NOAA Technical Report ERL 377-APCL 39 A Three-Dimensional Simulation of Winds and Non-Precipitating Orographic Clouds Over Hawaii Everett C. Nickerson Elemer L. Magaziner September 1976 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration Environmental Research Laboratories V V • ■ *i ' !j*£JJ222^ NORfl NOAA Technical Report ERL 377-APCL 39 A Three-Dimensional WA Simulation of Winds and — . Non-Precipitating Orographic Clouds Over Hawaii Everett C. Nickerson Elemer L. Magaziner Atmospheric Physics and Chemistry Laboratory Boulder, Colorado September 1976 P ft t 9 U.S. DEPARTMENT OF COMMERCE Elliot Richardson. Secretary National Oceanic and Atmospheric Administration Robert M. White. Administrator Environmental Research Laboratories Wilmot Hess. Director t< o*- UT, Ov .% 'JS*** S Plates Plate 1. Photograph of a model of Hawaii at the City of Refuge National Historical Park, Hawaii. The color shading indicates the approximate distribution of vegetation and lava flows. Plate 2. Photograph of the windswept trees southwest of Waimea. Plate 3. Photograph of the Kohala cloud from a loca- tion southwest of Waimea. Plate 4. Photograph of Hawaii taken from Apollo 9. The NASA caption reads: "Here the island of Hawaii looks 1 i ke a dark-visaged monster peering up out of the sea beyond the lunar module in the foreground. The snow-tufted peaks of Mauna Loa and Mauna Kea resemble eyes, and the wedge-shaped cloud bank where the trade wind was flowing over the saddle between those peaks suggests a brow. There are often moisture-laden clouds on the wind- ward side of a high tropical island and scarcely any on the leeward side." Arrows A-A point to the summit of Mauna Kea; arrows B-B point to the summit of Mauna Loa. m Plate, 1 Plate. 2, iv Plato. 3, Plate. 4, v CONTENTS Page Plate 1 iv Plate 2 iv Plate 3 v Plate 4 v Abstract 1 1. INTRODUCTION 1 2. BASIC EQUATIONS 2 3. LOWER BOUNDARY CONDITIONS 5 4. PLANETARY BOUNDARY LAYER 7 5. VERTICAL GRID 9 6. HORIZONTAL GRID 10 7. TIME DIFFERENCING 11 8. SPACE DIFFERENCING 11 9. SMOOTHING OPERATOR 15 10. BOUNDARY CONDITIONS 15 11. DISCUSSION 15 12. INITIAL CONDITIONS 19 13. MODEL RESULTS 19 14. CONCLUSION 26 15. ACKNOWLEDGMENTS 27 16. REFERENCES 27 APPENDIX A. BASIC NOMENCLATURE 29 APPENDIX B. TERRAIN PARAMETERS 31 APPENDIX C. EXCERPT FROM U.S.W.B. CLIMATOLOGICAL SUMMARY 34 VI 1 A THREE-DIMENSIONAL SIMULATION OE WINDS AND NON-PRECIPITATING OROGRAPHIC CLOUDS OVER HAWAII Everett C. Nickerson and Elemer L. Magaziner A mesoscale model is presented, designed with the capability of simulating moist, non-precipitating, three-dimensional flow over mountainous terrain. The model makes use of a transformed sigma coordinate system to assure adequate resolution in the bound- ary layer. A preliminary evaluation of the performance of the model has been obtained from simulations of airflow and orograph- ically induced cloud cover over the island of Hawaii. 1. INTRODUCTION Numerical models on an operational basis are now capable of reasonably accurate 48-hour forecasts of synoptic scale disturbances. However, those significant weather events that greatly affect our daily lives, such as heavy precipitation and damaging winds, are usually on a much smaller scale and therefore not directly resolvable by the large scale models. Detailed fore- casts for specific areas make use of information contained in the fields of large scale convergence, flow patterns, moisture supply, and atmospheric stability as well as local climatological information. That is, synoptic scale predictions of wind, temperature, moisture, and pressure fields do not in themselves indicate the occurrence of severe weather in a given locality, but their structure may indicate to an experienced person the likelihood of such occurrences. If you examine the status of current modeling efforts on the scale of individual convective clouds, you find that three-dimensional time dependent simulations are now possible. Many microphysical questions need to be re- solved and sub-grid scale closure poses a difficult challenge; nevertheless, the numerical simulation of isolated convective clouds is technically feas- ible. The fact that significant progress has been made in the numerical model- ing of these two dissimilar scales of motion is not fortuitous but is rather a manifestation of the energetics that characterize atmospheric motions. Current evidence suggests the existence of a bimodal distribution in the atmospheric kinetic energy spectrum, with peaks roughly at 4 days and at 1 minute(see Fiedler and Panofsky, 1970). The first peak is associated with synoptic scale baroclinic disturbances and the second with convective activity, Between these two peaks lies a broad minimum or "mesoscale gap." Strong controls on mesoscale systems are therefore exerted by the synoptic scale. These controls, manifesting themselves as time-dependent lateral boundary conditions for mesoscale models, also provide the forcing that leads to energy conversions on smaller scales. The specification of sub-grid scale processes is even more difficult for mesoscale than for synoptic scale mo- tions since the percentage of area covered by active updrafts may not be as small as is presumed in some parameterization schemes. A mesoscale model must therefore provide realistically for larger scale forcing as well as for sub-grid scale convective transports of a rather general nature. Significant progress in the modeling of mesoscale disturbances is essen- tial for more accurate forecasts of the damaging aspects of severe storms. Mesoscale models are also needed for the planning and evaluation of future attempts to modify severe storms, since fiscal constraints would prohibit the gathering of statistically significant data. Cumulus parameterization theory is being developed on a very sophisti- cated level for general circulation models, and yet the local recycling of mass and momentum by individual clouds is still not well understood. Despite the complexity of the problem, significant progress in developing a mesoscale model can still be made by formulating the basic equations to treat stable precipitation and other important physical processes. The details of a three-dimensional terrain model designed to simulate three-dimensional air- flow over complex terrain are presented in sections 2 to 10 of this report. Preliminary results using the island of Hawaii as a test case are contained in sections 11 to 14. 2. BASIC EQUATIONS In the presence of favorable synoptic scale conditions, the vertical fluxes of heat, momentum, and moisture within the atmospheric surface layer can exert a strong influence on the location, duration, and intensity of severe convective storms. Satellite pictures have clearly shown that pre- existing cloud cover can effectively suppress surface heating over a wide area and thereby prevent the development of convective activity (Weiss and Purdom, 1974). Moreover, variations in terrain, coupled with unevenness in surface heating, can produce slope winds, mountain-valley winds, and sea breeze circulation systems. A constant flux layer 100 m or more in depth is not at all uncommon during periods of strong insolation; however, the depth of that layer may decrease by an order of magnitude during nighttime hours. In order to provide for increased vertical resolution at the lower boundary and still to retain the advantages of a uniform grid, the following coordinate transformation was incorporated into the model: a = (4v-v 4 )/3 (1) where a is the conventional normalized pressure coordinate, 2 The basic equations for the Nu coordinate system are as follows: 3U_ 3t 3(Uu) 3(Uv) 1 a(o^Uv) fV / RT*aTT \ 3tt 3(ir uq VS q v = W/tt q = ) W ■; Trq TW ^VS T = T uns The saturation vapor pressure with respect to water, e s , used to compute q vs is taken from Murray (1967), e = 6.11 exp [17.27(T-273.16)/(T-35.86)1 . (21) 3. LOWER BOUNDARY CONDITIONS A constant flux layer is presumed to exist between the lower boundary (with roughness length z ) , and the first grid point above the surface, z = h. The friction velocity and surface fluxes of sensible and latent heat are computed using the Bussinger-Dyer surface layer formulation (see Nicker- son and Smiley, 1975) . ^= F (r 0), the functions F and G have the form kF = In (c-lKco+D" -(ctD(co-i'). + 2 tan" 1 c -2 tan" 1 5, (30) kG = R In (n 2 -D(nH) L(n 2 +D(n§-l)J (31) where ? = (1 - yz/L) 1 ^ , (32) Co ■ (1 - YZ /L) l/,f , (33) n = (1 - y"z/L) 1/t+ , (34) no = (1 - Y"z /L) lA , (35) For mildly stable conditions (i.e., z/L < 1), the functions F and G have the form kF = In (z/z ) + Bz/L , (36) kG = R In (z/z ) + 6z/L . (37) For z/L > 1, the functions F and G are obtained from profiles given by Webb (1970). kF = 3 In (z/L) + In (z/z ) + e (38) kG = (1 + 3 - R) In (z/L) + R In (z/z ) + B . (39) At the present stage of model development the surface temperature and mixing ratio are specified initially and not allowed to change during the course of the model run. 4. PLANETARY BOUNDARY LAYER A boundary layer is presumed to exist between zg, the first grid point above the surface and z/\, some specified height where the winds become de- coupled from surface layer effects. At the present time z/\ is set equal to one kilometer. Exchange coefficients for momentum, K(u), and for the temp- erature and moisture, K(T), between z/\ and Z3 are computed in accordance with the profile given by O'Brien (1970). K = K A + [(z-z a )/(az)2J [k b - K A + (z-z B )(Kg+2 (K R -K A )/Az)] (40) The prime denotes a derivative with respect to z, and az = (z/\ - zg). Both K(u) and K(T) are set equal to zero for heights greater than z/\. More- over, horizontal friction is not explicitly included in the model except at the top and on the lateral boundaries (see section 10). The frictional terms in (2), (3), (4), and (5) are written as F u " A h (a«u) &) (41) K = A It: (ak(U) §£ ) (42) V 3v F s ■ A k ( AK(T) #) (43) F w ■ A h ( AK(T) Hf) • (44) where A = ar^? • At the lower boundary (v = a = 1), AK(u) |ij-= tt U 2 cos a (45) AK(u) |^= tt U 2 sin a (46) AK(T) || =n (V + ^e) (46) 3W AK(T)^=ttQ 6 (48) where tan a = V^/U^, the ratio of the two wind components at the first grid point above the surface, and where •"Q s = (e -e h ) /u h + v h /FG < 49 ) 7rQ e = (q -q h ) /U2 + V2 /FG (50) 5. VERTICAL GRID The vertical grid consists of 15 equally spaced levels in the Nu coor- dinate system. With reference to Figure 1, the variable v is defined at the circled levels, and all other variables are defined at the crosses. Vertical distances in the Nu coordinate system range from zero to unity, just as in the sigma system. Furthermore, equation (1) satisfies the condi- tion noted by De Rivas (1972), that da/dv must be finite over the entire do- main and must be equal to zero at v = a = 1 in order to assure second order accuracy in the discretization scheme. The vertial grid corresponding to the 15 levels in the model is shown in Table 1, together with the corresponding sigma levels. The first grid point is located approximately 18 meters above the lower boundary. Table. 7. CompaJvUon ofa the. Two Cooh.cU.nate. SyAtemi' UeAticat GiicU 0.0000 0.0000 0.0333 0.0444 0.1000 0.1333 0.1667 0.2220 0.2333 0.3101 0.3000 0.3973 0.3667 0.4829 0.4333 0.5660 0.5000 0.6458 0.5667 0.7212 0.6333 0.7908 0.7000 0.8533 0.7667 0.9071 0.8333 0.9504 0.9000 0.9813 0.9667 0.9978 1.0000 1.0000 6. HORIZONTAL GRID A staggered grid (fig. 2) is used to reduce the truncation errors on the level Nu surfaces (see Anthes and Warner, 1974). The variables u and v are defined at the crosses, while all thermodynamic variables are defined at the dots. The variable v is not defined on this horizontal level but at dis- tances Av/2 above or below this level and at positions corresponding to the dots. For the runs reported on in this paper, Ax and Ay have both been set equal to 10 km. With 26 grid points in each direction, this corresponds to an area 250 km on a side. wmmmmmmm. Grid Level Condition on Nu i/=0 X 1 o X 2 o X 3 o X 4 • • • X 13 o X 14 o Aj/=1/15 X 15 v=\ i-1,j+1 i-1. I-1.J-1 Ax i,j + 1 i + 1.j + 1 i,j i+1, i.j-1 i + 1.j-l , A V FiguAe. I. VeAllcal qfuA used In the. VIquJkl 1. Horizontal, qhld u^ed In model calculation*. the. model calculations. 10 7. TIME DIFFERENCING Centered differences are used to represent the time derivative, except that every fifth time-step a Time and Space Uncentered Matsuno procedure is introduced (see Arakawa and Mintz, 1974). If, for example, we are dealing with the equation oT' F - < 51 > then the TASU - Matsuno scheme is as follows A* (n+1) A n n A At ' F C< A ' A"*' - A" = F (A *(n*D, At ur x ' A *(n+2) . fl n+l _ . At c v ' Forward Centered fl n+2 fl n+l *f n +:M A A + = F ni (A ^ n+6 >) Backward Lower-Left At Time Space Forward Centered Backward Upper-Right 8. SPACE DIFFERENCING The horizontal grid and the discretization scheme that follows are based on Anthes and Warner (1974). The staggered grid has a smaller trunca- tion error than a non-staggered grid. Perhaps more important, however, is the fact that boundary values of the velocity components appear only in the flux terms of the momentum equations and not in calculations using the con- tinuity equation. The finite difference analogs of equations (2) to (8) are (w) = D i(U) + D,(U) + D 3 (U) + fV + D 6 (U) (52) 11 (if) = °i (V) + D ^ y) + D 3(V) - fU + D 7 (V) (53) I) = D 2 (S) + DJS) + D 5 (S) / i+^j+^k (54) (t-£) = D 2 (W) + DJW) + D 5 (W) (55) / \ 15 (|f) = - Av Z a'[D 8 (U) + D 9 (V) (56) (vaM i+v . i+i , ,^ = - Av £ a' f£ + D 8 (U) + D 9 (V) i T isJ T iji\ — a k*=l *- 3tt (57) *H%,j-tfs,k * ♦l-H s ,j-H Sf k+l + C P [e k+ ,(l + 0.6 lVk+!2 (f k+1 -P k ) i+ ^ + J(58) where the following definitions have been made: Di(x) = - 4a k [< U 1+1J^1 J> (X 1*jV X i+k,3-k ] - W L (V i,J+l + V i,j )(r i^,j^a + V 2 ,W i-^J-^J - (V. . + V. . ,)(X.,, • , + A j_ r a'Av |_ D 2 (X) = ^777 |(*a'A) k+Jg - ( va A*] 12 D 3 (X) (Av) 2 (X k+1 - A k )(AK(u)) - (A k - X^flWu)^. D ^ (A) = " 4 [< U i + l,j + l + U 1 + 1J« X W/2. W + V^W ■ (U 1.W + U 1.J )(X 14%.W + ^ H |J4I|)] (V i + l,j + l +V i, j + l^ A i^, j + 3/2^ i+ , 2 , j+ , 2 ) 4Ay D * (u) = 2Z7 ['n^j* ' ¥ 1-W* + '1-Mk.JJk " "i-H,3-H - ( ^i-^j-*] D 7 (V) = 2Ay p+Jg.j+Js " "l+^j-Jg + ^-JS.J^ " *1-%,J-%"J " W [ (7r * ) 1^.J+ J s - ( *Wj-% + ^l-W* " ^i-W-* D 8 (U) 2ax J i + ij + i- u ij + i + u i + i,j- u i,j 13 Mv>-^ [ v i+i.j+i " V M.J. + v i.j« " v u] U = TTU V = TTV \+k 2 ^ X k + X k+1^ X i+ia,J+% = 4 (x i+l,j+l + X i+l,j + X i,j+1 + X i,j } C = $ - RT*cnr/P The vertical velocity, dz/dt, is also calculated in order to assist in the interpretation of the results. This is especially important for computa- tions above steep mountain slopes where the usual approximation w = - u/pg can even give the wrong sign. The vertical velocity is given by w = 1 i+%,j+%,k g , (<|, i+3/2,j+is_" ^yWk - 2ax i+^j+^k , ( *i+%,j+3/2 ' *i+3 s ,j-3g ) k - 2Ay i+^j+^k ■(ttI^^W,^] 14 9. SMOOTHING OPERATOR In order to suppress 2ax noise in the flow fields, a smoother (see Shapiro, 1970) is applied to the predicted fields of ttu, and ttv. A smooth value of a variable, A, is defined by: *-F< X 1+lJ+l +A i-l,j + l +A i + l,j-l +X i-l,j-l +4> i,j ) 10. BOUNDARY CONDITIONS At the upper boundary (a = v = 0), v is set to zero. This is a reason- able condition when Pj = 0, but may lead to difficulties when Pj is greater than 100 mb. Quantities such as wind and temperature are set equal to values at the grid point immediately below the top of the model. Actual values chosen are not of great importance, however, since the equations are written in flux form (i.e. v-rru) and the terms containing boundary values vanish on the upper boundary due to the condition on v. The thermodynamic variables are specified at all boundaries. Winds are specified on inflow boundaries and extrapolated from interior values on out- flow boundaries. In addition, a viscous term is added to the predictive equations for U and V on the lateral and top boundaries. This is applied only at the first two interior grid points and is a simple Laplacian operator evaluated at the (n-1) time level. 11. DISCUSSION Model validation is an important part of any model development program. Before a model can be accepted as a useful operational or research tool, it must first be demonstrated that the model is capable of simulating signifi- cant features of the prototype data. The selection of a validation site is an especially important step in the evaluation of a complex meteorological model, because a site with well-defined patterns in the winds, precipitation, and cloud cover can provide many opportunities for comparison between model results and prototype data. The more features that a model is able to repro- duce, the greater the confidence that can be placed in calculations for loca- tions lacking a good data base. Moreover, in view of the numerous options available to the numerical modeler, such a site can also assist in the formu- lation and specification of computational boundary conditions, smoothing operators, and sub-gridscale parameterizations. Within this context, the island of Hawaii (Plate 1, Figs. 3-4) appears to be an ideal prototype for a mesoscale model development program. The 15 220 190 | 160 100 - raphy (km) 5015 70 60 Viga/ie 3. Elevation tioplethi ion. Hawaii. ContouA linei> wene dnawn aiing linean. interpolation between the value* given in Appendix B. Tku> fiiguAe aLso 6 hom the location* ofi the vertical caoaa section* coKxehpond- ing to {iawteh 19-23. 120 150 East-West (Kilometers) 180 210 relatively smooth topography, extending from sea level to 4.2 km, is charac- terized by barren lava fields, dry grasslands, rolling fields of sugar cane, and tropical rain forests. Considerable data have been gathered on rainfall and other meteorological parameters during Project Shower (Mordy et al . , 1957), and also during the Warm Rain Project (Lavoie, 1966, and Lavoie et al., 1967). Part of a concise introduction to the meteorology of the island (NOAA, 1972) is reproduced in Appendix C. The importance of the twin peaks Mauna Loa and Mauna Kea to the island meteorology cannot be overemphasized. Leeward convergence of the anti- cyclonic flow south of Mauna Loa and the cyclonic flow north of Mauna Kea results in a suppression of the vortex streets commonly observed in the cloud patterns downwind of tall islands. This zone of horizontal convergence (see Fig. 5, taken from Patzert, 1970) apparently interacts with the sea breeze to produce a secondary rainfall maximum on the leeward (Kona) coast (Chopra, 1973). The Kona rainfall maximum and the sharp demarcation between wet and dry areas are vividly illustrated in Fig. 6, which shows Lavoie' s (1966) VigvJie 4. TenJuzin peupec- tive oft Hawati. Locations ol individual. gnJA points coincide with the inteh.- i>ectioni> ofa north-south and eaAt-weAt lines. The view ij> ^rom northeast (50°) looking southwest. 16 154 F-uju/ie 5. Compo&itz iu/ifiace. uxlnd faeZd ovqa Hawaiian ioatc\b taken hnom anpubtuked IfJeatheA Buaqjclu. Vata, Honolulu (796$), niton PatzoAt [1970) . 17 5 10 Mites Vi.QuJi2. 6. RcUnfiaLt ion. the, poAlod 11 July to 24 AuguAt 1965 inom Lavoln [1966). l&ohyetA an.2. labeJLzd -In -lnchej>. summer rainfall distribution. Despite the occurrence of different wind regimes associated with winter storm systems, the locations of the precipi tation maxima and minima in Fig. 6 also coincide with those of the annual pattern. 18 12. INITIAL CONDITIONS During the fully developed and undisturbed trade wind regime, the air in the lowest few kilometers of the Eastern Pacific high has ample time to adjust to conditions at the ocean surface before encountering land. Under those conditions, topographical features or other sources of horizontal variability outside the computational domain ought to have little influence on the island weather. Representative inflow boundary data could therefore be obtained from an undisturbed trade wind sounding. The sounding station at Lihue, Kauai would probably provide the most representative data on a routine basis. Initial conditions of wind, temperature, and mixing ratio for the numer- ical calculations are shown in Figs. 7, 8, and 9. The initial wind field contains easterlies at low levels and westerlies aloft. The temperature at the lower boundary remains constant in time, and decreases linearly with altitude from its sea level value of 299°K (see Fig. 10). The temperature at the lower boundary has not been related to land use or vegetation; however, limited use of that information has been made in specifying the roughness length and mixing ratio at the surface. Vegetation and soil conditions were estimated from U.S.G.S. woodland print topographic charts. Roughness lengths were then obtained from Priestley (1959), Sellers (1966), and Sutton (1953). Terrain heights for individual grid points to- gether with the corresponding surface parameters are listed in Appendix B. 13. MODEL RESULTS All model results reported in this section were calculated in a terrain- following coordinate system. However, it is often more convenient to display the data using height as the vertical coordinate. Since the height of each grid point is known, the value of a guantity at some intermediate location is calculated using linear interpolation between two computational levels. Horizontal windfields corresponding to elevations of 100, 1000, 1500, 2500, and 4000 meters above sea level are shown in Figs. 11-15. The low level convergence zone on the left side of Fig. 11 is of particular interest, and an expanded view of the convergence zone is shown in Fig. 16. Although the computed winds do not contain all the details of the composite wind field shown in Fig. 5, the model did succeed in reproducing some of the primary features including the leeward convergence zone. The numerical computations indicate that the effects of the island on the airflow may be felt 50 to 100 km from land. A more accurate determina- tion of the island influence region would reguire the calculations to be performed over a larger horizontal domain. 19 .m/se: ;k::se:;- 5015 Vtguh.2. 7. Initial, itiind ph.o{ s tZ2. oveA opm wateA. f aj> inAjtlcdUiy zeAo £V2AywheA2., and both. compon2.nti> a/12. £>2t zquat to zqjio at the. titand &u/i- &ac.2.. Z(KMJ J 1 L. V-igun.2. S. Initial tompwatuJKL bounding ovqa opm watoA.. Sounding* ov2A Zand bzgtn at the, 2Z2.vatA.on* con- taA,n2.d tn Tabl.2. B3 In accon.danc.2. with th.2. AuA{ j ac2. t2mp2AatuA2. di&tAt- bution £>kown -in Fig. 10. 220 2d0 260 VAPOR MIXiNG R-T;0:3IVKGM) Tlf^!SEC)= 5015 flgun.2, 9. Initial mixing Aatio ■bounding. Th.2. mixing Aatio at th.2. Island huA^ao.2. li> d2t2Amln2.d {nom th.2. K.2JLath)2. hwnldltl2A glvm In App2.ndlx B. 12 16 20 20 ■n-^- K ■.; . . 1 - 5015 F^gu/te 7 0. l6o£heAm& along the ■Li land AuA^ace.. 100 150 E-WIKW) For the particular set of initial conditions used, Fig. 17 indicates that the strongest surface winds are located near the summit of Mauna Kea. At lower elevations, the area of strongest surface winds is along the north shore area and in the saddle region between Mauna Kea and the Kohala moun- tains (hereafter called the Kohala saddle). The asymmetry in the wind field is very apparent to the casual observer of Hawaiian weather. With the pos- sible exception of some promontories, nowhere along the entire main highway that circles the island is one more aware of the presence of persistent strong winds than in the region just south of Waimea (the leeward edge of the Kohala saddle). Strong winds may occur at other locations but the trees of that area vividly portray their struggle for survival in the face of the persistent and forceful northeasterlies (see Plate 2). Primary regions of convergence and divergence are quite apparent in the horizontal wind fields shown in Figs. 11-15. And yet there must be other, more subtle horizontal convergence patterns responsible for the extreme variability in the precipitation distribution of Fig. 6. Those terrain- induced regions of horizontal convergence become much more obvious when we examine the vertical motion field at an elevation of 1000 m above sea level as shown in Figs. 18a, b,c. An examination of Fig. 6 shows that all the major precipitation maxima are well correlated with the areas of rising motion in Figure 18a. Areas of strong descent in Fig. 18b also have their counterparts in the precipitation minima of Fig. 6. The Kau and South Kohala desert areas in particular are under the influence of sinking air, although the model calculations seem to have the North Kona rising motion intruding upon the South Kohala desert. The orographically forced upward motion on the windward slopes is relatively shallow (see Fig. 19), but in Fig. 20 we see that the leeward convergence zone is associated with a broad area of gradual ascent. The correlation between the vertical motion field and observed precipi- tation does not carry over to the calculated cloud cover distribution nearly as well. A vertical cross section through the Mauna Loa-Mauna Kea saddle cloud (Fig. 21) shows a cloud only half the thickness of that observed by 21 • *- *" *- ** V ■ *- «- * *■ If *" ****** *" ** * ******* *" S^ 4 .N. "s V V K «^ *-.<—<— « «^_ « f. «. *. *. *-<— < «. » ««.*.«- 4-4—' *■ < f *«-«-«-«— «- k* **«-«-«-• ** * * **■*■*-*- *- ^ ****■*■ <^*~ JJ*-*****'*'*'*' *- / / / ea. level a{tex 5015 6 o£ model, two,. See F^g. 76 tfo^i an expanded vtew oft the. Akade.d ajiea. V-iguAe. 11. \)e.c£oK. wind ^ietd at 1000 m above. &e,a levet a{tex 5015 4 ol model, time.. 4- «- V- V It* *• *'*'*' *~>r~^^ *■«■*■**■*****■ »«V*-^^ ♦ S «. »- •» ««. ♦- «-*-*-*r«'k'ifk'*'*' f^f- — «. * «. t. *. «~ ********** *^-~ ******** ^jx* «_»«««.«-«- ***-**fc**i «r < * * * *" *" ******kk*± *■ < * ♦ *■ *~ t + 4-4-4-4-****. ^ tf K K * *" «" 4>4-4-4-4-«-«-«-«-« /*■*****■*- *■*■*■+■*■*•*■*•*■*• 4JJ***'**'''' 4. 4- «. «. «. 4. 4. ♦. ♦. «- ,//*■*■*■<**■*■ *"^ «_«.«-«- ♦-♦-^♦-«~4- > _- t ^^ ((/K/ ^ y *■*-*■ *-*- «- *- ♦.«-«-.«-«•-«- *-'^-*-r' l s*'*' S S iS SV *""" 'S*'*'*'*'*'*' «— 4- « 4 "V— *-4c— ** *~ *<*>*** *~ *-*-*~*'*'*'*'ie'*'*-*r- *-*-*'*'*■*•*•*'*'*' •*-*-*-*•*•*•*•*'*•*' ■*-*-*-*-*■*• * * * *■ ■*'*■*-*-*■*•*' ^ * ■ 4- 4- 4- 4- 4- 4- «- * ^ Figu/ie. 13. Ve.ctox. wind ^ield at 1500 m above. -6ea £eve£ a&te/t 5075 A ojj modeZ time.. ViguAe. 14. Me.eX.on. wind {ield at 1500 m above, iea level afiteA 5015 4 ofa model time,. 22 (L55 -IS: ££L J77- APCL3 ofa model, time. The triangle indicates the approximate, location ofa Plate 1. E 150 100 - 100 150 200 East- West (Kilometers) 250 23 WiClVSEC: *;re.sE::« 5015 wicrvSEO Ti^tSECi- 5015 250 100 150 E-HOtMl OL. 250 100 150 E-W(KM) 250 b. wicrvsEO l^ISEC)- 5015 100 150 E-W(KM) C. ViguAe IS. a. Voiijtive ventrical velocity {ield 1000 m above. i>ea level a^tex 5015 6 ofi model time.. Shaded a/ieaA indicate valaei in exceAA 0^ 5 am/6, b. Negative vertical velocity iield 1000 m above tea level OL^tQA 5015 4 o{> model, time.. Shaded aAeat, indi- cjat.2. 6i.nking motion in ex.cei>i> o{, 5 cm/6, c. Vertical velocity at 1000 m above hea level afcteh. 5015 -6 o{ model twe. Mordy et al. (1957). It would probably be necessary to include warm rain processes and a convective parameterization scheme in order to simulate the observed cloud cover more realistically. The lack of any cloud cover in the Kona convergence zone should probably not cause any great surprise in view of the model's neglect of cumulus processes. The calculated North Kohala cloud (Figs. 22-23), on' the other hand, appears deeper than the saddle cloud and may be representative of the actual Kohala cloud (Plate 3). 24 W (CM/SEC) TIM(SEC)- 5015 WICM/SECJ ZIKWJ 1(10 150 E-W(KM) 80 90 E-W(KM) VlguAC 7 9. ltiQJ>t-ZCLi>t Ch.066 6ectix>n o{ vertical veloctty thAough the saddle, cloud a^ten. 5015 6 ofi model time.. Shaded an.ea6 indicate valuer In exce66 oh 20 cm/6. Tlguxe 10. We6t-ea6t cn.o66 section o& vertical velocity through the Kona convcn.go.nc2. zone a^ten. 5015 6 0^ model time. Shaded a/iea6 Indi- cate valuer -in exce66 o£ 20 cm/6. cloud water m;x;n6 rat; 3 : 'jvr~jn: t;me:se:;- 5015 ZtKMl CLOUD WATER MIXING RATIOIGM/KGM) TIMEISEO- 5015 IflO 150 E-W(KM) 130 160 N-S(KH) 220 Vigu/ie 21. Cloud wateA mixing latio thAough the 6addle cloud cokac6- pondlng to the 6ame cao66 6ection o6 Vig. 7 9. Shaded aAeo6 Indicate valuc6 In exce66 ojJ 7 gm/kgm. FiguAe 22. HoAth-6outh CA066 6ectlon o^ cloud uxuteA 6houiing a pontic n o& the Kolvxla cloud a^tcA 5015 6 o\ model tone. Shaded oAea6 indi- cate value* in exce66 OjJ 7 gm/kgm. 25 CLOU) WATER MIXING RATIO (GM/KGM) Tilt (SEC)- 5015 Z(KM) -i r VIquaz 23. An expanded vim oft the Kohala cZoud oxobh 4 cation Ahown In Fig. 22. The 6 haded axea Indicate* value* In exceA* o{ 1 gm/kgm. 190 200 210 N-S(KM) 220 230 14. CONCLUSION Terrain-following coordinate systems, such as the one used in the pres- ent calculations, eliminate the need for special boundary treatments where quasi -horizontal computational surfaces intersect the surface topography. However, one seldom gets something for nothing. In the case of a non-uni- formly spaced terrain-following system, the modeler must contend with vari- able vertical resolution. The vertical resolution shown in Table 1 is prob- ably sufficient for most applications in the lower troposphere. On the other hand, upper tropospheric or stratospheric applications would require a differ- ent coordinate transformation in order to match the region of primary inter- est with the region of improved vertical resolution, which occurs near the lower boundary in the Nu system. In spite of the above caveat, preliminary results indicate that the Nu coordinate system model is capable of simulating not only the significant features of the wind field (i.e. the Kona convergence zone), but also the smaller scale features that correspond well with observed precipitation pat- terns. Apparently the smoothing of the pressure weighted wind field still allows the model to resolve those important smaller scale features. The importance of the island of Hawaii as a prototype for mesoscale model development cannot be overemphasized. Considerable data already exist, and reasonably comprehensive sets of initial and boundary data could be ob- tained at relatively small additional cost. However, new interactive pro- grams combining numerical modeling studies and field investigations would derive considerable benefit from the remote sensing capabilities of aircraft and satellites (see Plates 1 and 4). Comprehensive studies of the spatially varying but semi -permanent wind, cloud, and precipitation regimes of Hawaii would (1) assist in the evaluation of mesoscale models, and (2) provide the physical insight required for improved parameterization of sub-grid scale transport processes. 26 15. ACKNOWLEDGMENTS The authors wish to thank Dr. H. K. Weickmann and Dr. C. F. Chappell of the Atmospheric Physics and Chemistry Laboratory, NOAA, for their encourage- ment, support, and guidance in developing the model and the graphics software necessary for viewing the output of a three-dimensional model. Professors T. Schroeder and C. S. Ramage of the University of Hawaii at Manoa also contri- buted to our knowledge of Hawaiian meteorology. This work was sponsored in part by the Bureau of Reclamation under Contract No. 14-06-D-7676. 16. REFERENCES Anthes, R. A. and T. T. Warner, 1974: Prediction of mesoscale flows over complex terrain. ECOM Technical Report No. 5532, White Sands Missile Range, New Mexico 88002. Arakawa, A. and Y. Mintz, 1974: The UCLA general circulation model. Notes distributed at the workshop, 25 March-4 April 1974, Dept. of Meteor- ology, UCLA. Chopra, K. P., 1973: Atmospheric and oceanic flow problems introduced by islands. Advances In Gzophy&i-CA, 76:297-421. De Rivas, E. K. , 1972: On the use of nonuniform grids in finite difference equations. J. Comp. ?ky&. t 70:202-210. Fiedler, F., and H. Panofsky, 1970: Atmospheric scales and spectral gaps. Bull. Am. Mztaoi. Soc, 57:1114-1119. Lavoie, R. L., 1966: The warm rain project in Hilo, Hawaii, summer 1965. H.I.G.-66-5, Univ. of Hawaii. Lavoie, R. L., et al . , 1967: The warm rain project in Hawaii. TelZuA, 79: 347-461. Mordy, W. A., et al., 1957: Project Shower, an investigation on warm rain- fall in Hawaii. TMuA, 9:471-590. Murray, F. W., 1967: On the computation of saturation vapor pressure. J. Appl. Mutton.., 6:203-204. Nickerson, E. C. and V. E. Smiley, 1975: Surface layer and energy budget parameteri zations for mesoscale models. J. Appl. Me-teot., 74 : 297-300 . NOAA, 1972: Local cl imatological data; annual summary...; Hilo, Hawaii. U.S. Dept. of Commerce, National Oceanic and Atmospheric Administration, Environmental Data Service. 27 O'Brien, J., 1970: On the vertical structure of the eddy exchange coeffi- cient in the planetary boundary layer. J. Atmo*. Sec., 27:1213-1215. Patzert, W. C. , 1970: Eddies in Hawaiian waters. H.I.G.-69-8, University of Hawaii. Priestley, C. H. B., 1959: TuAbuJLznt TKmu>{oA in. the. LoweA AtmotpheAe.. Univ. of Chicago Press, Chicago, 111. Sellers, W. D., 1966: Physical Ctimatology. Univ. of Chicago Press, Chicago, 111. Shapiro, R. , 1970: Smoothing, filtering and boundary effects. Rev. Go.oph.y6. Space. Phy*., 5:359-387. Sutton, 0. G., 1953: Ikioxometeo oology, McGraw-Hill, New York. Webb, E. K. , 1970: Profile relationships: The log-linear range and exten- sion to strong stability. Q. J. Roy. Wete.on.. Soc. , 96:67-90. Weiss, C. E., and J. F. Purdom, 1974: The effect of early-morning cloudiness on squall line activity. Hon. UJ&a. Rev., 7 02:400-402. 28 APPENDIX A. BASIC NOMENCLATURE w G K(u) K(T) L L P P P R R R. *l S T T. uns Unspecified variable at time level n Specific heat at constant pressure Normalized surface layer wind speed Friction term in the entropy equation Friction term in the x equation of motion Friction term in the y equation of motion Fraction term in the moisture equation Normalized surface layer heat flux Exchange coefficient for momentum Exchange coefficient for thermodynamic variables Monin-Obukhov length Latent heat of vaporization Pressure Normalized pressure Reference pressure uie, -m kg w'8g Pressure at the ground (v = 1) Pressure at the top of the atmosphere (v = 0) Moisture flux Sensible heat flux Gas constant [= 287.04] Boundary layer constant [= .74) Bulk Richardson number Entropy variable [= it(ln (T/p) + Lq /C p T; Temperature Temperature corresponding to saturated conditions Temperature corresponding to unsaturated conditions Virtual temperature [= T ( 1+0. 61 q )] Pressure weighted wind component [=nu] Pressure weighted wind component at z=h Pressure weighted wind component |=nv] Pressure weighted wind component at z=h Pressure weighted water mixinq ratio [= n(q +q ) ] 1 VM v M cw f g h i.j.k k q cw q h % \ \s t u u h V w x,y z Z A Z B z o Wq6 w'ej a B Y y" c CO n no Saturation vapor pressure with respect to water Coriolis parameter [= 5x10"^ ] Acceleration of gravity [= 9.8062] Heiqht of the first grid point above the lower boundary Grid locations in the (x,y,v) coordinate system Von Karman constant [= .35] Cloud water mixinq ratio Water vapor mixing ratio at z=h Water vapor mixing ratio at z=z Q Water vapor mixing ratio Saturated mixing ratio Time Velocity component in the x direction Friction velocity Wind speed at z = h Velocity component in the y direction Velocity component in the z direction Horizontal coordinates in the Nu coord- inate system Vertical coordinate Height of the planetary boundary layer Heiqht of the surface layer Surface roughness length Surface moisture flux Surface heat flux Surface wind direction [= tan -1 (V h /U h ) ] Surface layer constant [= 4.7] Surface layer constant (= 15] Surface layer constant [= 9] Surface layer function [= (1-yz/L)' Surface layer function [= (l->z : /L)' Surface layer function [■ ( 1 - . " z / L ) ' . Surface layer function [■ (l-.'z-'L' 29 e Potential temperature [= T/P] 6. Potential temperature at z = h e Potential temperature at z = z k Thermodynamic constant [= 2/7] X Dummy variable v Transformed vertical coordinate v Time derivative of v;v = dv/dt tt Pressure variable; tt = P_ - P_ o Vertical coordinate; a = a (v) = (p-p-r)/iT F Function a' Vertical derivative of o;o' = da/dv d Time derivative of o;6 - da/dt Geopotential ; 4> = gz 10 Vertical motion in the P system; uj=dp/dt At Time step [= 10 sec, except that the last time step was 15 sec] ax, Ay Horizontal grid length [= 10 km] Az Boundary layer thickness Az = z A - z g Av Vertical grid length [= 1/15] 30 APPENDIX B. TERRAIN PARAMETERS A soil parameter and a vegetation parameter have been assigned to each grid point at the lower boundary of the model. The two surface parameters are further subdivided and a number is assigned to each category. Those num- bers are then used to estimate the surface roughness length and mixing ratio at the individual grid points. Table. 87. SoU. PoAameteJU Soil Type Coded Form Sea 1 Dry 2 Semi moist 3 Wet A Lava 5 Sand 6 Table 82. Vegetation VahjmeteAA Vegetation Type Coded Form None 1 Short Grass 2 Tal 1 Grass 3 Shrub k Forest 5 The relative humidity at the surface was set equal to 1002 except for soil types 2 and 3, which were assigned the values 80"' and 90' respectively, 31 Table. B3. RoughnzAA Pcviam&teAA Soil Type Vegetation Type Roughness Length (meters) Sea None 0.0001 Lava None 0.01 Dry Grass 0.01 Dry Shrub 0.2 Semimoist Grass 0.05 Semimoist Shrub 0.5 Wet Grass 0.1 Wet Shrub 1.0 Wet Forest 3-0 All points on the 26 by 26 not listed below are over water. Their sur- face parameters are coded (0,1,1), where the first digit represent height above sea level in meters, the second the soil parameter, and the third the vegetation parameter. 32 Table. B4 . TeAAain H(Ugkt6 and SuA^ace. ?aAamoJ:&a> fan. Hawaii 1 J Z S V 1 J Z S V 12 8 1 6 1 10 16 1510 4 4 11 9 60 5 1 11 16 1580 3 3 12 9 340 5 1 12 16 1940 5 1 10 10 240 5 1 13 16 2150 5 1 11 10 700 3 4 14 16 2150 5 1 12 10 840 3 5 15 16 1830 5 1 13 10 430 3 3 16 16 1330 4 5 10 550 4 5 17 16 720 4 5 11 1650 4 4 18 16 260 4 5 12 1580 4 5 19 16 30 4 4 13 820 4 5 8 17 20 3 3 14 120 4 3 9 17 880 4 4 10 12 670 4 5 10 17 1300 3 3 n 12 2130 5 1 11 17 1340 5 1 12 12 2290 5 1 12 17 1600 5 1 13 12 i860 4 5 13 17 1810 5 1 14 12 700 4 3 14 17 2130 2 4 15 12 590 5 1 15 17 1910 4 4 16 12 210 2 2 16 17 1250 4 5 17 12 30 2 2 17 17 670 4 5 10 13 580 4 5 18 17 60 4 5 11 13 2100 5 1 19 17 20 4 3 12 13 3020 5 1 9 18 150 5 1 13 13 2760 5 1 10 18 500 5 1 1/* 13 1650 4 5 11 18 900 5 1 15 13 930 4 4 12 18 1310 2 3 16 13 1000 3 4 13 18 2470 2 3 17 13 930 4 4 14 18 3990 5 1 18 13 640 4 4 15 18 2680 3 4 19 13 90 3 4 16 18 1580 4 5 10 14 850 4 5 17 18 820 4 5 11 14 1900 4 4 18 18 90 4 4 12 14 3200 5 1 10 19 90 5 1 13 14 3960 5 1 1 1 19 430 2 4 14 14 2650 5 1 12 19 1010 2 4 15 14 1650 4 4 13 19 1430 2 4 16 14 1280 4 5 14 19 2190 2 2 17 14 910 4 5 15 19 2130 4 4 18 14 620 4 5 16 19 1370 4 5 19 14 410 4 5 17 19 520 4 5 20 14 170 4 5 11 20 400 2 2 9 15 30 2 2 12 20 820 2 2 10 15 1 160 4 4 13 20 1010 3 4 11 15 1740 5 14 20 1 1 10 4 4 12 15 2600 5 15 20 850 4 5 13 15 3230 5 16 20 380 4 5 14 15 3020 5 10 21 110 2 2 15 15 2270 5 1 1 21 1070 3 4 16 15 1490 4 5 12 21 1040 4 5 17 15 880 4 5 13 21 580 4 5 18 15 430 4 4 14 21 270 4 5 19 15 170 4 4 10 22 340 3 3 20 15 80 4 4 11 22 590 4 5 9 16 370 4 4 12 22 1 4 4 10 23 3 4 3 33 APPENDIX C. EXCERPT FROM U.S.W.B. CLIMATOLOGICAL SUMMARY (NOAA, 1972) The city of Hi lo is located near the midpoint of the eastern shore of the Island of Hawaii. This island is by far the largest of the Hawaiian group, with an area of 4,038 square miles, more than twice that of all the other islands combined. Its topography is dominated by the great volcanic masses of Mauna Loa and Mauna Kea, both of which exceed 13,600 feet in ele- vation, and of Hualalai (8,271 feet), the Kohala Mountains (5,480 feet), and Kilauea (4,090 feet). In fact the island consists entirely of the slopes of these mountains and of the broad saddles between them. Hawaii is diamond-shaped, about 93 miles long, from north to south, and 76 miles wide. Its highest point is the summit of Mauna Kea at 13,796 feet. Mauna Loa and Kilauea, which occupy the southern half of the island, are still active volcanoes and hence smooth-sloped, in contrast to the deeply eroded valleys that indent portions of Mauna Kea and the Kohala Mountains. Hawaii lies well within the belt of northeasterly trade winds generated by the semi -permanent Pacific high pressure cell to the north and east. The climate of the island is greatly influenced by terrain. Its outstanding fea- tures are the marked variations in rainfall with elevation and from place to place, the persistent northeasterly trade winds in areas exposed to them, and the equable temperatures from day to day and season to season in localities near sea level . Over the island's windward slopes, rainfall occurs principally in the form of showers within the ascending moist trade winds. Mean annual rainfall increases from 100 inches or more along the coasts to a maximum of over 300 inches at elevations of 2,000 to 3,000 feet, and then declines to about 15 inches at the summits of Mauna Kea and Mauna Loa. In general, leeward (southern and western) areas are topographically sheltered from the trades — hence from trade-wind showers — and are therefore drier; although sea breezes created by daytime heating of the land move onshore and upslope, causing afternoon and evening cloudiness and showers. Where mountain slopes are steeper, mean annual rainfall may range from 30 inches along the coast to 120 inches at elevations of 2,500 to 3,000 feet. The driest locality on the island — and in the State — with an average annual rainfall of less than 10 inches, is the coastal strip just leeward of the southern portion of the Kohala Mountains and of the saddle between the Kohalas and Mauna Kea. These marked contrasts in rainfall are reflected in soil and vegetation, with frequent abrupt transitions from lush tropical growth to near-desert conditions, such as occurs between Kilauea's wet windward slopes and the Kau Desert just to the south. Within the city of Hi lo itself, average rainfall varies from about 130 inches a year near the shore to as much as 200 inches in mountain sections. The wettest part of the island, with a mean annual rainfall exceeding 300 inches, lies about 6 miles upslope from the city limits. Rain falls on about 280 days a year in the Hilo area. 34 Hawaii's equable temperatures are associated with its mid-ocean location and the small seasonal variation in the amount of energy received from the sun. At Hilo, the range in average temperature from February and March, the coldest months, to August, the warmest, is only 5.2° F and the average daily range, 15.1° F. The highest temperature of record at Hilo Airport is 94° F; the lowest 53° F. Greater variations occur in localities with less rain and cloud, but temperatures in the mid-90's and low 50's are uncommon anywhere on the island near sea level. The trade winds prevail throughout the year (although they may be absent for days or even weeks at a time) and profoundly influence the climate. How- ever, the island's entire western coast is sheltered from the trades by high mountains, except that unusually strong trade winds may sweep through the relatively low (2,600-foot) saddle between the Kohala Mountains and Mauna Kea and reach the areas to the lee. But even places exposed to the trades may be affected by local mountain circulations. For example, the prevailing wind at Hilo Airport is not the northeasterly trade, but the southwesterly wind that drifts downslope off Mauna Loa during the night and early morning hours. Except for heavy rain, really bad weather seldom occurs. Thunderstorms average only 8 per year, and are rarely severe. During the winter, cold fronts or the cyclonic storms of subtropical origin (the so-called Kona storms) may bring blizzards to the upper slopes of Mauna Loa and Mauna Kea, with snow extending at times to 9,000 feet or below and icing nearer the summit. Storms crossing the Pacific a thousand miles to the north, or Kona storms closer by, may generate seas that cause heavy swell and surf along the northern, eastern, and southwestern shores of the island. 35 * U.S. Government Printing Office: 1976 -7 77 •? 06 r LABOR AT DRIES The mission of the Environmental Research Laboratories (ERL) is to conduct an integrated program of fundamental research, related technology development, and services to improve understanding and prediction of the geophysical environment comprising the oceans and inland waters, the lower and upper atmosphere, the space environment and the Earth. The following participate in the ERL missions: MESA Marine EcoSystems Analysis Program. 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