ranged from 2 feet to 10 feet in length and from 0.2 foot to 1 foot in height, whereas those in the Mississippi River described by Carey and Keller (1957) were several hundred feet long and as much as 40 feet high. The maximum amplitude to which dunes can develop is approximately the average depth. Hence, in contrast with ripples, the amplitude of dunes can increase with increasing depth, so that the relative roughness can remain essentially constant or even increase with increasing depth of flow. Figure 7.—Upstream view of dune bed. Bed material is sand II: slope = 0.00167, depth = 0.94 foot, discharge = 15.6 cfs, C/y/g = 9.3.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J7 Figure 8.—Side view of the dune bed shown in figure 7. Note the bottom-set, foreset, and top-set layering of the sand. Field observations by the authors indicated that dunes form in any channel, irrespective of the size of bed material, if the stream power is sufficiently large to cause general transport of the bed material without exceeding a Froude number of unity. Also, the length and shape of the dunes are functions of the fall velocity of the bed material; dune height did not appear to be a function of fall velocity of the bed material. The length of the dunes increased and the angle of the upstream and downstream faces decreased as fall velocity decreased. Dunes formed of fine sand (d-l0 < 0.4 mm) were longer and less angular than those formed of coarser sand. Resistance to flow caused by dunes is large, but not as large as that caused by ripples formed of finer sand and at shallow depth. For dunes, the discharge coefficient C/Vg varied from 8 to 12. Resistance to flow increased with an increase in depth for coarser sands (dM > 0.3 mm) and decreased with an increase in depth for finer sands (dr,o < 0.3 mm). Also, when dunes were more than 10 times longer than they were high, the grain roughness on the back of the dunes influenced resistance to flow. Knoroz (1959) observed that resistance to flow for dunes depended on the grain roughness, in addition to the form roughness, whereas, that for ripples did not depend on grain roughness. Dunes cause large separation zones in the flow. These zones, in turn, cause large boils to form on the surface of the stream. Measurements of flow velocities within the zone of separation showed that velocities in the upstream direction existed that were 1—1 the average stream velocity. Boundary shear stress was sometimes sufficient to cause formation of ripples oriented in a direction opposite to that of the primary flow in the channel. With dunes, as with any tranquil flow over an obstruction, the water surface is always out of phase with the bed surface. The flow accelerates over the crest of dunes and decelerates over the trough, thereby contracting the flow’ over the crest and expanding it over the trough. Some investigators do not agree that there is a difference between ripple- and dune-bed configurations. Vanoni, Brooks, and Kennedy (1961), for example, saw little reason for distinguishing between ripples and dunes because the mechanisms by which they are formed and by which they move are similar. The following factors, however, indicate that there are major differences, in addition to the difference in size, between the two bed configurations: 1. The effects of a change in depth on resistance to flow are opposite: an increase in depth causes a decrease in the resistance to flow for flow over a ripple bed, but an increase in depth causes an increase in the resistance to flow for flow over a dune bed when the bed material is coarser than 0.3 mm, and a decrease in the resistance to flow when the bed material is finer than 0.3 mm. 2. Ripples do not form if the median diameter of the bed material is larger than 0.6 mm. 3. Resistance to flow caused by ripples is inde- pendent of the grain size of the bed material, whereas that caused by dunes is dependent on the grain size. This question might seem academic if it were not for the fact that ripple beds are dominant in flume investigations and dune beds are dominant in the field. Taylor and Brooks (1962) pointed out that if there is a fundamental difference between ripples and dunes, the problems of roughness analysis and of modeling alluvial channels cannot be resolved by small-scale laboratory studies. PLANE BED WITH SEDIMENT MOVEMENT A plane bed is a bed without elevations or depressions larger than the largest grains of the bed material (fig. 9). The resistance to flow for flow over a plane bed results largely from grain roughness and C/Vg is large, ranging from 14 to 23. Grain roughness is not of the usual type, however, because grains roll, hop, and slide along the bed. For flow over a plane bed with sediment movement, the resistance to flow is slightly less than that for flow over a static plane bed, which is essentially an artificial rigidboundary condition that exists after screeding when stream power is insufficient to cause significantJ8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS transport of the bed material. The difference in resistance to flow between that on a plane bed with sediment moving and that on a static plane is shown in figure 10. This lower resistance to flow for flow over a plane bed with motion has been observed by many experimenters and was attributed by Vanoni and Nonicos (1960) to dampening of the turbulence by the suspended sediment. Elata and Ippen (1961) indicated that the structure of the turbulence was not damped by the suspended bed material but that the structure was changed by the movement of the sediment at the boundary. The magnitude of the stream power (r0V) at which the dunes or transition roughness changes to the plane bed depends mainly on the fall velocity of Figure 9.—Upstream view of plane bed. Bed material is sand II ; slope — 0.00153, depth = 0.60 foot, discharge 14.9 cfs, C/\/(j = 18.1. 17 ----------------------—--------—I— 0.2 0.4 0.6 0.8 1.0 DEPTH,IN FEET the bed material. Dunes of fine sand (low fall velocity) are washed out at lower values of stream power than are dunes of coarser sand. Consequently, in the flume experiments, where the depths were shallow (0.5-1.0 ft), the plane bed formed with finer sands at smaller slopes than it did with coarser sands; the result was smaller velocities and smaller Froude numbers (F). The plane bed with the fine sand occurred at slopes such that the Froude number, based on average depth and average velocity, was as small as 0.3, and it existed until the Froude number increased to about 0.8. If coarse sands are involved, larger slopes are required to effect the change from transition to the plane bed; the result is larger velocities and larger Froude numbers. Hence, in a flume containing fine sand, the plane-bed condition commonly exists after the transition and persists over a wide range of Froude numbers (0.3 < F < 0.8). If the sand is coarse and the depth is shallow, however, transition may not terminate until the Froude number is so large that the subsequent bed form may be antidunes rather than plane bed. In natural streams, because of their greater depths, the change from transition to plane bed may occur at a much lower Froude number than in the flumes. ANTIDUNES Antidunes form as a series or train of inphase (coupled) symmetrical sand and water waves. (See figs. 11 and 12.) The height and length of these waves depend on the scale of the flow system and the characteristics of the fluid and the bed material. In Figure 10.—Comparison of resistance to flow for flow over a plane bed with and without movement of the bed material. (Sand II.) IflGUKE 11.—Downstream view of antidune flow. Bed material is sand II; slope = 0.0059, depth = 0.56 foot, discharge = 21.2 cfs, C/y/fj = 14.6.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J9 Fiouke 12.—Side view of the antidune flow shown in figure 11. the 8-foot-wide flume, the height of the sand waves (from the bottom of the trough to the top of the crest) ranged from 0.03 foot to 0.5 foot, and the height of the water waves was 1.5-2 times the height of the sand waves. The length of the waves, from crest to crest or trough to trough, ranged from 5 to 10 feet. In natural streams, such as the Rio Grande or the Colorado River, much larger antidunes form. In these streams surface waves 2-3 feet high and 10-20 feet long have been observed. Antidunes do not exist as a continuous train of waves that never change shape; rather, they form as trains of'waves that gradually build up from a plane bed and a plane water surface. The waves may grow in height until they become unstable and break like the sea surf, or they may gradually subside. The former have been called breaking antidunes, or antidunes ; and the latter, standing waves. As the antidunes form and increase in height, they may move upstream, downstream, or remain stationary. Their upstream movement led Gilbert (1914) to name them antidunes. Resistance to flow due to antidunes depends on how often the antidunes form, the area of the reach that they occupy, and the violence and frequency of their breaking. If the antidunes do not break, resistance to flow is about the same as for a plane bed, and C/ V g ranges from 14 to 23. The acceleration and deceleration of the flow through the nonbreaking antidunes (frequently called standing waves) causes resistance to flow to be slightly more than that for flow over a plane bed. If many antidunes break, resistance to flow can be very large because the breaking waves dissipate a considerable amount of energy. With breaking waves, C/ Vg may range from 10 to 20. With antidune flow, the fact that the water and bed surfaces are inphase is a positive indication that the flow is rapid (F > 1). With dunes, the fact that the water surface is out of phase with the bed surface is a positive indication that the flow is tranquil (F < 1). In both instances the Froude number, which is based on the ratio of the mean velocity of the flow to the velocity of a gravity wave (C), must take into account the wave length (L). That is, C'1 = tanh ^~P- and cannot be approximated Zz Lj by C? = VgD, as is often done in open-channel flow, unless L/D > 10. Also, the mean velocity must be for the flow in the section of the stream where the antidunes are forming and not for the entire cross section of the stream. Kennedy (1961) made a detailed study of antidune flow. His conclusions were: 1. The wave length of antidunes is given by L - 2ttV2/g. 2. Where a limited range of depth and velocity exists, antidunes will form only if an initial surface wave is introduced. 3. The Froude number (V/C) for the occurrence of antidunes decreases as the depth of flow increases or as the size of the sand grains decreases. 4. Two-dimensional waves (those that occupy the full width of the flume) break when the ratio of wave height to wave length reaches a value of approximately 0.14. This value of 0.14 agrees with the theoretical value for deepwater waves. From Kennedy’s first conclusion it is obvious that the wave length is independent of depth of flow and, hence, the Froude number. However, Kennedy (1963) revised his first conclusion and stated instead that only the minimum length is given by 2irV2/g and that, for each bed material, there is a characteristic wave length dependent on velocity and depth. CHUTES AND POOLS At very steep slopes, alluvial-channel flow changes to what has been called chutes and pools. In the large, 8-foot-wide flume, this type of flow and bed configuration occurred only in runs using sands I and II. Chute-and-pool flow could not be attained when coarser sands were used because the steeper slope required could not be set up in the flume. This type of flow consisted of a long chute (10-30 ft) in which the flow accelerated rapidly, a hydraulic jump at the end of the chute, and then a long pool (10-30J10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS ft) in which the flow was tranquil but was accelerating. (See figs. 13, 14, and 15.) The chutes and pools moved upstream at velocities of about 1-2 fpm (feet per minute). The elevation of the sand bed varied within wide limits, although at no time was the flume floor exposed. Resistance to flow was large, and C/Vg ranged from 9 to 16. This type of flow was frequently accompanied by a decrease in the mean velocity of the flow in the flume even though there was an increase in stream power (fig. 16). Figure 13.—Downstream view of chute-and-pool flow. Note the pool in the foreground and the breaking wave in the background, at the downstream end of the chute. Bed material is sand II ; slope = 0.0095, depth = 0.45 foot, discharge = 15.4 cfs, C/y/q = 11.5. Figure 14.—Downstream view of chute (foreground) and breaking antidune shown in figure 13. Standing waves are terminating in the breaking antidune. Figure 15.—Side view of the breaking antidune illustrated in figure 14. Note the large quantity of bed material in suspension. REGIMES OF FLOW IN ALLUVIAL CHANNELS The flow in alluvial channels is divided into two flow regimes with a transition zone between (Simons and Richardson, 1963). These two flow regimes are AVERAGE VELOCITY (V), IN FEET PER SECOND Figure 16.—Relation of average velocity to stream power for runs with sand I as the bed material.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS Jll characterized by similarities in the shape of the bed configuration, mode of sediment transport, process of energy dissipation, and phase relation between the bed and water surfaces. The two regimes and their associated bed configurations, discussed in the following sections, are listed as follows: Classification of bed forms into flow regimes Lower flow regime (smallest values of shear, yDS, or stream power, VyDS) : 1. Ripples 2. Dunes with ripples superposed 3. Dunes Transition zone (bed roughness ranges from dunes to plane bed or antidunes). Upper flow regime (largest values of shear, 7DS, or stream power, Vy DS) : 1. Plane bed 2. Antidunes a. Standing waves b. Breaking antidunes 3. Chutes and pools LOWER FLOW REGIME In the lower flow regime, resistance to flow is large and sediment transport is small. The bed form is either ripples or dunes or some combination of the two. The water-surface undulations are out of phase with the bed surface, and there is a relatively large separation zone downstream from the crest of each ripple or dune. Total resistance to flow is the result partly of grain roughness and dominantly roughness of form. The most common mode of bed-material transport is for the individual grains to move up the back of the ripple or dune and avalanche down its face. After coming to rest on the downstream face of the ripple or dune, the particles remain there until exposed by the downstream movement of the dunes; then the cycle of moving up the back of the dune, avalanching, and storage is repeated. Thus, most movement of the bed-material particles is in steps, of the order of magnitude of a ripple or dune length, separated by rest periods dependent on ripple or dune height and velocity. The velocity of the downstream movement of the ripples or dunes depends on their height and the velocity of the grains moving up their backs. In flume studies, the ripple bed is the configuration usually formed in the lower flow regime; in natural streams and rivers, dunes or dunes with ripples superposed are the dominant bed forms in the lower flow regime. UPPER FLOW REGIME In the upper flow regime, resistance to flow is small and sediment transport is large. The usual bed forms are plane bed or antidunes. The water surface is inphase with the bed surface except when an antidune breaks, and normally the fluid does not separate from the boundary. A small separation zone may exist downstream from the crest of an antidune prior to breaking. Resistance to flow is the result of grain roughness with the grains moving, of wave formation and subsidence, and of energy dissipation when the antidunes break. The mode of sediment transport is for the individual grains to roll almost continuously downstream in sheets one or two grain diameters thick; when antidunes break, however, much bed material is briefly suspended, then movement stops temporarily and there is some storage of the particles in the bed. TRANSITION The bed configuration in the transition zone is erratic. It may range from that typical of the lower flow regime to that typical of the upper flow regime, depending mainly on antecedent conditions. If the bed configuration is dunes, the depth or slope can be increased to values more consistent with those of the upper flow regime without changing the bed form; or, conversely, if the bed is plane, depth and slope can be decreased to values more consistent with those of the lower flow regime without changing the bed form. Often in the transition from the lower to the upper flow regime, the dunes will decrease in amplitude and increase in length before the bed becomes plane (washed-out dunes). Resistance to flow and sediment transport also have the same variability as the bed configuration in the transition. It was the transition zone, which unfortunately covers a wide range of shear values, that Brooks (1958) was investigating when he concluded that a singlevalued function does not exist between velocity or sediment transport and the shear stress on the bed. In many instances when the flow conditions are such that the bed form is in the transition zone, the bed configuration will oscillate between dunes and plane bed. This phenomenon can be explained by the changes in resistance to flow and, consequently, the changes in depth and slope as the bed form changes. Resistance to flow is small for flow over a plane bed; so the shear stress on the bed decreases to a value that is incompatible with the shear stress for a plane bed but compatible with the shear stress for a dune bed, and the bed form changes to dunes. The increase in resistance to flow causes an increase in the shear stress on the bed so that the dunes wash out and the bed becomes plane, and the cycle continues.J12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS COMPARISON OF FLUME AND FIELD CONDITIONS The preceding comments were based on observations of bed configurations and flow phenomena that occurred in the flume experiments and in natural rivers (field conditions) and are equally true for both situations. However, there are major differences between flume and field conditions. In the usual flumes only a limited range of depth and discharge can be investigated, but slope and velocity can be varied within a wide range. In the field, a larger range of depth and discharge is common, but slope of a particular channel reach is virtually constant. Consequently, the variation in shear stress (yDS) and stream power (VyDS) is principally the result of slope variation in flume studies and of depth variation in a particular natural stream. The walls of a flume are nonerodible, and, hence, the width is constant; but in most alluvial channels the width depends on flow and bank conditions. Larger Froude numbers (V/VgD) can be achieved in flume studies than will occur in most natural alluvial channels because natural banks cannot withstand prolonged high-velocity flow without eroding. This erosion increases the cross-sectional area and this causes a reduction in the average velocity and the Froude number. Rarely does a Froude number, based on average velocity and depth, exceed unity for any extended time period in a natural stream with erodi-ble banks. In the field, where the slope of the energy grade line is constant, the Froude number is also constant unless there is a change in the resistance to flow (F = -s/S). Flow is more nearly two dimensional in flumes than in natural streams. However, the main current meanders from side to side in a large flume, as it does in the field, and bars of small amplitude but large area develop in an alternating pattern adjacent to the walls of the flume. (It is on these bars that the bed forms shown in fig. 3 superpose themselves.) If very large width-depth ratios are maintained in the flume by keeping depth of flow shallow, these bars may grow to the water surface. Figure 17 shows the bars that formed in the 8-foot-wide flume when the width-depth ratio was about 22, the slope was 0.0020, the total discharge was 4.0 cfs, and the bed material had a median fall diameter of 0.19 mm. In the field it is even more obvious that the flow meanders between the parallel banks of a straight channel, and the alternate bars which form opposite the apex of the meanders are easier to distinguish. As in a flume, if the depth is decreased the alternate bars increase in amplitude until they are close to the water surface, or even exposed. (See fig. 18.) In fact, scour in the main current adjacent to a large bar may cause the water surface there to drop slightly so that the top of the bar is exposed. This has been observed in the Rio Grande (R. K. Fahnestock, oral commun., 1962) and in other natural channels. If the banks are not stable, erosion occurs where the high-velocity water impinges, and deposition occurs on the opposite bank. The ultimate development is a meandering stream if other factors such as slope, discharge, and size of bed material are compatible (Leopold and Maddock, 1953). VARIABLES Resistance to flow in alluvial channels is complicated by the large number of variables. It is further complicated by the interdependency, either real or apparent, of the variables. In fact, some variables may be altered or even determined by the flow, and changes in flow conditions may change the role of a dependent variable into that of an independent one. It is difficult, especially in field studies, to differentiate between the independent and dependent variables. The slope of the energy grade line of an alluvial stream illustrates the changing role of a variable and the difficulty of selecting the dependent variable. If a stream is in equilibrium with its environment, slope is an independent variable. In such a stream the average slope over a period of years has adjusted so that the flow is capable of transporting only the amount of sediment supplied at the upper end of the stream and by the tributaries. If ^or some reason a tributary or upstream reach supplied a larger or smaller quantity of sediment than the stream was capable of transporting, the slope would change and would be dependent on the amount of sediment supplied. Clearly, it is difficult to determine if a stream is in equilibrium. Lane (1955) and Leopold and Langbein (1962) pointed out that a stream reacts quickly to any change and that the slope of the stream, although it may never reach true equilibrium (geologically, it never does), will approach its final equilibrium value in a short time. If during a runoff event in a natural stream the water and sediment discharges do not change rapidly, the slope of the water surface and the energy gradient for a reach of the stream also will not change significantly. If the discharge does change rapidly (surge), so that there is an appreciable difference in discharge between the ends of a relatively short reach, the slopeRESISTANCE to flow in alluvial channels J13 LONGITUDINAL STATION, IN FEET O O Ll. of the water surface and the energy gradient will vary. We have given much thought to the significant variables in an alluvial channel, as have others (Rouse, 1959; Vanoni, 1946; Brooks, 1958; Einstein, 1950; and Bagnold, 1956, to mention a few), and have presented our ideas in several publications (Simons and others, 1961; Simons and Richardson, 1962a; and Richardson and others, 1962)—publications which have provoked stimulating discussions. In the following sections the variables affecting resistance to flow are listed and their dependency or independency is discussed. Also discussed are some conditions whereby a dependent variable may become an independent one; which variables may be eliminated as a first approximation to simplify the problem; and, in general terms, the effect or importance of a variable on resistance to flow. The measure of resistance to flow used in these discussions (and in this report) will be either the Chezy discharge coefficient (C/^g) or the Darcy-Weisbach resistance coefficient (/ = 8/ (C/ Vg)2). Under the restraint of equilibrium flow, dCr dCr _ dV dV _ q dt dx dt dx the variables that determine resistance to flow are: [V, D, S, P, (A, g, d, a, p„ Sp, Sifr Sn /»] = 0, (1)J14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS where Cr = concentration of bed-material discharge, V = velocity, D = depth, S = slope of the energy grade line, p = density of water-sediment mixture, :p. = apparent dynamic viscosity of the water- sediment mixture, g = gravitational constant, d = representative fall diameter of the bed material, ■a — measure of the size distribution of the bed material, p, = density of sediment, Sp = shape factor of the particles, Sx = shape faetor of the reach of the stream, Sc = shape factor of the cross section of the stream, and js = seepage force in the bed of the stream. Figure 1&.—Large alternate bar in Marala Ravi Canal, Pakistan. CONCENTRATION OF BED-MATERIAL DISCHARGE The concentration of bed-material discharge (CT), which some investigators have suggested is a variable, has been left out of equation 1. Admittedly, the concentration of bed material affects the fluid properties by increasing the apparent viscosity and the density of the water-sediment mixture (Einstein, 1906; Bingham, 1922; Bagnold, 1956; and Elata and Ippen, 1961). The effect of the sediment on viscosity (/1) and density (p) in any resistance to flow relation is accounted for by using their values for the water-sediment mixture instead of their values for pure water (Elata and Ippen, 1961; Simons, and others, 1963). Also, as Vanoni and Nomicos (1960), Vanoni and Brooks (1957), and Elata and Ippen (1961) demonstrated, the presence of sediment in the flow causes a change in the turbulence charac- teristics and in the velocity distribution. However, Elata and Ippen found that the effects of the particles on the resistance to flow, by the change in turbulence and velocity distribution, can be accounted for by using the apparent jviscosity of the mixture to evaluate R in the C/Vg versus R diagram. Furthermore, both Vanoni and Brooks (1957) and Elata and Ippen (1961) found that the effect of the sediment concentration on resistance to flow was small. Elata and Ippen found about a 5-percent difference in C/Vg between clear-water runs and runs in which as much as 27 percent by volume of the water-sediment mixture was neutrally buoyant granular material in suspension. A more fundamental reason for not including the concentration of bed-material discharge in equation 1 is that, for equilibrium flow, it is a dependent variable. If, however, the flow is not in equilibrium— that is, the concentration of bed material entering the flume or reach of a stream is larger or smaller than the concentration leaving—the concentration is an independent variable and should be included as a variable. Needless to say, for nonequilibrium flow the concentration of bed-material discharge would have to be known, but determining the concentration is almost an impossibility. FINE-SEDIMENT CONCENTRATION Fine sediment is that part of the total sediment discharge that is not found in appreciable quantities in the bed material. If much fine sediment is in suspension, its effect on the viscosity of the water-sediment mixture should be taken into account (Simons and others, 1963). The effect of fine sediment on resistance to flow is a result of its effect on the apparent viscosity and the density of the water-sediment mixture; so the fine-sediment factor is not included in the list of variables, but it is included in the viscosity and density terms and is discussed in a later section. The fundamental differences between fine sediment and bed material are in the availability of the sediment, the source of the sediment, and the capacity of the flow to carry the sediment. Fine sediment is not available in appreciable quantities from the bed, and it generally is not transported in concentrations that approach the fine-sediment transport capacity of the stream; bed material, however, is always available and is transported at the capacity of the stream. That is, fine-sediment concentration is dependent upon the availability of the sediment to the stream; and bed-material concentration is dependent upon the ability of the stream to transport a particularRESISTANCE TO FLOW IN ALLUVIAL CHANNELS J15 size in a conglomeration of sizes and, hence, upon the variables of equation 1. Fine sediment that is in suspension at one discharge or in one stream may be bed material at another discharge or in another stream. Generally the fine sediment is uniformly distributed in the stream cross section. DEPENDENT VARIABLE The velocity (F) was selected as the dependent variable in the present study, and the other variables were assumed to be independent. If the study had had other objectives, any one of the other variables could have been selected as the dependent variable, if the conditions of the experiment and the research objectives were properly treated. For example, the size distribution of the bed material could have been the dependent variable; then, the experiment would have been designed to determine the change in size distribution of the bed material with changes in F, D, S, and the other variables. To study resistance to flow in an alluvial channel, either V, D, or S may be selected as the dependent variable. In the present study velocity was selected because in a natural stream it is generally the unknown quantity and the other variables are measured quantities. In our experiments with given bed materials, the slope was varied and the resultant depth measured for different values of discharge. The velocity was then computed from the discharge and depth. Bed-material discharge and bed configuration were also measured, and either could be substituted in equation 1 for one of the other variables. Whether they should be included depends on the objectives of the study and the availability of data both in the laboratory and in the field. If the objective of the investigation is sediment transport or bed configuration under equilibrium conditions, then obviously the concentration or the bed configuration should replace one of the other variables. Including either one or both as independent variables in a resistance to flow study, however, might make the analysis more difficult by obscuring an essential relationship. Another reason for not including them is that they are often unknown quantities in natural streams, although by visual observation of the water surface or of the bed configuration and stratifications in the bed material after a runoff event, the bed configuration can usually be estimated. Later in this report, bed configuration is used as an independent variable in determining resistance to flow. CHANGE OF VARIABLES In equation 1, fluid viscosity, particle shape, and density can be eliminated if the fall velocity of the bed material is included in the list of variables, giving equation 2: V = [D, S, p, g, d, (.(, SR„ S'c,.fs, cr]; (2) where fall velocity, <■>, depends on several variables, (•) = [dr ps, p, g, Sp, [x, s].. (3) Viscosity can be eliminated because flow over an alluvial bed after the beginning of motion is fully developed, rough, turbulent flow. The laminar sublayer, since it is very much smaller than the grain diameters, should not exist. Resistance to flow for this type of flow is independent of the flow Reynolds number (R) and the viscosity. The changes in resistance to flow over sand beds which have: been observed with changes in viscosity (Straub, 1954; Hubbell, 1956; Hubbell and Al-Shaikh Ali, 1961; Vanoni and Brooks, 1957) are the result of changes in the fall velocity of the bed material, which cause a corresponding change in the bed configuration (Hubbell and Ali, 1961; Simons and others, 1963; and Simons and Richardson, 1963). The density and shape factors of the particles were included in equation 1 with the size of the bed material and the viscosity to account for the interaction of the fluid and the bed material. To have kept them in equation 2 with fall velocity would have resulted in a redundancy. Furthermore, it has frequently been demonstrated that fall velocity is more representative of the interaction of the fluid and the bed material than any other single variable. The size and gradation of the bed material cannot be replaced by the fall velocity because, in addition to determining the fall velocity, they determine the grain roughness of the bed. Gradation parameters based on the physical sizes of the bed material and on fall velocities may be necessary. This Is true because of the nonlinearity of the relation between the coefficient of particle drag versus SHAPE FACTOR FOR THE REACH AND CROSS SECTION The shape factor for the reach (SR) was included in equations 1 and 2 to focus attention on the energy losses resulting from the nonuniformity of the flow in a natural stream caused by the bends and the nonuniformity of the banks. Study of these losses in natural channels has long been neglected, although, as Ippen and Drinker (1962) stated, there is no net excess energy loss in bends with straight approach and exit reaches of sufficient length. In straight flumes with parallel walls, the shape factor of the reach is a constant and can be disregarded.J16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS The shape factor for the cross section (Sc) is a constant for the flume and can be disregarded. However, as is explained in the following paragraph, a shape factor for the cross section is needed when data from flumes of different widths are compared. Under these conditions, flow phenomena, bed configuration, and resistance to flow vary with the width of the flume. Simons, Richardson, and Haushild (1963) noted that: 1. Ripples observed in the 8-foot-wide flume with sand III did not occur with the same sand in the 2-foot-wide flume. 2. Dune fronts in the 8-foot-wide flume seldom were continuous from one wall to the other, whereas those in the 2-foot-wide flume were continuous. 3. Antidunes in the 8-foot-wide flume could be three- dimensional and could occur in more than one row, whereas those in the 2-foot-wide flume were always two-dimensional and extended from wall to wall. Hence, the resistance to flow was different in the two flumes for the same conditions (depth, slope, fall velocity, and so on). Vanoni and Brooks (1957) found a difference in results for their 10.5-inch-wide and 33.3-inch-wide flumes which could not be accounted for by a sidewall correction. SEEPAGE FORCE The seepage force (F.„), which occurs whenever there is inflow or outflow through the bed material and banks of a channel in permeable alluvium, may have a significant affect on bed configuration and resistance to flow (Simons and Richardson, 1962b). The inflow or outflow through the interface between water and the bed and the bank depends on the difference in pressure across the interface and the permeability of the bed material. For flow in a flume with a rigid bed and walls, the pressure difference and the permeability of the bed material depend only on the variables in equation 1; thus, the seepage force is a dependent variable and can be eliminated from equation 2. For flow in a natural channel, however, the pressure difference will depend on the location of the water table in the alluvium and also on the variables in equation 1; now, seepage force is an independent variable insofar as resistance to flow is concerned. If there is inflow, the seepage force acts to reduce the effective weight of the sand and, consequently, the stability of the bed material. If there is outflow, the seepage force acts in the direction of gravity and increases the effective weight of the sand and the stability of the bed material. As a direct result of changing the effective weight, the seepage forces can influence the form of bed roughness and the resistance to flow for a given channel slope, channel shape, bed material, and discharge. For example, under shallow flow a bed material with median diameter of 0.5 mm will be molded into the following forms as shear stress is increased: Ripples, dunes, transition, standing sand and water waves, and antidunes. If this same material was subjected to a seepage force that reduced its effective weight to a value consistent with that of medium sand (median diameter, d = 0.3 mm), the forms of bed roughness would be ripples, dunes, transition, plane bed, and antidunes for the same range of flow conditions. The reason for this difference in the forms of bed roughness is the reduction in effective weight and increased mobility of the bed material. A rather common field condition is outflow from the channel during the rising stage; this process builds up bank storage and increases the stability of the bed and bank material. In the falling stage, the situation is reversed; inflow to the channel reduces the effective weight and stability of the bed and bank material and influences the form of bed roughness and the resistance to flow. RESISTANCE COEFFICIENT With the proposed simplifications and by grouping V, D, S, P, and g into the Darcy-Weisbach resistance coefficient or into the Chezy discharge coefficient, equation 2 becomes: ^ = / = 8/ (f)2 = * IS, D, d, [S, D, d, o>, a, g, p]. (4a) The variables were not grouped into dimensionless parameters to present more clearly the essential role of each. Because of the interdependency of slope, depth, bed-material characteristics, bed configuration, and resistance to flow, it is difficult to isolate the effect of any one variable. For example, an increase in fall velocity may increase resistance to flow at one slope and decrease it at another slope. The relation of the variables in equations 4 and 4a to the bed configuration and resistance to flow are discussed in the following sections. SI.OPK The effect of slope on bed configuration and resistance to flow for two depths and four bed materials is shown in figure 19. With constant depth and in-RESISTANCE COEFFICIENT RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J17 Sand IV 0.48 < 03 < 0.49 ft per sec / ■ / / f 1 / / | / ✓ / / I / / n 1 / I f / / / / a 8 pi Ui ✓ I / I f f f / / i B ® * 1 0.02 0.005 0.01 0.05 0.10 0.50 1.0 1.5 SLOPE (S), IN PERCENT EXPLANATION o )epth from 0.90 Antidunes Depth from 0.41 to 1.3 ft to 0.59 ft O □ • ■ © B Ripples Dunes Planes € E 9 n Q B Small dunes Transition Standing waves □ Figure 19.—Change in Darcy-Weisbach resistance coefficient (/) with slope, depth, and fall velocity of the bed material in the 8-foot-wide flume.J18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS creasing slope, the bed configurations for the bed material having a median fall velocity equal to or less than 0.22 fps (sand III) were (1) an artificial plane bed without sediment movement, (2) ripples, (3) ripples on dunes, (4) dunes, (5) transition (where the bed configuration was either dunes, plane bed, or a combination of them), (6) plane bed with sediment movement, and (7) antidunes. For the bed material having a median fall velocity of 0.48 fps (sand IV), the bed forms were the same but ripples did not form. Resistance to flow is a function of slope even when the class of bed configurations given on page J4 does not change. For example: 1. With shallow depths and the ripple-bed configura- tion, resistance to flow increased with an increase in slope; at greater depths, resistance to flow did not change. At greater depths, ripples gave way to dunes; but with an increase in slope at the shallow depths, dunes did not form. The shallow depth appeared to inhibit the formation of dunes. 2. With the dune-bed configuration, an increase in slope increased resistance to flow for bed material having fall velocities greater than 0.20 fps. For those bed materials having fall velocities less than 0.20 fps, an increase in slope decreased resistance to flow at shallow depths and slightly increased resistance to flow at greater depths. 3. The slope (where the transition from flow in the lower flow regime to the upper flow regime occurred) depended on the fall velocity of the bed material and the depth of flow. Constant depth required an increase in slope with an increase in fall velocity for the occurrence of the transition. With fall velocity constant, an increase in slope and a decrease in depth were necessary for the occurrence of the transition. 4. With antidunes, resistances to flow increased with increasing slope; the magnitude of the increase of resistance to flow with an increase in slope depended on the fall velocity of the bed material. The smaller the fall velocity, the larger the increase in resistance to flow with an increase in slope. DErTH The effect of depth on bed form and resistance to flow is not well defined because only a limited range in depths have been studied. However, just as changes in slope can change the bed configuration when depth and size of bed material are constant, so will changes in depth change the bed configuration when slope and bed material are constant. With a constant slope and bed material, an increase in depth can change a plane bed without movement to ripples, and a ripple-bed configuration to dunes, as is illustrated in figure 19. Although not clearly shown in figure 19, flume studies (Simons and Richardson, 1962b) and field studies (Colby, 1960; Dawdy, 1961; and Culbertson and Dawdy, 1964) have demonstrated that an increase in depth, without varying slope and bed material, may cause a dune bed to change to a plane bed or antidunes, and that a decrease in depth may cause a plane bed or antidunes to change to a dune-bed configuration. Figure 20 shows a typical break in a depth-discharge relation caused by a change in bed form from dunes to plane bed or from plane bed to dunes (Beckman and Furness, 1962). Limiting the variables to discharge and depth, resistance to flow will also vary with depth even when the bed configurations given on page J4 do not change. When the bed configuration is plane bed, either with or without sediment movement, there is a decrease in resistance to flow with an increase in depth—that is, a relative roughness effect. When the bed configuration is ripples, because their form is independent of depth, there is also a relative roughness effect—resistance to flow decreases with an increase in depth, as shown in figures 6 and 19. When the bed configuration is dune bed, although dunes increase in size when depth is increased, resistance to flow increases with an increase in depth only for sands coarser than 0.3 mm; for sands finer than 0.3 mm, resistance to flow decreases with an increase in depth. This effect is the result of a decrease in angularity of the dunes as they increase in size. Field studies indicate that at large depths, resistance to flow may decrease with an increase in depth even when the bed material is composed of sands coarser than 0.3 mm. This is true even though the size of the dunes continues to increase with an increase in depth. The anomalism may be explained by the facts that (1) the long dunes with mobile grain roughness on their backs compensate for the form effects of the separation zones downstream from the crests, and (2) at large depths the dunes, although large, may not cause appreciable nonuniformity of the flow. In the flume studies, dunes of the coarser sands created much acceleration and deceleration of the flow. Studies are needed to determine the conditions where resistance to flow for flow over a dune bed ceases to increase with an increase in depth.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J19 500 1000 2000 4000 6000 10,000 DISCHARGE (Q), IN CUBIC FEET PER SECOND Figure 20.—Relation of depth to discharge for Elkhorn River near Waterloo, Nebr. When the bed configuration is antidunes, resistance to flow increases with an increase in depth to some maximum value, then decreases as depth is increased further. This increase or decrease in flow resistance is directly related to changes in length, amplitude, and activity of the antidunes as depth is increased. SIZE OF BED MATERIAL The effects of the physical size of the bed material on resistance to flow are (1) its influence on the fall velocity, which is a measure of the interaction of the fluid and the particle in the formation of the bed configurations, (2) its effect as grain roughness, and (3) its effect on the turbulent structure and the velocity field of the flow. The physical size of the bed material, as measured by the fall diameter (Colby and Christensen, 1956) or by sieve diameter, is a primary factor in determining fall velocity, although only in relation to the other variables in equation 3. Use of the fall diameter instead of the sieve diameter is advantageous because the shape factor and density of the particle can be eliminated as variables. That is, if only the fall diameter is known, the fall velocity of the par- ticle in any fluid at any temperature can be computed ; whereas, to make the same computation when the sieve diameter is known, knowledge of the shape factor and density of the particle are also required. The physical size of the bed material is the chief factor in determining the friction factor for the plane-bed condition and for antidunes when they are not actively breaking. The additional dissipation of energy when antidunes are present is caused by the formation and breaking of the waves, which increases with an increase in energy input or a decrease in the fall velocity of the bed material. The physical size of the bed material for a dune-bed configuration also has an effect on resistance to flow. The flow of fluid over the back of dunes is affected by grain roughness, although the dissipation of energy by the form roughness is the major factor. The form of the dunes is related to the fall velocity of the bed material. FALL VELOCITY Fall velocity is the primary variable that determines the interaction between the bed material and the fluid. For a given depth and slope, it determines the bed form that will occur, the actual dimensionsJ20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Table 1.—Comparison of the various characteristics of ripples and dunes with fall velocity of the bed material Sand Ripples Dunes d 60 Fall Velocity Depth c/v7 L h Depth C/Vg L h No. (mm) (fps) (to (to (fo (f« (to (ft) i 0.19 0.064-.073 1.01 13.1 0.63 0.04 0.91 12.0 8.4 0.30 ii 0.27, 0.28 .10 -.12 .96 11.1 .95 .04 .93 10.6 7.6 .30 hi.... 0.45, 0.47 04 1 O OJ .81 11.2 1.13 .10 .81 9.6 5.9 .30 iv_... 0.93 .48 -.49 ... ... ... ... 1.02 10.8 4.7 .21 of the bed form (table 1) , and, except for the contribution of the grain roughness, the resistance to flow. The significance of fall velocity of the bed material on the spacing or length of dunes for various sizes of bed material at virtually constant depth is shown in figure 21. The dunes are not only shorter in length but are also much more angular when the fall velocity of the bed material is relatively large. For example, with sand IV (fall velocity = 0.48 fps) the average length of dunes was 6.5 feet as compared to 11.6 feet for sand I (fall velocity « 0.068 fps). The differences in resistance to flow for the four sands in figure 19 are primarily the result of differences in fall velocity. The differences in resistance to flow when the bed form is plane bed, however, are caused by the differences in grain roughness for the different diameters of bed material. Nevertheless, the magnitude of depth and slope at the beginning of motion (when a plane bed without motion changes to ripples or dunes) and that when a dune bed changes to a plane bed with motion or antidunes depends primarily on the fall velocity. An increase in fall velocity requires an increase in the product of depth times slope (that is, shear stress) for the change from static plane bed to ripples or dunes, or the change from dunes to plane bed and antidunes. Fall velocity is the critical factor in determining (1) whether ripples or dunes will form after the beginning of motion, (2) the shear stress at the beginning of motion when ripples or dunes begin to form from a plane bed, and (3) the shear stress at which ripples change to dunes. Ripples will not form in material having a fall diameter larger than 0.6 mm or a fall velocity higher than about 0.22 fps; there is some evidence, however, that the size of the bed material is a factor in determining whether ripples form. The magnitude of the shear stress required for the formation of ripples increases with an increase in fall velocity. When the bed form is dunes, the slower the median fall velocity of the bed material is, the longer and less angular the dunes will be and, also, the smaller Figure 21.—Approximate relation of length of dunes to a median fall velocity of the bed material, at constant depth. the range in shear stress or stream power will be within which a dune bed occurs. For bed material having a median fall velocity equal to or greater than 0.2 fps, the range of stream power within which dunes occur is large and that within which a plane bed occurs is small. In fact, whether or not a plane bed does occur between dunes and standing waves is intimately related to fall velocity and depth. If the depth is shallow and the fall velocity of the bed material is fairly high, the bed configuration may change from transition to standing waves without the development of a plane bed. The magnitude of the shear required for the formation of antidunes and the amount of antidune activity (formation and breaking of the antidunes) that will result from a given shear depends on the fall velocity. With a decrease in fall velocity there is a corresponding decrease in the magnitude of the shear required for the formation of the antidunes and for an increase in antidune activity. The in-RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J21 crease in antidune activity with a decrease in fall velocity causes an increase in resistance to flow. Special flume studies verified that the changes in bed form and resistance to flow are primarily the result of changes in the fall velocity. These special studies were: 1. The fall velocity of a given bed material was varied by changing the viscosity of the water-sediment mixture. Viscosity was varied by changing either the water temperature or the concentration of fine sediment in the water-sediment mixture. A change in fall velocity caused the bed configuration and resistance to flow to change (Hubbell and Al-Shaikh Ali, 1961; Simons and others, 1963). A decrease in fall velocity resulting from an increase in fluid viscosity caused resistance to flow to decrease for a dune bed and to increase for the antidune bed. (See figs. 22 and 23.) Thus, by o.n 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 p. \ ' '''*—, \ \ \ -> / / / / / / 0 NT \ \ \ \ / / X / □ / ' / / / Oo,/ A \ \ 1 ^ \ V 7 N/ T' / 1 / A ■x x. A-., S \ N \ \ s \ X T4 / \ _ a-y o' \ \ v \ ^A 1 EXPLAr NATION O Ripples □ Transition A Standing waves • Dunes having ripples ■ Plane A Breaking antidunes © Dunes __________l----------1----------1__________I----------I-------- 5 10 15 20 25 30 35 TEMPERATURE, IN DEGREES CELSIUS decreasing fall velocity, a dune bed could be changed to a plane bed or a plane bed could be changed to antidunes. 2. The density of sand V was 1.87, the median fall diameter was 0.36 mm, and the median sieve diameter was 0.68 mm. (See fig. 2.) The bed configurations and the magnitude of the shear stress for the various bed configurations and for a change in bed configurations corresponded to those for a bed material with the same fall velocity but not for material with the same sieve diameter. In fact, ripples formed with this bed material, which is expected for a 0.36-mm sand but not for a 0.68-mm sand. In addition to the flume experiments, observations of natural streams have shown that the bed con- 0.06 0.04 Ri n 17 Z Q=7.9cfs J 1 10 102 103 1Q4 iq5 Cf. IN PARTS PER MILLION EXPLANATION O □ ■ • A Dune Transition dune Transition plane Standing wave Antidune Figure 22.—Change in resistance to flow with temperature. Connected | points represent paired runs during which temperature was the only i independent variable (Hubbell and Al-Shaikh Ali, 1961). Figure 23.—Variation in resistance to flow (/) with concentration of fine sediment (Cf). Each run number represents a sequence in which concentration of fine sediment was the only independent variable.J22 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS figuration and resistance to flow will change with changes in fall velocity when the discharge and bed material are constant (Hubbell and others, 1956). For example, the Loup River near Dunning, Nebr., has bed roughness in the form of dunes in the summer when the stream fluid is warm and less viscous but has a nearly plane bed during the cold winter months. Similarly, two sets of data collected by R. K. Fahnestock (written commun., 1962) on a stable reach of the Rio Grande at similar discharges show that when the water was cold, the bed of the stream was plane, the resistance to flow was small, the depth was relatively shallow, and the velocity was large; but when the water was warm, the bed roughness was dunes, the resistance to flow was large, the depth was deeper, and the velocity was smaller. (See table 2.) Table 2.—Data from a stable reach of the Rio Grande at similar discharges but different temperatures Temperature (°F) Velocity (fps) Depth (ft) Slope Bed material (dso) Bed roughness c/V7 50° 4.25 2.45 0.00049 0.24 21.7 80° 2.53 3.66 .00053 .24 Dunes 10.2 APPARENT VISCOSITY AND DENSITY The effect of the various factors given in equation 3 on fall velocity are well known. However, the effects of fine sediment in suspension on fluid viscosity and fall velocity are less well known (Simons and others, 1963). Figure 24 shows the effect of fine TEMPERATURE, IN DEGREES CELSIUS sediment (bentonite) on the apparent kinematic viscosity of the mixture. The magnitude of the effect of fine sediment on viscosity is large and depends on the chemical makeup of the fine sediment. In addition to changing the viscosity, fine sediment suspended in water increases the mass density (p) and, consequently, the specific weight (y) of the mixture. The specific weight (y) of a sediment-water mixture can be computed from the relation, (Simons and others, 1963) T JU) T 8______ Y *? Cs(y s T w) (5) where yw = specific weight of the water, about 62.4 lb per cu ft (pounds per cubic foot) ; ys = specific weight of the sediment, about 164.5 lb per cu ft; and C8 = concentration (in percent by weight) of the suspended sediment. A sediment-water mixture, where Cs = 10 percent, has a specific weight (y) of about 66.6 lb per cu ft, and any change in y affects the boundary shear stress and the stream power. Changes in the fall velocity of a particle caused by changes in the viscosity and the fluid density are evident in figure 25. For comparative purposes, the 0 2 4 6 8 10 PERCENTAGE OF BENTONITE, BY WEIGHT Figure 24.—Apparent kinematic viscosity of water-bentonite dispersions. Figure 25.—Variation of fall velocity with percent bentonite in water.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J23 effect of temperature on fall velocity is given in figure 26. The details of the determination of fall velocity by computation and by visual accumulation (VA) tube analysis are given in the report of Simons and others (1963). GRADATION OF BBI) MATERIAL The effect of gradation of the bed material on flow in alluvial channels was investigated by Daranan-dana (1962) under the supervision of the authors and the sponsorship of the U.S. Geological Survey. His study was conducted in the 2-foot by 60-foot flume using two quartz sands with the same d50 Figure 26.—Variation of fall velocity with temperature. (sands VI in fig. 2; dr,0 == 0.33 mm). Sand VIi had a uniform gradation coefficient (a) of 1.27, and sand VI2 was graded with a = 2.07, where In this study, depth and water temperature were kept constant, and the change in bed configuration, resistance to flow, and sediment transport were observed and measured for various slopes and discharges of the water-sediment mixture. The effect of the gradation of the bed material on bed form and resistance to flow is shown in figure 27. The resistance to flow for flow over a plane bed with- 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1.0 SLOPE rsx w2) Figure 27.—Variation of resistance to flow with slope as a function of the size distribution of the bed material. out motion was slightly larger for the uniform sand, whereas that on a plane bed with motion was nearly the same for the two sands. Resistance to flow was much larger for the uniform sand on beds having ripples, dunes, or antidunes. The transition from a dune bed to plane bed occurred over a narrower range of shear values for the uniform sand than for the graded sand. Visual observation of the bed configuration provided the reason why resistance to flow was different for the two sands, but it did not show why the bed configurations were different. In the studies using the graded bed material, however, the continual sorting and remixing which took place suggested that the representative fall velocity and gradation also varied. In essence, then, the characteristics of the bed material were continually changing.J24 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS For flow over a plane bed without movement, the finer particles in the graded sand filled the voids, so that the boundary was smoother than the plane bed formed in graded sand. On the plane bed with movement, however, the effect of sediment on the turbulent structure of the flow probably had a counterinfluence on resistance to flow that compensated for any filling of the large voids by fine said, and resistance to flow was similar for the two sands. The average height and length of ripples in runs using uniform sand were larger than those in runs using graded sand, which increased resistance to flow. Ripples formed of uniform sand averaged 0.032 foot in height and 0.86 foot in length; those formed of graded sand averaged 0.025 foot in height and 0.46 foot in length. Dunes of the uniform sands had ripples superposed on them, whereas those of the graded sand did not. Although the dunes of the uniform sand were smaller, the effect of the ripples on their back increased resistance to flow. Dune length and height were 3.33 feet and 0.095 foot, respectively, in the uniform sand and 4.00 feet and 0.14 foot in the graded sand. Also, the longer dunes, crest to crest, decreased resistance to flow for the graded sand. The greater resistance to flow in runs using the uniform sands cannot be explained by the differences in antidune length and amplitude. The average antidune lengths and heights were greater for the uniform sands (5.05 ft and 0.15 ft, respectively) than for the graded sand (4.3 ft and 0.085 ft, respectively) . These changes do not adequately explain the large differences in resistance to flow. Daranandana’s investigation proved that there is a gradation effect on resistance to flow and emphasized that natural river sands should be used in flume experiments if the results are to be extrapolated to the field. The importance of using natural river sands in flume experiments is further emphasized by Blench’s (1952) conclusion that alluvial bed material is always adjusted by the flow so that its distribution conforms to a fairly definite normal distribution law. PREDICTION OF FORM OF BED ROUGHNESS A completely satisfactory method for predicting form of bed roughness has not been developed. Various methods of predicting form roughness (Albertson and others, 1958; Simons and Richardson, 1963; and Garde, 1959) have been proposed, but none have been suitable for both laboratory and field conditions. A simple relation (fig. 28) was developed by Simons and Richardson (1964) to relate stream power, 0.2 0.3 0.4 0.5 0.6 0.7 0.8 MEDIAN FALL DIAMETER, IN MILLIMETERS Figure 28.—Relation of form of bed roughness to stream power and median fall diameter of the bed material. median fall velocity of bed material, and form roughness. This relation gives an indication of the form of bed roughness one can anticipate if the depth, slope, velocity, and fall diameter of bed material are known. Flume data were utilized to establish the boundaries separating plane bed and ripples, ripples and dunes for all sizes of bed material, and dunes and transition for the 0.93-mm bed material. The lines dividing dunes and transition and dividing transition and upper regime are based on the following field data: (1) Elkhorn River, near Waterloo, Nebr. (Beckman and Furness, 1962), (2) Rio Grande 20 miles above El Paso, Tex. (see table 2), (3) Middle Loup River at Dunning, Nebr. (Hubbell and Matejka, 1959), (4) Rio Grande at Cochiti, near Bernalillo, and at Angostura heading, N. Mex. (Culbertson and Dawdy, 1964), (5) Punjab canal dataRESISTANCE TO FLOW IN ALLUVIAL CHANNELS J25 (Simons, 1957), and (6) Harza Engineering Co., International (1963) canal data. If just the flume data were used, the dividing line between dunes and transition would occur at about 10 percent less stream power than the field data indicates. Figure 28 shows that the range of stream power in which dunes occur becomes smaller with decreasing fall diameter of bed material. Thus, a small change in stream power can change the bed form and resistance to flow when the median fall diameter of the bed material is small. Using additional alluvial canal and river data, Tipton and Kalmbach, Inc. (written eommun., 1963) further validated the stream-power relation for predicting form of bed roughness. They also suggested using figure 29 to illustrate the large number of V* IN FEET PER SECOND Figure 29.—Relation of ra2 V/R1/4, V*, and regime of flow for laboratory and field conditions. The data are from the Rio Grande (Culbertson and Dawdy, 1964), Elkhorn River (Beckman and Furness, 1962), Pakistan canals (Harza Engineering Co., Internat., 1963), and the 8-foot-wide flume.J26 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS upper regime flows that have been observed in large irrigation canals that have fine-sand beds. Knowledge of what the bed form is or may be under given fluid, flow, and sediment conditions is important in designing channels that will neither erode nor fill up with sediment (stable channel), determining the stage-discharge relation for natural channels, and estimating the water and sediment discharge of a stream. For example, in stable-channel, design, if dunes with a large resistance to flow are anticipated, then a small average velocity, fairly large depth, and steep slope must exist for a given discharge. With fine bed material, a small change in stream power can change the bed form, and the channel may not function as designed. The bed form may not be dunes as anticipated but may be transitional or plane and result in small resistance to flow, high average velocity, relatively shallow depth, and an unstable channel having high transport capacity. The Marala-Ravi Canal in Pakistan is an example. This canal has bed material with a median diameter of approximately 0.16 mm and was designed to carry 21,000 cfs. When the canal was put in operation, the slope steepened from 1:10,000 to 1:8,000 because of excess sediment from the river. This resulted in a combination of slope, depth, and velocity which gave a stream power of 0.71. A stream power of this magnitude was sufficient to cause development of a plane bed and cause channel instability, large changes in channel geometry, and a bed-material discharge of 132 tons per day per foot of width. VELOCITY DISTRIBUTION The velocity distribution in a wide alluvial channel is as complex as the bed forms that occur. With dunes and antidunes, the velocity distribution is constantly changing with time and space. The variation of the velocity distribution in the vertical with these bed forms is so great that a detailed statistical study would be required to describe them. This would require many more profiles, covering a longer period of time over a larger area of the flume, than are available. Typical velocity profiles for ripples, dunes, and plane-bed configurations are given in figure 30. With both plane bed and ripples, the velocity profiles are constant with time and space outside a zone near the boundary roughness. (See fig. 30.) In the following section the velocity distribution for the plane bed with sediment moving is analyzed. From this analysis an equation for the mean velocity and the roughness coefficient for the plane bed with sediment movement is evolved. PLANE BED WITH SEDIMENT MOVEMENT The velocity distribution in the vertical for two-dimensional flow over a plane bed with sediment movement can be described by Vy = A* In y + B*, (7) where A* and B* are the slope and the intercept of the vy versus Iny plot. The values of A* and B* for the various runs are given in table 3 along with other pertinent information for the run. Figure 30.—Typical vcrtical-velocity-distribution curves for an alluvial channel. Data are from the runs in the 8-foot-wide flume with sand III as the bed material. Station refers to the normal distance from the flume wall.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J27 Table 3.—Slope (A*) and intercept (B*) of the velocity profiles for a plane bed with sediment movement Sand dm Run V, A* B* K Ct I 0.19 10 0.167 0.561 4.22 0.30 2480 15 .169 .595 4.50 2000 16 .190 .610 4.92 2750 II .27 46 .199 .608 4.74 .30 1670 .28 22 .172 .548 3.91 .29 1540 25 .215 .589 4.89 .34 2710 28 .201 .545 4.69 .34 2760 Ill 45 26 .200 .700 5.03 .27 4580 47 60 .261 1.005 6.05 .22 3290 61 .264 .782 5.90 .31 3390 71 .234 .826 5.43 .27 5250 72 .238 .813 5.20 .28 5680 70 .249 .590 5.21 .41 6310 100 .360 1.170 7.94 .29 8440 IV .93 27 .095 .261 2.02 .34 3 32 .103 .244 2.30 .40 26 24 .104 .261 2.30 .38 1 Velocity varies with In y, so it is natural to assume that the velocity distribution will have a form similar to that of the Von Karman-Prandtl velocity-distribution equation for flow near hydraulically rough boundaries: Vy/Vt = Aln (y/() + B. (8) With flow over a rigid boundary, A (A = 1/k, where k is Von Karman’s kappa) has the constant value 2.5, B varies with the roughness and form of the cross section, $ is some roughness height, and V* is the shear velocity ( VgDS). Experiments with flow over an alluvial bed have revealed that A is a variable and can exceed 2.5 (Vanoni, 1946; Ismail, 1952; Vanoni and Brooks, 1957). The difference in A, comparing flow between movable alluvial boundaries and rigid boundaries, was attributed by these investigators to a damping effect on the turbulence by the suspended sediment. Laursen and Lin (1952), in a discussion of Ismail’s paper, disputed this view and held that the change in A resulted from the change in bed roughness. Vanoni and Brooks 11957) showed a good correlation between the change in A and the ratio of the power to suspend the sediment in a thin layer near the bed (AyCs <» Ay) and stream power to overcome friction (r0V). Elata and Ippen (1961) used neutrally buoyant particles to show that suspended sediment causes an increases in A. Also, by measuring turbulence they found that the change in A was not the result of damping the turbulence but of a change in its structure. Elata and Ippen’s investigation indicated that the size and density of the bed material, in addition to its concentration near the boundary, affected A. Thus, the size and concentration of the bed material moving in suspension and in contact with an alluvial bed fairly conclusively affect A, but so also might changes in bed form. Also, the relative importance of the three factors (suspended sediment, contact load, or bed form) affecting A is not known. To determine A and B, equation 7 is equated to equation 8, from which A = p (9) and B = + A In 5. (10) V * Plotting A* versus V* (fig. 31), A, which is the v, Figure 31.—Variation of the slope of the vertical-velocity distribution M*) with shear velocity (V*) for a plane bed with sediment movement. slope of the line, does not vary systematically with concentration, size of bed material, or any other variable. Thus, it appears that the slope of an average line through the plotted data represents A for the plane-bed data for sands I, II, and III. The three points representing sand IV are for plane-bed runs in which shear values were slightly larger than were required for beginning of motion, and A for this sand should be about the same as that in runs in rigid-boundary channels. The capabilities of the flume system were inadequate to obtain a plane bed with high rates of sediment movement (after the dune-bed configuration) with sand IV. The average value of A for sands I, II, and III in a channel with a moving boundary was 3.20, and that for sand IV was 2.64; these values are equivalent to kappa values of 0.31 and 0.38, respectively. The absence of a systematic variation of A with sand size for flow over a plane bed having a moving boundary can be explained by the nature of the experiments. In experiments conducted under equilibrium plane-bed conditions, the concentration of suspended sediment and the movement of the contact bed-material discharge depend on the magnitude of the shear velocity and the size of the bed material.J28 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Similarly, A*, the slope of the vy versus In y plot, depends on the existence of the plane bed, the concentration of the suspended sediment, the magnitude of the contact bed-material discharge, and thus on the shear velocity and the size of the bed material. If a 1:1 correspondence exists between the shear velocity for the plane bed and the effect of the shear velocity on the slope of the velocity profile irrespec- tive of the size of the bed material, then A = ~ should be a constant. That is, the increase in shear velocity required by larger size bed material for the formation of a plane bed resulted in a compensating increase in the slope of the vy versus In y relation so that A remained constant. To determine B and 4 in equation 10, B* (the intercept of the plot of v„ versus In y) is plotted versus V* in figure 32. Either B or 4 can be selected and the other solved with the aid of figure 32. Except 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 V, =VjDS Table 4.—Values of B when £ = d.;« and of £ when B = A. [Calculated values of B assume e = cfeo, and those of £ assume B = A.] Sand dw dss (ft) B»/V» B (fps) £ mm ft ft mm i 0.19 0.000613 0.00078 25.9 2.3 0.00083 0.25 ii... 0.27, 0.28 .00090 .0015 23.3 .9 .0018 .55 hi... 0.45, 0.47 .00151 .003 21.9 1.1 .0029 .88 When the d50 size of the bed material was substituted for 4 in equation 10, B was not a constant; but when the value of B in equation 8 was assumed to be equal to A, then 4 was approximately equal to the dsr> size (table 4). Using B equal to A and integrating equation 8 gives a simple mean-velocity relation V = U* [A ln(D/S) + (B - A)}. (11) However, because B varies with the shape of the cross section as well as with the grain roughness, other values of B or of 5 may have to be selected for other cross sections. With B = A, equation 8 becomes »i/V, = 3.2 [Iniy/S) + 1] (12) and Z- = 3.2 ln(D/?) = 7.4 log D/5; (13) * * where V* = VgDS (in eq 12) = shear velocity for the position where the velocity distribution in the vertical was measured, Figure 32.—Relation between the intercept of the vertical-velocity distribution (R*) and shear velocity (V*). for sand IV, the slope M of the B* versus V* relation depends on the size of the bed material. The relation between 5* and V* for sand IV at low rates of sediment movement is not the same as for high rates. Consequently, the slope of the B* versus V* relation is considered to be a function of the size of the bed material for all sand sizes. For each bed material having a large range in sand sizes, there are many possible values of the roughness size (£) and, thus, many combinations of B and 4. From the original concept of equation 8, B should be a constant for a given roughness and channel shape. Also, B should have such a value that 4 should be a size which always occurs with the same frequency in the bed material. That is, 4 should equal the d-Mt dr,5, d, from v„/V* = 3.2 [In (*//£) + 1]. 0.1 0.5 1.0 DEPTH (D), IN FEET Figure 34.—Comparison of the equation C/\^g = 7.4 log (Z?/£) with the relation between the Chezy discharge coefficient and depth for plane bed with sediment movement. flow in alluvial channels. The Manning and Chezy equations developed for channels having rigid boundaries, the various regime equations (Inglis, 1948), and Einstein and Barbarosa’s (1952) treatment of alluvial river-channel roughness have all been used to estimate channel resistance and average velocity. The data for the 8-foot-wide flume follow those plotted in Einstein and Barbarosa’s (1952) river curve, which relates V/ Vg<\RS to Ayd/yR'S, reasonably well within the range of Ayd/yR'S values for which ripples and dunes occur. However, for both large and small values of Ayd/yR'S, the data departs systematically and radically from the proposed curve. Therefore, some modification is required when Einstein and Barbarosa’s relation is applied in the upper flow regime. (See fig. 35.) Resistance to flow in alluvial channels is the result of one of the following processes or a combination of them: 1. Surface resistance (grain roughness). Where surface resistance occurs, the flow does not separate from the macroboundary but does separate from the grains, or microroughness. This type of resistance occurs on a plane bed, on the back of dunes, and in antidune flow. 2. Form resistance. Where form resistance occurs, the flow separates from the macroboundary. The result is a pressure reduction in the separation zone (form drag) and the generation of large-scale eddies; both processes dissipateJ30 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Md/yR'S Figure 35.—Comparison of resistance data from the 8-foot-wide flume to Einstein and Barbarosa’s (1952) bar-resistance relation for rivers (curve 1) and with Einstein and Kalkanis’ (1959) relation for flumes (curve 2). energy. This type of resistance occurs with ripples, with dunes, and, to a limited extent, with antidunes. 3. Acceleration and deceleration of the flow (non- uniform flow). With the growth and subsidence of the inphase water- and bed-surface waves, acceleration and deceleration of flow occurs. This is a source of energy dissipation in addition to that caused by the grain roughness. This type of energy dissipation occurs with antidune flow, although with dunes there is also some acceleration and deceleration of the out-of-phase flow over the roughness elements. 4. Breaking waves. The inphase water- and bed- surface waves reach an instability point and break. The eddies and turbulence generated by the breaking waves dissipate energy of the same order of magnitude as does a hydraulic jump with a Froude number slightly larger than unity. Fortunately, a breaking wave, except in chute-and-pool flow, occupies a limited amount of time and space and causes a rather small increase in resistance to flow. This type of energy dissipation occurs with antidunes and with chutes and pools. It is very difficult to separate the processes or to mathematically describe the forms of bed roughness which occur on the bed of a sand channel. The spacing, amplitude, and shape of these roughness elements are affected by many interrelated variables. Because of this, it is very difficult to write a generalized function to determine resistance to flow. A generalized function may not exist, because (1) i more than one resistance to flow may occur for a | given slope, depth, and bed material; (2) hysteresis | exists in the change in bed configuration so that the bed configuration and resistance to flow depend on preceding flow conditions; and (3) the bed configuration will oscillate between a dune bed and a plane bed for a given bed material at certain slopes and discharges. The problem is further complicated by three-dimensional flow, varying depth, varying bank roughness, and nonuniformity of flow in alluvial channels. The net result is varying bed configuration from point to point on the stream bed, both longitudinally and laterally, and, consequently, varying resistance to flow. Hence, average resistance to flow is the result of the combined effects of many different roughness elements and the turbulence they create. In a straight sand channel it is not uncommon to find a plane bed under the main current of the stream, transition roughness adjacent to the bed plane, and dunes near the banks. Meanders and bars also complicate the problem; meanders are present to some extent even in straight reaches of sand-bed streams. Various methods of treating resistance to flow and of determining the average velocity of flow are presented in the following sections. The methods are based either on adjusting the measured slope to compensate for the increase in energy loss caused by the different bed configurations, or by adjusting depth or hydraulic radius to compensate for the energy loss and the increase in flow cross section caused by the form roughness. The latter method results from the fact that there is no flow through part of the measured depth. Two methods of depth adjustment are discussed: (1) by adjusting the depth to the equivalent depth for a plane bed with grain roughness and, (2) by adjusting the depth to an equivalent depth for a smooth boundary as defined by the equation of Tracy and Lester (1961). These methods depend on knowledge of the bed configuration. If the bed configuration is not known, an estimation is made of what it was or may be, and the velocity is computed using one of the foregoing methods. With the computed velocity, shear stress, and bed-material size, figure 28 is used to determine if the selected bed configuration would occur. If the selected bed configuration would not occur, another trial is made and the velocity is recomputed. EVALUATING RESISTANCE TO FLOW BY ADJUSTING SLOPE Equation 13 can be used to calculate resistance to flow and average velocity for flow over a plane bed upon which there is sediment movement. Therefore,RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J31 it is proposed that an adjustment term be applied to the plane-bed equation to obtain the resistance to flow for the ripple, dune, antidune, or chute and pool bed configurations. This adjustment term would take into account the increase in resistance to flow caused by the form roughness, wave acceleration and deceleration, and wave breaking (all these effects will be referred to hereafter as form roughness effects). The resistance to flow is in terms of /' and A/, where f is the resistance to flow due to grain roughness and A/ is the increase in resistance to flow caused by various form roughness effects. This implies that / = /' + A/. (14) From equation 14, the total-resistance factor is / = 8{gR/V*){S' + AS) (15) or / = 8(gRS'/V*)(l + A S') = 8(gRS'/V*)(S/S'), where S' is the slope the channel would have for the same depth and velocity if the resistance to flow was only grain roughness, and AS is the increase in slope required to compensate for all other types of energy dissipation. From equation 14, 5 = 5' + AS. (16) In terms of the Chezy discharge coefficient, C/ Vg = V/ V^RS, f = 8/(C/ Vg)* = 8IC'I VgY(S/S’), (17) where C7 Vg = 7.4 log Z?/5 (18) or C/ Vg = C'/ Vg V'575 = (7.4 log Z)/5)C*, (19) and C* is the correction factor applied to equation 18 to obtain the discharge coefficient for bed forms other than a plane bed. F* is based on the hydraulic radius R and not the depth D. To determine C* it is first necessary to determine AS, the increase in slope of the energy grade line which results from the form roughness. The increment of slope (AS) should be a function of the bed form, shear stress on the bed, and size of bed material. Therefore, AS = S — S' was calculated for all the data and was plotted versus the shear stress for each sand with bed form and depth as additional variables. For sands finer than sand IV, a linear relation existed between the logarithms of AS and the shear stress (depth and bed form as additional variables). (See figs. 36-38.) Thus, AS = KDM t0n, (20) where K, M, and N are constants that are functions of the bed form and sand size. N is the slope of the AS versus r„ line for the different bed forms, and K and M are determined from / = KDM = (21) To aS by plotting I as determined from —r for each depth TO versus that depth on log paper and determining the slope M of the line and the intercept K. The equations expressing A5 as a function of depth and shear are given in the preceding three figures. These figures indicate that, given bed roughness of ripples and dunes, AS is independent of depth of flow for sands I and II and that it has a minor effect for sand III if the hydraulic radius is approximately equal to the depth—that is, when r0 = yRS ~ yDS. Therefore, A5 was plotted against the slope. (See figs. 39, 40, and 41.) The correction factor C* for each sand for the plane-bed equation was determined from these figures and is given in table 5. The flume data indicate that AS and C* are independent of depth for the ripple- and dune-bed configurations in the finer sands. This requires further verification because of the small range in depth investigated. To determine if AS was still independent of depth for the dune-bed configuration at large depths, field data from the Elkhorn River (Beckman and Furness, 1962), which was the source for sand II, were studied. Runs having fully developed dunes (indicated by the depth-discharge relation in fig. 20) were plotted in figure 40. As indicated in figure 40, AS was also independent of depth for the Elkhorn River, where the depth ranged from 1.4 feet to 5 feet. This tends to confirm that AS is independent of depth for dunes. However, additional field and laboratory studies are needed to conclusively determine whether or not C* is independent of depth for ripples and dunes in fine sands. Table 5.—Equation for C* as a function of bed form and grain size Bed form Grain size I (0.19 mm) II ■ (0.27 mm) III (0.45 mm) Ripples Dunes Plane bed _. . Antidunes Vl - 1.5S-»» Vl - .62 Vl - 2.7S-J Vl - 1.2S-09 1 Vl - .26/S*14 1 Vl - 2.6D-*S-t 1 Jl - 34S-» 1 D-1 Vl - 700S>-5 J1 - 2.9 X 104S3 1 D25J32 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 0.004 0.008 0.02 0.04 0.06 0.1 0.2 0.4 SHEAR STRESS (Tg='iDS), IN POUNDS PER SQUARE FOOT Figure 36.—Relation between slope adjustment (AS), shear stress (to), bed configuration, and depth (D) for sand I. Numeral by the point is the depth. The lack of variation of AS with depth is in conflict with prior statements concerning the effect of depth of flow on resistance to flow and with the changes in resistance to flow shown in figures 5 and 19. The only explanation is that the effect of depth is taken care of by the D/i term in equation 18 from which S' is determined. With sand IV, AS was not linearly related to the shear stress. (See fig. 42.) Therefore, C* was computed for each run and plotted against the shear stress as shown in figure 43. With the dune-bed configuration, C* was inversely related to the shear stress. An increase in shear stress, either by an increase in slope or an increase in depth, caused a decrease in (7* (that is, an increase in roughness). The variation of (7* with depth and slope in the lower flow regime is given in figure 44. When the bed form was in transition from the lower to upper flow regimes, (7* was affected by depth and the shear stress. Also, there was a systematic change in C* with depth and shear stress in the transition which was contrary to the results of runs using finer sands. (7* was inversely related to slope and directly related to slope and directly related to depth when flow was in the upper regime. An increase in slope resulted in a decrease in (7* (an increase in resistance to flow), and an increase in depth resulted in an increase in (7* (a decrease in resistance to flow). The effect of depth and slope on (7* has been qualitatively verified by field observations. Antidune activity and resistance to flow in natural alluvial channels will decrease with an increase in depth when slope is constant, and they will increase with an increase in slope when depth is constant. StudiesSLOPE ADJUSTMENT (AS X 102) RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J33 SHEAR STRESS (T0=yDSJ, IN POUNDS PER SQUARE FOOT Figure 37.—Relation between slope adjustment (AS), shear stress (to), bed configuration, and depth (Z>) for sand II. Numeral by the point is the depth.SLOPE ADJUSTMENT (ASX102) J34 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 38.—Relation between slope adjustment (AS), shear stress (to) . bed configuration, and depth (D) for sand III. depth. Numeral by the point is theSLOPE ADJUSTMENT ^ASKIO2; RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J35 Figure 39.—Relation between slope adjustment (AS), slope (S), bed configuration, and depth (D) for sand I. Numeral by the point is the depth.SLOPE ADJUSTMENT(ASX102J J36 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1.0 SLOPE (SXIO2) Figure 40.—Relation between slope adjustment (A S), slope (S), bed configuration, and depth (Z>) for sand II. Numeral by the point is the depth.SLOPE ADJUSTMENT (ASX102) RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J37 Figure 41.—Relation between slope adjustment (AS), slope (S), bed configuration, and depth (D) for sand III. Numeral by the point is the depth.J38 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 0.01 0.02 0.04 0.06 0.1 0.2 0.4 SHEAR STRESS (T0 = 7DS), IN POUNDS PER SQUARE FOOT Figure 42.—Relation between slope adjustment (AS), shear stress (to). bed configuration, and depth (D) for sand IV. Numeral by the point is the depth. in the field and laboratory are needed to quantitatively determine the relation between C*, slope, and depth for different sand sizes. Using equation 19 and the values of C* from either table 5 or figure 43, the velocity was computed for all the flume runs. The computed velocity and the measured velocity are compared in figure 45. In summary: 1. The discharge coefficient C'/Vg for a plane bed can be determined for the flume runs by C7 Vg = 7.4 log (D/S), (13) in which £ is the cf85 size of the sand in the flume (may vary for other streams), because £ will be a function of the form of the cross section. 2. The plane-bed equation can be multiplied by a parameter C* to obtain the discharge coefficient for the other bed configurations. 3. The parameter C* depends on the bed form, sand size, slope, and depth. A. In the lower flow regime: a. The coefficient C* for the finer sands (dao < 0.4 mm) was independent of depth and decreased (an increase in resistance to flow) with an increase in slope. The magnitude of the decrease in C* with an increase in slope depended on the sand size and whether the bed configuration was ripples or dunes. b. For the coarser sands (d > 0.4 mm), C* was dependent on the depth and the slope and decreased with an increase in depth and (or) an increase in slope. The coarser the sand, the greater was the effect of depth on the value of C*. B. In the upper flow regime: a. The value of C* for sands finer than 0.93 mm decreased with an increase in slope and increased with an increase in depth. The magnitude of the change depended on the sand size. C. In the transition zone: a. The value of C* was undetermined for all but the coarser sands. With the 0.93-mm sand, C* increased with an increase in slope or a decrease in depth. 4. The value of C* can be determined from equa- tions for sands I, II, and III (table 5), and from figure 43 for sand IV. EVALUATING RESISTANCE TO FLOW BY ADJUSTING DEPTH OF FLOW Resistance to flow or mean velocity can be determined by adjusting either slope or depth of flow. The depth adjustment takes into account the increase in energy dissipation resulting from the form roughness and the possible error in depth measurement resulting from inclusion of the separation zones downstream from ripples and dunes in the total area of flow. The measured depth of flow includes part of the separation zones downstream from ripples and dunes. This part does not actually convey the fluid downstream. The roughness elements and their associated separation zones were observed, measured, and photographed. Together with these measurements, much attention was given to the flow patternRESISTANCE TO FLOW IN ALLUVIAL CHANNELS J39 SHEAR STRESS (T0 = 7DS;,IN POUNDS PER SQUARE FOOT Figure 43.—Relation between form roughness correction (C*), shear stress (to), and depth (D) for sand IV. Numeral by the point is the depth. Figure 44.—Relation between form roughness correction (C*), depth (D), and slope (S) for dune-bed configuration in sand IV. Numeral by the point is the slope X 102. over the individual roughness elements. For example, downstream from the crest of each ripple and dune is a separation zone in which a part of the fluid rotates about a horizontal axis which passes through the mass of fluid within the zone of separation, as is shown in figure 46. By careful measurement, it was determined that the length of these separation zones conformed reasonably well with the length of separation zones observed downstream from baffles placed in a wind tunnel (Nagabhushanaiah, 1961). These lengths were approximately 10-12 times the amplitude of the roughness element measured from trough to crest. A study of these separation zones suggested that average velocity based on Q/A and average depth based on the average distance from the water surface to the bed surface are perhaps misleading. An effective depth (De) may be defined as the average depth (D) reduced to compensate for the zonesJ40 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS MEASURED VELOCITY (V ), IN FEET PER SECOND Figure 45.—Comparison of the computed velocity and the measured velocity. Figure 46.—Typical flow pattern over two-dimensional ripple and dune roughness elements. of separation, and an effective velocity may be defined correspondingly by applying the principle of continuity. The concept of an effective velocity and depth is further clarified by figure 47, from which it can be seen that the effective depth, D = D1+D2 + P3±............... + D-’ (22) n and the effective velocity, V,, = q/Dr. The effective depth and velocity given in equation 22 can be approximated in the flumes by mapping the form roughness. This is done using sonic equipment and ejecting dye to define the limits of the separation zone. Knowing effective depth and velocity, the average velocity in the channel can be determined from D - (23)RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J41 V'—q/De ---------*■ Water surface Figure 47.—Sketch showing effective depth (De) and effective velocity (V = q/De). This procedure is extremely difficult and is generally impossible to use. In addition, effective depth and velocity do not consider the increase in energy dissipation resulting from the form roughness. Therefore, in the next sections an adjusted depth (D') and velocity (V') are defined that take into account both correction factors. They are also relatively simple to use to evaluate resistance to flow and velocity. ADJUSTING DEPTH TO AVERAGE AEEUVIAE GRAIN ROUGHNESS The resistance to flow and the average velocity are evaluated by adjusting the measured depth (D) to the depth of flow of an equivalent channel with the same slope, discharge, and average grain roughness. That is, V'D' = VD (24) and D' = D - AD, where D' = depth of the channel would have for the same slope and discharge if the resistance to flow was the result of an average grain roughness, D = increase in depth resulting from the form roughness, V' = mean velocity of the equivalent plane-bed channel, V' = C7 Vg V^WS (25) To obtain D' it is first necessary to define the average grain roughness. This is done by arbitrarily using the average discharge coefficient (eq 25) obtained for each sand in the plane-bed runs made in the 8-foot-wide flume. The magnitude of C'/ Vg is a function of the median diameter of the bed material: C'/Vs Median diameter of sand (mm) 20.60 _________________________________________ 0.19 18.28_________________________________________ .28 17.10___________________________________________ .45 16.25___________________________________________ .93 As the tabulation shows, C'/ Vg is largest for the finest bed material, because resistance to flow for the plane bed is a function of the magnitude of the grain roughness. These C'/ Vg values ignore the effect of relative roughness (depth range from 0.4 ft to 1.0 ft) and the difference in resistance to flow between static-sand-grain roughness and moving-sand-grain roughness. Average C'/V~g values rather than values obtained from figure 34 are used for the grain roughness because both the relative roughness and the moving-sand-grain effects for the plane-bed condition will be accounted for by the adjustment of the measured depth to the adjusted depth (D'). ALLUVIAL SAND-BED ROUGHNESS Development of method To determine the adjusted depth (D') for an alluvial sand-bed roughness, consider the points plotted in figure 48 for sand I. The points for ripples and dunes fall below and to the right of those for the relation of average grain roughness. Similar figures can be drawn for the other sands. The adjusted depth (D') can be determined by finding the difference necessary to adjust a run with form roughness to the average grain-roughness line. This difference, AD, between the measured depth (D) and the adjusted depth (D') is a measure of the effect of the form roughness on resistance to flow. For the plane bed, AD is small; for ripples and dunes, AD is relatively large; for antidunes, AD is relatively small; and in the transition zone, AD varies from large to relatively small values. As was indicated previously, AD should be a function of the depth, slope, size of Figure 48.—Effect of bed configuration on resistance to flow for sand I.J42 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS the bed material, and the bed configuration. If a functional relation can be determined for AD, then knowing the average depth of flow (D), the adjusted depth (D') can be determined. Knowing D', the adjusted velocity (V') can be calculated from equation 25 and the mean velocity (V) can be determined from the continuity equation, equation 24. A method of estimating AD was established by computing it for the ripple and dune runs for each bed material studied in the 8-foot-wide flume. To compute AD, equation 24 was combined with equation 25 to obtain aD - D -[(Cv4 vj <26) The computed AD’s were plotted against depth, with slope as a third variable for each bed material and bed configuration. (See figs. 49-52.) These plots verified that there was a relation between the depth adjustment (aD) , depth, and bed form for each of the bed materials. The relation between AD and D is not a function of the slope for the ripple bed but is a function of slope for the dune bed formed in < 0.50 0.60 0.70 0.80 0.90 1.00 K DEPTH (D). IN FEET DEPTH (D), IN FEET Figure 49.—Relation between depth adjustment (AD), depth (D), and slope (S X 104) for ripples (upper part) and dunes (lower part) ; sand I. 0.40 0.50 0.60 0.70 0.80 0.90 1.00 DEPTH (D), IN FEET UJ l— (/) Figure 50.—Relation between depth adjustment (AD), depth (D), and slope (S X 104) for ripples (upper part) and dunes (lower part); sand II. the coarser (sand IV) bed material. The AD versus D relation is a function of slope for the dune-bed configuration for all sands if field data are considered. With a significant relation between AD, D, and S for the different bed forms and bed materials, it is now possible to determine the average velocity by adjusting the depth of flow to an equivalent plane-bed depth. This method of determining average velocity can be summarized as follows. If the depth, the median fall diameter of the bed material, and the bed configuration are known, AD can be estimated; then, knowing D, the term D’ = D — AD can be computed. Next, if the slope of the energy gradient is known, the average plane-bed shear velocity, V'* = V gD’S, can be evaluated, and the value of V’ can be computed using equation 25, V' = C'/Vjj V^WS. (25) From the continuity relation, V = VD'/D (24) the average velocity can be computed. The size of bed material and the corresponding text figure to be used in determining AD for other bed materials withRESISTANCE TO FLOW IN ALLUVIAL CHANNELS J43 O' DEPTH (D), IN FEET <1 DEPTH (D), IN FEET Figure 51.—Relation between depth adjustment (AD), depth (D), and slope (S X 10‘) for ripples (upper part) and dunes (lower part) ; sand III. DEPTH (D), IN FEET Figure 52.—Relation between depth adjustment (AD), depth (D), and slope (S X 104) for dunes ; sand IV. either the ripple- or dune-bed configuration are given in table 6. Similar relations can also be developed for transition or antidune roughness. Using the data for canals with a dune bed presented by Simons (1957) and by Harza Engineering Co., International (1963) and the average grain-roughness relations developed using data from the 8-foot-wide flume, values of AD, D, and S for various size ranges of bed material were plotted to give figures 53 and 54. For field conditions, in which Table 6.—Bed configurations and range in size of bed material for which figures 49-52 can be used to determine the AD adjustment based on an average grain roughness Bed Figure configuration Range of median fall diameters of the bed material for which each figure can be used (mm) 49 l Ripples 1 i Dunes J 0.14-4.25 50 ^ Ripples . - 1 Dunes j 0.25-0.35 51 Ripples 1 0.35-0.55 Dunes 52 Dunes. ... © CO 1 © depths are greater, slope is a major variable in the AD, D, and S relations. For flume conditions, in which depths are shallower, slope is not a major variable. However, the flume data are compatible with the field data. The usefulness of such relations requires further discussion. Applications Velocity and discharge are determined by using figures 53 and 54 and making an initial estimate of form roughness and then using the other known variables, D, S, and drM, computing the average velocity (V). As a check, it is necessary to enter figure 28 with the stream power, based on computed velocity, and median fall diameter of bed material (d30) to see if, in fact, the assumed bed roughness will actually occur. This procedure must be repeated until agreement exists. In the design of stable channels, the water discharge is known; the form of bed roughness is selected (usually dunes for sand-bed channels) ; the size of the bed material of the proposed channel is estimated by studying the river characteristics, the diversion conditions, and the natural material in which the channel is to be constructed; and an estimate is made of the bed-material discharge from the river to the canal. A depth of flow is selected from D versus Q or R versus Q relations based on stable-channel data. These are the regime-type relations described by Simons (1957) and Simons and Albertson (1963). A figure, such as 53 or 54, that is appropriate for the anticipated size of bed material in the canal and form of bed roughness can be entered with the selected depth of flow. A limited choice of slope is available. If the canal must carry a relatively large bed-material discharge, select a steep slope. The depth of flow and the tentative slope fixes the depth adjustment (AD). The average velocity is computed using equations 24 and 25. Stream power is then computed. Using the d50 of the bed material, determine if the assumed bed roughness agrees withJ44 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS C) <1 z LU 5 H (/) => > Q < I H CL LU Q DEPTH IN FEET Figure 53.—Relation between the depth adjustment (AD), depth (D), and slope (S X 10*) for dunes (flume data and field data; d-00 ranges from 0.15 to 0.25 mm). that given in figure 28. If there is disagreement, the geometry of the channel must be modified until there is agreement. When this phase of the design is completed, the bed-material-discharge capacity of the canal is computed using some appropriate procedure, such as that proposed by Colby (1964). If the bed-material discharge differs from the required capacity based on river and canal conditions, another revision is made in the design, such as a change in depth and (or) slope, to increase or decrease the bed-material discharge to as near the required value as possible without changing the bed form. In many channels, conditions for stability can only be achieved by including exclusion and (or) ejection structures to control the quantity of bed material entering the canal from the river. Finally, by dividing the known discharge by the selected depth and computed velocity, the average width of canal is calculated. If the depth relations and figure 28 are used to design stable channels to convey a given discharge, it will be necessary to avoid designs which have a combination of stream power and median fall diameter that plots close to the dividing line between dunes and upper regime (usually plane bed) flow. In this region a slight error, a change in fall velocity of bed material, or just the inherent error in figure 28 may be sufficient to give an altogether different form roughness than is desired. This can be a very serious error because the change in resistance to flow between plane bed and dunes is generally on the order of 100 percent. If one designs a channel anticipating dunes as the bed roughness (C/Vp value of about 12) but a plane bed develops (C'/Vg value of about 20), the velocity in the channel would be almost double that anticipated. Because the depth would be fairly great, it would be much more difficult to divert flow from the channel, maintain stable banks in the channel, and supply bed material at a rate equal to the channel’s new transport capacity. The accuracy of predicting average velocity by-use of the foregoing method for both laboratory and field conditions is indicated in figure 55. The average error is about 5 percent for the laboratory data and about 10 percent for the field data. To show the usefulness of this approach, resistance diagrams relating C/ V g and R for different values of AD/D or AR/R can be developed. In these diagrams the ratio A D/D or A R/R is similar to the measure of relative roughness £/D or £/R in the resistance diagram for pipes. A typical relationRESISTANCE TO FLOW IN ALLUVIAL CHANNELS J45 <3 (/) D —> O < X \— CL EXPLANATION • (Harza Engineering Co., International, 1963) O (Simons 1957 data) X Flume data, sand II • 3.33 #2.86 o1-30 O 3.30 O2-30 1.40 O^ XJ o <3, oo o • 2.00 1.82 •• 1.82 • 1.49 o1'20 X X X X X X to 2.00 o 0o 02.80 2 00 O2O600° 0220 2.00° 1.70° ^ 1.60 1.50 #1-60 O OO O1-50 0 2.20 n O 01.50° !-40 01-30 • 1.32 2 4 6 8 10 12 DEPTH (D), IN FEET Figure 54.—Relation between the depth adjustment (AD), depth (D), and slope (S X 101) for dunes (flume data and field data; dr,<> ranges from 0.25 to 0.35 mm). between C/ Vg, R, and AD/D for sand II is shown in figure 56. The AD/D ratio is a function of bed configuration and size of the bed material, so that a general equation cannot be developed as in studies of channels with rigid boundaries. However, this procedure can be applied to rigid-boundary problems. BATFLE AND CUBE ROUGHNESS The applicability of the foregoing concept to rigid channels having baffles or cubes as the roughness I elements was investigated to test the generality of the method. The baffles and cubes, like dunes, cause zones of separation, turbulence, changes in velocity distribution, and considerable dissipation of energy. A constant resistance coefficient for an average grain roughness can be assumed, and the measured depth in the channel containing baffles and cubes can be adjusted to an equivalent depth in a channel having the same slope and discharge. Then, as in studies of alluvial channels, a relation between the depth adjustment AD and D can be determined. With this relation and the assumed resistance coefficient, the velocity can be determined for flow over a boundary having baffle and cube roughness. If the average grain roughness for a channel containing baffles and cubes is the same as for the 0.19-mm sand, then C'/VgT = V/VgWS = 18.91 (27) or D' = yV(18.912firS) and AD = D - D' = D - ¥7(18.91 *gS). (28) The depth adjustment (AD) is an indirect measure of the total effect of the form roughness in question. If for a given form roughness, values of AD are plotted against average depth (D), a unique relation results for each specific type of roughness. (See fig. 57.) Then, for this type of roughness, if the depth (D) and slope (&) are known, one can evaluate AD, D', VgD'S, V, and finally, the average velocity, V'D' V = The whole scheme is shown in figure 57, which is based on data from studies by Sayre and Albertson (1963) and Koloseus and DavidianJ46 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS MEASURED VELOCITY (V), IN FEET PER SECOND Figure 55.—Comparison of the computed velocity with the measured velocity. The velocity was computed by adjusting depth to the equivalent depth for a channel having average grain roughness. Roughness is ripples and dunes, and the data were from the 8-foot-wide flume, Punjab canals (Simons, 1957), and Pakistan canals (Harza Engineering Co., Internat., 1963). Figure 57.—Relation between depth adjustment (AD) and depth (D) for the type of form roughness in the rigid-boundary flume. £3 refers to Sayre and Albertson’s (1963) roughness coefficient, and £4 refers to Koloseus and Davidian’s (1961) roughness coefficient. The general equation for the family of lines in the AD versus D relation in figure 57 is A D = MD + b, (29) and the general solution for C/ Vg for these roughness elements is Z UJ O o 0 UJ cr < 1 o co Q c ,0.39 ^0.35 ¥L = 035no.3e °0.35 ^35_ 0.34C S29 O0.32 0.32OO032 0.32 O0 31 „0.30° O0-30 0.30° n0.29 o 0.28 O 0.27 °0 28 0.32 °°-29 Oq.28 O 0.28 PR /-n®-27 0,28 0.26 O O ^0.25 ¥L=025 0.4 0.6 0.8 1.0 R X105 2.0 3.0 C/Vg = 18.91 D - (MD + by/2 D or V = V'D' D ' D — MD — b3/2 /----- 18.91 —-----^---------- VgDS, (30) where V = the average velocity, D = the average depth, M = the slope of the lines in the aD versus D relation, b = a function of the Y-intercept in the aD versus D relation, and S = the slope of the energy-grade line. ADJUSTING DEPTH TO SMOOTH BOUNDARY ROUGHNESS Figure 56.—Relation between discharge coefficient (C/\/g), and Reynolds number (R) for different AD/D ratios. Bed configuration is ripples and dunes in sand II. (1961). The upper six lines in the AD versus D relation are for Sayre’s Au Bu A2, B2, Cx and C2 roughnesses formed by baffles at different spacings. The lower four curves are for Koloseus and Davidian’s A = 1/8, A = 1/32, A = 1/128, and A = 1/512 cube roughnesses. Depth adjustments were determined by adjusting depth and velocity to an equivalent average grain roughness. The same effect is accomplished by correcting data for hydraulically rough channels, both rigid and alluvial, to fit the rigid-channel hydraulically smooth boundary relation presented by Tracy and Lester (1961). That is, C7 Vg = 5.75 log (^F + 2.38, (31)RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J47 where C'/ v g = Chezy discharge coefficient for hydraulically smooth rigid open channels, and R = Reynolds number, VR/v, which becomes VD/y for channels where sidewall effect can be neglected. Equation 31 indicates that the discharge coefficient is a function of depth, slope, and temperature. C'/Vg = 4>(D's,S,T) (32) where D's = equivalent smooth-channel depth for the same slope (S) and discharge, and T = temperature of the flow. If both discharge and temperature are constant, a a relation between C'/ Vg, D\, and S can be developed. (See fig. 58.) A quantitative relation can be written which expresses C'/ Vg versus D's and, holding the slope (S') and the unit discharge (q) constant, figure 59 is obtained. Using the results shown in figures 58 and 59, the smooth-boundary resistance {C'/ Vg) given by equation 31 can be expressed as C7 y/g = S, T) = 8.4 j^log 3.52 - 1.415 log T + 4.132 (33) To develop a general method of adjusting the hydraulic radius of a particular run to the smooth- j boundary relation, first assume that the change in ! the wetted perimeter (P) is incorporated into the j DEPTH (D')f IN FEET Figure 58.—Relation between discharge coefficient (C'/V^i/), depth (D'), and slope (S). Unit discharge (q) and temperature (T) are constant. lg ./..I/--------------------------------------------------------L_ 0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 DEPTH (D'), IN FEET Figure 59.- Relation between discharge coefficient (C'/Vy), depth (D')t and temperature (T). Unit discharge (q) and slope (S) are constant. adjusted hydraulic radius. Then, from the continuity equation, Q = AV = A'V', RPV = R'P'V' RV = R'—U' = R'SV, (34) V'R' V'R and for constant temperature, ------ = ----. V V The Chezy relation for smooth or equivalent boundary flow is V' = C7 Vg VgR'JS. (35) Solving equations 34 and 35 simultaneously for R’s, *'• - Mrvd "• « The correction to the hydraulic radius (AR) required to plot a particular run on the smooth boundary relation is A R = R-R'S = R- r VR _i«/, C'/Vg VgSj ’ or for channel conditions such that R « D, (37) a d = d ~ r—VD . n v* = d ~ Lev Vg VjfSj ----/- /—I 2/3. (38) LC'/Vg VgSj Values of AR or AD can be computed using either equation 37 or 38, and from these values AR versus R or AD versus D relations can be developed. For a particular roughness it appears that using AR versus R or AD versus D relations to adjust RJ48 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS or D to R's or D's, and using equations 33, 34, and 35, the average velocity can be estimated. This procedure will be discussed in detail after the required relations are developed. An alternative for determining the average velocity is suggested by referring to the resistance diagram in figure 56, which relates C/Vgr, R, and aD/D. If in such a resistance diagram a new parameter, is adopted, a new resistance diagram relating C/Vff, V*D/v, and AD/D can be developed. It is then possible to use the AD versus D relation and the new resistance diagram to establish average velocity for a given boundary roughness. This concept will also be tested and discussed in more detail. BAFFLE AND CUBE ROUGHNESS Using the data of Sayre and Albertson (1961) and Koloseus and Davidian (1961), in which the bed roughness and the width of channel are large so that R ~ D, the value of AD for each run for each roughness was computed using equation 38. The relations between AD and D for each of the families of roughness patterns studied are shown in figure 60. Resistance diagrams relating C/ Vg, R* = V^D/v, and A D/D were prepared as outlined in the preceding section for both Sayre and Albertson’s (1961) and Koloseus and Davidian’s (1961) data. These diagrams are presented in figures 61 and 62, respectively. As with the AD versus D relations, where AD was based on plane-bed roughness, the results based on the smooth bed are excellent. GRAVEL AND COBBLE ROUGHNESS The flume data of Kharrufa (1962) and equation 38 were used to compute AD. The relations between AD and D for Kharrufa’s type B and H rock roughnesses, which have diameters of 4 and 2 inches, respectively, are given in figure 63. Also, a resistance diagram was developed using the flume rock DEPTH (D). IN FEET Figure 60.—Relation between the depth adjustment (AD) and depth (D) for baffle and cube roughness elements. £$ refers to Sayre and Albertson’s roughness coefficient, and £, refers to Koloseus and Davidian’s (1961) roughness coefficient.RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J49 Figure 61.—Resistance diagram relating C/X g, R*. and AD/D for the baffle roughness (Sayre and Albertson, 1963). roughness for a limited number of these runs. (See fig. 64.) Study of natural cobble roughness was possible using data from Lane and Carlson’s (1953) examination of irrigation canals in the San Luis Valley, Colo. (fig. 65). The resultant roughness, a paved cobble bed, resulted from the interaction of the bed material and the flow. Values of AD for these canals were determined using equation 38. The canals were divided into three groups on the basis of the median size of the bed material at the streambed-water interface. The bed material in one group had a median diameter of about 3 inches, that in another group, about 2 inches, and that in the last group, about 1Y2 inches. The AD versus D relations for each of these groups of canals are given in figure 66. The corresponding resistance diagram is given in figure 67. As with the rigid-boundary roughnesses previously considered, AD is independent of slope; but in canals it can remain independent of slope only as long as the bed material is stable. If a large dis- charge was turned into any of these canals, a boundary shear could develop that would cause movement of bed material. The bed material would ultimately become a little coarser through transportation of some of the smaller particles and movement of larger roughness elements into new positions. The previous AD versus D relation may no longer hold after such a flow condition, because a new boundary condition is developed by the flow. Also, a dune-bed configuration may develop that would be residual at lower discharge. ALLUVIAL SAND-BED ROUGHNESS Sand-bed roughness is much more complex than the baffle, cube, and rock roughnesses. Form roughness is a function of flow. A small change in any variable affecting the form of bed roughness will cause a change in the bed roughness and the resistance to flow. However, AR versus R or AD versus D relations can be developed for the bed form shown in figure 3. For the flume data, D ^ R, and the relations are in terms of AD and D. As can be seen in equation 32, C'/ Vg and, hence, AD, for a constant (q) are a function of the slope and depth of the channel. Also, AD is more strongly dependent on slope when the bed form is dunes in the coarser sands. (See figs. 68 and 69.) In contrast to the previous study for sand bed roughness where AD was determined by correcting to an average sand-grain roughness, in this section AD is determined by correcting to the hydraulically smooth, rigid-boundary conditions, and it will not be zero for the plane bed. The AD versus D relation for plane-bed runs using sand I is given in figure 70. A similar relation using slope as the abscissa for plotting antidune flow is given in figure 71. Similar relations for the other forms of bed roughness and other bed materials can be developed. Although i^he resistance diagram for the flume data could be more precise if separate diagrams were made for each form roughness and size of bed material, only one diagram has been prepared. (See fig. 72.) This diagram relates C/^g, R, and A D/D for all the ripple and dune flume data. To show the possibilities of using this procedure to estimate the average velocity in alluvial channels, the foregoing analysis was applied to the Punjab canal data reported by Simons (1957) and the more recently collected Pakistan canal data reported by Harza Engineering Co., International (1963). Relations based on these data were prepared in terms of the hydraulicJ50 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Vi D , R,xio fR =——) Figuke 62—Resistance diagram relating C/Vu, R*. and AD/D for cube roughness (Koloseus and Davidian, 1961). Figure 63.—Relation between depth adjustment (AD) and D for rock roughness in a flume. B and H rock roughness of Kharrufa (1962). Figure 64.—Resistance diagram relating C/y/U, R*. and AD/D for rock Slope ranged from 0.001 to 0.09 foot per foot. , roughness in a flume. B and H rock roughness of Kharrufa (1962).DEPTH ADJUSTMENT (AD), IN FEET RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J51 Figure 65.—Typical San Luis Valley, Colo., canal with cobble roughness. 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 DEPTH (D), IN FEET Figure 66.—Relation between depth adjustment (AD) and depth (D) for the San Luis Valley, Colo., canals with cobble roughness (Lane and Carlson, 1953). The third variable is median diameter of the bed material, in inches. 0.3 0.4 0.6 0.8 1.0 2.0 3.0 R* X10"5 (R* =V*D/v) Figure 67.—Resistance diagram relating C/y/H, R*. and AD/D for the San Luis Valley, Colo., canals with cobble roughness. z 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 DEPTH (D), IN FEET Figure 68.—Relation between depth adjustment (AD), depth (D), and slope (S X 104) for the ripple-bed configuration in sands I, II, and III.J52 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS DEPTH (D), IN FEET Figure 69.—Relation between depth adjustment (AD), depth (D), and slope (S X 104) for the dune-bed configuration in sands I, II, m and IV. Sand I 15.6 11.2 1.6 ° ^ QJ .0 °°> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 DEPTH (D), IN FEET Figure 70.—Relation between depth adjustment (AD), depth (D) and slope (S X 104) for a plane bed in sand I. Figure 71.—Relation between depth adjustment (AD), depth (D), and slope (S X 102) for antidune flow in sand II. radius, although the relations would be equally valid if expressed in terms of depth and a depth correction. Figure 73 presents the AR versus R relation for the Punjab canals, which, according to the relationship given in figure 28, have a dune-bed roughness. The slope in the canals is a very significant third variable. The accompanying resistance relation for the Punjab canals is presented in figure 74. Following an identical procedure, the AR versus R relation for the Pakistan canals is presented in figure 75, and the resistance diagram is given in figure 76. Similar relations were tested for field conditions using river-channel data. Very good results were obtained for channels having the same pattern throughout—that is, dunes, plane bed, or antidunes. However, if multiple roughness patterns and other complicating factors occur, such as a braided stream, the accuracy of the method is reduced. DETERMINATION OF AVERAGE VELOCITY The preceding relations and equations offer two methods of estimating average velocity by using the appropriate AD versus D or AR versus R relations, equations, and resistance diagrams. The procedures are outlined in the following paragraphs. It is assumed that the depth or hydraulic radius is known or can be determined from D versus Q and R versus Q relations (Simons 1957), that the slope is known or can be assumed on the basis of limits set by the AD versus D relations, and that the form of roughness is known (rigid or alluvial). If the boundary is alluvial, the median fall diameter of the bed material 900 ft. Valley gradient: Low, <0.020; intermediate, 0.020-0.025; high >0.025; based on data on plate 2. Mean velocity of flood water: Low, <6 fps; intermediate, 6-12 fps; high >12 fps; based on estimated peak discharge of the flood at the site of the gaging station and cross-sectional area of the flood waters (pi. 2). Length of valley measured approximately along centerline] Reach Valley width Valley gradient Mean velocity of flood Number of flood channels Area of net scour— Area of net fill— water Per mile of valley, in thousands of sq ft per mile Mile— 4.1-5.3 2-4 900 1,800 25 700 3.2-4.1 High High 120 400 1.9-3.2 Intermediate to narrow 1-2 0.8-1.9 2-3 1,000 160 2,000 3,200 0-0.8 1-2 Material gained (+) or lost (—) per mile of valley, in thousands of cu ft per mile -800 -320 -190 +700 +6,000EFFECTS OF FLOOD OF DECEMBER 1964, COFFEE CREEK, CALIFORNIA Kll consolidated sand composing the colluvial and alluvial fan material in the upper 3.4 miles of the mapped area seemingly offered little resistance to the flood waters, and probably explains much of the net scour calculated for this area. In any one reach, the amount of material lost from an area where the postflood surface was below the preflood surface (net scour) tended to be matched by a corresponding gain of material in nearby areas where the postflood surface is above the preflood surface (net fill). In the reach from mile 4.1 to 5.3, for example, the total volume of material gained in areas of net fill is 4.4 million cubic feet, whereas the volume of material lost in areas of net scour is 5.4 million cubic feet; there was thus a net loss, but a minor one, in the reach as a whole. The reach from mile 0.8 to 1.9 and from mile 1.9 to 3.2 shows a corresponding similarity in the estimates of volume of net scour and fill. The amount of erosion and deposition are distinctly unbalanced, however, in the reach from mile 3.2 to 4.1, where the amount of net scour was twice as great as the net fill. Here the valley is narrow and the valley gradient and stream velocity high, and the whole flooded area was virtually one flood channel. In the reach from mile 0.0 to 0.8, on the other hand, where the valley is wide and the valley gradient and flood velocities low, deposition far exceeded erosion. DEPOSITION CAUSED BY THE FLOOD Sand and gravel deposited during the flood cover at least 70 percent of the flooded area. In some places these deposits were laid down directly on the preflood surface (fig. 9) or in the preflood stream channel (figs. 10 and 11), whereas in other places the deposits formed in a two-stage cycle consisting of erosion of the preflood Figure 9.—Flood gravels, preflood surface, and cutbank in preflood deposits. View north toward locality 6 (pi. 1). Pack on top of gravel about 15 inches high. terrane followed by deposition of sand and gravel. Locally the sand and gravel was deposited in several episodes of erosion and deposition, during which flood deposits were laid down, eroded, and followed by the deposition of new sand and gravel. Sand occurs at the surface on more than half of the area occupied by flood deposits (pi. 1). It is mostly in areas away from the main flood channel, or channels, and commonly within forests and brushy areas. In places, such as at Mad Meadows (mile 4.2 to 4.3) and Seymour’s Ranch (mile 0.6), sand covered large areas B Figure 10.—Preflood and postflood photographs taken from approximately the same position near mile 5.3 ; upstream view (north). House (1) is the same in both photographs (also loc. 1, pi. 1'). Comparison of A and B indicates that 5 feet or more of deposition occurred during flood in the preflood channel. A, Photograph taken during the 1950’s. According to residents, the stream channel looked approximately like this immediately prior to December 1964 flood. Picture supplied by J. H.* Jessup. B, Photograph taken in May 1965\. The stream channel here is not the same as that shown on plate 1 as a natural diversion has caused a shift in the stream channel between the date (April 24, 1965) the photographs used in preparing plate 1 were taken and the date of this photograph.K12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS of meadowland. It also occurs locally as a thin (less than 1 ft) cover over gravel, where it may have been deposited in relatively slack waiter after the crest of the flood had passed or after a shift in the position of the main flood channel. The average grain size of the sand deposits is variable, ranging, in the samples studied (fig. 12, table 3), from fine to very coarse. The sand is moderately sorted and commonly contains some fine gravel. The amount of silt and clay mixed with the sand is small, being less than 6 percent in all but one of the samples (sample at loc. 29); this latter sample contains 23 percent silt and clay and was specially selected to represent fine-grained material. In most places, the sand deposits are structureless or nearly structureless, although a crude horizontal layering or indistinct cross-stratification occurs rarely. The sand deposits are generally from a few inches to 2 feet thick, although locally they are 4 feet or more thick (pl.l). The sand deposits commonly occur in flat-topped and steep-sided bars (fig. 13) that are irregular in shape and elongate in the direction of streamflow. They lie between minor flood channels and apparently formed in relatively slack water between channels of more rapid flow. The top surface of these bars must have been commonly within a few’ inches of the mean high-water surface. The gravel deposited by the flood is widespread, although not as widespread as sand, and occurs near or within flood channels, where the velocity of the water presumably was high. The gravel deposits are broadest in areas where the stream channel is braided, such as mile 0.8 to 1.6 and mile 4.9 to 5.2, reaching a width of more than 500 feet near mile 5.0. At the surface, where the coarse flood deposits ap-EFFECTS OF FLOOD OF DECEMBER 1964, COFFEE CREEK, CALIFORNIA K13 Figure 12.—Cumulative curves of grain-size distributions of food deposits. Number indicates locality (pi. 1) from which sample taken. Sample described in table 3. parently have been “washed” and the sand removed, the gravel contains little, if any, sand, and is well sorted; but below the surface it is poorly sorted and contains 13-34 percent sand. The subsurface deposits also contain a large proportion of boulders and cobbles, as much as 42 percent in one sample (loc. 2, fig. 12). The gravel is mostly structureless; rarely, a crude horizontal layering occurs. The gravel is commonly thicker than the sand deposits, being, in places, as much as 8 feet thick; it may be even thicker where gravel has filled preflood stream channels. The largest material transported by the flood (table 4) was more than 6 feet in maximum diameter and almost 5 feet in intermediate diameter, and many boulders 3-5 feet in maximum diameter and 2-4 feet in intermediate diameter were moved. Movement of these boulders is indicated by their occurrence within or at the top of gravel that rests on a preflood terrane and that thus was clearly deposited during the flood. At other places, where the identification of flood deposits is obscure, movement is indicated by the presence of transported plant debris directly below the boulders.K14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Table 3.—Size analysis of flood deposits Locality (pi. l) Median diameter D50 (mm) First quartile D25 (mm) Third quartile D75 (mm) Sorting coefficient (VD75/D25) Description of sample 2 46 4. 8 91 4. 3 In natural levee deposit, 3 ft below top. Top of natural levee com- 12 30 3. 7 90 4. 9 posed of large boulders, the largest 5.5 ft in maximum diameter. Sample near and in material similar to that shown in fig. 17. )/2 ft above base of 2-ft-thick layer of gravel at side of small tributary 13 . 70 . 46 1. 3 1. 7 flood channel bordering major flood channel. Gravel layer rests on uneroded preflood surface. Largest boulder in gravel near sample is 1.4 by 0.8 by 0.8 ft. 2 ft below top of 6-ft-thick flood deposit. Lower 3 ft of deposit 25 11 1. 3 26 4. 5 is sandy gravel, upper 3 ft is sand. Deposit borders 3-ft-deep plunge pool cut into preflood terrane. ) From 5-ft-thick bedded flood deposit. At base is 2 ft of sandy cobble gravel, which is overlain by 2 ft of bedded coarse sand 26 . 42 . 28 . 58 1. 4 > (sample 26) that in turn is overlain by 1 ft of sandy gravel (sample 25). At top of section, only J/£ ft below peak water level, is a 27 28 . 31 1. 6 . 12 3. 5 . 22 1. 0 . 072 . 8 . 44 2. 8 16 1. 4 1 7 J pavement of cobbles as large as 4 in. in diameter. ) Samples from deposits in area of heavy brush and young forests; deposits in form of flat-topped bars separated by channels. Samples 29 1 5 > 27 and 28 are representative of most deposits in area. Sample 30 6. 4 2. 8 29 is a finer grained deposit in a willow thicket. Sample 30 is ) gravelly sand in a flood channel. Figure 13.—Flat-topped bar formed during flood and composed of sand lind fine gravel. View east-northeast from locality 31 near mile 0.3. Dark area in lower left corner is grass-covered preflood surface that elsewhere in the area photographed lies 2-3 feet below the new gravel. Scale indicated by pick in center of photograph. The extremely poor sorting of the flood sand and gravel seems to indicate rapid deposition from water containing a large quantity of sediment. At times the manner of sediment transport may have been similar to that of a debris flow or mudflow. The deposits are textur-ally similar to mudflow sequences illustrated by Fahnestock (1963, fig. 15-17), but the deposits on Coffee Creek apparently cannot be considered to be true mudflows, Table 4.—Same maximum sizes of transported material Locality pi. 1 Size (feet) Type of material Note 3 4.4 by 3.8 by 1.8 5.5 by 3.4 by 2.2 3.5 by 3.3 by 2.4 On top of natural levee 8 ft above adjacent streambed. 3.5 by 3.2 by 2.5 - 2.9 by 2.3 by 1.7 3.5 by 2.8 by 1.6 2.9 by 2.0 by 1.3 7 5.0 by 4.0 by 2.9 8 3.5 by 2.1 by 2.1 9 3.0 by 2.3 by 2.2 10 natural levee deposit. 14.. 5.0 by 3.5 by 3.0 4.2 by 2.9 by 2.8 2.3 by 1.4 by 1.2 15 Granitic 1 Lodged in tree 6 ft above 1 ground surface. Esti-[ mated mean velocity across j flood area 15 FPS. 17 6.2 by 4.8 by 3.3.. 4.4 by 3.2 by 1.9 4.3 by 3.0 by 2.1 On top of natural levee. 4.5 by 4.2 by 2.2 4.8 by 2.9 by 1.3± 4.8 by 4.3 by 1.7 Peridotite 22 2.8 by 2.7 by 1.8 On top of 5-ft-thick natural levee deposit, 15 ft above streambed. Bridge abutment probably from vicinity of Coffee Creek Campground (site) 1.7 miles upstream. 24 5.6 by 3.9 by 2.7 because they lie below the highwater marks of the flood and in places contain current featues that indicate a subaqueous origin. Furthermore, no source area, such as a landslide, that could produce a debris flow or mudflow was found in the reach studied. Some landslides occur upstream from the study area and must have contributed sediment to the flood waters, but the size of the slides is insignificant in relation to the amount of material moved by the flood. Features considered to be natural levees commonly occur along the edge of the main flood channel. These features apparently formed during the flood whenEFFECTS OF FLOOD OF DECEMBER 1964, COFFEE CREEK, CALIFORNIA K15 coarse material was transported out of the channel, where the water velocity was high, and deposited in the area of relatively low velocity flow to the side of the main channel. Fahnestock (1963, p. A50) has described apparently similar features in coarse gravel along a glacial stream on Mount Eainier, Wash. The natural levees are generally 30-50 feet wide and have a steep slope into the adjacent channel and a gentle slope (commonly 3°-4°) away from the channel (figs. 14,15, and 16). The crest of the levee is, in most places, topographically higher than the surrounding terrane and lies 8-10 feet above the streambed of the adjacent channels and 2-4 feet above the general level of the valley bottom away from the channel and levee. The sur- face of many of these natural levees was probably only about 0.5-3 feet below the high water of the flood, on the basis of the projection of the water level from nearby wash marks or other features indicating the height of the flood crest. In places, some of the large boulders may have projected above the mean high-water surface. The natural levees are composed of poorly sorted gravel (sample at loc. 2, fig. 12; fig. 17) and have a fairly high content of sand. The surfaces of the levees, however, are composed mainly of coarse gravel with little, if any, sand. The gravel at the surface appears coarser than that within the deposit, but this difference cannot be conclusively demonstrated. The natural levees on the outside of the bends in the stream are composed Figure 14.—Generalized map and section showing flow patterns and natural levees formed during flood.K16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS Figure 15.—Natural levee at The Cedars (mile 5.0) formed in flood. Large boulders occur at surface of levee, and levee slopes gently to left away from stream channel. View is downstream toward southeast. Detailed photograph (fig. 17) of levee deposit taken at locality 2 (also loc. 2, pi. 1). Figure 16.—Natural levees formed in the flood, near mile 3.1, looking southeastward. Stream is confined by the levees in this reach. The natural levees slope gently away from the stream. Locality 18 shown on plate 1. of coarser gravel than the natural levees along straight reaches, whereas the material on the inside of the bend is composed of finer material. Deposition was greatest in the lower 1.9 miles of the valley, and in this area a net gain of material occurred during the flood (table 2). In the lower 0.8 mile studied, this net gain was about 6 million cubic feet per mile of valley, a volume equivalent to a layer 1.3 feet thick across the 1,100-foot-wide valley bottom. Deposition was most conspicuous in broad valley areas where the valley gradient and stream velocity were low. This relationship is clearly shown in the lower 0.8 mile studied, where the valley is broad and gently sloping, and stream velocities were low. Figure 17.—Sandy gravel in natural levee at The Cedars (mile 5.0; loc. 2, pi. 1, fig. 15). Length of scale is 2 feet. Plant debris in deposit was transported during flood. FLOOD CHANNELS The type and character of the flood channels differ considerably within the 6 miles mapped along Coffee Creek and appear to be determined by such interrelated factors as the width of the valley, the valley slope, and the stream velocity. In the widest reach of the stream (about 1,100 ft), from mile 0 to 0.6, the water spread out into a broad, relatively shallow stream. Part of the main flow was in the area of the preflood channel along the south side of the valley bottom, and part wTas in the northern valley bottom near Treasure Creek, but compared to other reaches of the stream the flow was relatively unconcentrated. At section G-G' (pi. 2) in this reach, the mean velocity of the flood waters was 5.1 fps (feet per second) on the basis of estimated peak discharge of the flood at the site of the gaging station and the cross-sectional area of the flood waters. This mean velocity is the lowest indicated in any of the measured cross sections. The average valley slope near section G-G' is 0.13, the lowest slope measured along the valley. As indicated, this reach of the valley contains the largest accumulations of flood deposits. In areas where the valley has intermediate widths (500-800 ft), such as near mile 2.1 and from mile 5.0 to 5.1, the flood-water was concentrated into two or more major flood channels, commonly braided. Three majorEFFECTS OF FLOOD OF DECEMBER 1964, COFFEE CREEK, CALIFORNIA K17 flood channels occur at mile 5.0 to 5.1 (section A-A', pi. 2) and two at mile 2.1 (section F-F', pi. 2). In these areas, where the valley bottom is of intermediate width, the mean velocity of the flood waters has an intermediate value (6.5 to 7.0 fps at mile 5.0 to 5.1 and 7.9 fps at mile 2.1). The valley slope at mile 5.0 to 5.1 (approx. 0.024) and at mile 2.1 (approx. 0.022) is also intermediate between the slope of the wide-valley reaches and narrow-valley reaches. In areas where the valley is narrow (100-200 ft), such as from mile 3.7 to 3.8 (sections C-C and D-D', pi. 2), the flood water occupied a single channel. At mile 3.7 to 3.8, the mean velocity of the flood waters (14.7 fps) and the slope of the valley (approx. 0.032) are the highest estimated anywhere along the lower 6 miles of Coffee Creek. SHIFTS IN STREAM CHANNELS Changes in the position of the channel of Coffee Creek appear to have been greater during the December 1964 flood than during at least the previous 110 years of historical record. During the 1850’s, a trail which in many places was along the stream, was built up Coffee Creek valley. This trail had withstood all previous high water until the 1964 flood (Roy Connelly, written commun., 1965). In addition, the position of the stream channel on photographs taken in August 1944 is virtually the same as on photographs taken in August 1960, except in the braided reach of the stream between the Coffee Creek Campground (mile 2.1) and Seymour’s Ranch (mile 0.6), where some conspicuous channel changes took place between August 1944 and August 1960 (%• 18). Many new channels were established during the flood. In some places, such as the braided reaches from mile 4.9 to 5.3 and from mile 0.8 to 1.6, the old channels were filled and the postflood stream follows an entirely different course. Elsewhere, the flood water occupied both the preflood channel and newly formed channels. In some places the flow in the new channel became the permanent stream, whereas in other places the old channel was reoccupied. A conspicuous new channel system formed on the north side of the valley from mile 4.1 to 4.8, in an area where the valley is of intermediate width. In places this system consists of more than one channel, but from mile 4.1 to 4.5 the water formed a single channel about 150 feet wide and 8 feet deep. No water course had existed in this area before the flood. After the flood, according to residents, the flow reverted naturally to the old preflood channel, although a low levee (manmade) was later constructed to assure this flow pattern. Channel shifts also occurred from mile 1.9 to 2.2 and from mile 2.4 to 2.6. In these places, according to residents, the water flow persisted in the new channel after the flood until it was diverted back to the preflood channel by manmade levees. The shifting of channels, as well as the location of scour, was influenced by the piling up of logs and other debris. Much of this debris was removed before the authors visited Coffee Creek after the flood and before the postflood aerial photographs were taken. No attempt was made to show the location of debris on the map. Nonetheless, log jams commonly formed at the upstream limit of the forests and deflected the main flow of the flood to other parts of the valley. The residents also described examples of debris piles that blocked main channels and diverted the flow to other areas. Base from uncontrolled aerial photographs Figure 18.—Channel changes between site of Coffee Creek Campground (mile 2.1) and Seymour’s Ranch (mile 0.6).K18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS FLOOD FREQUENCY The problem of the frequency of occurrence of catastrophic floods such as that on Coffee Creek is of both geologic and hydrologic interest. The time that has elapsed since an event of similar magnitude can be estimated from historical, botanical, and geologic evidence. Alternatively, a measure of the recurrence interval of extreme floods can be obtained through flood-frequency analysis based on regional hydrologic data. Although accurate flood records are available for the past 7 years only, accounts of local residents indicate that the December 1964 flood greatly exceeded any previous flood on Coffee Creek during the 110-year period since original settlement of the area. Even the December 1955 floods, the greatest since the nearlegendary deluge of 1861 in most of the region, were not of great magnitude in the upper Trinity River basin. The gage height of the 1955 flood peak on Coffee Creek (estimated by U.S. Geol. Survey from flood marks at the time of gaging-station installation) was equaled by a flood in February 1958 and one in October 1962 (U.S. Geol. Survey, 1962, p. 443-444). In 1955 and in 1958, and in the flood of 1937 as well, flood waters covered the road (U.S. Geol. Survey, 1960, p. 638-639) in the lower Coffee Creek valley to a shallow depth. Some bank erosion and local channel changes took place, but the overall effects were insignificant compared with those of the catastrophic flood of December 1964. The discharge of 17,800 cfs on December 22, 1964, is more than five times greater than that of October 12, 1962 (3,360 cfs) (U.S. Geol. Survey, 1962, p. 443-444), the highest flood previously recorded. Other evidence suggests that recent floods have been relatively less severe in Coffee Creek than on most nearby streams. For example, comparison of floods related to winter storm runoff (fig. 19) shows that during the 7-year period prior to the December 1964 flood, the peak discharge per square mile of drainage area has been two to three times greater on Trinity River (drainage area 149 sq mi) than on Coffee Creek (drainage area 107 sq mi). Further, the preflood appearance of Coffee Creek valley differed from that of nearby streams of comparable size. The channel of Coffee Creek had been bordered for much of its length by large mature trees, whereas the low-water channels of the Trinity River and lower Swift Creek were flanked by broad, barren gravel flats. Part of the reason for the differences in frequency and magnitude of flooding on Coffee Creek in comparison with nearby streams may be the relatively high altitude of its drainage basin in the Trinity Alps. Although the mouth of Coffee Creek lies at an altitude of less than 2,500 feet, 40 percent of the drainage basin is above 6,000 feet. Only about 20 percent of the basin of the Trinity River above Coffee Creek is at comparable altitude. Winter snowfall and snowmelt runoff would thus be expected to be more important in the hydro-logic response of the higher basin. For example, in 1960 the peak discharge for the year occurred in February on the Trinity River. On Coffee Creek, the peak discharge associated with the same winter storm was exceeded by snowmelt runoff in June. Thus, the combination of prolonged high temperatures and intense rainfall, which were features of the December 1964 storm, and melting of much of the preexisting deep snowpack in the Trinity Alps triggered the tremendous runoff from the Coffee Creek basin. The flood-frequency on Coffee Creek cannot be analyzed directly, as a single-station analysis as described by Dalrymple (1960), because of the brevity (7-year period) of the streamflow record. However, an attempt was made to extend this record synthetically through correlation of the flood peaks with those recorded at the long-term station on the Trinity River at Lewiston. Because the correlation is very poor within the period of overlap in the records, a regional-analysis approach was used. In a comparative study of six methods of flood-frequency analysis applied to coastal basins in California, Cruff and Rantz (1964) concluded that the two Geological Survey methods, the index-flood method and the multiple-regression method, give better results where historical evidence of great floods is available. They concluded that of these two, the multiple-regression method was superior. A series of regression equations, based on correlation of peak discharges of a given recurrence interval with the watershed variables of area, mean basin altitude and mean annual basinwide precipitation, has been developed from analysis of records of long-term gaging stations in northern California that include the December 1964 flood peaks (R. W. Cruff, written commun., 1965). The calculated flood-frequency curve for Coffee Creek, together with the confidence limits, is shown on figure 20. This indirect analysis indicates that a flood discharge similar to that of the 1964 flood will probably occur, on the average, about once in 100 years in a drainage basin having the location, and gross physical characteristics of the basin of Coffee Creek. A minimum estimate of the time that has elapsed since a flood of a magnitude comparable to that of December 1964 is provided by the rings of many of the trees uprooted in the flood. Although individual old trees along a stream course may be undermined from time to time by lesser floods, extensive destruction of large areas of pine-fir forest such as occurred in CoffeeEFFECTS OF FLOOD OF DECEMBER 1964, COFFEE CREEK, CALIFORNIA K19 140 120 cr < o o - z cr i-< — i- z 20 O 2 1 1 1 1 1 ^ 1 * 12-22-64 #2-24-58 '12-22-55 (Peak estimated from floodmarks) - #l-12-59 2-18-58 _ #10-12-62 - #4-14-63 • 2-10-61 to 2-11-61 #2-3-63 to 2-4-63 12-15-62 1-31-61#* #5-ll-58 1-*-58,2;?:|?.63 , 7 co* * 11-13-57 2-7-58 S-S-6^ • _ 3 *6-1-58 10 6-2-58 4-14-62 12-2-62# • 10-10-57 4-5-59. •* *6-1-61 4-5-59. 4_2!_58 - 4-24-59. . 6-19-58 5-12-59 1 1 1 1 1 1 1 0 20 40 60 80 100 120 140 160 180 MOMENTARY PEAK DISCHARGE, IN CUBIC FEET PER SECOND PER SQUARE MILE (Coffee Creek near Trinity Center. Drainage area 107 sq mi) Figure 19.—Concurrent peak discharge per square mile of drainage area of Trinity River as compared with Coffee Creek. Numbers indicate month, day, and year of event. Data from U.S. Geological Survey (1960a, b; 1961a, b; 1962; 1963; 1964a; written commun., 1965). Creek valley is much rarer. The dating of logs reclaimed from trees in debris piles near mile 1.7 to 1.8 shows that many of them were 200-300 years old. A fir on the streambank between mile 3.4 to 3.5 had attained the age of 420 years when it was toppled across the stream during the flood. The site of Coffee Creek Campground (mile 2.1) was almost entirely forested before the flood, but now only a few pines 2-3 feet in diameter, said to be representative of the preflood forest, remain between the main flood channels. Dating of three of these through counting of rings exposed in increment cores gives ages of 150, 160, and 275 years. The largest trees near the north bank of the flood channel have similar ages. Upstream, near mile 5.2, two trees were found to have ages of 180 years. The tree-ring dating of both overturned and standing trees, together with general observations of the dimensions of trunks in the debris, clearly indicates destruction of extensive forested areas that have survived all the floods within at least the previous 200 years. The widespread erosion of thick alluvial and colluvial deposits along the margins of the valley is a geologic event of rare but unknown frequency. Deposits such as the alluvial fans of granitic sand at mile 3.8 and 5.0 are known to have changed little in the 100 years prior to the 1964 flood. The sparsely wooded, flat or gently sloping areas of alluvial soil overlying deeper gravels “* * * alike inviting to the miner or grazier * * *” (Cox, 1858), long a feature of Coffee Creek valley, no longer exist. The depth of soil and rock debris from adjacent slopes, 10-20 feet in many places, indicates that this debris must have been accumulating undis-K20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS turbed for several or many hundred years * 1 2 3 4 prior to its removal by deep lateral erosion during the 1964 flood. Indirect calculations thus suggest that a flood similar in magnitude to the 1964 event might be expected to occur no more than 10 times in 1,000 years, perhaps much less often. Botanical and geologic evidence indicates that the lower valley of Coffee Creek has not been similarly inundated for several or many hundred years. Although a great destructive flood is very infrequent event, it is apparently not unprecedented in the geological history of Coffee Creek. The ridgelike deposits of older gravel at several localities in the valley are clearly ancient natural levees formed in much the same way as those of the recent flood. It is important to note that the highest of the older deposits were not covered by the 1964 flood water but were as much as 2 feet above the level of the water surface in nearby channels. This implies a previous flood or floods that reached stages comparable to that of the immense flood runoff of December 1964 unless the stream had progressively degraded its bed during the intervening period. 1 Subsequent to preparation of this report, four radiocarbon dates were obtained by the writers on charcoal in alluvial fan deposits in the valley of Coffee Creek. The charcoal was exposed in cutbanks formed in the flood. These dates are : 1. <205 years B.P., 4,7 ft. below preflood surface, near mile 3.2, north side of valley (I-2i390) 2. 830 ±90 years B.P., 5.8 ft. below preflood surface, near mile 3.2, north side of stream, 30 ft. east and 7 ft. below sample 1-2390 (1-2389) 3. 1,650 ±100 years B.P., 10.0 ft. below preflood surface, total depth of alluvium 16.6 ft., near mile 3.8, north side of valley (1-2391) 4. 1,710±100 years B.P., 12.4 ft. below preflood surface, total depth of alluvium 14 ft., near mile 5.2, south side of valley (1-2392). SUMMARY AND CONCLUSIONS The December 1964 flood on Coffee Creek was of rare frequency and unprecedented in historic time. Erosion and deposition during the flood were catastrophic and significantly changed the character of the valley. Some of this erosion and deposition was similar to that described by Hack and Goodlett (1960, p. 48-51) for northern Virginia. Large areas of colluvial and alluvial material along the sides of the valley were eroded. Within the valley, the preflood channel was commonly filled, and new channels formed at entirely different locations. Deposition of sand and gravel occurred over a large part of the valley floor, covering, in places, large areas of meadowland. Along much of the valley, the volume of material lost in areas of net scour was about the same as that gained in nearby areas of net fill. Such a balance of erosion and deposition, however, did not occur in narrow steep reaches where the flood-water velocity was high; in these reaches, erosion far exceeded deposition. At the opposite extreme, in broad gently sloping reaches where the flood-water velocity was low, deposition far exceeded erosion. Although the entire 6 miles of the valley studied apparently gained more material than it lost, this gain was small. Nonetheless, a vast amount of material, some of it large boulders, was moved during the flood. The flood was primarily a transporting event during which erosional and depositional factors tended to be balanced. Physiographic changes were most pronounced where the valley is of intermediate width and moderate gradi-EFFECTS OF FLOOD OF DECEMBER 1964, COFFEE CREEK, CALIFORNIA K21 ent. In these reaches, erosion significantly widened the valley bottom by removing colluvium and alluvial fan deposits that had accumulated over a long period along the margins of the valley, encroaching on the valley bottom. Physiographic changes were minor in narrow high-gradient reaches and in wide low-gradient reaches, and were caused largely by scour in the former and by aggradation in the latter. There was little tendency for lateral cutting where the valley is wide. The contrast in effects of the flood in relation to the character of the different reaches may reflect different stages in valley evolution. Downcutting of the channel first establishes a smooth longitudinal profile; this downcutting is followed by lateral erosion that widens the valley floor and permits a flat valley to form. The flood was competent to transport material as large as 6 feet in maximum diameter and more than 5 feet in intermediate diameter. The highest mean velocity for the stream, based on the cross-sectional area of the stream and an estimate of the peak discharge, is 14.7 fps at the Coffee Creek Ranch. Maximum velocities were undoubtedly higher, but the velocity near the streambed could, in many places, be lower than the mean value. In any case, a velocity of 14.7 fps seems to be adequate to move the large boulders. Fahnestock (1963, fig. 30 and p. A30) indicates that velocities of 7 fps were sufficient to transport material having an intermediate diameter of 1.8 feet. Projection of his data to fragments of large size indicates that velocities of about 12 fps are needed to transport material as large as that in Coffee Creek. Some of the material transported on Coffee Creek is comparable in size to that transported in the August 1955 flood in Connecticut (Wolman and Eiler, 1958, table 1). Natural levees are one of the prominent depositional features produced by the flood. They are 30-50 feet wide, 2-4 feet above the general level of the adjoining flood deposits, and 8-10 feet above the adjoining stream-bed. In most places the levees are topographically the highest depositional features of the flood. The surface of the levees is composed of coarse gravel; in places, some of the largest boulders that were transported in the flood lie on these levees. The depth of the water over the levees, however, was relatively shallow, probably about 0.5-3 feet. In places, some of the large boulders may even have projected above the mean high-water level of the flood. Old natural-levee deposits that are widespread on the valley bottom indicate former floods of large magnitude. They are elongate bouldery areas topographically higher than the general level of the valley. Many of the old natural-levee deposits were islands within the area flooded in December 1964. Channel changes and sediment deposition on a high mountain stream such as Coffee Creek seem to occur in a manner different from that described in many other streams. Wolman and Leopold (1957) have emphasized that the flood-plain deposits that they studied consist predominantly of channel deposits (point bars) formed by the slow lateral migration of the stream across the valley bottom. They further emphasize that overbank deposition accounts for only a small part of the flood-plain deposit. On Coffee Creek, on the other hand, lateral migration of the stream seems to be negligible, except in the highest floods, and overbank flow lays down most of the coarse “channel” deposits on the valley bottom. Slow lateral migration seems to be nearly impossible on Coffee Creek because the natural levees are composed of coarse material that apparently can be transported only in the largest floods. The natural levees that form in great floods retain the water in the channel during lesser floods. Overbank flow, which apparently is a rare event along much of Coffee Creek but did occur in the December 1964 flood, caused a marked change in the entire character of the valley. Areas that were formerly dry land were converted into stream channels, and preflood stream channels were in places filled and abandoned. During this catastrophic change, coarse “channel” deposits were laid down on the valley bottom in areas far removed from the preflood channel. The recurrence interval of floods on Coffee Creek of a size comparable to that of December 1964 is estimated to be about 100 years on the basis of analysis of the expectable flood-frequency of a basin of the size, altitude, and rainfall of Coffee Creek in relation to other streams in northern California. This 100 years indicates the estimated average interval between floods, but the actual interval between any two floods of comparable size may be shorter or longer than the average, depending on chance conditions. On Coffee Creek, no flood of comparable size has occurred within the 110 years of historical record, and destruction of large areas of land containing 200-year-old trees and of alluvial fan material as much as 1700 years old suggests that the valley has not been similarly flooded for many hundred years. The morphological and sedimentary features of the valley of Coffee Creek are apparently in adjustment to present climatic conditions although other geologists have questioned such an adjustment in many glaciated mountain streams. Wolman and Miller suggested (1960, p. 67) that coarse gravel in many glaciated mountain areas may be a relic of glacial times when the stream had a high competence and that such gravel may beK22 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS too large to be moved by modern floods. Alternately, Wolman and Miller (1960, p. 67) suggested that modern catastrophic floods may have been competent to move the coarse material, although they thought this explanation less likely. The record from Coffee Creek is clear, however, that a modern flood is competent to move all but a few exceptionally large boulders. But without the knowledge of the 1964 flood, the coarse gravel in the stream channel and on the valley bottom of Coffee Creek might have been interpreted as a relic of glacial time, and the large colluvial deposits along the sides of the valley might have been considered to have accumulated since glacial time. The catastrophic flood of 1964 on Coffee Creek largely determined valley morphology, channel pattern and location, and the character of alluvial deposition, and apparently floods controlled the formation of these features in the past. Only in extreme events can the coarse material that makes up these features be transported. Wolman and Miller (1960) have suggested that fairly frequent events of moderate magnitude commonly do more “work” and are more important in establishing specific landscape features than are catastrophic events. The total amount of sediment transported during the 1964 flood cannot be determined from information available, but extreme events clearly are more important than lesser ones in the formation of the landscape features on Coffee Creek. REFERENCES CITED California Department of Water Resources, 1965, Flood! December 1964-January 1965: California Dept. Water Resources Bull. 161,48 p. Colby, B. R., 1964, Scour and fill in sand-bed streams: U.S. Geol. Survey Prof. Paper 462-D, p. D1-D32. Cox, Isaac, 1858, The annuals of Trinity County: San Francisco, Calif., Commercial Book and Job Steam Printing Establishment, 206 p. Cruff, R. V., and Rantz, S. E., 1964, A comparison of flood-frequency studies for coastal basins in California: U.S. Geol. Survey Water Resources Div. [rept.], 116 [p.]. Dalrymple, Tate, 1960, Flood-frequency analyses: U.S. Geol. Survey Water-Supply Paper 1543-A, p. 1-80. Fahnestock, R. K., 1963, Morphology and hydrology of a glacial stream—White River, Mount Rainier, Washington: U.S. Geol. Survey Prof. Paper 422-A, p. A1-A70. Hack, J. T., and Goodlett, J. C., 1960, Geomorphology and forest ecology of a mountain region in the central Appalachians: U.S. Geol. Survey Prof. Paper 347, 66 p. Irwin, W. P., 1960, Geologic reconnaissance of the northern Coast Ranges and Klamath Mountains, California, with a summary of mineral resources: California Div. Mines Bull. 179, 80 p. Posey, J. W., 1965, The weather and circulation of December 1964—record-breaking floods in the northwest: U.S. Weather Bur. Monthly Weather Review, v. 93, no. 3, p. 189-194. Rantz, S. E., and Moore, A. M., 1965, Floods of December 1964 in the far western states: U.S. Geol. Survey open-file report, 112 p. Sharp, R. P., 1960, Pleistocene glaciation in the Trinity Alps of northern California: Am. Jour. Sci., v. 258, no. 5, p. 305-340. Strand, R. G., 1964, Geologic map of California (Olaf P. Jenkins ed., Weed sheet) : California Div. Mines and Geology, scale 1:250,000. U.S. Geological Survey, 1960a, Pacific slope basins in California, pt. 11 of Surface water supply of the United States, 1958: U.S. Geol. Survey Water-Supply Paper 1565, 681 p. ------1960b, Pacific slope basins in California, pt. 11 of Surface water supply of the United States, 1959: U.S. Geol. Survey Water-Supply Paper 1635, 748 p. ------ 1961a, Pacific slope basins in California, pt. 11 of Surface water supply of the United States, 1960: U.S. Geol. Survey Water-Supply Paper 1715, 751 p. ------ 1961b, Colorado River basin, southern Great Basin and Pacific slope basins excluding Central Valley: U.S. Geol. Survey Surface Water Records of California, v. 1, 448 p. ------ 1962, Colorado River basin, southern Great Basin and Pacific slope basins excluding Central Valley: U.S. Geol. Survey Water Records of California, v. 1, 479 p. ------ 1963, Colorado River basin, southern Great Basin and Pacific slope basins excluding Central Valley: U.S. Geol. Survey Surface Water Records of California, v. 1, 463 p. ------1964a, Colorado River basin, southern Great Basin and Pacific slope basins excluding Central Valley: U.S. Geol. Survey Surface Water Records of California, v. 1, 498 p. ------1964b, Compilation of records of surface waters of the United States, October 1950 to September 1960—pt. 11, Pacific Slope basins in California: U.S. Geol. Survey Water-Supply Paper 1735, 715 p. U.S. Weather Bureau, 1964, Climatological data, California, December 1964, v. 68, no. 12, p. 410—453. Wolman, M. G., and Eiler, J. P., 1958, Reconnaissance study of erosion and deposition produced by the flood of August 1955 in Connecticut: Am. Geophys. Union Trans., v. 39, no. 1, p. 1-14. Wolman, M. G., and Leopold, L. B., 1957, River flood plains— some observations on their formation: U.S. Geol. Survey Prof. Paper 282-C, p. 87-109. Wolman, M. G., and Miller, J. P., 1960, Magnitude and frequency of forces in geomorphic processes: Jour. Geology, v. 68, no. 1, p. 54-74.PROFESSIONAL PAPER 422-K PLATE 1 UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY Some,V >.C::??TO gravel j;... :;'Mad.1 V;- Meadows EXPLANATION . Some :' gravel . . ■ Some: ./■' gravel:'/.'.-.' EROSIONAL AND DEPOS1TIONAL FEATURES Coffee Creek Guard Station \ (US Eorest Service) Fill indicates deposits formed during flood or, locally, since flood. Scour indicates erosion into preflood terrain . Some • '£) gravel^. 1 The L Cedars Gravel deposit Sand deposit Fill, little or no scour Depth in feet Gravel deposit Sand deposit Scour followed by fill Fill greater than scour Sand deposit Gravel deposit Scour followed by fill Fill less than scour. In places at base of flood channels may include lag gravels of preflood material Scour, little or no fill Depth in feet. Locally includes areas of slumped banks above high-water level. Limit of scour shown as solid line where well defined REFERENCE MILES Cutbank into preflood terrain Height in feet. In places deposits formed during flood extend back from top of cutbank as well as away from base roduced by ibsequent to flood Net scour or fill not determinable Net scour Net fill Area occupied by postflood stream channel US Geological Survey gaging ''-^station a/id cableway HYDROLOGIC FEATURES O’Brien Ranch High-water line of flood and depth of water Depth of water is distance, in feet, from post flood surface to level of flood crest Goldfield Campground Covered by water but unaffected or little affected by flood Coffee Creek Ranch Unflooded land within valley bottom Preflood (August 8,1960) Postflood (April 24, 1965) Stream channel Flood channel In places high-water line of flood is bank of flood channel Small permanent water channel Direction of flow during flood OTHER SURFACE EFFECTS Coffee* Creek \Ranch' Unaffected by flood Destroyed by flood Forest cover Seymour’s —1 Ranch Automobile bridge 122°45’ INDEX MAP SHOWING LOCATION OF FOUR PARTS OF MAP 2 MILES Main post flood channel before diversion .by man made levee Building Main post flood channel before diversion by man made levee, \ Coffee Creek ' /..Campground (site) OTHER FEATURES V-i;/Modified since flood. >>>>>> Postflood manmade levee Outcrop of bedrock Locality number Older v cut yy Line of cross section Shown on plate 2 APPROXIMATE SCALE REFERENCE MILES Seymour’s Ranch Numerous small exposures'of preflood deposits •Iv'U:and scoured areas not mapped• ^V.v.v.T £st Anthony’sit o ■°Creek Older' cut School. >•>>>> Modified Modified since flood Base from uncontrolled aerial photographs INTERIOR—GEOLOGICAL SURVEY, WASHINGTON, D. C. —1967—G66404 Geology by J. H. Stewart and V. C. LaMarche, Jr., May 1965 SHOWING EROSION AND DEPOSITION PRODUCED BY FLOOD OF DECEMBER 1964 ON COFFEE CREEK, TRINITY COUNTY, CALIFORNIA 0 Li 0 o o o 0 03 0 0 0 0 0 0 0 0 0 0 0 O ( \ # ^Morrison \\ / /UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-K PLATE 2 B B' s =0.032 D D' 60 FEET 40 EXPLANATION - 20 Sand Gravel Flood deposits <— o 2.5X VERTICAL EXAGGERATION Sand Gravel Older deposits v =7.9 fps E E' s =0.022 F' * > v c> t> v ^ V „ A <, Colluvium or locally derived alluvium ~+---=F U + + + + + + + Bedrock Net scour April 1965 Flood peak, December 196Jt Water line Restored, preflood ground or channel surface Dotted where uncertain s =0.024 Slope of floodwater surface between sections v =7.0 fps Mean flood velocity (feet per second) calculated from discharge estimate Location of sections shown on plate 1 Note: Greater accuracy of surveying and larger scale permits more detail on sections than shown on map (plate 1). Some minor changes occurred in stream channel between time of survey and date of aerial photography used as base in plate 1 G G' CROSS SECTIONS OF VALLEY OF COFFEE CREEK, TRINITY COUNTY, CALIFORNIA Level stadia survey by V. C. LaMarche, Jr., and J. Mullen, April 1965 50 50 100 150 200 FEET 10 10 20 30 40 METERS 257-295 0 - 67 (In pocket)//SARTH SCIENOU UBRAW pc; River Channel Bars and Dunes— Theory of Kinematic Waves i GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-L IGUJnNIS DLPaR S iVitrJ! MPV 2 i; issi L»Bi?ARY U*HVS,R$liV 0-- CAU'QRJjWRiver Channel Bars and Dunes— Theory of Kinematic Waves By WALTER B. LANGBEIN and LUNA B. LEOPOLD PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-L UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1968UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 20 cents (paper cover)CONTENTS Page Abstract_________________________________________________ LI Introduction______________________________________________ 1 Flux-concentration relations______________________________ 2 Relation of particle speed to spacing—a flume experiment______________________________________________________ 4 Transport of sand in pipes and flumes_____________________ 5 Flux-concentration curve for pipes___________________ 6 Flume transport of sand______________________________ 7 Page Effect of rock spacing on rock movement in an ephemeral stream_______________________________________________ L9 Waves in bed form_______________________________________ 12 General features________________*_______________ 12 Kinematic properties_______________________________ 15 Gravel bars________________________________________ 17 Summary_________________________________________________ 19 References______________________________________________ 19 ILLUSTRATIONS Page Figure 1. Flux-concentration curve for traffic_____________________________________________________________________________ L3 2. Sketch of flume____________________________________________________________________________________________ 4 3. Graph showing relation between speed of beads and linear concentration____________________________________ 5 4, 5. Flux-concentration curves for— 4. Beads_______________________________________________________________________________________________ 5 5. Transport of sand in a 1-inch pipe_________________________________________________________________ 6 6. Graph showing relation between mean water velocity and mean particle speed________________________________ 8 7. Flux-concentration curve for transport of sand____________________________________________________________ 9 8. Location map showing Arroyo de los Frijoles near Santa Fe, N. Mex__________________________________________ 10 9, 10. Graphs showing— 9. Data on rock movement and percentage of rocks moved during two individual flows___________________ 11 10. Representative plots of the effect of spacing on the percentage of rocks moved by different discharges_____________________________________________________________________________________________ 13 11. Generalized diagram showing discharge required to move all rocks of given size as a function of spacing_ 14 12. Graphs showing field data for rock movement in Arroyo de los Frijoles and data for bead experiments in a flume___________________________________I______________________________________________________________ 15 13. Sketch of tilting flume used to observe movement of beads at different spacings___________________________ 16 14-18. Graphs showing— 14. Variation in car density along a highway__________________________________________________________ 16 15. Generation of waves in a random model_____________________________________________________________ 17 16. Length of sand waves in a flume in relation to mean water velocity, mean wave amplitude, and depth of water________________________________________________________________________________ 18 17. Suggested transport relations in tranquil flow_____________________________________________________ 18 18. Suggested transport relations in rapid flow________________________________________________________ 18 19. Suggested flux-concentration curves for river gravel________________________ 19 TABLES Page Table 1. Observations on 0.185-inch beads_________________________________________________________________________________ L5 2. Data from Blatch’s experiments on transport of 0.6-millimeter sand in a 1-inch pipe____________________________ 6 3. Data on flume transport of 0.19-millimeter sand________________________________________________________________ 8 4. Observations in miniature flume of slopes at which 25 percent of beads moved_______•__________________________ 14 inPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES By Walter B. Langbein and Luna B. Leopold ABSTRACT A kinematic wave is a grouping cf moving objects in zones along a flow path and through which the objects pass. These concentrations may be characterized by a simple relation between the speed of the moving objects and their spacing as a result of interaction between them. Vehicular traffic has long been known to have such properties. Data are introduced to show that beads carried by flowing water in a narrow flume behave in an analogous way. The flux or transport of objects in a single lane of traffic is greatest when the objects are spaced about two diameters apart; beads in a singlelane flume as well as highway traffic conform to this property. By considering the sand in a pipe or flume to a depth affected by dune movement, it is shown that flux-concentration curves similar to the previously known cases can be constructed from experimental data. From the kinematic point of view, concentration of particles in dunes and other wave bed forms results when particles in transport become more numerous or closely spaced and interact to reduce the effectiveness of the ambient water to move them. Field observations over a 5-year period are reported in which individual rocks were painted for identification and placed at various spacings on the bed of ephemeral stream in New Mexico, to study the effect of storm flows on rock movement. The data on about 14,000 rocks so observed show the effect of variable spacing which is quantitatively as well as qualitatively comparable to the spacing effect on small glass beads in a flume. Dunes and gravel bars may be considered kinematic waves caused by particle interaction, and certain of their properties can be related to the characteristics of the flux-concentration curve. INTRODUCTION Natural river channels are neither smooth nor regular in form. As seen on a map or from an airplane, their sinuous curves convey only one aspect of their changing form. The streambed has a patterned or textured irregularity which is composed not only of the grains or cobbles making up the bed surface but also of wavelike undulations of larger magnitude. Ripples and dunes are characteristic of sand beds except for high flow. As for the majority of rivers which have beds composed of gravel of heterogeneous size, the alternation of deeps and shallows—what we have called pools and riffles—is ubiquitous. These forms result from the accumulation of gravel in bars spaced along the stream at distances equal to five to seven channel widths. Each of these bed forms—ripples, dunes, and gravel bars—is composed of groups of noncoherent particles piled up in some characteristic manner. As compared with the surface texture presented by the bed grains themselves, these bed forms provide a larger scale bed rugosity and account for an important part of the total roughness. The character of the bed end the overall pattern of sinuosity of the channel as well as the hydraulics of the system are influenced by the undulatory forms assumed by bed particles. These accumulations of grains, however, are by no means composed always of the same particles, for at those stages of flow when bed grains are in motion, there is a continual trading of particles as some are swept away only to be replaced by others. The form of ripple, dune, or bar exists independently of the grains which compose them and may move upstream, downstream, or remain stationary in the channel though the particles move into and out of it in their more rapid downstream progress. In a typical pool and riffle sequence of a gravel bed stream in eastern United States we painted individual cobbles on a riffle or bar at low flow. After a high discharge the painted rocks were gone, but the form and the position of the bar were unchanged. In flows of bankfull stage or within the banks, the trading process apparently involves only those particles lying at or very near the bar sin-face. In those rivers which we have studied in detail, the surface layer of gravel on a riffle bar will be in motion and participate in the trading process when the flow reaches about three-quarters bankfull stage. In a fundamental paper, Lighthill and Whitham (1955a) direct attention to a class of wave motions which they distinguish as being different from the classical wave motions found in dynamic systems. These waves they call kinematic because their major properties may be described by the equation of continu- L1L2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS ity and a velocity-relation. These define the association between the flux or transport of the objects or particles (quantity per unit of time) and the concentration (quantity per unit of space), which association will be called the flux-concentration relation. Although Lighthill and Whitham imply a contrast between these waves and dynamic waves, it is well to recognize, as they do, that dynamic considerations are involved in the flux-concentration relation, but that the latter relation may be defined experimentally without reference to the dynamics. Automobile traffic is one of the most easily visualized examples of kinematic waves cited by Lighthill and Whitham (1955b), because it is experienced in everyday living. How exasperatingly often we encounter a slowly moving line of cars. Even on the open highway concentrations of cars occur, for it is a usual driving experience to overtake and finally progress through or pass a considerable number of cars that are close to one another, even after one has driven several miles without passing any. Such concentrations of cars are a form of kinematic waves, for through them move the objects (automobiles in this example) composing them. These concentrations of automobiles are related to a well-known driving axiom that at high speeds a driver stays farther behind the car in front than he does when traffic is moving slowly. The spacing of cars, in other words, is dependent in part on the speed at which they are moving. It is an interaction between cars governed by the drivers’ judgments of braking distance as well as other conditions along the road, such as likelihood of stopping. The queue of persons waiting at a bus stop is also analogous. The queue may remain at the boarding point even though some passengers get on the bus, for other prospective passengers take a place at the end of the waiting line. A flood wave in a river is a hydrologic example of kinematic waves. The concentration of water particles is a function of water depth, for the latter is proportional to the volume of water per unit area of channel. The flux-concentration curve for riverflow is the familiar discharge rating, as defined from current-meter measurements. In addition to flood waves, other interesting wave phenomena in hydrology have been explained by the properties of a flux-concentration curve. Waves in glacial ice (Nye, 1958) are an example. In the preceding two examples—flood waves and waves in glacial ice—flux or discharge increases with increase in concentration or depth. In these examples, as the depth increases, the velocity also increases. Their flux-concentration curves are monotonically increasing and do not show the bell shape that is charac- teristic of the flow of discrete noncoherent particles. In the flow of discrete particles in a given environment, as their number per unit of space increases, their mutual interference increases and their velocity decreases. This paper, therefore, treats of the flow of noncoherent discrete particles. Several examples will be presented. The first is a simple experiment of the single-file movement of beads in a small flume. Others will deal with the transport of sand and gravel in pipes, flumes, and rivers. It is out thesis in the present paper that ripples, dunes, and bars on the bed of a river are kinematic waves composed of bed particles that concentrate temporarily in the bed form as in a queue and are later carried downstream. FLUX-CONCENTRATION RELATIONS Some properties of kinematic waves are readily, noted by inspection of a graph in which flux is plotted in relation to linear concentration. These terms find easiest introduction in the single-lane traffic case. Flux refers to the number of cars passing a particular point on the route per unit of time. Linear concentration or linear density in a given zone is the average number of cars per unit distance of traffic lane. The reciprocal of linear density is the traffic spacing or average distance between cars. Flux is plotted in relation to linear concentration in figure 1. The attributes of the relation shown in figure 1 have been presented by Lighthill and Whitham (1955b). If the cars on a highway are very far apart—that is, when linear concentration approaches zero—it seems clear that the flux approaches zero, and so the curve goes through the origin of the graph. Experience indicates further that, as traffic density has increased to the bumper-to-bumper condition, flux again becomes zero. Thus the curve relating flux to density is of the shape shown in figure 1. To illustrate more specifically, consider the movement of cars on the single-lane highway. Many data show that the mean speed, v, and mean linear concentration, k, follow a linear relationship (Highway Research Board, 1950) that has been expressed as follows: v=Vo[l-(k/k')], (1) where v0 is the maximum car speed for the given highway and k' is the maximum linear density which equals the reciprocal of car length. This equation states that mean speed decreases linearly from v0 when a car is alone on the highway to zero when cars are bumper-to-bumper, which exists when k/k'= 1.0. Since the product of mean speed by mean linearRIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES L3 Fioobe 1.—Flax-concentration curve for traffic. concentration equals mean rate of transport, T, or cars per unit of time, T=vk=v0k[l-(k/k')]. (2) This equation, which is called a flux-concentration relationship, is graphed on figure 1. The equation defines a maximum transport rate, which occurs when the spatial concentration is half that when traffic is bumper-to-bumper, or in other words, when the cars are spaced two car lengths apart. Haight (1963, p. 82) lists several theoretical and empirical relations, including the empirical equation 2, which is his Case VI. The following theoretical equation, also included in Haight’s list as Case III, is based on a car-following theory: T=vok In (j~). Although this relation does not satisfy the condition that v=v0 when k is zero, the function indicates that transport is a maximum when linear concentration, k/k', is 0.37 or when average car spacing equals 2.7 car lengths. In either model, transport rates less than the maximum value can, theoretically, be carried at one of two traffic densities. Only one of these rates, however, is stable. Consider the effect of a perturbation at a point A on the rising limb of the curve (fig. 1). If the concentration or density should increase locally, the transport capacity would exceed the actual average transport rate, and the concentration would tend to return toward the average transport rate. Similarly, if the concentration should decrease locally, the capacity would be less than the average transport rate, and concentration would tend to increase toward the average. Thus, the rising limb of the flux-concentration curve is stable. The highway is said to be uncrowded. A similar analysis on the descending curve, say point D, will show that a perturbation, which would locally increase the concentration, would decrease the flux and so tend to ultimately increase the concentration to a bumper-to-bumper condition. Similarly, a perturbation, which would locally decrease concentration, would tend toward further decrease in concentration, and the concentration would decrease toward the rising limb. Thus, the descending limb is unstable. The highway is said to be crowded. The instability characteristic of the descending limb of a flux-concentration curve is piquantly illustrated by the patronage of a honky-tonk bar. People who seek amusement of the sort offered are hardly attracted to the cold emptiness of Monday mid-morning. Rather, they want company. So, if an inspection from the door discloses a lively crowd, they go in and try to find a table; but if they find it virtually empty, they turn away and seek elsewhere. The more people a bar has gathered, the more inviting it appears to prospective customers, and thus it tends either to be crowded or empty. The proprietor, well aware of the effect if not the mathematics of the unstable queue, tries to dampen the oscillation by such devices as “ladies tables,” “cocktail hour specials,” or myriad mirrors to make the place look more crowded, or at least inviting, even when sparsely patronized. Note that it is the interaction among individual units that causes changes in concentration and thus gives these concentrations the properties of kinematic waves. In the traffic case interaction between automobiles follows from driver reaction to the longer braking and maneuvering distance required at high speeds. In the example of patronage at a bar, it is the interaction of present and potential patrons that gives the relation its wave properties. The discussions of waves in the transport of grains by fluids will indicate how interaction among grains leads to analogous wave properties. In figure 1, a line drawn from the origin to any point on the curve has a slope expressed by: Number of cars Unit time Number of cars _ Distance Unit distance Time Speed, or the slope is equal to the average speed of cars at that place represented by the point on the curve. In traffic flow there are, of course, local concentrations of vehicles. Behind and in front of such a concentration is a zone of lower density, and the alternation along theIA PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS path may be considered to be a wave, in this case a kinematic wave. The zones of low density are troughs on either side of a wave peak. Since the spatial concentration in the trough is less than at the wave peak, transport or flux is also less than at the peak. Thus, there is a loss in volume between trough and peak, and the “upstream” part of the wave therefore decreases in volume and in height because of the net transport of cars away from the region. The opposite is true of the “downstream” part, which increases in volume by the exact amount lost by the “upstream” part. Thus, the wave form progresses at the rate AT/Ale. The slope of a chord connecting two points on the flux-concentration curve equals the celerity (that is, speed) of a kinematic wave whose peak and trough are represented by the two points, as for example, points B and A on figure 1. The derivation of this statement follows from the equation of continuity exactly parallels the derivation of the celerity of a flood wave as described by Seddon (1900). A wave whose peak and trough correspond to B and A would move more slowly than the cars within the wave but in the same direction. In the example shown the wave celerity is 20 miles per hour. Car speed in the troughs (point A) is 44 miles per hour and at the peak (point B), 32 miles per hour. A tangent at point D has a negative slope, and so the wave moves “upstream” as a shock. A tangent at point C is horizontal, and so the wave is stationary relative to the roadway. RELATION OF PARTICLE SPEED TO SPACING— A FLUME EXPERIMENT The flume diagrammed on figure 2 was built only for determining whether the speed of discrete particles moved by water can be related to the distance between them and whether a flux-concentration curve can be developed for this kind of transport. Observations were made of spherical glass beads in a single-lane trough. The flume was 2 feet long, 2 inches high, and 0.27 inch wide and had one side of clear plastic. Glass spheres 0.185 inch in diameter were transported by the water flow on a smooth, painted bed. Thus, the channel was wide enough to permit easy passage of the glass spheres but not wide enough for the spheres to pass each other. Water was fed into the upper end through a wire-screen baffle; the flow and tailgate positions were adjusted to obtain a uniform depth of about seven-eighths inch. When so adjusted, the speed of a single bead on the bed was about 0.25 foot per second. Speeds were determined by timing over a 1-foot reach centrally located. The experiment was not designed to measure the relationship between water depth, velocity, or slope; so, no observations were made of these quantities. Water velocity, depth, and rate of flow remained constant throughout, as shown in headnote of table 1. A number of beads were dropped in at the upper end of the flume at approximately equalintervals. A sufficient number of beads was used in a run to cover at least an 8-inch reach at the indicated spacing. A bead selected at random was timed over a 1-foot reach. Different beads moved with different speeds depending on the size of their group or as they left one group and moved ahead to overtake another. To obtain an average speed for a given average spacing each “run” was repeated several times. Table 1 lists the data obtained for beads of 0.185-inch diameter. Figure 3 shows a plot of linear concentration, k/k', in relation to observed mean speed of beads. The points establish a well-defined line until the linear concentration reached about 0.5 bead per diameter. As will be explained, no observations could be obtained for closer spacings. The line is extended to zero speed observed when the beads would be in contact with each other. The data define the line v=v0 [1— (k/k')\, which corresponds to the linear equation for highway traffic given previously. The value of v0, the speed of a single particle on the bed of a stream of flowing water, in relation to the hydraulic properties of the stream has been reported by Krumbein (1942) and Ippen and Verma (1953); so, no study was made of this relation. The retarding action of close spacing may be viewed as being due to the shielding action of one particle Figube 2—Sketch of flume.RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES L5 LINEAR CONCENTRATION, IN BEADS PER DIAMETER Figure 3.—Relation between speed of beads aDd linear concentration. Table 1.—Summary of observations with 0.186-inch beads [Hydraulic conditions: Discharge = 0.0057cfs; width=0.022 ft; depth=0.073ft; velocity =0.35 ft per sec] Mean linear concentration, k (beads per inch) (1) Spacing (inches) (2) Speed of beads (ft per sec) (3) Transport rate or flux (beads per sec) (4) Linear concentration, h/k' (beads per diam) (5) 2. 6 0. 39 0. 14 4. 3 0. 48 2. 5 . 4 . 11 3. 3 . 46 2. 2 . 46 . 13 3. 4 . 41 2. 0 . 50 . 167 4. 0 . 37 1. 67 . 6 . 18 3. 6 . 31 1. 67 . 60 . 185 3. 7 . 31 1. 25 . 80 . 20 3. 0 . 23 1. 25 . 80 . 20 3. 0 . 23 . 90 1. 1 . 215 2. 24 . 17 . 83 1. 2 . 22 2. 2 . 15 . 74 1. 35 . 21 1. 87 . 14 . 39 2. 6 . 25 1. 15 . 07 . 50 2. 0 . 235 1. 41 . 09 0 OO . 26 0 0 0 00 . 25 0 0 upon its neighbors downstream; that is, a particle tends to restrict the linkage of its neighbor with the water in which the particles are entrained. This shielding action was noted when two beads were placed in the flume about an inch apart. The upstream bead would move faster and overtake the downstream bead, and then both would move more slowly as a group. The speed of a bead selected at random at the beginning of the 1-foot reach depended on whether the bead became a part of a group or whether it remained separate over the 1-foot measured course. The beads did not move uniformly but formed in groups rather quickly. Single beads sometimes left the front of one group and overtook the group ahead. As the rate of feed was increased, the size of the groups increased; and the average speed decreased so that ultimately a point was reached where a group became so large that a jam formed and all motion halted. The analogy to the traffic case is evident. Figure 4 shows the results in the form of a flux-concentration graph. This graph shows transport rate SPACING OF BEADS, IN DIAMETERS 16 8 4.0 2.0 1.33 1.0 Figure 4—Flux-concentration curve for beads. in beads per second in relation to the average linear concentration, in beads per inch, as calculated from the curve on figure 3. The transport rate (col. 4, table 1) equals the product of the linear concentration, k, and the speed (col. 3) converted to inches per second. The peak of the graph on figure 4 represents the maximum rate of transport in the given flume under the given hydraulic factors. The descending branch of the curve represents only a kind of statistical average between cases when (1) concentrations of beads break up or become more open and transport takes place at the maximum rate and (2) the beads jam and transport is zero. The greater the spatial concentration, the greater the frequency of jamming; and when linear concentration is such that the beads are in continuous contact, transport is zero. Note that the transport rate increases in a nearly linear fashion with density or concentration until the concentration reaches 0.7 to 1.0 bead per inch, which corresponds to a bead spacing of five to seven diameters. At closer spacing, the transport rate increases at less than the linear rate. It can be said, then, that the effect of close spacing does not become very large until the particles are as close as about seven diameters apart. TRANSPORT OF SAND IN PIPES AND FLUMES The traffic example and bead experiments serve to illustrate the nature of the flux-concentration curve for kinematic waves composed by discrete particles. As already stated, these flux-concentration curves differ 296-757 0-68-2L6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS greatly from those of water waves. For example, the flux-concentration wave for water in a channel shows increasing velocity for increasing concentration ( = stage or depth). The same is true of the flow of glacial ice (Nye, 1958). In the traffic and bead examples, particles interact so as to reduce the mean speed with increasing concentration. As will be shown in this section, sand movement in pipes and flumes behaves similarly, even though as may be readily recognized, the movement of grains of sand is unlike that of cars on a single-lane road, or of beads in a single-lane flume. Sediment grains can readily pass one another; they start, stop, and exchange positions in rather complex fashion. The analogy is rather on bulk considerations. The interaction is considered not only between particular grains but between groups of grains. Grains grouped together move less readily than those widely or thinly spaced, and one type of group includes grains lying in a ripple or dune and thus temporarily covered by other grains. The analogy suggests that similar flux-concentration curves will exist, provided that concentration is interpreted in terms of weight per unit of space, rather than particles per unit of space. There is a further consideration, In the traffic and bead examples, environmental conditions remained unchanged ; as for example, in the bead experiment, the rate of flow and depth of water were held constant as transport was varied. However, the pipe and flume experiments were conducted for a different purpose: to define transport in relation to gradients, water velocity, and other hydraulic factors. In these experiments, therefore, constant environmental conditions could only be inferred from the data, by the methods explained. FLUX-CONCENTRATION CURVE FOR PIPES Data on sediment transport in pipes are available from several experiments by Blatch (1906), Howard (1939), and Durand (1953). Blatch, whose experiments are among the best available and who perhaps ranks as the Gilbert among those who have studied sand movement in pipes, prepared a diagram (Blatch, 1906, p. 401) showing loss of head in a 1-inch pipe for various velocities and various percentages of sand. The “percent sand” is the volume of sand transported in ratio to the total mixture discharged. Figure 5 shows the bulk transport of sand (product of percent sand X mean velocity X cross-sectional area of pipe) in relation to concentration estimated as described in the following paragraph. Data plotted in figure 5 are listed in table 2, as read from Blatch’s diagram for a constant head-loss rate of 30 feet per hundred, which Figure 5-Flux-concentration curve for transport of sand in a 1-inch pipe (data from Blatch, 1906). is sufficiently great to insure that any deposition within the pipe was local and temporary. In the single-lane flume, as on the highway, linear concentration meant the number of particles per unit length. In the transport of sand in pipes, one deals with bulk volumes of material, and so concentration will be defined in terms of weight of sand per unit length of pipe. As in other sediment examples discussed in the present paper, concentration is defined to include moving plus deposited sediment temporarily at rest. It is necessary to emphasize along with Guy and Simons (1964) that this definition of concentration is not the same as that usually ascribed to sediment in transport—mass of sediment per unit mass of water— sediment mixture. Table 2.—Data from Blotch’s experiments on transport of 0.6-millimeter sand in a 1-inch pipe for a head loss of 30 feet per 100 feet of pipe [From N. S. Blatch (1900, p. 401)] Velocity (ft per sec) Percent of sand Transport (cu ft per min) Linear concentration (lb per ft) i 16. 5 0. 06 0. 58 4 19 . 27 . 43 6. 7 15 . 36 . 27 8. 2 10 . 30 . 15 8. 95 5 . 16 . 06 9. 25 0 0 0 Note.—Pipe area 1.04 inch diam=0.006 sq ft; spatial concentration = [Tc+0.7(l-jj,)]o.# lb per ft, where o'=9.25 ft per sec and c = percent of sand/100.RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES L7 The concentration in the terms used in the present discussion was not measured, but it can be estimated. . . . 7 When all said is carried in turbulent suspension, the concentration is cAw, where c is the measured proportion of sand in the effluent per unit volume of mixture, A is the area of pipe, and w is the unit weight of the sediment material. At the other extreme, when the pipe is clogged, sediment concentration is (1—i\)Aw, the factor »j being the porosity, about 0.3 for sand particles. Within these limits the concentration is assumed to vary linearly with (1 —v/v'), where v is the reported mean velocity and v' is the mean velocity for the same rate of head loss for clear water. Other variations could have been assumed, but the results are not sensitive to this factor. When sand transport is light, v=v'; when the pipe is clogged, v—0. Hence, lineal concentration in the pipe has been estimated for medium sands by the following formula: *=[°-7(1_?)+v]Aw- At low rates of transport, v is near v', and the values of k approach the value of ^Aw. However, as v decreases relative to v', the value of k tends to exceed cAw and to approach 0.7Aw. A check is available from a curve showing distribution of medium sand across a 4-inch pipe (Howard, 1939, p. 1339). Integration indicates that the concentration, k, equals 0.28 Aw, value of the parameter, c, percentage of sands, was given as 14.4 percent and mean velocity of flow in pipe, as 8.88 feet per second. From the head-loss curves, the value of v' is about 11 feet per second. Equation 4 gives a computed Aw value of 0.25 whereas the actual value is 0.28. Data on the cross-sectional distribution of solids compiled by Durand (1953, p. 101-102) provide a corresponding check. Referring again to figure 5, we note that the data define a typical flux-concentration curve, although the mechanics are different from the traffic or bead flume cases. According to Blatch (1906), the data representing the points on the ascending branch of this curve correspond to complete suspension of the transported sand. The descending branch indicates that sand is also being dragged along the bottom of the pipe. A layer of sand is built up on the bottom, and the sand is transported much in the same manner as in an open flume. As load increases, there is spasmodically a blockage, and transport exists only in an average sense. Transport ceases entirely as spatial concentration approaches its maximum, 0.63 pound per foot. Significantly, however, maximum transport occurs when concentration is at about 45 .percent of its maximum value. Thus, the pipe-transport example and the previous bead example—two examples of the kinematic wave theory— demonstrate that the average speed of particle movement decreases with an increase in concentration. FLUME TRANSPORT OF SAND Transport of noncoherent particles expressed in terms of kinetic theory, as derived from the traffic analogy and the bead experiments, is given in equation 2. In this equation v0 is the velocity of a single particle, k is the linear concentration (particles per unit length), and k' is the reciprocal of the linear dimension of a single particle. To apply this formula to the transport of sand by flowing water in an open flume, T is defined in pounds of sand carried per second per foot of width and k, in terms of areal concentration—pounds of sand in motion per square foot. Thus, (5) where v0 is the velocity of particles as concentration approaches zero. The value of v0 is a function of the velocity of water (depth and size of particles are assumed to be constant), and W is the areal concentration—weight of sand per square foot of channel. According to the theory, the factor in parentheses of equation 5 decreases as concentration W increases, reaching zero as W=W’ where W’ is the areal concentration when transport ceases. The value of W' conceptually corresponds to a state represented by dunes whose height is so great as to block the flow in the flume. The value of W’ should therefore vary with water depth, but it is assumed to be a constant in a set of flume runs in which depths vary over relatively conservative range. As in the pipe example, concentration is the weight of moving plus deposited sediment temporarily at rest. Concentration includes sediment moving in suspension or in bed forms. In the flume moving bed forms account for nearly all transport. An estimate of W can be made from the reported height of dunes or sand waves on the flume bed (if suspended sediment is neglected); Inasmuch as the average depth of sand in motion is about half the dune height and the weight-volume ratio is about 100 pounds per cubic foot, therefore W in pounds per square foot numerically equals 50 times dune height in feet. That this estimate is correct may be judged from the experiments with radioactive tracer sands reported by Hubbell and Sayre (1964). They show that depth of zone of particle movement as determined by sampling the bed is closely the same as that computed from the bed forms. The validity of equation 5 can beL8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS tested using the data for 0.19 tested using the data for 0.19-millimeter sand as reported by Guy, Simons, and Richardson (1966) and listed in table 3. Transport is, in effect, the product of two factors— mean particle velocity and areal concentration, W. Mean particle velocity, as previously stated, is a function of the water velocity and, as in the previous examples, of the concentration as well. Thus, there are two independent factors—water velocity and concentration. Their effects can be separated by multiple regression analysis. The results of the regression analysis by a converging graphical process (Ezekiel and Fox, 1959) are shown in figures 6 and 7. Table 3.—Observed data for flume transport, 0.19-millimeter sand iSource: Guy, Simons, and Richardson (1966)] Run Velocity (ft per sec) Mean depth (feet) Mean amplitude of sand waves (feet) Mean length of sand waves (feet) Speed of sand waves (ft per sec) Transport T (lbs per sec per ft) 22a 0.78 0.48 0.028 0.57 _ 0. 25 .87 .93 .034 .44 .000015 20 1.11 1.00 .032 .62 0.0004 .00026 1 .74 .58 .023 .000014 .000026 31 1.30 1.02 .038 .58 .0024 27 .93 .55 .033 .53 . 000125 5 1.54 1.03 .045 .92 .0013 .012 23 .85 .44 .031 .45 .000049 .000046 32 1.79 .95 .096 3.97 .004 .03 8 1.99 .93 .18 5.39 .003 .06 28 1.04 .54 .038 .58 .0012 33 1.96 1.06 .65 8.0 .002 .11 29 1.13 .56 .038 .60 .0024 3 1.18 .55 .035 .70 .00053 .0034 11 2.35 1.09 .39 11.6 .0035 .21 13 3.09 .89 .13 13.4 .0033 .21 14 3.22 .86 .17 18.8 .26 15 3.46 .79 .04 20.5 .34 34 1.68 .52 .044 1.67 .0043 .027 12 2.69 1.02 .32 17.7 .0033 .22 6 1.67 .61 .18 5.06 .0017 .055 7 1.78 .68 .31 7.4 .004 .094 10 2.89 .51 .10 24.0 .013 .23 9 2.10 .49 .20 5.19 .0025 .077 17 4.14 .67 .10 4.4 .80 18 4.33 .64 .10 4.9 . 1.6 35 1.81 .52 .13 4.5 .004 .059 Figure 6 shows the variation in v0, the imputed velocity of particles as concentration approaches zero, with mean water velocity. Values of v0 equal the quantity T W — (1-—,). It should be kept in mind that the curve of W W figure 6 expresses a dynamic relation that applies only to the particular set of experiments with unigranular sands. Note that values of v0 are small relative to mean water velocity but somewhat greater than wave speeds listed in table 3. Figure 6 applies where transport is in the bed forms and where suspended sediment represents saltation of grains from one position of rest to another. The data listed in table 3 include all those runs made by D. B. Simons and E. V. Richardson for which bed-form data were reported. This information was not given for runs made at steep slopes. Descriptions of these runs indicate that the bed was composed of “chutes and pools” and that sediment transport was in dispersed or suspended form. In these steep-slope runs, spatial concentration was very low, and v0—v. Figure 7 shows the values of T/v0 corresponding to various values of areal concentration, W. The curve corresponds to an imputed flux-concentration curve if water velocity in each run were the same (»0=constant), but the runs differ in flux (transport) and in concentration (dune heights). Figure 7 is defined by multiple regression in that the values of v0 are read off the graph of figure 6. Thus, points in figure 7 correspond to different runs, each with different water velocities and therefore different conditions. Specifically, ordinate values of figure 7 represent the ratio of transport as measured in a given run to the value of MEAN PARTICLE SPEED (vo), IN FEET PER SECOND Figube 6.—Relation between mean water velocity and mean particle speed, 0.19-mUlimeter sand.RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES L9 AREAL CONCENTRATION (U/), IN POUNDS PER SQUARE FOOT Figure 7.—Flux-concentration curve for transport of sand. va read from figure 6 for the mean water velocity measured in that run. In this way the observed transport data are adjusted for different water velocities as mentioned above. The ordinate of figure 7 has the dimension of concentration as does the abscissa. The slope of the curve is 1 to 1 relative to the scales used at the origin. Thus, at the origin (concentration is low), mean particle speed equals v0. As concentration increases, mean particle speeds decrease and thus the curve deviates from a 1 to 1 slope and flattens; in other words, the rate of increase in transport lessens as concentration increases. There is a suggested maximum for a value of W of about 25 pounds per square foot, and W' may be estimated at 50 pounds per square foot. The data shown in figure 7 follow the curve Hence transport in this set of flume experiments may be described by T=vqW{1-^) (6) where v0 is defined by figure 6. The inference is that transport would be zero when W= 0 and when W=50. This equation indicates that transport for a given depth and size of material varies with the water velocity in accord with previous findings by Colby (1964), and with areal concentration as well. Mean particle speed, T/W, decreases linearly with IT to a value of zero when W=W'. However, in a sand-channel flume, unlike the bead flume, water velocities differ at troughs and crests of dunes; therefore different flux-concentration curves apply at such points. Therefore the imputed flux-concentration in figure 7 should not be used to define wave celerities. This matter is developed further in the section on “Waves in bed form.” THE EFFECT OF ROCK SPACING ON ROCK MOVEMENT IN AN EPHEMERAL STREAM 1 Even before the experiments were set up to test the effect of spacing of beads in a small flume, we had conjectured from consideration of the highway traffic analysis that the interaction between individual rocks must be an important factor in the formation of gravel bars on a streambed. Observations of rock movement on a streambed, which were made before some of the other research discussed in this paper had been conducted, suggested the mode of investigation to be followed. The design of the field experiment perhaps, in retrospect, could have been made somewhat more efficient for present purposes. However, despite their limitations, the results are in keeping with the observations of the bead and flume experiments. In streambeds that are covered by gravel of heterogeneous size, alternation of pools and riffles is an obvious characteristic. But the undulation of bed elevation is small or not apparent in those ephemeral channels that so commonly rise in the foothills in the semiarid countries and that either debouch in alluvial basins or reach master streams of a perennial character rising in nearby mountains. These sandy washes range in size from a few feet to several hundred feet in width and characteristically exhibit a bed profile of nearly uniform slope through reaches of moderate length. They have no deep hollows or deposited bars typical of gravelly streams of perennial flow. Walking along a typical sandy arroyo, one has the impression of walking on a nearly level surface free of bed undulations either along the length or across a main channel. Such washes are ubiquitous in New Mexico in a broad area that is drained westward from foothills along the base of the Sangre de Cristo Range. One of these washes—the Arroyo de los Frijoles, a tributary of the Santa Fe River and thereby of the Rio Grande—was chosen for an intensive study. Its location, about 10 miles northwest of Santa Fe, N. Mex., is shown in figure 8. It is in a semarid area having about 12 inches of precipitation annually. Other investigations in the 1 The fieldwork described in this section was initiated by L. B. Leopold with the help of the late John P. Miller and was pursued with the assistance of Robert M. Myrick, William W. Emmett, and personnel of the U.S. Geological Survey at Santa Fe, N. Mex.; all are sincerely thanked for this effort. The interest and the work of Wilbur L. Heckler, Leon A. Wiard, Leo G. Stearns, Louis J. Reiland, Kyle D. Medina, and Charlie R. Siebei are particularly acknowledged for they not only helped with the painted rock experiments but made the field surveys aod office computations of the flow discharges.Other aspects of the field experiment are discussed by Leopold < Leopold and others, 1966).L10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS same area are reported by Leopold and Miller (1956). The arroyo rises in low and rounded foothills that are underlain by Pleistocene fan gravels and the Santa Fe Formation, a poorly consolidated mixture of rounded gravel and sand. Despite the large amounts of gravel which tend to pave some of the hillslopes nearby, the ephemeral washes that drain the area are composed of sand and only minor amounts of gravel. Detailed planetable mapping shows gravel accumulations occurring in a regular pattern along the channel length at a spacing comparable to that observed in the pool and riffle sequence of perennial streams. Furthermore, remapping at yearly intervals showed the location of these gravelly accumulations remained more or less constant from year to year. A large number of holes were dug in the streambed, both on the gravel accumulations which we will here call bars and on the' intergravel sandy areas. Nearly without exception it was found that the gravel occurred only as a thin skin, 2 to 4 cobble diameters in thickness, lying on top of the sand. Flows which occur in these washes average in number about three per year and are caused by summer thunderstorm rainfall only. Snowmelt in spring produces no surface runoff. During a flow the channel typically scours to a depth averaging at least one-quarter of the maximum depth of the flowing water. Individual rocks were painted and placed on the streambed in groups with varied spacing between the individual cobbles. The experiment was designed to determine the effect of spacing as well as of discharge and rock size on the ability of flowing water to move the rocks downstream. The rocks selected to be painted were obtained from the streambed itself. They were individually weighed and placed in groups of 24 rocks each. On each rock was painted a number that represented its weight in grams. Because the numbers generally consisted of four digits, the number also identified each rock. Six categories of rock size were chosen, ranging from 300 to 13,000 grams, and each individual group of 24 rocks contained four of each of the six weight categories. The 24 rocks were arranged in a rectangle—six rocks in the direction of the channel and four rocks in the cross-channel direction. The rocks within each group were arranged in a latin square so that in a given row two rocks of the same size did not occur and so that is each column of six rocks all the size classes were represented. Each group was arranged with one of three spacings; that is, in a given group the rocks were placed a distance apart equal to 0.5 foot, 1 foot, or 2 feet. Typical rock groups placed on the streambed and rock movement resulting from a small flow are shown in figure 152 of Professional Paper 352-G. l~T", INDEX MAP 36” 0 10 Figure 8— Arroyo de los Frijolesnear Santa Fe, N. Mex.RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES Lll The groups were placed in three reaches of the channel—the most upstream reach was 9 feet wide, the intermediate reach was 35 feet, and the downstream reach was 100 feet. The number of rocks in place before a given flow averaged about 900 during the 5 years of experiment. After each flow the streambed was searched the whole distance affected by that flow. The location of each rock found was recorded and related to its position before the flow occurred. After a small flow, 70* 90 percent of the rocks was usually recovered; and after a large flow, 50 to 80 percent recovery was usual. In one big flood, however, 98 percent was not found and was presumably carried out of the 7-mile study reach. f In the period from 1958 to 1962, a total of 14 flows occurred; but because of the scattered nature of thunderstorm rainfall, each flow did not necessarily affect all rock groups nor each of the three principal locations where the rock groups were placed. Discharges were measured by indirect or slope-area methods, the cross-sectional area being determined by measurement of scour depth registered by buried chains. Because of the difference in channel width at the various locations, discharge was tabulated for purposes of analysis as cubic feet per second per foot of channel width. In the experiment, therefore, there were four variables—discharge, rock size, rock spacing, and percentage of rocks moved by a given flow. Owing to the fact that even in a single cross section a given flow may not have a uniform effect on each rock group, the data are understandably scattered, and the problem in analysis was to choose a method of plotting which smoothed the scatter in some consistent and reasonable way. Smoothing of the data is justified because the total number of observations of painted rocks affected by a flow during the 5-year period was about 14,000. The analysis began by treating each flow event separately and for a given flow plotting the percentage of rocks of a given size which moved from their original positions as a function of rock size and spacing. Two typical plots of this relationship are shown in figure 9 which shows that in a given flow a larger percentage of small rocks was moved than of large and Figure 9.—Original data on rock movement plotted as percentage of rocks moved during two Individual flows represented in the two separate graphs. Each graph represents data from 648 painted rocks affected by the flow.L12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS that rocks in groups having wider spacing (smaller value of linear concentration) moved more readily than rocks which were close together. By considering all flows and only those rocks of a given size, we constructed plots from values read off the smooth curves of figure 9; the percentage of rocks moved was expressed as a function of discharge with rock spacing as a third variable. Examples for two rock sizes are presented in figure 10. The lines in figure 10 were drawn as envelopes so that, other than groups where 100 percent of the rocks moved, at least 80 percent of the individual points lies above the respective line. For example, for rocks averaging 1,000 grams, a line representing a spacing of one-half foot envelops or lies below all points except two. The abscissa value, where a curve intersects the ordinate value of 100 percent of rocks moved, is an estimate of the discharge required to move all rocks of a given weight and at a given spacing. Though the positions of these lines must perforce be subjective, a pattern that emerged in figure 11 was obtained from the intercepts on the discharge scale for 100 percent of rocks moving. The abscissa on the figure has two scales—the 6-axis diameter and weight of particles. The ordinate also has two scales—a scale of discharge in cubic feet per second per foot of width required to move 100 percent of the rocks and a scale of shear corresponding to the values of discharge using field measurements of stream slope. The family of lines represents spacing converted from feet to particle diameter. The graph shows that 100 percent of the rocks of a particular size will be moved only by increasingly larger discharges as the spacing between the rocks decreases. In agreement with the data on glass beads, particle interaction is negligible for spacing greater than about eight diameters. Another plot can be constructed from straight-line graphs exemplified by those in figure 11. For different values of particle size the discharge required for movement of 100 percent of the rocks was plotted in relation to rock concentration (rocks per diameter), as shown in figure 12A. If these lines are extrapolated to the point where each intersects the ordinate axis, the values then represent the theoretical discharge necessary to move all the rocks of a given size at zero concentration— that is, for a single particle. The values of these discharges for zero concentration were used as the denominator in the ratio Discharge for the individual event Discharge for zero concentration (single particle) which will be referred to as the discharge ratio. This discharge ratio could then be plotted against spacing in diameters to result in the nondimensional diagram shown in figure \2B. Comparable date were obtained in a miniature tilting flume shown on figure 13 by using glass beads of 0.12-and 0.18-inch diameter. (See table 4.) Beads were placed at various spacings, and the flume was filled with flowing water sufficient to cover them by a uniform depth of at least three diameters. With discharge constant, the flume was then slowly tilted until the beads began to move. Readings were made of the slope at which 25 percent of the beads moved at various linear concentrations for a fixed discharge. A 25-percent value was used because the single-track flume could not be tilted sufficiently to obtain 100-percent movement when beads were closely spaced. Values of slope representing zero concentration were estimated from the data and used to compute values of a slope ratio, defined as Slope of flume in individual experiment Slope for zero concentration (single particle)' The data are plotted in relation to the square root of this ratio in figure 12B. The rationale for the square root is that discharge varies at constant depth as the square root of slope, as indicated in the Chezy equation. The ratio of the square roots of the slope is thus comparable to the discharge ratio in the arroyo. The comparison between movement of rocks in the arroyos and glass beads in the miniature flume is presented in figure 12 B. The rough agreement between the comparable plots for these two very different experiments may be partly a matter of coincidence. But, two types of quantitative data—rock movement in a sandy ephemeral wash and experimental findings concerning beadr» moving in a small flume—both show that, as the par tides are spaced more closely, the slippage is greater or the linkage is weaker between the flowing water and the grains. A larger shear must be administered to move individual particles closely spaced. The relation expresses a nearly linear tendency for particles to be moved as their spacing increases. WAVES IN BED FORM GENERAL FEATURES Each of the three kinds of waves—dunes, antidunes, and gravel bars—that has been observed in flumes and rivers complies with the kinematic properties of the flux-concentration curves. Dunes can result from random variations in spatial concentrations which give rise to wave forms such as traveling queues or platoons on the highway. These dunes can be explainedPERCENTAGE OF PARTICLES MOVING RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES L13 Figure 10.—Representative plots of the effect of spacing on the percentage of rocks moved by different discharges. The respective graphs apply to rocks of two sire classes. Data on these graphs read from smooth curves of which figure 9 shows two examples.L14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS PARTICLE SIZE (b AXIS), IN FEET 0.23 0.30 0.39 0.51 0.64 Figure 11.—GeDeralized diagram showing discharge required to move all rocks of given size as a function of spacing. Derived from values read from graphs typified by figure 10 where lines intercept ordinate value of 100 percent. Table 4.—Observations in miniature flume of slopes at which %6 percent of beads moved [Discharge constant for each run, but not measured] Linear concentration (beads per diameter) Slope Slope ratio Run 1 (0.18-inch beads) 0. 50 0. 053 4. 6 . 32 . 044 3. 8 . 20 . 023 2. 0 . 13 . 02 1. 75 . 034 . 0125 1. 1 0 . 0115 1. 0 Run 2 (0.12-inch beads) 0. 475 0. 047 4. 9 . 26 . 025 2. 6 . 053 . Oil 1. 15 0 . 0096 1. 0 Run 3 (0.18-inch beads) 0. 90 0. 10 12. 0 . 40 . 028 3. 3 . 03 . 0093 1. 1 0 . 0085 1. 0 Run 4 (0.12-inch beads) 0. 90 0. 087 12. 0 . 53 . 043 5. 8 . 014 . 0078 1. 05 0 . 0075 1. 0 on the basis of kinematic relationships. As shown by field and laboratory evidence, the probability of a particle being moved increases as the spatial concentration decreases. Therefore, areas of low concentration (thin bed forms) tend to become thinner as areas of high concentration increase. Waves are formed that take on characteristic lengths and heights. One of the well-known properties of highway traffic is the Poisson distribution of vehicular spacing, but even here waves—or platoons, as they are called—are formed. Figure 14, for example, shows “waves” in traffic along a 50-mile stretch of highway in Arizona. The example was obtained from a part of strip aerial photographs made along the highway from Tucson to Eloy. The vehicles measured were in a single lane in a divided four-lane highway. Such waves in highway traffic are common enough, but crossings, trucks, and towns may cause them to be very irregular. Fairly regular waves can be generated by a random process, as illustrated by the following model. Consider initially a distribution of particles along a line with average concentration of k particles per unit length. These particles are subject to the condition that, where particles happen to be far apart, they are more easily moved than where particles are close. Thus, the movements will collect the particles into groups having wavelike form. The results of a trial are shown in figure 15. Particles that have an average concentration k areRIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES LI 5 LINEAR CONCENTRATION, IN PARTICLES PER DIAMETER A LINEAR CONCENTRATION, IN PARTICLES PER DIAMETER B Figure 12.—A, Field data for rock movement in Arroyo de los Frijoles. B, Comparison of movement of rocks in arroyo with beads in flume. arranged in random position in the first column (marked “Time interval 0”). Only one particle is permitted in a single position or box. Each column represents a successive unit of time, and the direction of motion is down the sheet. The diagram shown is merely the central part of a longer sheet on which a very long sequence of wave groups was developed. The whole suite of particles may be moving with some speed which does not concern this analysis of relative speeds and positions. The following rule governs this relative movement. A particle cannot move if there is one in the space immediately ahead. A particle at the front of a group is free to move; and if it does so, it will move and overtake a particle in the group ahead. A particle free to move in a particluar time interval is selected to do so at random, its probability of selection being p. The process here of a particle taking off at random and overtaking the one ahead is not unlike that observed in the single-lane flume. (See p. L5.) Figure 15 shows an experimental example where &=0.33 and p=0.50. After 10 time intervals, a well-defined pattern of groups is established owing to the interaction between particles. The lengths of such wave patterns would be proportional to the mean concentration and speed. Figure 16 shows, for example, the relation of length of sand waves to water velocity and to the ratio of amplitude of waves to depth of water over dunes as observed in the set of experiments on 0.19-millimeter sand in a flume at Fort Collins, Colo., listed in table 3 (Guy and others, 1966). Because these relations are of a random nature rather than of dynamic origin like the unique relation between velocity and length of antidunes also shown in figure 16, precise relationships cannot be found. KINEMATIC PROPERTIES The flux-concentration curve, shown in figure 7, for transport of sand in a flume describes the relation between transport and average concentration along the flume for different runs. Consider a single run. Transport and concentration differ along the flume. Unfortunately, flume experiments only report averages for each run, and so the nature of the flux-concentration curves along the flume can only be inferred as on figures 17 and 18. Flux-concentration curves, as suggested in figures 17 and 18, can be useful in explaining the development of modes of transport, dunes, plane bed, or antidunes as usually observed in flume experiments. A perturbation in the bed creates a difference in depth of sediment, in velocity, and in rate of transport at that point. When velocities are subcritical (that is, less than VgD), the velocity over the top of a protuberance in the bed is greater than in a trough. One may infer a different flux-concentration curve for the crest and for the trough. However, the actual relationship is probably a loop as shown. The relations for dunes are indicated by points 1 and 2 in figure 17. At point 1, representing the crest, velocities are higher than at point 2, representing the trough, and transport is greater over the crest than in the trough. Thus, the action tends to decrease theL16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 0.8" —(<- M VZZZZZZA Section A-A' Figure 13.—Tilting flume used to observe movement of beads at different spacings. DISTANCE, IN MILES Figure 14.—Variation in car density along a highway. crest of the protuberance and to deposit particles in the trough. However, because of the forward motion, the cutting and filling action is continuous and leads to the formation of dunes in forward motion as previously explained. The rate of forward motion of the dunes equals the slope AT/Ak of the chord 1-2 in figure 17, which is positive—that is, downstreamward— in this phase. When transport is low, chord 1-2 is short; when transport increases, the chord moves up the graph and increases in steepness. Dune speed increases, but the dune height (the horizontal projection of chord 1-2) decreases. Ultimately, dune height decreases to zero, and the plane-bed phase is reached. As velocity increases and approaches -JgD, standing hydraulic waves are formed. However, it should be noted that, although the lengths of antidunes or standing waves are controlled dynamically (Kennedy, 1963), their speed can be inferred from the kinetic properties of the flux-concentration curve. These waves induce perturbations in the bed, as shown in figure 18. When velocities are supercritical (that is, greater than VyD), the depth of water over the tops of the bed forms is greater than the depth of water over the troughs; therefore, the water velocity over the tops is less than in the troughs. Hence, the slope of the chord connecting points 1 (crest) and 2 (trough) will be either flat (standing waves with zero speed) or negative (antidunes). The corresponding points are shown, in figure 18. Transport is greater in the trough than at the peaks, and the sand waves grow in height, eventually become unstable and break, and throw large parts of the load into turbulent suspension. The slope of chord 1-2 is negative; hence, wave movement is upstream, and the waves are therefore called antidunes. As explained on page L4, the rate of movement of dunes equals the slope of a chord connecting points on the flux-concentration curves for troughs and the peaks. Dune speed, c, therefore equals the ratio r Tt-T, C~W1-W2’ where the subscripts 1 and 2 refer to transport, T, and areal concentration, W, at the crest and trough, respectively. Since Wi—W2=w(hi—h2) and hi—h2=h, mean amplitude of dunes, then chvy=Ti— T2 where w is the unit weight of the sand. For those runs in which wave speed, c, and mean height of dunes were observed, the left-hand side of the equation may be calculated, but data on Tt and T2 are not available. It would be possible to report these quantities by observing changing transport rates as dunes move out of the flume.0 5 10 15 20 LU O z < 25 co o 30 35 40 45 50 RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES LI 7 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XXX x X X X X X X X X X X X X X X X X X X X X X X X X X X XXX > o o UJ ec. o XX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X "1 1 I 1 I 1 I 1 I 1 I 0123456789 10 TIME INTERVAL 15.—Generation of waves in a random model. Letting Tx=b T, where T is the mean rate of transport which is also assumed here to be the arithmetic mean of Tx and Ts, then 6=1—(c/vp), where vP, the mean particle speed, equals T/W. The factor, 6, can be calculated in the set of experiments cited in table 3 for those runs in which dune speed, c, was observed. It ranges from a negative quantity for those runs with low velocity and increases to over 0.7 for those runs with high velocity. The significance of this range is that for low-velocity runs the interdune transport (net transport from one dune to the dune ahead) is zero or even upstream, but it increases as velocity increases. The interdune transport, for any given velocity also increases as dune height decreases. These results are in accord with the observation that as velocities increase the difference between crest and transport decreases and residence time in the dunes decreases. GRAVEL BARS Because of the limited supply of gravel, the spatial concentration of gravel is almost entirely in “wave” form. The “crests” of these waves are riffles or channel bars. The flux-concentration curve for the bar may be as suggested in figure 19. The flux is derived entirely by erosion of the bar deposits, where, in the kinetic terminology, spatial concentration decreases as river flow increases—points 1, 2, and 3 in figure 19. The flux-concentration curve for low riverflow, zero gravel transport, coincides with the horizontal axis. On the other hand, the flux-concentration curve for the troughs (where concentration is near zero) virtually coincides with the vertical axis. The flux is indicated to be the same in the troughs and the bars, and so the slopes of the chords 1—1, 2—2, and 3-3 are horizontal; these slopes indicate that wave velocity is zero. That is, the bars are fixed relative to the channel. Mean particle velocity in the troughs is high; it is low in the bar. (Again, when we speak of mean particle velocity, it is the mean speed of all particles having concentration, k, even though not all are in motion at a particular time.) A small-flume experiment confirmed earlier observations by Gilbert (1914, p. 243). A supply of sand formed dunelike deposits separated by bare reaches approximately equal in length. In the flume, particles moved rapidly from the toe of one dune to the heel of the oneWATER VELOCITY, IN FEET PER SECOND L18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 16.—Length of sand waves in a flume in relation to water velocity and to ratio of mean amplitude of dunes (0.19-mm sand) to mean depth of water over dunes. Flux-concentration curves Figure 17.—Suggested transport relations in tranquil flow. k Flux-concentration curves / 2 Figure 18.—Suggested transport relations in rapid flow.RIVER CHANNEL BARS AND DUNES—THEORY OF KINEMATIC WAVES L19 Figure 19.—Suggested flux-concentration curves for river gravel. downstream. In this steady state of water and sediment input, the dunes moved slowly downstream. However, when the flow of water was increased, the dunes were slowly eroded away, but they remained fixed in position and did not then migrate down streamward. As observed in rivers, gravel bars are spaced more or less irregularly at distances that average five to seven channel widths. The spacing of bars is seemingly controlled by the channel dimensions and remains fixed even though the velocity changes. To the extent that the theory relating to dunes applies, an increase in the velocity would tend to increase the spacing. However, the accompanying increase in depth of water and a decrease in height of the bar by scour has the opposite effect; therefore, the two conditions may combine so as to maintain bars at fixed distances apart. SUMMARY The theory and experimental evidence presented show that interaction among the individual units moving in a continuous flow pattern causes velocity of the units to vary with their distance apart. Concentrations of these units have certain wave properties encompassed under the term “kinematic waves.” With these properties it is unlikely that such particles can remain uniformly distributed for any distance along the flow path, for random perturbations lead to the formation of groups or waves which take the form of the commonly observed dunes in sandy channels and the pool and riffle sequence in gravel river beds. The rationale may be summarized as follows. Particles of sand or gravel which can be moved by the flowing water are seldom all in motion simultaneously unless their number is very small. There is some probability of one being set in motion instead of its neighbor. Once set in motion, the speed of a given particle is affected by its proximity to neighboring moving grains; the mean grain speed is slower when the grain density is higher. The result of this interaction is to intensify any initial random grouping whereas open spaces or places where grains are sparse tend to become even more open—have even smaller density of grains. Thus, any random influx of grains into a reach of channel will not continue to be random, for as the grains travel downstream their interaction will set up waves—that is, groups of grains separated by relatively open spaces will tend to attain a more or less regular spacing along the direction of flow. By such a process sand collects in dunes which are small compared with channel width. The same principle will apply in a transverse section as in the longitudinal profile just described. Transverse to the current the dunes will also alternate with open spaces or troughs. Thus, there will be built the pattern of dunes so characteristic of sandy channels, in which the whole surface is covered with dunes separated by troughs; the dunes will be arranged in a rather uniform and random pattern, with some dunes arranged in echelon but others appearing to be uniformly irregular. In the situations where the particles composing the kinematic wave are only one to several diameters thick, as in beads in a single lane flume or a veneer of cobbles on the sandy bed of an arroyo, the linear concentration is easily visualized because nearly all the particles potentially in motion are on or near the surface and visible. Concentration can then be measured by the spacing of particles on the surface. In the case of dunes and riffle bars, the concentration is not so apparent at a glance because it is measured by the weight or number of particles on a unit area to a depth of the sand or gravel above the plane representing the base of the moving dune—that is, the depth down to which particles are participating in the motion as the dune or bar is eroded or moves. With regard to pool and riffle sequences in gravel-bed streams, the kinematics suggest that spacing of the riffles is related in part to the thin veneer of gravel set in motion by the flow. The bar represents an interaction between two opposing factors: increasing water velocity, which tends to increase wavelength, and decreasing amplitude, due to erosion, which tends to decrease wavelength. Such a balance results in riffle bars which do not appreciably move downstream. REFERENCES Blatch, N. S., 1906, Works for the purification of the water supply of Washington, D.C.: Am. Soc. Civil Engineers Trans., v. 57, discussion, p. 400-408. Colby, B. R., 1964, Discharge of sands and mean-velocity relationships in sand-bed streams: U.S. Geol. Survey Prof. Paper 462-A, 47 p. Durand, R., 1953, Basic relationships of the transportation of solids in pipes: Internat. Hydraulics Convention, St. Anthony Falls Hydraulic Lab., Minneapolis, Minn., Proc., p. 89-103.L20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Ezekiel, M. J., and Fox, K. A., 1959, Methods of correlation and regression analysis, linear and curvillinear [3d ed.]: New York, John Wiley & Sons, Inc., 548 p. Gilbert, G. K., 1914, The transportation of debris by running water: U.S. Geol. Survey Prof. Paper 86, 263 p. Guy, H. P., and Simons, D. B., 1964, Dissimilarity between spatial and velocity-weighted sediment concentrations: U.S. Geol. Survey Prof. Paper 475-D, p. 134-137. Guy, H. P., Simons, D. B., and Richardson, E. V., 1966, Summary of alluvial channel data from flume experiments, 1956-1961: U.S. Geol. Survey Prof. Paper 462-1, 96 p. Haight, F. A., 1963, Mathematical theories of traffic flow: New York, Academic Press, 242 p. Highway Research Board, 1950, Highway capacity manual: U.S. Bur. of Public Roads, 147 p. Howard, G. W., 1939, Transportation of sand and gravel in a 4-inch pipe: Am. Soc. Civil Engineers Trans., v. 104, p. 1334-1380. Hubbell, D. W., and Sayre, W. W., 1964, Sand Transport studies with radioactive tracers: Jour. Hydraulics Div., Am. Soc. Civil Engineers Proc., v. 90, Hy 3, p. 39-68. Ippen, A. T., and Verma, R. P., 1953, The motion of discrete particles along the bed of a turbulent stream: Internat. Hydraulics Convention, St. Anthony Falls Hydraulic Lab., Minneapolis, Minn., Proc., p. 7-20. Kennedy, J. F., 1963, Mechanics of dunes and antidunes in erodible bed channels: Jour. Fluid Mechanics, p. 521-544 Krumbein, W. C., 1942, Settling-velocity and flume-behavior of non-spherical particles: Am. Geophys. Union Trans., pt. 2, Nov., p. 621-632. Leopold, L. B., Emmett, W. W., and Myrick, R. M. 1966, Channel and hillslope processes in a semiarid area, New Mexico: U.S. Geol. Survey Prof., Paper 352-G, p. 193-253. Leopold, L. B., and Miller, J. P., 1956, Ephemeral streams— hydraulic factors and their relation to the drainage net: U.S. Geol. Survey Prof. Paper 282-A, 37 p. Lighthill, J. J., and Whitham, G, B., 1955a, On kinematic waves I. Flood movement in long rivers: Royal Soc. [London] Proc., v. 229A, p. 281-316. ------- 1955b, On kinematic waves II. A theory of traffic flow on long crowded roads: Royal Soc. [London] Proc., v. 229A, p. 317-345. Nye, J. F., 1958, Surges in glaciers: Nature, v. 181 May, p. 1450-1451 Seddon, J., 1900, River hydraulics: Am Soc. Civil Engineers Trans., v. 43, p. 217-229 U. S. GOVERNMENT PRINTING OFFICE : 1968 O - 296-757P- lb , <- / 5 ? 'A| Flood Surge on the Rubicon River, California Hydrology, Hydraulics and Boulder Transport GEOLOGICAL SURVEY PROFESSIONAL Prepared in cooperation with the California Department of Water Resources PAPER 422-MFlood Surge on the Rubicon River, California— Hydrology, Hydraulics and Boulder Transport By KEVIN M. SCOTT and GEORGE C. GRAVLEE, Jr. PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-M Prepared in cooperation with the California Department of IVater Resources UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1968UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY William T. Pecora, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 45 cents (paper cover)CONTENTS Page Abstract_______________________________________________ Ml Introduction__________________________________________ 1 Acknowledgments_________________________________________ 2 Physical setting_____________________________________ 2 Location and physical features______________________ 2 Climate and vegetation______________________________ 3 Bedrock geology_____________________________________ 4 Geomorphic history and stratigraphy of the Rubicon canyon_______________________________________. 5 Embankment failure and release and downstream passage of surge__________________________________ 6 Flood of December 21-23, 1964, in the Rubicon River basin_________________________________________________ 8 The storm_________________________________________ 8 The flood________________________________________ 8 Inflow and outflow at Hell Hole Dam----------- 10 Peak discharge of the surge downstream from Hell Hole Dam------------------------------ 12 Attenuation of surge wave--------------------- 12 Erosional effects of the surge_________________________ 12 Effects on channel morphology-------------------------- 13 Mass movements caused by the surge--------------------- 14 Depositional features__________________________________ 17 Methods of study___________________________________ 17 Depositional features—Continued Page Lateral berms or terraces__________________________ M18 Boulder bars________________________________________ 19 Terrace accretion___________________________________ 19 Boulder fronts______________________________________ 20 Episodic boulder movement___________________________ 23 Pool and riffle pattern_____________________________ 25 Bed-material forms and internal sedimentary structures______________________________________________ 26 Transport and deposition of bed material in reach downstream from damsite_____________________________________ 27 Downstream changes in patterns of bars and berms. 27 Composition_________________________________________ 27 Particle size_______________________________________ 27 Distribution of particle size_______________________ 29 Competence__________________________________________ 31 Roundness___________________________________________ 32 Cause of downstream decline in particle size________ 33 Macroturbulence_____________________________________ 34 Effects of the surge relative to normal stream processes______________________________________________ 35 Summary and conclusions_________________________________ 36 References cited________________________________________ 37 Index___________________________________________________ 39 ILLUSTRATIONS Page Figure 1. Index map showing drainage areas in study area_______________________________________________________________ M3 2. Longitudinal profile of the flood route_______________________________________________________________ 4 3. Map showing location of physical features in the reaches below the Hell Hole damsite pertaining to the study _ 5 4. Composite stratigraphic column of the deposits in the upper Rubicon River canyon---------------------- 6 5. Cross section of the completed Hell Hole Dam and the stage of construction at the time of failure_____ 7 6. Photograph showing breaching of the rockfill embankment at the Hell Hole damsite near time of peak dis- charge__,___________________________________________________________________________________________ 7 7. Graphs showing hourly precipitation at stations nearest the Hell Hole damsite for the period Decem- ber 18-26, 1964..................................................................................... 9 8. Graphs showing flow and storage data associated with the failure of Hell Hole Dam in December 1964___ 11 9. Photograph showing lateral supply of sediment to the flood channel from a deposit of terrace gravel and till, 3.3 miles downstream from the damsite_______________________________________________________________ 13 mIV CONTENTS Page Figure 10. Photograph of trees showing maximum abrasion level 3-5 feat above present ground surface---------------- M14 11. Prefailure and postfailure cross profiles of the channel at the site of the first gaging station reached by the surge............................................................................................ 15 12-16. Photographs showing— 12. Base of the largest slide in the Rubicon gorge________________________________________________ 16 13. Boulder berms preserved 0.7 mile downstream from the damsite__________________________________ 17 14. Section of berm parallel to stream channel in reach below Hell Hole damsite___________________ 18 15. Lobate bar formed downstream from a bedrock projection into channel___________________________ 20 16. Large gravel bar formed 1.2 miles below Hell Hole damsite_____________________________________ 21 17. Map showing radial pattern of flow and deposition at the downstream end of Parsley Bar________________ 22 18. Photograph showing view across flood channel of boulder front on Parsley Bar__________________________ 23 19. Map of the part of Parsley Bar containing the largest boulder front________________________________ 24 20. Map of reach showing sequence of bedload movement__________________________________________________ 25 21. Photograph showing contribution of sediment to the flood channel by a landslide, 2.5 miles upstream from the junction of the Rubicon River with the Middle Fork American River____________________________ 26 22. Graph showing relation of bed-material composition, in terms of the percentage of diorite, to the channel distance downstream from the damsite_____________________________________________________________ 27 23. Photograph showing bed material of Parsley Bar, extending from 1.6 to 2.9 miles downstream from the Hell Hole damsite________________________________________________________________________________ 28 24-33. Graphs showing relation of— 24. Mean particle size to distance downstream from damsite---------------------------------------- 28 25. Mean particle size of samples in pools and riffles to distance downstream from damsite________ 28 26. Diameter of larger cored boulders to distance downstream from damsite_________________________ 29 27. Dispersion in particle size to distance downstream from damsite_______________________________ 30 28. Skewness to distance downstream from damsite__________________________________________________ 30 29. Indirectly measured tractive force to distance downstream from damsite________________________ 31 30. Maximum particle size to indirectly measured tractive force___________________________________ 32 31. Particle roundness to distance downstream from damsite---------------------------------------- 33 32. Roundness of cored boulders to distance downstream from damsite______________________________ 33 33. Mean particle size to indirectly measured tractive force-------------------------------------- 34 TABLES Page Table 1. Summary of flood stages and discharges_____________________________________________________________________ M10 2. Description of channel cross profiles_______________________________________________-___________________ 15 3. Summary of sediment data in channel below damsite_______________________________________________________ 29 4. Summary of maximum particle-size data___________________________________________________________________ 32 5. Size and height above present thalweg and distance from point of origin of boulders deposited by macrotur- bulence_________________________________________________________________________________________________ 35 SYMBOLS A Cross-sectional area of channel R Hydraulic radius d(f> Phi skewness S Slope of water-surface gradient •D 1,1)3 Quartile size SKX Inclusive graphic skewness d Depth of channel S0 Trask sorting coefficient K Conveyance y Specific weight Mz Graphic mean diameter Inclusive graphic standard deviation N Size a* Phi or graphic standard deviation n Manning roughness coefficient T Shear stress Q Discharge — Log2AfPHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA—HYDROLOGY, HYDRAULICS, AND BOULDER TRANSPORT By Kevin M. Scott and George C. Gravlee, Jr. ABSTRACT The failure of the partly completed Hell Hole Dam December 23, 1964, released a surge with discharge greatly in excess of any recorded flow on the upper part of the Rubicon River, a westward-drainage of the Sierra Nevada. Extensive erosion of glacial-outwash terraces along the steep, bedrock course of the stream indicates that the surge was probably greater than any post-Pleistocene discharge. Such a unique event, during which more than 700,000 cubic yards of rockfill from the breached dam embankment was washed downstream, permits documentation of the sedimentologic and geomorphie effects of a single catastrophic flow. A torrential rainfall, 22 inches in the basin upstream from the damsiite during the 5 days preceding the failure, produced dramatic and unprecedented runoff. Natural peak discharges in the region were generally equivalent to the maximum discharges attained in previous floods for which there is record. The surge release, however, produced peak discharges substantially in excess of previously recorded flows along the entire 61-mile route downstream from the damsite, along the Rubicon, Middle Fork American, and North Fork American Rivers. The discharge was still 3.3 times the magnitude of the 100-year flood on the Middle Fork American River, 36 miles downstream from the damsite. Average velocity of the flood wave was approximately 22 feet per second. Erosion of detritus from steep, thickly mantled canyon walls resulted in thalweg aggradation at five cross-profile sites. Stripping of the lower valley side-slopes may have triggered a period of increased mass movement in the gorge of the Rubicon River. Depositional forms and flow dynamics were strongly influenced by sediment sources that included colluvium, terrace remnants, till, and landslides directly triggered by the surge. Terracelike boulder 'berms, probably associated with macrotur-bulent transport of boulders in suspension, formed in backwater areas in the uppermost Rubicon River canyon. Boulders were piled to a depth of 5 feet on a terrace 28 feet above the thalweg at a peak stage of 45 feet. Boulder fronts as much as 7 feet high that formed lobate scarps transverse to the channel in an expanding reach indicate that locally bed material moved as viscous subaqueous rockflows. Movement of coarse detritus in large gravel waves may also have occurred. The diorite rockfill in the dam embankment acted as a point source of boulders distinguishable downstream, analogous to a natural point source of coarse particles, and allowed determination of downstream changes in sedimentological parameters. RoundneSs changes were rapid owing to the extreme coarseness of the material. Transition from angular to subangular occurred almost immediately after initiation of movement, and change from subrounded to rounded took place within approximately 1.5 miles of transport from the damsite. Pronounced downstream decrease in mean particle size was mainly due to progressive sorting. The effects of abrasion and breakage were relatively minor and caused less than 10 percent of the overall size decline in the section of channel from 0.4 to 1.3 miles below the damsite. Sorting improved irregularly downstream with respect to the damfill components. Competency of the flow was approximated by a method in which tractive force is determined indirectly at the deepest point in a cross section. Tractive force was then plotted against the mean diameter of the 10 largest boulders deposited at each point where observations were made to extend the relation between tractive force and particle size into the range of extremely coarse detritus. Competency generally decreased downstream but fluctuated greatly. Great quantities of coarse material were swept laterally into the channel by the surge, but were carried only short distances downstream, as shown by variation in competence, marked decrease in size of boulders of upstream lithologic types downstream from a geologic contact, and by distance of transport of material from the dam embankment. Maximum distance of transport of any identifiable particle from the damsite whs 2.1 miles. Assessing the effects of the surge passage in terms of a natural geomorphie event, catastrophic floods, possibly resulting from landslide or ice damming as well as from unusual runoff, may strongly influence the morphology of mountain streams yet be relatively unimportant in terms of total sediment transport. Such rare floods may set the modify pool-riffle patterns, cause cycles of increased mass movement, and trigger mass flow of coarse detritus in the channel. They may cause extensive lateral supply of extremely coarse sediment to the channel where, however, the material is dispersed by flows of lesser magnitude in combination with the size-reduction effects of weathering in place. INTRODUCTION On December 23, 1964, impoundment of runoff from an intense and prolonged storm caused the failure of a partly completed rockfill dam on the western slope of the Sierra Nevada. A surge was released which traveled down the Rubicon River into the Middle Fork American River and into North Fork American River before final containment in Folsom Lake. Emphasis in this study is on the hydraulics of the surge flow, the move- MlM2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS ment of bed material, the resulting depositional forms, and erosional and other geomorphic effects—in short, the events following release of the surge at the Hell Hole damsite. A description of the precipitation intensity and distribution of the storm that caused the failure, synthesis of hydrographs at the damsite based on available flow and storage records in the basin, and a description of the failure of the rockfill are included to give as full a documentation of the event as possible. What happened both before and after release of the flood wave is of interest from the standpoint of dam planning and construction on the many rivers of the West with precipitous, rock-walled canyons and steep gradients. Progressive downstream breaching of dams could effect considerable destruction, whether caused by catastrophic natural floods or possible wartime action. The passage of the flood wave also furnished the opportunity to study the effects of an unusual hydrologic event and the role of such extraordinary floods in determining stream morphology, sedimentation, and landscape evolution. Although such after-the-fact studies of floods are necessarily limited in scope to largely qualitative observations, several aspects of this event are noteworthy. The surge on the Rubicon River was greatly in excess of any recorded natural flood discharge throughout its entire course and may have been the largest post-Pleistocene discharge to traverse the upper reaches of the Rubicon River canyon. The distinctive lithology of the rockfill from the partly completed dam provided a tracer for study of bed-material movement and changes in sedimentological parameters resulting from particle movement from a point source by a single large flow. ACKNOWLEDGMENTS J. C. Brice, Washington University, St. Louis, Mo., accompanied the senior writer on a preliminary reconnaissance trip into the Rubicon River canyon and made several helpful suggestions as to the conduct of the study. G. O. Balding, U.S. Geological Survey, provided valuable field assistance during most of the 18 days of fieldwork during August and September 1965. The writers are also grateful to McCreary-Koretsky Engineers, supervisors of the construction of Hell Hole Dam, for providing reservoir-stage readings and estimates of the volume of rock washed from the breached embankment. The cooperation of I. L. Van Patten, assistant resident engineer at Hell Hole damsite, greatly expedited the fieldwork. W. A. Scott, sheriff-coroner of Placer County, provided a log of officers’ observations of the downstream passage of the surge. Unpublished precipitation data were supplied by the Sacramento Munic- ipal Utility District. Moore and Taber, Inc., permitted use of seismic measurements of one of the largest landslides triggered by the surge in the Rubicon River canyon. The manuscript benefited from criticism by J. C. Brice, Washington University; R. K. Fahnestock, University of Texas; and George Porterfield and S. E. Rantz of the Geological Survey. The study was part of a cooperative program with the California Department of Water Resources. PHYSICAL SETTING LOCATION AND PHYSICAL FEATURES The Rubicon and American Rivers drain a part of the western flank of the Sierra Nevada in Placer and El Dorado Counties, Calif, (fig. 1), at about the latitude of Lake Tahoe. The Rubicon rises northeast of Pyramid Peak in El Dorado County at an elevation of 8,700 feet above mean sea level and flows through a broad glaciated valley before reaching the meadow known as Hell Hole, which was the site of dam construction and release of the flood wave. Hell Hole is 65 airline miles northeast of Sacramento. The surge traversed 61.1 channel miles, descended 3,800 feet (fig. 2), and finally debouched into Folsom Lake, a reservoir 20 miles northeast of Sacramento. During its downstream movement, the flood wave passed initially through the upper glaciated part of the Rubicon River canyon, which has a length of 7.0 miles as measured from the damsite (elev. 4,250 ft) to the lowest probable Pleistocene ice terminus (3,650 ft) and an average slope of 86 feet per mile. This section of the canyon has an average width of about 2 miles and a depth of approximately 1,800 feet below the undulating, dissected erosion surface that forms the gently west-sloping flank of the Sierra Nevada. Downstream from the glaciated reaches, the flow entered the gorge of the Rubicon, a steep-walled V-shaped incision 22.7 miles in length, as much as 2,400 feet deep, and commonly less than 2 miles wide. The channel descends at a rate of 110 feet per mile between canyon walls having slopes greater than 90 percent. Both relief and valley side-slope diminish rapidly downstream from the junction of the Rubicon with the Middle Fork American River. Several bedrock meanders of large amplitude and small radius are immediately below the junction. Canyon depth in this part of the flood route diminishes from 2,200 to 800 feet in a distance of 26.5 channel miles, and the channel slope averages 23 feet per mile. The last channel entered by the surge was the North Fork American River which slopes 19 feet per mile in the final 5.0-mile passage into Folsom Lake.FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M3 Recording precipitation 1 Blue Canyon Airport (U.S. Weather Bureau) 2 Union Valley Reservoir (U.S. Weather Bureau) 3 Picket Pen Creek near Kyburz (U.S. Geological Survey) 4 Kyburz-Strawberry (U.S. Weather Bureau) Figure 1.—Drainage areas in study area. CLIMATE AND VEGETATION Because of strong relief, the local climatic variations of the western Sierra Nevada are great, but some generalizations are possible. The area is characterized by a pattern of summer drought, punctuated by thunderstorms, and winter rain related to changes in the general circulation caused by migrating Pacific Ocean pressure centers. Rapid increase in precipitation with elevation results from rising and cooling of moist airmasses having prevailing west-to-east movement as they reach the great Sierra Nevada barrier. Mean annual precipitation ranges from about 25 inches in the vicinity of Folsom Lake to about 60 inches at Hell Hole. The difference in precipitation total reflects orographic effects in the region. Although there is a general relation between precipitation and elevation, the orientation and exposure of an area with respect to the direction of air movement likewise affect the quantity of precipitation received, and consequently, depar- tures from the general precipitation-elevation relation are common. Most of the total annual precipitation occurs during the period November through March. Above an elevation of 5,000 feet the precipitation is usually in the form of snow, most of which is stored in mountain snowpacks. Consequently, annual peak discharges for streams draining the higher elevations generally occur in late spring during melting of the snowpack. The peak discharges of late spring usually are not excessively large. The severe floods that infrequently occur do so during heavy winter storms when meteorologic conditions are such that the freezing level is 8,000 feet or higher, with the result that rain rather than snow occurs over most of the mountainous terrain. The catastrophic flood of December 1964 occurred under such conditions. The vegetation pattern is highly irregular owing to strong relief and associated microclimatic changes. The canyon bottom traversed by the surge is in the AridM4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Rubicon River Glacial Valley Rubicon Gorge Middle Fork American River--------*4*Norths ' Fork American River -86- -no- 23 19- GRADIENT, IN FEET PER MILE Figure 2.—Longitudinal profile of the flood route. Transition life zone above an elevation of about 3,000 feet, which is substantially lower than the regional Sonoran Transition boundary because down-canyon drainage of colder air admits intrusion of higher elevation vegetation into the lower life zone. A striking difference in vegetation between opposing canyon walls was noted and is a function of isolation differences. The southerly shaded side of the canyon supports a much larger percentage of high-elevation flora. BEDROCK GEOLOGY Bedrock exposed in the immediate vicinity of the damsite and along the first 6 miles of channel downstream from the dam is part of a typical sierran hybrid diorite pluton. The texture of the rock is variable but generally is fairly coarse and devoid of phenocrysts. Local phases of granodiorite, gabbro, and diabase were observed. Within the reach that extends 4 miles from the damsite to the south end of Parsley Bar (fig. 3), a general increase in the quantity of mafic minerals and the number of mafic inclusions occurs. Downstream, the pluton becomes more gneissic and passes gradationally into a small roof pendant of gneiss and sericite schist, probably part of the Calaveras Formation of late Paleo- zoic age that forms the bedrock floor of the channel for approximately 100 yards 2.5 miles below the damsite. Downstream from the pendant, the rock is a biotite quartz diorite. Platy flow structure, schlieren, and alined mafic inclusions and segregations are present throughout the plutonic mass. All the material used in the dam was excavated from the spillway immediately adjacent to the damsite and is a uniform massive fine- to medium-grained light-gray diorite. It is possible to differentiate the fresh hackly pieces of fill diorite washed downstream at the time of failure from the rock types naturally present as clasts in the channel. The clasts are normal dioritic alluvial boulders, substantially better rounded than the dam material, and have a white weathering patina. Angular blocks of bedrock which are derived from valley sides can be differentiated by the presence of iron-stained sheared surfaces locally crossed by dikelets. Downstream from the junction with the South Fork Rubicon River (fig. 1), the channel is cut mainly in schist, slate, and metasandstone of the Calaveras Formation. Below the confluence of the Rubicon River with the Middle Fork American River, the channel is incised into highly sheared metasedimentary rock ofFLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M5 Figure 3.—Location of physical features in the reaches below the Hell Hole damsite pertaining to the study. Evidence of flow episodes shown on figure 20 ; site of channel cross profile, figure 11; bar with radial flow pattern, figure 17. the Calaveras Formation of late Paleozoic age and, farther downstream, into the Mariposa Slate of Mesozoic age which is complexly faulted against amphibolite, chlorite schist, and assorted greenstones of the Amador Group of Mesozoic age. Major structural trends are northwest-southeast and are part of the Mother Lode belt in the western half of the area, but are more nearly east-west in the reaches close to the dam. Shear zones and fractures trend slightly north of west and show no determinable direction of slip. A conjugate set of joints and extension fractures into which aplite and pegmatite dikes have been introduced is apparently the basic structural control of the channel trends. Conspicuous joint sets trend approximately east-west and north-northeast. GEOMORPHIC HISTORY AND STRATIGRAPHY OE THE RUBICON CANYON The Rubicon and Middle Fork American Rivers are bedrock gorges incised within a west-sloping erosion surface of Tertiary age, remnants of Which form the divides on the west flank of the Sierra Nevada. Incision that accompanied the Pliocene and Pleistocene uplift and tilting of the range allowed the major trunk streams, such as the Rubicon, to maintain a westerly course normal to the dominant structural lineaments. Minor tributaries, however, show a high degree of structural control, particularly along the western flank of the range where their flow is parallel to the north and northwesterly shear system. Some apparent bedrock meanders in upper reaches of the Rubicon are attributed to the combined influence of regional slope and transverse joint sets. Various strath terrace levels on gorge sides represent fluctuations in the process of valley incision. A poorly defined remnant of what Matthes (1930, p. 32) calls the “mountain-valley” stage of Sierra Nevada development is present as a series of matching slope inflections that probably represent terrace remnants at a level of approximately 1,600 feet above the river and 800 feet below the plateau surface in the vicinity of the downstream end of the Rubicon River. Within the upper section of the Rubicon are remnants of two former levels, one at 15-30 feet and one at 45-60 feet above the present channel thalweg. The lower of these, and possibly the upper as well, represent valley widening by glaciation, followed by cutting of the notch that locally confines the present channel. The Rubicon River canyon was glaciated downslope at least to an elevation of 3,975 feet, three-fourths of a mile below Parsley Bar, as indicated by glacial stria-tions and glacially shaped asymmetrical bosses on the 293-922 O—61 -2M6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS bedrock surface. The valley form and gradual transition from a broader to a narrower channel suggest glaciation to an elevation of 3,650 feet. Striations are also present on bedrock spurs at the head of Parsley Bar and 700 feet below its downstream end. The lowest of the glacial benches is marked by glacial flutings and large, meltwater-scoured potholes. Parsley Bar, an alluviated valley flat 1,900 feet in maximum width, is the most notable of many small areas of glacial and glaciofluvial deposition present in wider reaches. A composite stratigraphic column of the deposits in the cutbanks of Parsley Bar and the rest of the glaciated upper part of the flooded channel is presented in figure 4. These deposits contain an abundance of extremely coarse material, supply of which has been a controlling factor in the depositional pattern resulting from the flood surge. THICK- LITHOLOGY NESS, IN FEET DESCRIPTION 16.'- .<=>• (• p o 0 . • O 0_o 1 Ci: o • o ' U° ■ fv;../- 0-20 Cobble-boulder gravel, sand matrix; commonly horizontally stratified, some large-scale crossbedding; coarsest material at top; scour channels, sand lenses. / r- Mudflow or till, similar to lower till; 0-3 thinner weathering rinds; great lateral extent. P' c-v p'o. p100-yr flood) Rubicon River 4280..-- Rubicon River at Rubicon Dec. 1955.. 13.0 9,270 1.20 Springs. Feb. 1963.. . 14.28 11,500 1.49 Dec. 1964.. 1 11,300 1.47 4310...- Rubicon River near Dec. 1955.. 18. 76 51,000 1.46 Georgetown. Feb. 1963.. 25.8 58,000 1.66 Dec. 1964.. 71.1± 4332.... Rubicon River near Feb. 1963- 35 83,000 1.54 Foresthill. Dec. 1964.. 78 Middle Fork American River 4333.... Middle Fork American Feb. 1963.. 38 113,000 1.20 River near Foresthill. Dec. 1964.. 61 2 310,000 3.30 4335.... Middle Fork American Dec. 1955- 33.9 3 79,000 .34 River near Auburn. Feb. 1963.. 43.1 3 121,000 1.06 Dec. 1964.. <83,600 .38 60.4 •253,000 2.22 1 Adjusted for storage and diversion. 2 1964 discharge conputed by slope-conveyance method. 2 Old site; about 1,000 ft upstream from present site. * Natural flow prior to surge. * Surge. Of the two gaging stations on the Middle Fork American River, that near Foresthill, just below the mouth of the Ribucon River, was established after 1955 and destroyed in 1964. Consequently, the only one of the three flood hydrographs available for this station is that for 1963. The station on the Middle Fork American River near Auburn operated during the floods of 1955 and 1964, and flood hydrographs are therefore available for those years. The station was destroyed, however, by the flood of 1963. Because there are no concurrent hydrographs for any of the three major floods at successive gaging stations on the Rubicon River mainstem and lower Middle Fork American River, flood-routing procedures could not be developed to compute the hydrographs for the flood wave of December 1964 at the destroyed gaging stations. Furthermore, the lack of precipitation stations in the affected basins precludes the use of rainfall-runoff relations for reliable computation of inflow into the reaches between gaging stations, for use with any flood-routing procedures that might be developed. Some rather crude methods were used, therefore, to compute the inflow and outflow hydrographs at Hell Hole Dam and the peak discharge at stations downstream from the damsite. INFLOW AND OUTFLOW AT HELL HOLE DAM The only data available for computing inflow and outflow at Hell Hole Dam were a record of storage of impounded water behind the dam during the flood period (fig. 8) and a chronology of the dam failure, both supplied by McCreary-Koretsky Engineers. It was essential that the inflow hydrograph be synthesized, because with an inflow and storage record, the outflow hydrograph oould be computed from the equation for conservation of mass: Outflow=Inflow ± change in storage. The obvious method of computing the inflow hydrograph would be by means of a rainfall-runoff relation, such as the unit hydrograph, which could not be used because of the lack of rainfall records in the basin. The method finally used to compute inflow involved a relation between natural discharge of the Rubicon River at Rubicon Springs (31.4 sq mi) and at Hell Hole Dam (120 sq mi). This relation, in turn, was derived from a relation between natural discharge of the Rubicon River at Rubicon Springs and discharge of the Rubicon River above the mouth of South Fork Rubicon River (135 sq mi). The discharge of Rubicon River above the mouth of South Fork Rubicon River was obtained by subtracting the gaged discharge of South Fork from the gaged discharge of the Rubicon River near Georgetown. The net result of the analysis was the equation: Inflow at Hell Hole Dam=3.5 X (discharge at Rubicon Springs, adjusted for storage and diversion) From this equation, the inflow hydrograph at Hell Hole Dam was computed (fig. 8), its peak discharge being 39.000 cfs. A check of the inflow hydrograph was made by examining the area under the hydrograph for the storm period December 20-27. A basic premise implied in the derivation of this hydrograph was that storm runoff, in inches, of the Rubicon River at Hell Hole damsite was closely equivalent to the storm runoff of the Rubicon River near Georgetown. Furthermore, runoff records for the past 15 years show that during major storms, the storm runoff at the Georgetown station was 1.25 times as great as the storm runoff of Silver Creek at Union Valley Dam (fig. 1). For the period December 20-27, 1964, a storm runoff of 16.97 inches for Silver Creek indicates a corresponding storm runoff of 21.21 inches for the Rubicon River near Georgetown. This runoff value for the Georgetown station closely checks the volume of inflow—21.24 inches—under the hydrograph for Hell Hole Dam. The computed inflow hydrograph and the record of storage at Hell Hole Dam were used, as explained above, to compute the outflow hydrograph. The outflow hydrograph, also shown in figure 8, shows that the discharge attained a maximum 1-hour mean discharge of 260.000 cfs when the embankment was breached.STORAGE, IN ACRE-FEET DISCHARGE, IN CUBIC FEET PER SECOND FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA Mil DECEMBER 280,000 O O 240,000 O UJ (/> K w 200,000 160,000 O CD D U z 120.000 j ou,uuu o CD o 40,000 He 1 Hole out low hydro graph, 19C \» 4 r -m r- h 21 22 23 DECEMBER A. Inflow and outflow hydrographs at Hell Hole Reservoir B. Hydrograph of peak surge outflow at Hell Hole Reservoir DECEMBER DECEMBER C. Record of storage in Hell Hole Reservoir D. Hydrographs at Middle Fork American River near Auburn Figure 8.—Flow and storage data associated with the failure of Hell Hole Dam in December 1964.M12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS PEAK DISCHARGE OF THE SURGE DOWNSTREAM FROM HELL HOLE DAM In the absence of a reliable procedure to route the flood wave from Hell Hole Dam to downstream gaging stations, the slope-conveyance method was used to compute peak discharge. The basic equation in the method is the Manning equation: Q=KS», (l) where Q is peak discharge, in cubic feet per second, K is conveyance, in cubic feet per second, and S is the slope of the energy gradient. The conveyance, K, is determined from the geometry and roughness of the channel, by means of the equation: ^1^486 ARH ( n where n is the Manning roughness coefficient, A is cross-sectional area, in square feet, and R is hydraulic radius, in feet, or cross-sectional area divided by cross-sectional wetted perimeter. In the slope-conveyance method, the assumption is made that the energy slope at a site is constant for all extremely high stages. Therefore, if the discharge at some high stage (Qi) is known, the discharge at some higher stage (Q2) can be computed from the ratio of the conveyances at the two stages, where Qi=Qx{KiIKl). (3) If there is no overbank flow at either of the two discharges, the values of n2 and n2 will probably be equal to each other in which case equation 3 can be further simplified to: Q2=Q2{A2R&/AJIP). (4) Equation 4 was used to compute the 1964 peak discharge (Q2) at the gaging station on the Middle Fork American River near Foresthill (table 1) which was destroyed by the flood. At this station, the 1963 peak stage and discharge (Qr) were known, as well as the 1964 peak stage. The gage on the Middle Fork American River near Auburn operated throughout the flood and strikingly shows the surge caused by the failure (fig. 8). The computed volume of water in the surge is 24,800 acre-feet at this station. ATTENUATION OF SURGE WAVE Numerous approximations of stage, made by hand level throughout the flood course, indicate that the height of the flood wave decreased at a fairly constant rate between the dam and Parsley Bar and again in the reaches between the Rubicon gorge and Folsom Lake. Wave Height increased below Parsley Bar, reaching a maximum in the confining narrows of the Rubicon gorge and reflecting the gradual narrowing of the channel downstream from Parsley Bar. The peak discharge of the surge also decreased downstream. The peak on the Middle Fork American River near Foresthill, computed by the slope-conveyance method, was 310,000 cfs (table 1). Downstream, peak flow near Auburn was 253,000 cfs, including a natural flow of approximately 60,000 cfs in the river just before the surge. Trial application of the slope-conveyance technique at the three upstream gaging stations which were destroyed by the surge suggested that larger discharges occurred there. There is no way, unfortunately, of estimating the momentary peak release at the dam-site which was certainly substantially greater than the maximum 1-hour average flow of 260,000 cfs (fig 8). Because breaching of the dam was not instantaneous, release from the reservoir was sufficiently retarded so that the surge did not immediately attain its peak discharge. There was apparently no development of a bore or breaking front to the wave during any part of the downstream flow. Average velocity of the wave, approximately 22 feet per second (or 15 miles per hour) over the entire route, was determined by map measurement of longitudinal distance in the channel thalweg and from the time that the main mass of forest debris uprooted by the surge entered Folsom (Lake. EROSIONAL EFFECTS OF THE SURGE Sedimentary deposits in and along the channel were severely eroded by the surge. Deposits affected by the flood wave include, in approximate order of magnitude: (1) Alluvial deposits, of both Pleistocene and Recent age, (2) colluvium, including material from landslides triggered by the surge, (3) soil, and (4) bedrock. Erosion of the terrace gravel of Pleistocene age was dominantly by lateral cutting rather than surface scour, probably because the surface of the deposits was armored by a mat of vegetation (fig. 9). A surprisingly small amount of bedrock plucking occurred, but notable expanses of bare scoured bedrock surfaces, particularly in the gorge of the Rubicon, give the impression of extensive removal of joint-bounded blocks. Wherever preflood control is present, as at bridge and gaging-station sites and as evidenced by comparison of preflood photographs with the present channel, only a minor amount of bedrock removal can be proved. Striations and percussion marks formed by cobbles and boulders carried by macroturbulence are common on the bedrock surfaces up to the high-water mark in the intervalFLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M13 Figure 9—Lateral snuoly of sediment to the flood channel from a deposit of terrace gravel and till, 3.3 miles downstream from the } damsite. from the damsite to approximately 1,200 feet down the channel. Stands of mature yellow pine, cedar, and Douglas-fir were removed by the surge, the maximum loss of timber occurring in the Parsley Bar area. Many of the trees were 250^100 years old. A large yellow pine and a cedar felled by the flood wave were 351 and 390 years old, respectively. Jams of timber within the channel were relatively rare. Several formed in the 4-mile reach below Parsely Bar, but most of the vegetal debris traveled the complete route to Folsom Lake. A few trees in the upper reaches were killed by bark removal. The maximum abrasion of tree trunks by suspended rock fragments was generally in a zone from 3 to 5 feet above the present aggradational surface (fig. 10). Below this level, the trunks may have been protected by temporary burial. Even in the lower reaches of the flood course, the discharge was clearly of a high recurrence interval. Possibly late 19th century placer tailings were removed from Poverty Bar, Cherokee Bar, and uppermost Oregon Bar, all on the Middle Fork American River. 293-922 O—68--3 EFFECTS ON CHANNEL MORPHOLOGY The five gaging-stations sites along the flood course at which channel cross profiles had been previously measured wrere resurveyed. Changes in channel morphology wrought by the surge decreased downstream as discharge and recurrence interval of the surge decreased. Comparison of two presurge surveys at each of two of the cross-profile sites shows that the bulk of the observed change was in fact related to passage of the surge and not to normal annual fluctuations. A summary of the channel modifications and stage at each station is presented in table 2. Change is most evident at the station farthest upstream where the pronounced aggradation in the thalweg is related to downstream movement of rockfill from the dam (fig. 11). Deposits on the glacial-outwash terrace at this station, however, contain very little material derived from the dam, and therefore occurred at an early stage of the surge. The other cross profiles exhibit thalweg aggradation that is unrelated to deposition of the dam rockfill. The cross profiles and field observations suggest that much colluvium, locally containing very coarse material,M14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 10.—Trees showing maximum abrasion level 3-5 feet above present ground surface. Perched boulder is 3.4 feet in intermediate diameter. View is looking downstream, approximately 4 miles below the damsite. has been stripped from the base of steep valley side-slopes adjoining the bedrock channel which probably have not been awash since glacial recession. Slumping of boundary terrace deposits into the channel resulted from undercutting. Most of this locally derived sediment was apparently introduced into the flow on the receding stage when competency was continuously decreasing. Therefore, far more material was supplied to the channel than could be transported, and much of the extremely coarse material supplied from the sides of the channel was either not moved or moved only short distances. Thus, a chief effect of this short-period catastrophic surge was to introduce coarse sediment from lateral deposits into the active stream channel with a resulting net aggradation in the thalweg. In short, the effect in terms of cross profile was to modify a V-shaped channel to a more nearly U-shaped channel. The surge altered the channel radically in confined reaches and moved, at least for short distances, a large proportion of the material previously present in the channel. Although only five cross profiles are a meager record on which to generalize, deposition in the thalweg may have been nearly continuous along the entire flood route. MASS MOVEMENTS CAUSED BY THE SURGE The surge triggered massive landslides as it progressed downstream and undercut valley side-slopes. Most of the slope failures were at the outer bank of channel bends where velocity and water depth were greatest. Large slides were confined to the steep-sided gorge of the Rubicon, which is cut in slaty metamorphic rocks of the Calaveras Formation. The steepness of these slopes, which have a deep regolith, is attributed partly to the nearly vertical attitude of the bedrock. At the time of surge passage, the slopes had been exposed to approximately 48 hours of continuous rain and were saturated so that conditions were ripe for slope failures. Although evidence of soil creep is general, there are virtually no recent scars of previous major earthslides in the canyon. The slope failures are commonly triangular in plan view and the largest has a maximum toe width of 630 feet. Slide scars reach a maximum elevation of 500 feetFLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA Table 2—Description of channel cross profiles M15 Location of stream-gaging station Distance downstream Peak stage (ft) Description of flow cross profile and channel Approximate change in thalweg elevation (ft) Changes in cross profile site (miles) 1963 1964 Rubicon River below Hell Hole Dam. (See fig. 11.). 1. 3 21. 5 53. 5 Lateral bedrock control, 675 ft wide; alluvial fill, 375 ft wide. Glacial-outwash terrace comprises left half of fill. + 7. 0 Aggradation in thalweg; 125-ft lateral movement of thalweg; accretion on terrace surface; probable bedrock removal on right bank. Rubicon River near Georgetown. 8. 9 25. 8 71. 1 Lateral bedrock control, 400 ft wide; alluvial fill, 250 ft wide. Left bank cut in glaciofiuvial and colluvial deposits. + 1. 5 Aggradation in thalweg; 50-ft lateral movement of thalweg; deposition of large bculder bar on right side of channel. No erosion of bedrock. Rubicon River near Foresthill. 26. 3 35. 0 78. 2 Lateral bedrock control, 255 ft wide; alluvial fill, 135 ft wide and probably quite shallow. + 2. 5 Aggradation in thalweg; no lateral displacement of thalweg; addition to boulder bar on left side of channel. Possible slight removal of bedrock. Middle Fork American River near Foresthill. 35. 6 38. 0 61. 1 Lateral bedrock control, 378 ft wide; alluvial fill, 260 ft wide. + 3. 0 Aggradation in thalweg; 20-ft lateral displacement of thalweg; addition to coarse bar on left side of channel. No removal of bedrock. Middle Fork American River near Auburn. 54. 5 1 43. 1 60. 4 Only basal part of section was resurveyed. + . 2 Slight aggradation in thalweg but as much as 3.0 ft of aggradation in other parts of channel, addition to coarse bar on left side of channel. No determinable removal of bedrock. 1 Old site; about 1,000 ft upstream from present site. Figure 11.—Prefailure and postfailure cross profiles of the channel at the site of the first gaging station reached by the surge, Rubicon River below Hell Hole Dam, 1.3 miles downstream.M16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 12.—Base of the largest slide ia the Rubicon gorge, 0.5 mile downstream from the mouth of Little Grizzly Canyon, 18.7 miles downstream from Hell Hole damsite. Note scour due to surge at right, the downstream direction. Helicopter, right, indicates scale. above the channel. The slide that involves the greatest volume of displaced material is linear in plan view and has a nearly constant width of about 300 feet. It originated in a spoon-shaped rupture approximately 400 feet above the bottom of the gorge (fig. 12). In addition to the large slides, nearly continuous sliding and slumping of colluvium on a smaller scale occurred along channel banks within the gorge. Slides are identified as the slow-earthflow and the debris-avalanche type described by Varnes (1958). At the heads of several slides, individual blocks of bedrock moved by slip, but such block slumping was transitional to movement by earthflow and debris avalanche in the lower part of the slides. Most of the slides began as mass movements of semiconsolidated material and were soon transformed downslope to flows. The largest slides occurred after the surge peak and thus did not contribute significantly to the sediment load of the flood. Movement of some slides probably began before arrival of the flood peak, and the removal of detritus from the toe of the slide allowed renewal of movement. Most of the smaller slides were probably di- rectly triggered by erosion near the flood peak and were virtually coincident with passage of the surge. Presence of highest scour lines on some slides at levels substantially below the high-water mark shows that sliding continued during and after flood recession. Virtually all the landslips were closely related to the surge passage, and most of the slides were directly triggered by the flood wave. Probably comparable saturation conditions resulting from storms of 1955 and 1963 produced little or no sliding. As discussed on page M36, the destruction of slope equilibrium by the surge of 1964 also resulted in renewed sliding a year later during the next rainy season. Even the largest slide did not appreciably dam the channel. The present low-water channel cuts through as much as 7 feet of slide detritus and lateral levees of slide detritus (fig. 12) are present downstream from the slide. Measurements of floodmarks immediately upstream showed a maximum depth of 62.3 feet and those downstream, 63.5 feet, in a comparable channel. Thus, the peak flow apparently was not affected byFLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M17 the slide, and most of the movement was during the recession phase. Determination of the surface area of three of the largest slides and estimation of the depth, for slumped parts of slides as described by Philbrick and Cleaves (1958), permitted an estimate to be made of the volume of material in each slide. Seismic measurements of depth were available for one of the three slides. Extension of these estimates to 29 other slides, on the basis of relative size as measured on aerial photographs, yielded an order-of-magnitude approximation of the total amount of material involved in landslides resulting from the flood—800,000 cubic yards—a surprisingly small figure and an estimate subject to large sources of error. Only a small part of this volume was actually contributed as sediment to the flood. The time relation of sliding to the flood crest, armoring of the debris avalanches by extremely coarse material at the surface, and coherence of the mud snouts of the flows prevented much erosion by the flood. The percentage of material eroded from each slide by the surge was estimated principally on the basis of aerial photographs and, for about a third, on the basis of field observations. Less than 30 percent of the total slide volume was removed by the flood. Eroded colluvium and the inestimable quantity of detritus supplied by continuous small-scale sliding that occurred in steep-walled reaches were almost certainly more important as sources of sediment. Virtually all the latter material was moved by the flood. DEPOSITIONAL FEATURES METHODS OF STUDY Depositional forms were studied in the field and by analysis of large scale (1:1,200) aerial photographs, obtained for the 4 miles of channel downstream from the damsite. Additional photography was made of the remainder of the flood channel at a scale of 1: 6,000. The photography was made between August 28 and September 10, 1965. A comparison with two sets of preflood aerial photographs was possible, one flown in September and October 1948, at a scale of 1: 42,000, and another taken during July and August 1962, at a scale of 1: 20,000. Approximately 12 miles of the poorly accessible canyon was covered on foot and the rest was traversed by helicopter. Channel forms from the dam Figure 13___B id berms preserved 0.7 mile downstream from the damsite. Note greater coarseness of material in center of ou er channel. Flow is from left to right. Photograph by J. C. Brice.M18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS to a point 5 miles downstream were studied and mapped by use of the large-scale photographs and T^-minute topographic maps enlarged three times as base maps. LATERAL BERMS OR TERRACES The most striking depositional forms resulting from passage of the flood surge are terracelike berms composed of boulders of diorite from the dam rockfill (fig. 13). The nearly flat-topped berms occur continuously on both sides of the channel in the uppermost reach, but downstream they are confined to protected areas behind bedrock projections or the inner bank of bends. Berms composed of rockfill derived at the damsite are restricted to the upper part of the flood route, which ends 1.1 miles downstream from the dam toe. The berm surface is as much as 28 feet above the present channel thalweg and decreases generally, but not regularly, downstream. Although subsequent artificial removal of material prevents exact determination, the level of the berms probably rose upstream nearly to the level of the dam core, 50 feet above the channel. At a point approximately 1,600 feet downstream from the dam, a pronounced inflection in the berm surface occurs. The surface does not correlate with a change in slope in the present thalweg, which is extensively aggraded in that reach, but may correspond to a slope change in the original channel or to a constriction in the temporarily clogged channel. In transverse section, the surface of the berms usually slopes gently toward the present channel, and wdiere berms are present on both sides of the channel they occur at approximately the same level. Another characteristic of the berms is the presence of the coarsest detritus at or near the surface (fig. 14). They are similar in this respect to the glacial-outwash terraces in the Rubicon River basin. This reverse grading of the deposits relates to the depositional dynamics, possibly to dispersive stress forcing larger particles to the surface during movement (Leopold and others, 1964, p. 211) rather than to occurrence of the coarser particles remaining as a lag deposit after removal of fines. Generalization on the relation of the coarseness of the berms to the coarseness of material in the channel is not possible. In the reach immediately downstream from the dam, coarsest debris occurs in the center of the channel. Through the middle of the area of berm formation, in which the pool-riffle pattern becomes more pronounced, Figure 14.—Section of berm parallel to stream channel In reach below Hell Hole damsite. Rucksack is at the base of the berm section. Note the coarse material at the top of the berm surface and the contrast in size with coarser material in center of channel, foreground. All material is diorite from the damsite.FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M19 berms may be either coarser or finer than the adjacent channel material, but all the berms are coarser than pool components. The berms farthest downstream are distinctly finer than the channel material. The terracelike berms are superficially analogous to common alluvial terraces, formed by channel aggradation followed by incision of a lower channel. Generally, alluviation followed by erosion is a lengthy process. The fact that the terracelike berms were formed by a single flood event and that the bed material is almost entirely of boulder size, and thus, once deposited, would be unlikely to be eroded at a later, necessarily lower, stage suggests another origin for the coarse terracelike deposits. Macroturbulent conditions transported boulders in suspension well above the berm surface; the evidence of such transport, perched boulders of rockfill diorite, is most conspicuous in the uppermost reaches of the flood channel which show extensive berm formation. The material composing the berms, however, was deposited from bedload, which locally moved as thick subaqueous debris flows or a series of gravel waves, in this section of the channel. The top of the continuous berms may represent the approximate level of bedload flow or wave movement attained in the main channel rather than any static aggradational surface. The degree to which the present channel below the berm surface represents incision by erosion at a later stage must remain an unknown, as must the configuration of the surface, whether static or dynamic, represented by the berms. The berm surface probably was continuous across the channel at one period during the surge. Bedload movement may have been continuous in the center of the channel and, as the material moved out of the reach, the present low-water channel could have been left as basically a depositional rather than an erosional feature. In terms of the time sequence of events only, the situation may be similar to that occurring in the formation of mudflow levees in which, just after a confined mudflow has peaked, continuing flow in the center of the channel moves most of the flow out of the reach, leaving terraces marking the peak on each side of the watercourse. Local presence of the coarsest material in the center of the channel is due to lagging of the largest particles well behind the peak volume of bedload movement. At least in the lower reaches of berm formation, both formation and incision or removal of the berms probably occurred during the receding stage. This probability is evidenced by relations at the cross-profile site (fig. 11) where material carried by the surge peak and then deposited on a high-level terrace included very little rockfill detritus. All the berms described above are composed of dam-fill material. Downstream, similar but discontinuous isolated ridges of sand formed in areas of low velocity during the surge, particularly on vegetated banks. Such distinctly finer grained berms may be present within 10 feet of the high-water mark where the stage was approximately 50-60 feet. Also in a downstream direction, local terracelike berms formed by the surge and composed of normal alluvial gravel, including some detritus of boulder size, occur on point bars and behind bedrock projections on the Middle and North Forks of the American River. Thus, the berm-forming process is not solely related to the supply of sediment provided by the rockfill, but does relate generally to the large volume of sediment transported by the surge. BOULDER BARS The terracelike berms are transitionally replaced downstream by lobate bars. The bars are in part flat topped, consist of boulder-size material, and are the dominant depositional forms downstream. The upstream pattern of the bar trends is linear, parallel to the straight reaches between the damsite and Parsley Bar, and the apexes of the bars are directly downstream from bedrock projections within or lateral to the channel (fig. 15). As the channel becomes sinuous, the pitch of the surfaces of the bars alternates from one side of the channel to the other, as does the low-water channel (fig. 16). In part of the uppermost section of the channel and at the downstream end of Parsley Bar, there is a medial bar or ridge of coarse material. The bar terminates in a frondescent or radial pattern with flow lines deviating as much as 50° on each side of the axis of the channel. The flow lines are not visibly related to channel configuration, a slightly contracting reach, but are marked distinctly on the aerial photographs by felled and lodged trees, textural changes, and ridges of bed material (fig. 17). TERRACE ACCRETION Addition of material on the surface of a terrace remnant near the head of Parsley Bar suggests that the flood had a greater magnitude than flows, presumably of Pleistocene age, that were responsible for transport of the terrace fill. As much as 5 feet of boulder gravel was added to the terrace remnant on the left bank at the site of the former gaging station, Rubicon River below Hell Hole Dam, 1.3 miles downstream from the dam (fig. 11). The flood did not aggrade the extensive terrace deposits of Parsley Bar or any terrace farther downstream. The gravel added to the terrace upstream from Parsley Bar consists almost entirely of reworked terrace gravel derived from the margins of that terrace andM20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 15.—Lobate bar formed downstream from a bedrock projection into channel at right. Central part of bar is mainly reworked terrace and recently deposited alluvial detritus. Secondary bermlike formation in foreground is predominantly rockfill from the dam embankment. View looking downstream 1.1 miles below the damsite. from others upstream. However, two cored boulders of dam fill were observed in the deposit. The head of the terrace was eroded and a cutbank having a maximum height of 12 feet was formed, at the base of which the present channel fill is composed of diorite derived from the dam. The comparison of aerial photographs suggests that as much as 20 feet of lateral erosion occurred along the terrace margin. BOULDER FRONTS An arcuate, boulder front 3-7 feet high extends 250 feet across the flood channel in the middle of Parsley Bar (figs. 18,19). The front represents a point to which, at a late stage of the surge passage, boulders as much as 10.5 feet in intermediate diameter were transported, but beyond which little coarse material was moved. The boulder deposit transgressed across and only partly replaced a sand layer 2-8 feet thick that was deposited during an earlier interval of surge recession. Numerous small trees in growth position protrude through the sand layer which was molded into longitudinal and arcuate transverse dunes. Similar boulder fronts on a smaller scale occur at other localities on Parsley Bar. The term “boulder front” is used here to indicate a scarp or face of boulder gravel normal to the flow direction. The deposits are not similar to the bouldery snouts of mudflows, to which the term has been applied, in that the entire deposit in this case is composed of boulder gravel. Like the berms, the main boulder front apparently is characterized by presence of the coarsest clasts at the surface, although the coarseness of the deposit precluded thorough excavation. The front occurs in a rapidly expanding part of the Parsley Bar reach. During the flood recession, the front was incised at the position of the present low-water channel, and coarse material was moved a small distance downstream. The single cored boulder of dam fill observed beyond this point was probably transported by macroturbulence during the surge peak. The front represents the point in the channel at which the critical tractive force fell below the value necessary to move the entire flow, because of reduction of depth associated with the gradual but pronouncedFLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA. M21 Figure 16.—Large gravel bar formed 1.2 miles below Hell Hole damsite. Riffle in foreground is parallel to stream course. Bed material is predominantly rockfill from the dam. Width of channel is approximately 200 feet. expansion of the reach and reduction in slope. A stage measurement showed that maxium depth of water at this point in the channel was 5-6 feet above the top of the deposit and was probably substantially less at the time the boulder flow reached this point. Competency necessary to move the mass was clearly much less than that required to move the component particles individually. Two alternative hypotheses for formation and movement of the boulder front are proposed. According to one, the boulder front represents the front of a type of subaqueous viscous flow in which the bedload moved as a churning mass—fluid, yet sufficiently viscous to move as a discrete unit. Fronts that are possibly in part analogous have been reported on subaerial gravelly mudflows by Sharp and Noble (1953, p. 551) and Fahnestock (1963, p. 20) and on flood gravels by Krumbein (1942, p. 1364) who applied the term “boulder jams.” According to Krumbein, longitudinal segregation results in concentration fronts of very coarse debris which dam the channel. Alternatively, the boulder fronts may represent a large wave of gravelly material analogous to a sand 293-922 O—68---i wave or delta front. Movement is by rolling and saltation of the component material at the surface of the deposit until the foreset part or avalanche face of the form is reached, whereupon particles cascade down the front and are at least temporarily deposited owing to the sharp reduction in competence at the crest of the front. Migration of the dune form or delta takes place continuously by erosion of the upstream surface and deposition on the front. Tractive force necessary is that required to move the component particles as normal bedload. Dune or delta movement obviously requires particles with similar hydraulic behavior, and dune materials are thus generally fairly well sorted. Consideration of this mode of transport is based on two lines of evidence. The morphology of the front is remarkably similar to that of generally smaller scale sand waves or migrating microdeltas (Pettijohn and Potter, 1964, pi. 74). However, strongest evidence for this mode of formation is the presence of probable crossbedding of an amplitude similar to the height of the boulder front and exposed in a cutbank in glaciofluvial deposits of Pleistocene age at the same position in Parsley Bar.✓05 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 3 C 3 Q- EX PLAN ATI ON Scarp of sand or gravel Marked change in grain size o Boulders (X 2 scale size) Current direction (fallen trees) / 0) Flow lines / r+ Approximate high- / / ; CD water mark Q- X / / O Cutbank in terrace Op — deposits — 0 / 0 Oo / /° rSe ; ; / < Riffle in low-water 0 \ ; \ a> channel to Boundary of low-water O / r-+- channel a> o> Field sketches applied to an aerial-photograph base. Boulders are shown twice the actual scale size and each is the coarsest particle in its immediate vicinity. 100 ___|_ Figure 17.—Radial pattern of flow and deposition at the downstream end of Parsley Bar.FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M23 Figure 18.—View across flood channel of boulder front on Parsley Bar, Maximum height of the front is 7.2 feet. Note trees tilted in a downstream direction, to the right. this type has not been directly observed, and its general importance in fluvial transport cannot be evaluated at present. This species of subfluvial debris flow can be considered transitional between mudflow and normal bedload movement by sliding, rolling, and saltation. The large-scale crossbedding and smaller, dunelike fronts composed of better sorted material indicate that local movement of material in the form of dunes in which the front represents an avalanche face probably accompanied the mass flow. Both forms of movement probably occurred simultaneously. EPISODIC BOULDER MOVEMENT A series of three berm surfaces 1 mile below the dam-site shows that a series of boulder movements occurred during the surge. These surf aces are underlain by rock-fill from the dam (fig. 20). Adjacent to the channel wall at this site is a very coarse deposit in which no lateral variation in size was noted. This deposit abuts against a distinctly finer bermlike mass characterized by a marked fine-to-coarse gradation toward the center of the channel. In the central channel the material is distinctly finer than in the adjacent berm, but is similarly coarser toward the channel axis. Thus, during Crossbedding, inclined stratification in most places truncated at the upper surface, is preserved evidence of the migrating front or foreset area of dune or micro-delta. Thus, wave movement with large amplitude probably occurred at that point on the bar during Pleistocene time. The crossbedded unit, which will be described below, contains a few clasts of boulder size but is dominantly a sand, pebble, and cobble gravel. The material making up the boulder front deposit in the foreset region has a median diameter (74 mm) that is much in excess of any reported value for crossbedded deposits and sorting (a/=2.34; S0=3.36) that is substantially poorer than any previously recorded for deposits whose texture has not been altered by induration. A surge released by failure of a natural gravel fill formed large dunes with a modal class of granules (2-4 mm) in the foreset region (Thiel, 1932, p. 456), but the resulting deposits were substantially better sorted than the boulder fronts on Parsley Bar. The Parsely Bar example contains some extremely coarse boulders with diameters equal to the probable thickness of the deposit. The authors conclude therefore, that the deposit and the boulder front primarily represent the sudden cessation of in-mass viscous flow of bed material. Movement ofM24 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS EXPLANATION Boulder front Scarp of sand or gravel Dune forms Marked change in particle size CD Boulders (X 2 scale size) Current direction (fallen trees) Approximate high-water mark x Cutbank in terrace deposits Boundary of low-water channel 200 ___I_ 300 FEET __I Field sketches applied to an aerial-photograph base. Boulders are shown twice the actual scale size and each is the coarsest particle in its immediate vicinity. Bed material adjacent to the front downstream is sand Figure 19.—Part of Parsley Bar containing the largest boulder front. each of three episodes, successively finer materials were deposited, hut within each deposit the grain size probably coarsened toward the center of the channel. The relative difference in particle size between berms and channel is thus partly explained. In the steep-walled confining reach nearest the dam and at the downstream terminus of rockfill movement, only a single episode of movement is preserved and is recorded by deposits in which the coarsest material is in the middle of the channel. The sequence of berms noted in the dam-fill material and the boulder front on Parsley Bar both suggest that movement of bed material was episodic in response to a single flood wave. The possibly irregular supply of rockfill by periodic caving of the dam embankment may explain the berm sequence. According to an eyewitness,FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M25 however, failure of the structure was gradual and continuous. Movement of bed material as viscous debris flows and dune forms created longitudinal concentrations of boulders moving down the channel. In any reach of channel other than Parsley Bar, the boulder EXPLANATION Qr Predominantly rockfill from damsite Deposits are numbered in order of deposition Qt Terrace gravels Bedrock Contact Dashed where approximately located 5o0 Relative size of boulders 111111 m 11 Scarp Riffle in low-water channel Fioore 20.—Reach showing sequence of bedload movement. See figure 3 for location. fronts would have migrated to a point where locally increased competence, at a channel bend or constriction, would have removed material from the flow or wave front at the same rate it was supplied, and coarse material would tend to accumulate and form a riffle analogous to a kinematic wave (Leopold and others, 1964, p. 212). Movement in waves or as viscous flows in response to extraordinary floods may be a factor in controlling the pool-riffle pattern in boulder-bed streams. POOL AND RIFFLE PATTERN Most stream channels show longitudinally alternating deeps and shallows, described as pools and riffles. Pools are the quiet stretches of relatively smooth water, and riffles occur in steeper parts of the channel with more rapid flow. Such sequences are scarce in the upper section of the Rubicon River above Parsley Bar, and the comparison of preflood and postflood aerial photographs shows that there are now fewer rapids in this upper, heavily aggraded part of the flood course than were there before the flood surge. In the sinuous section of channel immediately above Parsley Bar, the typical pool-riffle pattern is well developed. Riffles are located where the low-water flow moves across the bar and impinges against the bedrock wall of the channel. In general, the preflood pattern of pools and riffles has not been greatly changed. The preflood pattern of Parsley Bar has been modified and pool-riffle sequences are now few. Creation of a few new riffles resulted from supply of coarse detritus by terrace erosion during the receding stage of the surge and from cessation of movement of the boulder fronts. The normal pattern of alternately pitching bars and pools and riffles predominates in the channel between Parsley Bar and the downstream end of the Rubicon River. Demonstrable modification of a pool and riffle has occurred at the site of the Rubicon River-Foresthill stream-gaging station where a riffle just downstream from the bridge at that location has been removed and is now the deepest part of a pool. The poor resolution of both sets of preflood aerial photographs in that part of the canyon prevents precise documentation of other changes in the pattern. Definite new riffles were formed by lateral supply from mass movements of material too large to be moved by the flood (fig. 21). The comparison of aerial photographs suggests, however, that there are now fewer riffles in that section of the surge channel. The flood removed numerous deltaic contributions of sediment by tributary streams and small preflood mass movements that had created riffles. No noticeable modification of pools and riffles occurred on the Middle and North Forks of the American River, although a few minor changes in configurationM26 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 21.—Contribution of sediment to the flood channel by a landslide, 2.5 miles upstream from the junction of the Rubicon River with the Middle Fork American River. Because the slide occurred after passage of the surge at a stage during which the competence was insufflcient to transport the coarses detritus of the slide, a riffle sequence was formed. Material with Intermediate diameters in excess of 10 feet remains in the channel. Base of the slide Is 535 feet across. occurred. Mass movements directly resulting from the flood are also relatively minor along that part of the channel. BED-MATERIAL FORMS AND INTERNAL SEDIMENTARY STRUCTURES Deposits of the flood surge in the upper reaches are noticeably lacking in preserved sedimentary structures owing to the extreme coarseness of the deposits. Stratification could not be observed in the exposed berm sections. Surficial forms in the sand deposits consist of longitudinal ridges, flat-topped bars which in some places are associated with fronts as much as 2.5 feet high, and a series of crescentic convex-upstream waveforms as much as 20 feet across, visible on aerial photographs but not readily noticeable on the ground. The fronts associated with the sandbars do not display internal stratification, either plane or crossbedded. Horizontal stratification, which is evidence of a plane-bed form, becomes abundant in the sand and gravel deposits of the lower reaches, particularly on the Middle and North Forks of the American River. Sand deposits of the lower reaches shown abundant crossbedding and ripple lamination, as well. Sets of crossbeds are tabular, with approximately parallel cutoffs and nontangential foresets, and have an amplitude of as much as 10 inches. Current-ripple lamination consists of climbing-ripple forms, as much as 0.8 inch in amplitude, of Walker’s type 1 and 2 (Walker, 1963, p. 175) in cosets as much as 4 inches in thickness interbedded with plane-bedded strata. Abundance of preserved bed forms in a downstream direction in both gravel and sand deposits reflects a general decrease in grain size downstream and more sustained flow in the lower reaches. Lack of structures in the sand deposits of the upper reaches, such as Parsley Bar, may reflect rapid deposition. Plane beds, preserved as horizontal stratification, are usually interpreted (Gwinn, 1964, fig. 1) as evidence of the upper rapid-turbulent flow regime of Simons and Richardson (1961) ; dunes and ripples, preserved as types of crossbedding, as evidence of the lower, tranquil-turbulent flow regime. The cutbanks of the glacial terrace deposits show a variety of crossbedding. Crossbedding with an ampli-FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M27 tude of as much as 7 feet occurs in the deposits of Parsley Bar. The inclined strata of these largest crossbeds consist of highly variable layers of sandy pebble and cobble gravel and are at the base of the youngest unit of the alluvial stratigraphic succession (fig. 4). Trough crossbedding, preserved evidence of crescentic dunes, is abundant in the units of white sand and silt and rust-colored granule gravel occurring above the main body of till. TRANSPORT AND DEPOSITION OF BED MATERIAL IN REACH DOWNSTREAM FROM DAMSITE DOWNSTREAM CHANGES IN PATTERNS OF BARS AND BERMS PARSLEY Berms of dam-fill diorite are present immediately below the dam; they grade downstream, at progressively lower elevations, into deposits in the present channel thalweg. The pattern of berms that sharply abut against valley walls is gradually replaced by a pattern of local flat-topped and lobate bars. The bars and the terrace-accretion deposits seem to have been molded of locally derived stream alluvium by the initial flood peak which had not entrained much coarse dam-fill material. There was a distinct time lag between formation of the lateral bars of stream alluvium and arrival of the bulk of the dam fill which predominates within the channel itself. Upon reaching Parsley Bar, flow expanded from the confining bedrock channel, cut into terrace deposits, and formed boulder fronts and a large medial bar with radial flow patterns. Movement of the dam fill ended abruptly at the head of the bar, 1.6 miles below the damsite. The dam fill moved as a nearly undiluted mass because the surge had virtually cleared the channel. Beginning with the bedrock channel at the lower end of Parsley Bar, the normal pattern of alternately pitching gravel bars occurs. COMPOSITION The percentage of diorite in streambed materials is plotted against distance from the original toe of the dam in figure 22. The sharp contrast between the bed material in the reach above Parsley Bar and in Parsley Bar itself is evident from a comparison of figures 14 and 23. A large majority of the diorite boulders in the deposits which contain more than 90 percent diorite (fig. 14) originated from the dam embankment. Even at the downstream terminus of the in-mass movement of fill where only 8 percent of other rock types dilute the deposit, probably less than 6 percent is diorite derived from within the channel. This conclusion is based on the proportion of diorite to other rock types in the Figure 22.—Relation of bed-material composition, in terms of the percentage of diorite, to the channel distance downstream from the damsite. All measurements, except those noted, were made in or near the thalweg. bars composed of locally derived material that were formed during an earlier stage of the surge. A small proportion of the diorite in those bars was derived from the dam fill, in all probability less than 10 percent. PARTICLE SIZE The size of particles moved by the flood was measured to obtain some idea of variation in dynamics along the stream channel and possible variations in competence in a downstream direction. The method described by Wolman (1954), in which the intermediate diameter of 100 clasts is measured in the field, was used because of the extreme coarseness of the deposits. A preliminary evaluation suggested that samples involving traverses perpendicular to the channel would show great variation. Therefore, a tape was used to construct a sampling grid in the area of each reach containing the coarsest particles. Several sampling problems occur in measurement of bed materials as coarse as those present in the surge channel. In some samples the intermediate axis must be approximated, but placement in the correct size class for calculation of a size distribution is generally possible. Boulders having a diameter greater than —9 on the phi grade scale (512 mm) are difficult to maneuver into a position where the true intermediate axis can be measured. Where large boulders are partly buried, the exposed minimum diameter was used as an approximation of the intermediate diameter. When the grid point fell over a crevice in boulder gravels with an openwork texture, not even an approximation of the size of the selected particle was possible. Such sampling points were excluded.M28 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Figure 23.—Bed material of Parsley Bar, extending from 1.6 to 2.9 miles downstream from the Hell Hole damsite. Very little rockfill moved this far. The material is predominantly reworked terrace gravel. Compare with figure 14. Weathered diorite, granodiorite, and granite are the main rock types. Figure 24 is a plot of the graphic mean diameter (Folk, 1964, p. 44) at localities sampled in the upper reaches, calculated from the formula: j^_«frl6+50+ Inclusive graphic deviation Inclusive graphic skewness SKi Slope S( ft per ft) Maximum stage (ft) Tractive force 1 (lb per sq ft) 1 0.42 338 -8.40 1.79 1.40 1.49 0.0 0.08 0.0304 58.0 110.0 2 .51 338 -8.53 1.79 1.30 1.30 -.15 .06 .0289 63.3 114.1 3 .56 416 -8.30 2.06 1.70 1.91 .35 .41 .0283 47.6 84.1 4 .61 315 -8.27 1.74 1.15 1.28 .04 .11 .0281 44.0 77.1 5 .76 416 -8. 57 1.86 1.30 1.23 .15 .18 .0214 31.1 41.5 6 .81 274 -7.80 2.07 1.75 2. 08 .26 .33 .0208 35.5 46.1 7 .99 239 -7.93 1.98 1.45 1.57 -.03 .11 .0207 37.0 47.8 8 1.10 274 -8.13 1.56 .95 .96 -.05 -.03 . 0175 40.3 44.0 9 1.16 147 -7.07 2.08 1.60 1.78 .13 .27 .0172 42.5 45.6 10 1.27 147 -7.17 1.47 1.20 1.15 .08 .13 .0169 43.0 45.4 11 1.32 208 -7. 57 1.63 1.00 1.00 .20 .21 .0165 41.0 42.2 12 1.45 169 -7. 30 2.14 1.75 1.71 .09 .13 .0158 37.2 36.7 13 1.48 239 -7. 77 2.14 1.70 1.73 .12 .16 .0158 36.0 35.5 14 1.56 208 -7. 70 1.57 .90 .95 .0 .08 .0125 30.5 23.8 15 1.60 89 -6.30 1.51 .95 .99 .16 .20 .0125 25.0 19.5 16 1.67 169 -7.17 2.22 2.05 2.22 .17 .27 .0121 22.5 17.0 17 1.90 74 -5.93 3.36 2.40 2.34 .17 .17 .0058 23.8 8.6 18 1.99 112 -6. 60 2.73 2.00 2.03 .15 .18 .0062 22.7 8.8 19 2.32 119 -6. 50 2. 37 2.00 2.00 .30 .30 .0061 29.6 11.3 20 2.50 137 -6.83 2.37 1.90 1.89 .21 . 19 .0068 25.3 10.7 21 3.10 512 -8. 77 2.13 1.75 1.79 .40 .37 .0134 34.0 28.4 22 3. 55 588 -8.93 2.07 1.70 1.74 .24 .24 .0080 47.6 23.8 23 4.10 388 -8.10 2.15 1.75 1.73 .43 .36 .0085 44.3 23.5 Pi .56 194 -7. 53 1.51 1.00 1.03 .10 .18 Ri .59 416 -8. 50 1.51 1.00 1.06 .30 .27 Pa .70 208 -7.40 1.63 1.05 1.08 .43 .44 Ra .73 315 -8.13 1.68 1.15 1.11 .22 .21 P3 1.09 89 -6. 67 1.87 1.30 1.20 -.31 -.29 Rj 1.15 256 -8.07 1.95 1.35 1.38 -.09 -.01 Pi 1.23 91 -6. 50 1.57 1.00 1.02 .0 -.03 R.--- 1.27 194 -7.67 1.46 .90 .95 -.11 -. 10 • Assuming 7—62.4 lb per cu ft. between the dam and Parsley Bar was excavated during a period of subsurface flow. Only a slight increase in coarseness was observed to a depth of 4 feet. Thus, the difference in grain size between the material in pools and that in riffles is probably not a surficial effect. Downstream size decrease is also shown by the intermediate diameters of clasts marked by shothole cor-ing (fig. 26) which positively identifies such boulders as being supplied to the surge at the damsite. The relative coarseness of the cored boulders and the size of the bed material confirms the interpretation that the bulk of the detritus in the channel downstream as far as Parsley Bar consists of dam fill. The sharp end to the tongue of dam fill at the head of Parsley Bar, the marked reduction in competence owing to slope and stage reduction at that point, and the apparent lack of dilution of Parsley Bar bed materials by dam fill all suggest that coarse fill was not transported beyond the head of the bar, 1.6 miles downstream from the dam. However, a definite boulder of fill, having an intermediate diameter of 1.4 feet and containing a shothole, was found near the middle of Parsley Bar, 2.1 miles from the damsite, and downstream from the boulder front. The boulder was probably one of a very few pieces of detritus transported that distance during conditions of maximum turbulence, with or shortly after the surge crest. The noticeable rounding of the specimen suggests that it was not rafted to this point by vegetation. DISTRIBUTION OF PARTICLE SIZE The distribution of particle size of clastic elements at each sampling locality is presented in table 3 in the form of a sorting coefficient S0 (Trask, 1932, p. 71), the phi or graphic standard deviation (Inman, 1952, p. 130), and the inclusive graphic standard deviation + 0-0095 — 05- -9.0 - | BAR I ( | | 0 i ii i 1 I- o ° ° o o o o EXPLANATION o o Boulders in or near thalweg 0 . • oD Boulders on lateral bars or terraces 1 1 1 1 1 1 1 1 1 2048 s 1024 z DISTANCE FROM DAMSITE, IN MILES Figure 26.—Relation of diameter of larger cored boulders to distance downstream from damsite.M30 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS where Zh = diameter, in millimeters, at which 25 percent of the material is larger, and D3=the diameter at which 75 percent is larger. Other subscripts also denote the percentages of material larger. For each of the measures, values are inversely proportional to the degree of perfection of sorting. Figure 27 is a plot of the inclusive graphic standard deviation against distance downstream from the dam-site. All the coarse deposits, regardless of location, can be characterized as poorly sorted relative to gravel found in most other geological environments, all values of 07 being above 0.9. The degree of sorting is also on the low side relative to other published values of sorting of fluvial gravel (Emery, 1955, p. 47). However, an irregular increase in degree of sorting is evident in the dam fill distributed along the stream course. Material in the blasted fill was probably originally distributed according to Rosin’s law, the geometrical relation between quantity of material in each size class produced by crushing. Some natural sediment sources, such as talus, glacial till, and mechanically and chemically weathered igneous rocks, approximate the distribution, in which each size interval contains about half the weight of material in the next larger interval (Krum-bein and Tisdel, 1940, p. 301-304). Thus, the changes in size and sorting in the distributed dam fill should approximate the results of transportation away from a natural point source of sediment supply. The values should be only slightly affected by contamination as shown in figure 23. Previous studies show that sorting commonly does not vary according to distance of transport when fairly short distances of transport are involved (Plumley, 1948, p. 548; Brush, 1961, p. 152). Normal stream gravel is the product of many probably short increments of movement accompanied by mixing with material in the channel, in-place weathering, and contributions from O Z 1—i r~ EXPLANATION i i 11 i i i OBoulder front • O Mainly diorite from damsite O „ o Mainly normal stream alluvium • e o o ■ • . • i i i i i 11 i i I 0.5 1.0 2.0 3.0 DISTANCE FROM DAMSITE, IN MILES Figure 27.—Relation of dispersion in particle size to distance downstream from damsite. A decrease in the inclusive graphic deviation reflects increased sorting. colluvium and tributaries. However, the effect on each individual supply of sediment, as shown in figure 28, is to increase the degree of sorting of the material as it is transported downstream. Sorting is not, as would be expected, better in the riffles than in the pools. The beds of both pools and riffles composed of fill material are presently of openwork texture. Accumulation and infiltration of fines would be expected in the pools but not in the riffles, where velocity is greater, but infiltrated fines were not included by the surficial measurements. Skewness, or degree of asymmetry of the distribution curve, was calculated as the phi skewness a* (Inman, 1952, p. 130) and the inclusive graphic skewness SKr (Folk, 1964, p. 46), as follows: ,/ (016 + 084)—050. at—72 ~ ) 07V- _016 + 084—205q . 05 + 095—205q 2(084 (f>]8) 2(095 — 5) Departure from 0.0 indicates the degree of asymmetry of the distribution, absolute limits ranging between + 1.0 and —1.0. Most of the deposits are positively skewed, a characteristic of coarse fluvial gravel (table 3). A plot of the inclusive graphic skewness (fig. 28) indicates approximate uniformity of skewness in a downstream direction. The apparent slight tendency for increase of positive skewness downstream may reflect only the fact that the deposits of normal stream gravel are more positively skewed than the deposits of fill material. No definable change in skewness resulted from sedimentation of the dam fill. The material in the fill was originally positively skewed if distributed according to Rosin’s law. £ (/) UJ J— H o _ K Z O ” 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 1—i—i— Mil 1 1 1 • 5 O O • ° 0 • • a # #° ^"^Boulder front • • » • • Mainly diorite from damsite o stream alluvium i i i llli_ i i i 0.5 1.0 2.0 DISTANCE FROM DAMSITE, IN MILES Figure 28.—Relation of skewness to distance downstream from damsite.FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M31 COMPETENCE The transporting power or ability of moving water to transport debris may be expressed as competence, a measure of the maximum size of particles of a certain specific gravity that a current is able to move. Competence is commonly expressed in terms of the intermediate diameter of the largest particle that can be transported at a given flow velocity. Because of the likelihood of unstable flow and the laborious nature of indirect velocity determinations, which could have been made for only a few suitable reaches, tractive force was used as an alternative measure of competence (Leliav-sky, 1955, chap. 5). Tractive force, r=ydS, the “boundary shear” of fluid mechanics, is the product of the specific weight of the transporting medium, y, water depth, d, and the slope of the hydraulic energy gradient, S, and represents the drag force per unit area of bed surface. Because the relation applies to uniform, steady flow and does not consider solid grain stress, it must be regarded as yielding an approximation of competence. The slope of the energy gradient may be approximated by the slope of the channel surface which was measured on an enlarged part of a Y^-minute topographic map with 40-foot contours. Depth of flow was measured as the distance between thalweg elevation and high-water marks by level at previously defined crossprofile locations and by a hand level, mounted on a Jacob’s staff, at other localities. Specific weight of the transporting medium, although doubtless variable, could only be treated as a constant. A plot of changes in tractive force in a downstream direction (fig. 29) shows that variation was extreme. Flow was probably sufficiently competent throughout the course of the surge to transport boulder-size material in terms of the tractive force-particle size relations summarized by Fahnestock (1963, fig. 31). However, extrapolation of Fahnestock’s data to the much coarser detritus of this study is certainly not warranted. In spite of high competence, the fact that most large boulders were transported only short distances is shown by the striking predominance of locally derived rock types in boulder deposits along the lower unglaciated parts of the Rubicon and American Rivers. The small amount of longitudinal movement was due to the short duration of the surge, the extreme macroturbulence that rapidly moved debris to lateral areas of lowered velocity, and the fact that much erosion and supply of coarse material to the flood occurred after the flood crest. Study of the depositional forms clearly indicates that local sources of supply were a major factor in controlling particle diameter at a point. In view of the large transport capability of the surge, an attempt was made to determine competency in terms of actual size of particle moved and to extend the rela- DISTANCE FROM DAMSITE. IN MILES Figure 29.—Relation of indirectly measured tractive force to distance downstream from damsite.M32 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS tion between shear and size of particle moved. Particle size was measured on the assumption that all boulders on or only moderately buried in the bed had been moved by the flood. The following criteria were employed to indicate specific boulder movement: (1) Buried vegetation, (2) fresh abrasion on angular corners of downstream sides of blocks, and (3) presence in or on a depo-sitional form clearly related to the flood wave, as indicated by buried or protruding vegetation and comparison of preflood and postflood aerial photographs. Such an indirect approach is obviously uncertain, in part because we are dealing just with the exposed depo-sitional remains of the flood detritus, which may be only a crude approximation of the actual maximum-particle transport past the sampling point. Furthermore, the velocity or tractive force permitting deposition of a particle should be less than that required to initiate or maintain movement. However, Kramer (1935, p. 824) found that under some circumstances deposition occurred at a higher value of tractive force than that at which movement began, because of roughness differences. In addition, rolling or a settling lag (Postma, 1961) may cause movement to continue into areas of substantially reduced tractive force. The average intermediate diameter of the 10 largest particles at each of nine localities was measured to reduce the possibility that an unmoved boulder was included in the data. Measurements were restricted to the lower nonglaciated part of the channel, and only particles showing effects of abrasion were included. Angular blocks without abrasion and located near channel sides were attributed to purely lateral movement and were excluded. Data are summarized in table 4. A plot of the mean intermediate diameter of the 10 largest particles against tractive force at each of the nine localities yielded a direct relation (fig. 30), similar to what would be expected in projecting the direct measurements of competence-size relations collected and summarized by Fahnestock (1963, fig. 31). Fahnestock’s White River data are based on measurements of particles being transported and on shear calculated from mean depths; other data compiled by Fahnestock are based largely on particle erosion. The indirect method used in the present study, which utilizes depositional remains of a flow and maximum shear at the location of each particle, yields compatible results. Such an indirect determination of competence assumes that a spectrum of particle sizes up to and above the size which the flood is capable of moving is available. Field relations are considerably more complicated than suggested by figure 30. There is great longitudinal variation in particle size, only in part related to the pool and rifle pattern, along a reach in which tractive 10 50 100 150 TRACTIVE FORCE, IN POUNDS PER SQUARE FOOT Figure 30.—Relation of maximum particle size to indirectly measured tractive force. force must have been fairly constant. Measurements of maximum particle size and associated stage were made at localities that contained the coarsest particles in a given reach. The extremely coarse material in the Rubicon gorge was clearly moved only very short distances. Additional evidence for small increments of transport of coarse material are particle-size changes at the point the flow left the plutonic terrain of the upper Rubicon and crossed the contact with metamorphic rocks of the Calaveras Formation. In the channel, just upstream from the contact, the average 6-axis (intermediate axis) of the 10 largest granitic boulders measures 10.8 feet. In the metamorphic terrain 50-60 yards downstream, the 10 largest boulders of granitic composition average only 4.7 feet in diameter. The contract is below the maximum extent of glaciation. ROUNDNESS Downstream changes in particle shape were determined by application of Krumbein’s visual comparison charts (1941a, pi. 1) to photographs of the bed material, as well as to individual particles observed in the field. Roimdness is expressed as the ratio of the average radius Table 4.—Summary of maximum particle-size data Location Distance downstream from dam (miles) Mean 6-axis particle diameter Stage (ft) Slope (ft per ft) Tractive force (lb per sq ft) mm Ft Rubicon gorge; change in canyon configuration, plutonic- metamorphic contact 12.1 3,290 10.80 71.6 0.0226 101.0 Rubicon gorge 15.6 2,380 7.80 63.9 .0290 115.7 Rubicon gorge 18.2 1,450 4.74 64.1 .0179 71.6 Rubicon gorge; base of largest slide. 18.5 1,830 6.00 61.1 .0169 64.5 Unnamed bar 3 miles below M iddle Fork American River at Foresthill gaging station 38.5 497 1.63 56.2 .0039 13.7 Oregon Bar (upper) 47.2 899 2.95 53.1 .0036 11.9 Philadelphia Bar 50.7 479 1.57 45.0 .0026 7.3 Diversion dam_ 60.2 457 1.50 36.2 .0030 6.8 Oregon Bar (lower) 61.0 625 2.05 25.8 .0034 5.5FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M33 r*. to u doc 1 1—1— ' f"TT I 1 1 1— s 1 EXPLANATION O n O § i ROUNDNESS 3 O O O C - k> co a c Mainly diorite < ° uo o O o O 0C Figure 31.—Relation of particle roundness to distance downstream from damsite. Each value represents the average of at least 20 determinations. The range of values is much greater in the normal stream alluvium relative to the rockfiill diorite. of curvature of particle edges to the radius of curvature of the maximum inscribed sphere. Thus, roundness is theoretically independent of sphericity—the degree of approximation of the particle to a spherical shape—and is less sensitive to the internal structure of particles than sphericity. Changes in roundness are illustrated in figure 31, in which the average roundness of at least 20 particles is plotted for each locality. Roundness of the rockfill particles increases rapidly downstream. Studies of roundness in streams and abrasions mills indicate that the rate of rounding decreases with distance and that roundness changes after a first small increment of transportation may be marked (Krumbein, 1941b, 1942, p. 1385; Plumley, 1948, p. 559). Experimental data indicate that rate of rounding may vary considerably with grain size. The downstream grain-size decrease accompanying the rounding increase probably causes the apparent rate of rounding interpreted from figure 31 to be too low. Projection of the rate of rounding indicates that the dam fill would have obtained an average roundness corresponding to the roundness of the normal alluvial material in less than approximately 5 miles of downstream movement. Roundness was also determined for the boulders containing shotholes (fig. 32) and the values are similar to those at the sampling localities, additional evidence that the bulk of diorite within the channel below the dam is Figure 32.—Relation of roundness of cored boulders to distance downstream from damsite. redistributed fill. Several cored boulders were discovered on and in the reworked alluvial and terrace gravel occurring as berms or bars along the main channel and as terrace accretions (fig. 11). Such boulders are perceptibly more angular than cored boulders within the central part of the channel. Distribution of the depositional forms and contacts between units indicates that the lateral bars, berms, and terrace accretions were formed earlier in the flood than the thalweg deposits. A lesser degree of rounding of boulders transpored during an earlier interval of the flood passage suggests that these boulders may have been carried in suspension by macroturbulence, or in part caught up and rafted by uprooted forest debris. Roundness of the cored boulders indicates that transition from angular (0-0.15) to subangular (0.15-0.25) occurred almost immediately downstream from the damsite, using the roundness class limits of Pettijohn (1957, p. 59). Transition from subrounded (0.25-0.40) to rounded (above 0.40) in both cored boulders and bed-material samples occurred approximately 1.5 miles downstream. In comparison with other geologic and experimental studies of rounding as related to time and distance, for comparable rock types, the greatest rounding per interval of distance transported was attained in this example. The greater particle size in this example is, of course, the major cause. Downstream dilution of the samples of dam fill by indigenous, previously rounded particles was assessed and did not influence the findings. The unusually rapid rounding occurred, however, in spite of the noteworthy uniformity and apparent durability of the freshly quarried diorite (fig. 6). An additional factor adding to the high rate of rounding probably is the intense turbulence and the resulting rigor of the transport environment. Schoklitsch (1933), for example, has shown that abrasion is intensified as velocity increases. Another factor is the possibility of bedload movement as subaqueous rock flows in which attrition by grinding could have been intense. As noted above, the deposits of rockfill from the damsite are openwork gravel in which the particles are dominantly boulders. The cushioning effect of fines was decidedly less than in normal bedload transport. CAUSE OF DOWNSTREAM DECLINE IN PARTICLE SIZE Most studies have emphasized abrasion as the fundamental cause of the often reported decline in size of fluvial sediment particles downstream. Experimental abrasion studies (Krumbein, 1941b; Kuenen, 1956) have shown that clasts become rapidly rounded under condi-M34 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS tions simulating stream transport. The rate of rounding is, among other parameters, highly sensitive to particle size and composition, sorting, and position of a given particle within the distribution. A second factor, progressive sorting, is difficult to assess, but is currently believed to be of at least moderate importance. Progressive sorting involves two aspects : (1) the lagging of larger particles, caused by the differences between particle velocity and flow velocity which are related to particle size under conditions of constant or increasing competency, and (2) the progressive deposition of the coarser grades during reduction in competency. As previously shown, the bed material in the channel to a point 1.6 miles downstream from the dam originated at the damsite and has a known degree of dilution. Therefore, the effects of abrasion and progressive sorting can be evaluated. Over an interval of 1.2 miles, beginning at a point 0.4 mile downstream where rock recovery operations have not disturbed continuity of the deposits, the graphic mean size, in terms of intermediate diameter, changes from —8.0 (256mm) to <£ — 6.5 (91mm), a reduction of approximately 65 percent. Over the same interval, average rounding increases form 0.25 to 0.40. Krumbein’s experimental results (1941b, fig. 3) show that limestone pebbles undergoing the same roundness change would be decreased in weight by only about 3 percent. Composition is not critical in extending these determinations to the diorite rock-fill. In general, a fragment may become moderately well rounded without any appreciable effect on size and shape. Translation of weight decrease to the percentage decrease in particle diameter would be highly variable, depending on the individual particle configuration, but the intermediate-diameter change can be conservatively estimated as less than 5 percent. Therefore, more than 90 percent of the size decline is the result of progressive sorting. Plumley (1948, p. 570), using a different line of reasoning, concluded that 75 percent of the size decline in Black Hill terrace gravel was due to progressive sorting. This determination pointedly neglects inclusion of a factor for size reduction by breakage. Thorough study of the rockfill deposits over their entire length revealed few broken rounds or fresh fracture surfaces. Consequently, breakage was not significant in the overall size decline, because of the tough, massive character of the fresh diorite. Smaller particles produced by flaking and chipping of coarser detritus were either carried past the area of fill deposition or infiltrated the openwork boulder deposits and did not appreciably influence the mean size of the downstream surficial samples. Fracturing may be considerably more important in the big-picture view of fluvial sediments, however. According to Bretz (in Pettijohn, 1957, p. 537) broken rounds do not normally exceed 15 percent of a gravel deposit, but they may dominate, as in glacial-outwash gravel of the Snake River (Bretz, 1929). In view of the rarity of splitting in the rockfill boulders and its prevalence in older gravels, particle breakage may be chiefly due to in-place weathering along planes of weakness. The internal lithologic fabric of the particles is naturally an important consideration. Figure 29 shows that size decline in the reach below the dam occurred under conditions of nearly continuously decreasing competency. However, when graphic mean size at each locality is plotted against tractive force (fig. 33), at best only a very crude association is evident. Such a relation emphasizes the importance of local channel configuration in controlling deposition and the longitudinal as well as lateral variation in particle size within the stream channel. Although the downstream decrease in particle size of the bed material derived at the damsite clearly relates to progressive sorting and, in part, to a concomitant decrease in competence, the lagging of the larger particles because of fluctuations in flow competency is probably a significant cause of size decline on the Rubicon and Middle Fork American Rivers. Large fluctuations in maximum competency shown in figure 29 were present along the flood course, and similar changes in competency would occur for any flow in the canyon. MACROTURBULENCE The dynamics of the surge were such that turbulence was extreme, particularly in the reach below the dam where virtually no fines had yet been incorporated in the flow to dampen turbulence effects. Table 5 illustrates the effects of the turbulence as indicated by the diameter and height above the streambed of boulders deposited by the surge. Only boulders which were 10 50 100 200 TRACTIVE FORCE, IN POUNDS PER SQUARE FOOT Figure 33.—Relation of mean particle size to indirectly measured tractive force.FLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M35 freshly deposited were measured. There is a general decrease in magnitude of turbulent effects throughout the mile of channel below the dam. Below Parsley Bar, which was a source of a great quantity of fines to the flood, boulder movement did not occur much above the level of the channel bottom. Matthes (1947) described forms of macroturbulence in streams, and emphasized the efficacy of spasmodic vortex actions which cause upward suction. Such phenomena, designated as kolks by Matthes, act in swift and deep water in a manner analogous to tornados in air, develop cavitation, and can lift bedload materials and pluck fragments from a jointed rock surface. Probably, such action was responsible for deposition of boulders well above the level of the channel and was effective in removal of some bedrock, particularly in reaches underlain by jointed granitic rocks. EFFECTS OF THE SURGE RELATIVE TO NORMAL STREAM PROCESSES Steep incised bedrock canyons of mountain rivers like the Rubicon may be blocked occasionally by massive landslips, the breaching of which could yield abrupt surges similar to that resulting from the embankment failure. The Rubicon River surge was more abrupt than most natural flood waves, and its effects would not expectedly duplicate those of ordinary floods. However, abrupt surges with unusually high discharges may be significant in rock-walled canyons along which damming by landslides or ice dams is a frequent geologic occurrence. The collapse of unstable ice dams probably produced many such surges in mountainous regions during the Pleistocene. Catastrophic floods, the jokulhlaups of Iceland (Thorarinsson, 1939), are also produced by the periodic glacier-damming of lakes, which are rapidly emptied when the impounded water buoys up the ice barrier. Regardless of duration, catastrophically high flood Table 5.—Size and height above present thalweg and distance from point of origin of boulders deposited by macroturbulence 6-axis particle-sire Height (ft) Distance from damsite (miles) mm Ft 1,550 5.1 28.0 0.07 396 1.3 35.0 .08 518 1.7 30.0 .17 1,158 3.8 34.0 .18 344 .8 41.0 .19 396 1.3 36.0 .20 213 .7 28.0 .31 183 .6 22.0 .37 683 .6 20.0 .41 274 .9 17.0 .42 518 1.7 33.0 .50 213 .7 40.0 .51 366 1.2 34.0 .51 683 .6 17.0 .57 213 .7 18.0 .85 305 1.0 15.0 .85 peaks must greatly influence the morphology and sediment regimen in streams having extremely coarse bed material. The coarse detritus may be derived from glacial deposits, from gravity movements on valley sides, or by hydraulic plucking of joint blocks. The surge probably had a continuous maximum competency sufficient to transport large boulders for considerable distances; yet it did not, as indicated by tracing of material from the point source of sediment at the damsite, predominance of locally derived rock types, and size changes in a given rock type across a geologic contact. Difference between surge velocity and the slower velocity of fractionally moved particles caused entrained boulders to be soon stranded as they were bypassed by the surge peak. Lateral as well as longitudinal variations in competency were pronounced. The usual annual floods are probably competent to move most materials in the nonglaciated parts of the canyon. Flows competent to move the coarsest material in the glaciated part of the canyon and the huge blocks supplied to the channel in the Rubicon gorge are probably so infrequent that decomposition, disintegration, and abrasion to a size transportable by unusual yet expectable floods outranks movement by catastrophic discharges in dispersal of the coarsest fraction. Many large blocks included in the till and terrace deposits of the upper reaches show deep weathering rinds. The surge wave was probably at least the equal of the highest discharge attained in any part of the Rubicon River since glacial recession (within aproximately the last 10,000 years), as indicated by the erosional effects on glaciofluvial deposits in even the lower part of the river. The peak discharge of at least 300,000 cfs attributable to the surge compares with an estimated natural peak of about 40,000 cfs in the Hell Hole area. Although not as important as a much greater number of annual floods in terms of volume of sediment transported and in denudation rates (Wolman and Miller, 1960), such catastrophic floods may control the morphology and character of depositional forms in mountain streams 'because of the extreme coarseness of the bed material. The surge passage totally modified and reestablished most depositional forms along the Rubicon River. This situation contrasts with alluvial channels in which both form and patterns are attributed to events of moderate frequency (Wolman and Miller, 1960, p. 67). The amount of actual landscape sculpture produced by catastrophic floods relative to normal stream processes, in both mountain and alluvial streams, is probably small. One possibly unique aspect of such catastrophic events is the mass and wave movement of coarse bed material, such as occurred in the Parsley Bar reach.M36 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Another effect of catastrophic floods may be the triggering of periods of increased mass movement by oversteepening of slope profiles. Tricart (1961) noted that a large flood in the French Alps activated a period of intensified soil creep and slumping. Such a cycle may have been initiated in the steep-walled yet thickly mantled Rubicon gorge. Observation of the flood route in April 1966 indicated that 38 new landslides, in addition to the 32 formed at the time of the surge, had occurred during the 1965-66 wet season, one of subnormal precipitation. The slides were generally smaller in size, having an aggregate volume probably less than that of the original slides. Soil creep not present the preceding year was evidenced by generally small areas of tilted trees adjacent to the flood channel. One large area of new slippage, 1,500 feet in length by 900 feet in height, was formed on the north side of Cock Robin Point, approximately 5 miles downstream from the gaging station, Middle Fork American River near Foresthill. Another possibly unique effect of catastrophic surges is the supply of coarse sediment to the channel by lateral erosion of colluvium. However, the effects of erosion and apparent transportation of this single event indicate that the bulk of even coarse-sediment disposal may be accomplished by more frequent but less spectacular events in combination with in-place particle weathering. SUMMARY AND CONCLUSIONS 1. The partly completed Hell Hole Dam failed on De- cember 23, 1964, in response to a torrential 5-day rainfall of approximately 22 inches in the basin upstream from the damsite. Natural peak discharges in the region were generally equivalent to the maximum discharges attained in previous floods for which there is record, but the surge re1 leased at the damsite was greatly in excess of previously recorded flows along its entire route—61 miles of the Rubicon, Middle Fork American, and North Fork American Rivers. The flood surge was released over a period of 1 hour, during which the mean dischage was 260,000 cfs. Probable continuous attenuation in discharge occurred, and with the exception of the narrowing section of channel between Parsley Bar and the Rubicon gorge, reduction in height of the flood wave also occurred. Wave velocity averaged 22 feet per second. 2. Natural sources of sediment contributed to the flood included, in relative order of magnitude, stream alluvium, colluvium, glacial-outwash terrace gravel, landslides, glacial deposits, soil, and bedrock. More than 30 discrete landslides were trig- gered as a result of disturbance of slope equilibrium by scour along channel sides. Most of the mass movement occurred after the crest of the wave had passed, so large slides were not a significant source of sediment to the surge. Substantially more important as a source of sediment were the nearly continuous removal of colluvium by scour and flood-induced mass movements on a small scale. 3. A pronounced change in channel morphology oc- curred throughout the bedrock-controlled flood course, from a V-shaped channnel to one more broadly U-shaped. Erosion of channel sides was concomitant with thalweg aggradation, which was measured at five cross-profile sites and probably was nearly continuous along the flood course. Extensive aggradation in the 1.6 miles of channel below the damsite is related to deposition of rock-fill from the breached dam embankment. 4. Depositional forms in the channel were strongly in- fluenced by local sources of sediment. Terracelike berms were formed along channel walls in the reach below the dam, in which flow was highly macroturbulent, turbulence was not damped by included fines, and boulders were transported in suspension. As much as 5 feet of boulder gravel was added to the surface of a glacial-outwash terrace 28 feet above the presurge thalweg at a peak stage of 45 feet relative to the presurge thalweg. 5. Boulder fronts as much as 7 feet high and 250 feet long were formed transverse to flow direction in an expanding reach with a declining gradient. Such forms are interpreted as the snouts of discrete flows of bed material rather than waveforms, but wave movement also occurred. The evidence for movement of bed material as viscous subaqueous rockflows includes: (a) The presence of boulder fronts in the channel, (b) the presence in the deposits associated with the boulder fronts of extremely coarse particles with diameter equal to the thickness of the deposit, (c) the unusually poor sorting of the deposits, (d) the occurrence of the boulder-front deposits in reaches where the maximum tractive force applied by the surge did not approach the critical tractive force necessary to move the coarser individual particles. The terracelike boulder berms are believed to represent backwater deposition of bedload moved in this manner or by wave movement. 6. Some change in the pool-riffle pattern resulted from the surge passage, but because of the scale of all preflood aerial photographs, change could be documented at only one locality. New riffles wereFLOOD SURGE ON THE RUBICON RIVER, CALIFORNIA M37 formed by lateral supply of coarse material by slides and by cutting of boulder fronts, and some presurge riffles were removed. 7. The rockfill from the damsite moved as mass flows or as a series of waves without mixing in the channel thalweg, as far as 1.6 miles from the damsite, and a single boulder traveled 2.1 miles. Deposits above and lateral to the channel in this interval are attributed to an early phase of the surge, probably near the crest, and consist in large part of normal stream gravel and reworked terrace gravel. Compositional contrast between the two episodes of deposition from a single flood wave is pronounced. 8. Movement of the rockfill from the damsite demon- strated a pronounced downstream decline in mean diameter as well as an imperfect increase in degree of sorting. No regular change in skewness was detected. 9. The downstream decline in particle size occurred under conditions of continuously decreasing competency, as indicated by indirect measurements of tractive force. Sporadic measurements of tractive force along the rest of the flood route show that longitudinal variation in maximum competency was substantial. The competency of the flood was also determined by measuring the size of boulders moved. Maximum tractive force attained at a locality is related to the maximum size of boulders whose movement could be attributed to the flood, and the relation was compatible with previous studies of tractive force and particle size. With the exception of coarse material supplied laterally to the channel at a late stage in the surge when flow was not sufficiently competent to cause movement, virtually all detritus in the channel was moved. Distances traveled by individual boulders were small, however. 10. Rounding of the rockfill occurred more rapidly over a much smaller increment of transport than was anticipated from previous field and experimental studies; this unusual rate of rounding is explained by the coarseness of the material, the unusual rigor of the transport conditions, and the possibility of transport in subaqueous rock flow. Boulders carried in part in suspension at or near the surge peak and deposited on lateral bars are not as well rounded as those deposited in the thalweg. The massive diorite boulders were rounded after only 1.5 miles of downstream movement. 11. Consideration of the relative importance of abrasion and fracturing versus sorting effects indicates that progressive sorting is the dominant cause of the downstream decline in particle size of the rockfill. 12. If the effects of the surge passage are analogous to natural catastrophic floods, jokulhlaups, and those caused by the breaching of landslide dams, such floods, in mountain streams with bed material of boulder size, may set and control pool-riffle patterns, initiate periods of increased mass movement, and give rise to subaqueous mass flow in the channel. Another striking result of the surge was the introduction into the channel of large quantities of valleyside material, a major source of coarse material. Greater dispersal of the coarse sediment fraction is accomplished by normal floods in combination with size reduction by weathering than by natural catastrophe floods, however. REFERENCES CITED Bretz, J H., 1929, Valley deposits immediately east of the channeled scabland of Washington: Jour. Geology, v. 37, p. 505-541. Brush, L. M., Jr., 1961, Drainage basins, channels, and flow characteristics of selected streams in central Pennsylvania: U.S. Geol. Survey Prof. Paper 282-F, p. 145-181. Emery, K. O., 1955, Grain size of marine beach gravels: Jour. Geology, v. 63, p. 39-49. Fahnestock, R. K., 1963, Morphology and hydrology of a glacial stream—White River, Mount Rainier, Washington: U.S. Geol. Survey Prof. Paper 422-A, 70 p. [1964], Folk, R. L., 1964, Petrology of sedimentary rocks: Austin, Tex., Hemphill’s, 154 p. Gwinn, V. E., 1964, Deduction of flow regime from bedding character in conglomerates and sandstones : Jour. Sed. Petrology, v. 34, p. 656-658. Inman, D. I., 1952, Measures for describing the size distribution of sediments: Jour. Sed. Petrology, v. 22, p. 125-145. Kramer, H., 1935, Sand mixtures and sand movement in fluvial models: Am. Soc. Civil Engineers Trans., v. 100, p. 798-838. Krumbein, W. C., 1941a, Measurement and geological significance of shape and roundness of sedimentary particles: Jour. Sed. Petrology, v. 11, p. 64-72. ------1941b, The effects of abrasion on the size, shape, and roundness of rock fragments: Jour. Geology, v. 49, p. 482-520. ------1942, Flood deposits of Arroyo Seco, Los Angeles County, California: Geol. Soc. America Bull., v. 53, p. 1355-1402. Krumbein, W. C., and Tisdel, F. W., 1940, Size distributions of source rocks of sediments: Am. Jour. Sei., v. 238, p. 296-305. Kuenen, Ph. H., 1956, Rolling by current, pt. 2 of Experimental abrasion of pebbles: Jour Geology, v. 64, p. 336-368. Leliavsky, S., 1955, An introduction to fluvial hydraulics: London, Constable and Co., Ltd., 257 p. Leopold, L. B., Wolman, M. G., and Miller, J. P., 1964, Fluvial processess in geomorphology: San Francisco, W. H. Freeman and Co., 522 p. Matthes, F. E., 1930, Geologic history of the Yosemite Valley: U.S. Geol. Survey Prof. Paper 160,137 p. Matthes, G. H., 1947, Macroturbulence in natural streamflow: Am. Geophys Union Trans., v. 28, p. 255-262. Pettijohn, F. J., 1957, Sedimentary rocks [2d ed.] : New York, Harper and Bros., 718 p.M38 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS Pettijohn, F. J., and Potter, P. E., 1964, Atlas and glossary of primary sedimentary structures: New York, Springer-Ver-lag, 370 p. Philbrick, S. S., and Cleaves, A. B., 1958, Field and laboratory investigations, in Landslides and engineering practices: Highway Research Board Spec. Rept. 29, Nat. Resources Council Pub. 544, p. 93-111. Plumley, W. J., 1948, Black Hills terrace gravels; a study in sediment transport: Jour. Geology, v. 58, p. 526-577. Postma, H., 1961, Transport and accumulation of suspended matter in the Dutch Wadden Sea : Netherlands Jour. Sea Research, v. 1 (2v.), p. 148-190. Schoklitsch, A., 1933, Ueber die Verkleinerung der Geschiebe in Flusslaufen: Akad. Wiss., Wien, v. 142, p. 343-366. Sharp, R. P., and Noble, L. H., 1953, Mudflow of 1941 at Wright-wood, southern California: Geol. Soc. America Bull., v. 64, p. 547-560. Simons, D. B., and Richardson, E. V., 1961, Forms of bed roughness in alluvial channels: Am. Soc. Civil Engineers Hydraulics Jour., v. 87, no. 3, p. 87-105. Thiel, G. S., 1932, Giant current ripples in coarse fluvial gravel: Jour. Geology, v. 40, p. 452-458. Thorarinsson, Sigurdur, 1939, The ice-dammed lakes of Iceland with particular reference to their values as indicators of glacier oscillations: Geog. Annaler, v. 21, p. 216-242. Trask, P. D., 1932, Origin and environment of source sediments of petroleum: Houston, Tex., Gulf Publishing Co., 323 p. Tricarte, J., et collaborateurs, 1961, Mdcanismes normaux et phteomenes catastrophiques dans 1’ Evolution des versants du bassin du Guil (Htes-Alpes, France) : Zeitschr. Geomor-phologie, v. 5, p. 277-301. Vames, D. J., 1958, Landslide types and processes, in Landslides and engineering practice: Highway Research Board Spec. Rept. 29, Nat. Resources Council Pub. 544, p. 20-47. Walker, R. G., 1963, Distinctive types of ripple-drift cross-lamination : Sedimentology, v. 2, p. 173-188. Wolman, M. G., 1954. A method of sampling coarse river-bed material: Am. Geophys. Union Trans., v. 35, p. 951-956. Wolman, M. G., and Miller, J. P., 1960, Magnitude and frequency of forces in geomorphic processes: Jour. Geology, v. 68, p. 54-74. Young, L. E., and Cruff, R. W., 1966, Magnitude and frequency of floods in the United States; pt. 11, Pacific slope basins in California—Coastal basins south of the Klamath River basin and Central Valley drainage from the west: U.S. Geol. Survey open-file rept, 70 p.INDEX [Italic numbers indicate major references] Page Alluvial deposits, erosion.......................................... M12 Amador Group............................................................ 5 American River, drainage area.............................................. 2 Amphibolite................................................................ 5 Arid Transition life zone.................................................. 3 Auburn gaging station, discharge......................................10,12 Bars, downstream changes in patterns...................................... 27 Bed material, transport and deposition.................................... 27 Bed-material forms........................................................ 26 Bedrock, erosion.......................................................... 12 Berms, deposition......................................................... 18 downstream changes in patterns........................................ 27 Biotite quartz diorite..................................................... 4 Boulder bars.............................................................. 19 Boulder fronts............................................................ 20 Boulder jams.............................................................. 21 Boulder movement, criteria indicating..................................... 32 episodic..........................................................- 23 Bridges, destruction................................................... 6 Calaveras Formation..................................................... 4,14 Channel morphology, effects of flood surge.............................. 13 Cherokee Bar, erosion..................................................... 13 Chlorite schist........................................................... 5 Climate................................................................... 3 Cock Robin Point, slippage................................................ 36 Colluvium, erosion................................................12,13,36 Competence, water......................................................... 31 Crossbedding.........................................................21,23,26 Current-ripple lamination................................................. 26 Delta movement............................................................ 21 Deposition, bed material.................................................. 27 D epositional features.................................................... 17 Diabase.................................................................... 4 Dikes...................................................................... 5 Diorite...........................................................4,18,27,33 Discharge, downstream from Hell Hole Dam.................................. 12 flood surge....................................................... 6,10 Rubicon River......................................................... 35 Dune movement............................................................. 21 Embankment failure......................................................... 6 Erosional effects of the surge............................................ 12 Floods, 1955-1963 ......................................................... 9 1964 .................................................................. 8 Folsom Lake, flood surge................................................... 1 precipitation ranges................................................... 3 Foresthill gaging station............................................. 9,12 French Alps, flood........................................................ 36 Gabbro..................................................................... 4 Geology.................................................................... 4 Geomorphic history, Rubicon Canyon......................................... 5 Georgetown, storm runoff.............................................. 10 Georgetown gaging station............................................... 9,10 Glaciation............................................................... 5 Grading, berms............................................................ 18 Granodiorite............................................................... 4 Greenstone............................................................... 5 Page Hell Hole, precipitation ranges............................................ M3 Hell Hole Dam, construction................................................. 4 failure................................................................. 6 inflow and outflow..................................................... 10 Hydrographs................................................................ 10 Inflow, Hell Hole Dam...................................................... 10 Joints...................................................................... 5 Kolks.................................................................... 35 Landslides.............................................................H, 36 Lateral berms.............................................................. 18 Location of area............................................................ 2 McCreary-Koretsky Engineers............................................ 6,10 Macroturbulence......................................................... 19,54 Manning equation........................................................... 12 Mariposa Slate.............................................................. 5 Mass movements............................................................. 14 Meteorological conditions, time of flood.................................... 8 Middle Fork American River, berms.......................................... 19 flood surge............................................................. 1 gaging stations........................................................ 10 peak discharge......................................................... 12 pool and riffle pattern................................................ 25 slope................................................................... 2 Middle Fork American River Project.......................................... 6 Mother Lode belt............................................................ 5 Mountain-valley stage, Sierra Nevada development............................ 5 North Fork American River, berms........................................... 19 flood surge............................................................. 1 pool and riffle pattern................................................ 25 slope................................................................... 2 Oregon Bar, erosion........................................................ 13 Outflow, Hell Hole Dam..................................................... 10 Parsley Bar, bed material.................................................. 27 boulder bars........................................................... 19 boulder front...................................................... 20,23, 24 geology................................................................. 4 loss of timber......................................................... 13 pool and riffle pattern................................................ 25 stratigraphy............................................................ 5 Particle size, cause of downstream decline.............................- - 33 flood material......................................................... 27 Photography, channel forms................................................. 17 Physical setting............................................................ 2 Pleistocene ice terminus, Rubicon River canyon.............................. 2 Pleistocene uplift.......................................................... 5 Pliocene uplift............................................................. 5 Pool and riffle pattern................................................- - 25 Portland-Pacific Cement Co. bridge.......................................... 6 Poverty Bar, erosion....................................................... 13 Precipitation, annual....................................................... 3 December 19-23.......................................................... 8 Pyramid Peak................................................................ 2 M39M40 INDEX Rosin’s law. ............................................. Roundness, flood material................................. Rubicon River, discharge.................................. drainage area.......................................... flood surge............................................ gaging stations........................................ pool and riffle pattern................................ Rubicon River canyon, geomorphic history and stratigraphy. landslides............................................. slope.................................................. Rubicon Springs, discharge................................ Rubicon Springs gaging station............................ Runoff, floods............................................ Sacramento................................................ Sand wave.................................................. Sedimentary structures, internal.......................... Seismic measurements, landslides.......................... Sierra Nevada............................................. Sierra Nevada floods, effect of snowpack.................. Silver Creek, storm runoff................................ Skewness, flood material.................................. Slope-conveyance method, discharge........................ Snow, effect on flood..................................... Soil, erosion............................................. Page Soil creep.........................................-..........-........... M36 Sonoran Transition life zone................................................. 4 Sorting, flood particles.................................................... 34 Stage, surge wave.........................................................- - 12 Storm, December 1964......................................................... 3 Stratigraphy, Rubicon Canyon................................................. 5 Streambed material, composition...............................-........... 27 Structure, internal......................................................... 26 major trends.............................................................. 5 Temperatures, D ecember 21-23.............................................. 8 Terrace accretion........................................................... 19 Terrace levels, Rubicon River canyon.......................................... 5 Timber, effects of flood..................................................... I3 Tractive force.............................................................. 31 Transport, bed material...................................................... 27 Trask sorting coefficient................................................. 29 Twin Lakes, snowpack...............................................-...... 8 Union Valley Dam, storm runoff............................................... 1° Vegetation.................................................................... 3 Wave height, flood surge..................................................... 12 Wave velocity, flood surge................................................ 8,12 Page M30 32 10 2 1 9 25 5 14 2 10 9 9 2 21 26 17 1 8 10 30 12 8 12