5 . LOcX- -j%, W«2219 PROCEEDING S 1983 TSUNAMI *» SYMPOSIUM ^£' H A M B U R G F • R G . -?* , u AUGUST 1983 Pacific Marine Environmental Laboratory National Oceanic and Atmospheric Administration 7600 Sand Point W a v N E Seattle, Washington 981 15 USA PROCEEDINGS TSUNAMI SYMPOSIUM HAMBURG FEDERAL REPUBLIC OF GERMANY ^ C 4 , ^ f AUGUST 1983 EDITED BY: E. N. BERNARD PACIFIC MARINE ENVIRONMENTAL LABORATORY 7600 SAND POINT WAY N.E. SEATTLE, WASHINGTON 98115 U. S. A. .upgwos^. UNITED STATES DEPARTMENT OF COMMERCE Malcolm Baldrfge, Secretary NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION John V. Byrne, Administrator Environmental Research Laboratories Vernon E. Derr Director U. S. Depository Copy NOTICE Mention of a commercial company or product does not constitute an endorsement by NOAA Environmental Research Laboratories. Use for publicity or advertising purposes of information from this publication concerning proprietary products or the tests of such products is not authorized. 11 CONTENTS Page PREFACE vii PRESENTED PAPERS: SESSION I - AUG. 19, 1983 1. DIFFUSIVE KINEMATIC WAVES VERSUS HYPERBOLIC LONG 1 WAVES IN TSUNAMI PROPAGATION T. S. MURTY 2. THE EXPERIMENTS ON THE SEICHE 23 H. NIYOSHI 3. SEICHES IN BAYS FORMING A COUPLED SYSTEM 37 Af. NAKANO, N. FUJIMOTO 4. TSUNAMI FLOOD CONTROL AT THE OPENING OF A BAY 65 OR HARBOUR S. NAKAMURA 5. EXCITATION MECHANISMS OF THE 'ABIKI' PHENOMENON 83 (A KIND OF SEICHE) IN NAGASAKI BAY T. HIBIYA 6. RECONSIDERATIONS ON THE HUGH TSUNAMIS - UTILITY 107 OF THE SEAWALL H. MIYOSHI 7. REGIONAL TSUNAMI WARNINGS USING SATELLITES 117 E. N. BERNARD, G. T. HEBENSTREIT, J. F. LANDER AND P. F. KRUMPE in THE ALASKA TSUNAMI WARNING CENTER'S AUTOMATIC EARTHQUAKE PROCESSING SYSTEM T. H. SOKOLOWSKI, N. E. BLACKFORD, G. W. FULLER AND W. J. JORGENSEN 131 A PRELIMINARY INVESTIGATION OF TSUNAMI HAZARD (ABSTRACT ONLY) C. C. TUNG 149 10 SYNTHESIS OF TSUNAMI WAVE EXCITATION BY NORMAL MODE SUMMATION (ABSTRACT ONLY) S. N. WARD 153 11 TSUNAMI -RESISTANT GAUGES FOR EPICENTRAL SEA-LEVEL STUDIES R. BILHAN 155 12, THE NONLINEAR RESPONSE OF A TIDE GAUGE TO A TSUNAMI H. G. LOONIS 177 13 TSUNAMI OF MAY 11, 1981 ON THE COAST OF SOUTH AFRICA S. 0. WIGEN, T. S. VIURTY, D. G. PHILIP 187 SESSION II - AUGUST 24, 1983 14. RUNUP OF A TRANSIENT WAVE ON A SLOPING BEACH (ABSTRACT ONLY) K. KAJIURA 203 15 RUN-UP OF LONG WATER WAVES (ABSTRACT ONLY) G. PEDERSEN, B. GJEVIK 205 IV 16. OPEN OCEAN SIGNATURE OF TSUNAMI ON THE SEA 207 FLOOR: OBSERVATION AND USEFULNESS (ABSTRACT ONLY) J. H. FILLOUX 17. LONG WAVE OBSERVATIONS NEAR THE GALAPAGOS ISLANDS 209 (ABSTRACT ONLY) E. N. BERNARD, H. 0. MOFJELD, H. B. NILBURN, AND E. G. WOOD 18. DIFFRACTED LONG WAVES ALONG CONTINENTAL SHELF 211 EDGES T. S. MURTY, H. G. LOONIS 19. NUMERICAL SIMULATION OF A TSUNAMI ON A 229 TRIANGULAR MESH H. G. LOOMIS 20. PROPAGATION OF TSUNAMI OVER THREE-DIMENSIONAL 239 SHELVES (ABSTRACT ONLY) r. Y. WU, H. SCHEMBER 21. ENERGY OF THE TSUNAMI CONVERGING ONTO AN ISLAND 241 H. miYOSHI 22. NUMERICAL SIMULATION OF THE 1975 SHIKOTAN 249 TSUNAMI (ABSTRACT ONLY) A. S. ALEKSEEV, V. K. GVSIAKOV, L. B. CHUBAROV AND Y. I. SHOKIN 23. SEICHE ON A PARABOLIC SEA SHELF 251 S. NAKAMURA 24. A HYDRIB FEM-MODEL FOR TSUNAMI AMPLIFICATION 265 IN NEARSHORE REGIONS L. BEHRENDT, I. G. JONS SON AND O. SKOVGAARD PREFACE During the Seventeenth Assembly of the IUGG in Hamburg, Federal Republic of Germany, two tsunami sessions were held. The first session, chaired by H. G. Loomis , was held on August 19 as the 9th session of the Inter-Disciplinary Symposium - Assessment of Natural Hazards. The second session, chaired by T. Y. Wu, was held on August 24 as IAPSO Symposium 12 - Tsunami Wave Propagation. Twenty-four scientific presentations were made during these two sessions. On August 24, K. Kajiura and H. Miyoshi led a panel discussion on the May 26, 1983 tsunami in Japan. They showed photographs and movies of the tsunami and subsequent damage. This document represents the proceedings of these tsunami sessions. Sixteen manuscripts are included along with eight abstracts to document the scientific information presented during the Assembly. Some of the abstracts contain references to more complete published works on the subject. These proceedings are published by the National Oceanic and Atmospheric Administration's Pacific Marine Environmental Laboratory. The preparation of this document for publication was done by Ryan Whitney of PMEL. Vll DIFFUSIVE KINEMATIC WAVES VERSUS HYPERBOLIC LONG WAVES IN TSUNAMI PROPAGATION T. S. Murty Institute of Ocean Sciences Department of Fisheries and Oceans Sidney, B. C. , CANADA ABSTRACT For the Chilean earthquake tsunami of May 1960, the travel-times computed from the long wave formula do not agree with observed travel times at some stations in the Queen Charlotte Strait near the west coast of Canada. For these stations there was about two hours of delay in the arrival of the tsunami. Making use of a concept that in shallow water the propagation of a long gravity wave is governed predominantly by friction, these delays were satisfactorily accounted for. 1. INTRODUCTION The terminology "long gravity waves" includes tsunamis which have periods in the range of a few minutes to about three hours. It is generally known that, to a first approximation, tsunamis travel with the speed C = v gH where g is gravity and H is the average water depth. Indeed tsunami travel-time charts used for practical prediction are based on this formula. However, there are instances in which this formula appears not be be valid. Specifically, when the tsunami is propagating in extremely shallow water, the travel times computed from this formula are much smaller than the observed travel times. Loucks (1962) studied the Chilean earthquake tsunami of May 1960 near the west coast of Canada, and he showed that at some stations on the Queen Charlotte Strait, the observed travel times are some two hours greater than the travel times computed using the long wave formula. In this paper we explain the tsunami delay by invoking that, in extremely shallow water, the tsunami travels not as a hyperbolic long wave but as a diffusive kinematic wave. 2. DESCRIPTION OF THE CHILEAN EARTHQUAKE TSUNAMI On May 22, 1960 a major earthquake occurred in Chile at 41° S, 73.5° W. The tsunami that resulted was recorded by seventeen tide gauges (out of a total of twenty one) on the west coast of Canada. Some seventeen hours after the occurrence of the earthquake, the tsunami arrived on the Canadian coast from a general south-west direction. Figure 1 shows the geography of the region consisting of the Dixon Entrance, Hecate Strait and Queen Charlotte Sound, whereas in Figure 2 is shown the area comprising of the Vancouver Island, the Strait of Georgia, and the Queen Charlotte Strait? A detailed * For a description of tides in Canadian waters, see Dohler (1964). TO ID a 14-1 O *J CO 03 O u 4-1 CO 01 3 cu M 03 a 0) -c 1-1 o c OJ 4-1 o -c C- 03 00 o o 35 g Ui t<3 o o Q 4! » 0) J= a a E 3 O ex S3 O 0) O en dJ t-> <4-l O a. CO u oo o (U o m o 10 IO — < 2 CM (/) O I (1) U~l 0) a m u *. o c •H 14-1 o H * -a c D o c/) rs 0) rH M u ed ca •u a) en rH 0) > 0) rH r-l • 0J -a 4-J 8 3 rH en T3 H 0) T3 r-l l~i 0) O a u a 0) o PBi u 1 J o ■i J O s 0) rH f*. 1-1 CU a 3 (£ Pi u in •5 5 * o ID K> (0 g rH >- < 2 CO rH C\J -~ IO >> CM CU . c — c — K ai O 3 O X o CO 0) CO rH CD > 0) • rH CO t- V-i cfl 0) CX •Ul ID crj 3 •5 13 14-1 a) •H ifi T3 l-l V-i o O CJ -a £ 8 — 0» a: z> 2 § 3 >-■ 3 o XI u cfl TJ M O iw H 3 M O 4J U •H > M cu ■u 3 T3 0) TD u o u S vjD H J O X> o en u g R CD (Xi c T3 o c en 3 c O x. 00 o •-> -C AJ 4-1 M 03 O 3 05 OJ H M OJ cu > S CD i— 1 T> l-i § OJ 4J 4J CO QJ 3 H c 'O H 01 •X3 QJ u N o •H u rH 0> CD peS pq 10 TABLE 1 (continued) STATION FEATURES Fulford Harbour Prince Rupert Caulfield McKenney Island Klemtu Griffin Passage Johnson Point Nugent Sound Mereworth Sound Seymour Inlet Belize Inlet Less tsunami action than at Victoria. Almost no tsunami action. Sheltered by the Queen Charlotte Islands. Almost no tsunami action. Sheltered by the Gulf Islands. Somewhat similar to Cape St. James record. Smoother compared to McKenney Island record. Evidence of a 25 minute seiche. Showed a surprising amount of tsunami action, Influenced only slightly by the tsunami. Power spectra showed more power in the 15 to 150 minute period range than in the 2 to 15 minute range. 3. EXPLANATION OF THE TSUNAMI DELAY USING RADIATION STRESS MECHANISM The arrival times at the tide gauges up-inlet from the Nakwakto Rapids (Figure 3) are approximately two hours greater than the times predicted using the long wave formula (Loucks, 1962). For example, the first wave of the tsunami took about 200 minutes to cover the distance between Cape Scott and Johnson Point (Figure 2) , whereas the predicted travel time is only 80 minutes. Loucks (1962) offered the following explanation: There was a strong tidal current directed seaward through Nakwakto Rapids, Slingsby Channel 11 (Figure 3) and Outer Narrows at about the time the first tsunami wave was moving inward through these waters. The current speed in Nakwakto Rapids during the time of ebb is about 15 knots which is about equal to the speed of travel of a long gravity wave in that rapids area. According to Loucks (1962) the tidal current held the tsunami wave to a standstill for an extended period of time. As for the mechanism of how the tidal current interacted with the tsunami wave, Loucks (1962) made use of the concept of radiation stress (Longuet- Higgins, 1952; Longuet-Higgins and Stewart, 1960, 1961). It can be shown that waves can extract energy from an opposing current and lose energy to a following current via Reynolds stresses. Throughout the approaches to Johnson Point, the Reynolds stresses would be acting so that the wave energy is increased. This is an important point to note because, with the restricted access to Johnson Point from the Queen Charlotte Strait, and based on the local water depths, there is little chance of the energy of the tsunami waves being directed into the entrance to the Slingsby Channel. Loucks (1962) found it amazing that the tsunami did progress through the narrows, channels and rapids to reach Johnson Point with appreciable amplitude. Another important point to note is that, out of the seventeen tide gauge records used in this study, the Cape St. James record is probably the most represen- tative of open ocean conditions, because this station lying at the southern tip of the Queen Charlotte Islands is directly exposed and there is little possibility for dissipation of tsunami energy. In addition, increase of wave amplitude due to shoaling is also minimal here because the continental shelf is very narrow here. 12 4. HYPERBOLIC LONG WAVES VERSUS DIFFUSIVE KINEMATIC WAVES A flow regime that is relevant for tidal propagation was considered by LeBlond (1978) , in which the momentum balance of the long gravity wave is best described by a set of equations somewhat intermediate between the frictionless long gravity wave equations (hyperbolic waves) and the Kinematic wave equations used in the study of flood waves (Whitham, 197 4)* This intermediate flow regime, referred to as "Diffusive Kinematic waves" is dominated by frictional terms in its momentum balance, and can account for the long phase lags associated with low water. LeBlond (1978) showed that the most relevant model for a long wave travelling in shallow water is a diffusive model (since the frictionally dominated equations are parabolic) rather than a standard wave propagation model. We will briefly summarize the relevant points from LeBlond' s (1978) work. Consider a narrow rectangular channel of uniform depth and uniform width. The momentum and continuity equations for one -dimensional tidal propagation are 3v , 9v , 8n kv v n , ... - r + v ~ r + g — r H hr- = (1) 3t 3x 8x (H+n) _3_ 9x ;1 +^ J 3t (H+n)v + -^± = (2) where the x coordinate is along the channel axis (positive towards upstream) , t is time, v is the depth-averaged current, H is the water depth, r\ is the deviation of the waterlevel from its equilibrium position, g is gravity, k is a friction coefficient, and the cap denotes that the variables are dimensional (note that on certain variables like g, H the caps are not used, even though * Also see Abbott (1956), Dronkers (1964) and Kreiss (1957) 13 they are dimensional. Let L, T and U respectively be characteristic scales for length, time and velocity. Define dimension less variables through - x t X = T~ » t = — L ' T _ v _ n V = V n = IH (3) where the scaling parameter e is the ratio of the tidal range to the meanwater depth. Use of equation (3) into (1) and (2) gives c 9v F 2 3v , 3n Rvlv| n , ,, M JL [ (1+en)v ] + |a . o ( 5 ) Here the dimensionless parameters S, F, R, M are defined as „ 2t U T q - F L M - ° s = I^ • M = T7 (6) One can identify F as the Froude number. Different ranges of the values for these four dimensionless parameters correspond to different flow regimes. For the frictional regime, the equations reduce to the following form 14 iHc^-Vcfe 1H. + K + A 3x 3t 3x u(z+C) + UC - < uC > = (8) where v = U(x) + u(x,t) (9) n E z(x) + ?(x,t) The flow regime described by (7) and (8) is intermediate to Dynamic and Kinematic waves. For Dynamic waves, S/R >> 1, the momentum and continuity equations are strongly coupled, and the system is Hyperbolic. For Kinematic Waves, S/R << 1, the momentum equation reduces to a relationship between flow speed and depth, obtained by balancing the mean slope of the free surface with friction. 5. DISCUSSION OF RESULTS In the theory outlined above in section 4, the long gravity wave is assumed to be periodic. Although tsunami waves are not as periodic as tides, they do exhibit certain predominant periods, as can be seen from Table 2 for the Chilean earthquake tsunami. It can be seen that near the west coast of the U.S.A. the dominant period is in the range of 30 to 60 minutes. These results are in agreement with those deduced from the Canadian tide gauge records. 15 TABLE 2 Maximum tsunami amplitude and corresponding period for the Chilean earth- quake tsunami of May 1960 (based on Takahasi and Hatori, 1961) * Out of scale. Location Maximum Amplitude Period (m) (min) > 2.8* 180 1.3 50, 100 2.0 180 0.9 35, 120 1.4 60 1.0 40 1.5 30, 60 >3.1* 40 1.1 50 0.5 50 >3.0* 40 0.4 40 0.7 35 0.4 50 1.1 30 1.5 45 1.6 60 >2.2* 45 >3.3* 45 >3.3* 30 continued . . . La Punta, Peru La Liberatad, Ecuador La Union, El Salvador San Jose, Guatemala San Diego, California San Pedro, Calfornia Los Angeles, Calfornia Santa Monica, California San Fransisco, California Alameda, California Crescent City, California Astoria, Oregon Neah Bay, Washington Juneau, Alaska Sitka, Alaska Yakutat , Alaska Seward, Alaska Kodiak, Alaska Attu, Alaska Hilo, Hawaii 16 Table 2 (continued) Location Maximum Amplitude (m) Period (min) Honolulu, Hawaii Johnston Island Pago Pago Christmas Island Canton Island Midway Island Wake Island Kwajalein Eniwetok Island Guam Island Hobart Port Mac Donnse Cap Cove Iluka Ballina Port Den is on Lord Howe Island Wellington, N.Z. Auckland, N.Z. Hong Kong >1.6* 1.0 >1.9* 0.4 0.1 0.6 1.0 0.8 0.4 0.4 0.4 0.3 0.3 0.7 0.3 0.8 0.6 1.7 0.3 0.5 35 50 20 40 30 15, 40 12, 50 30 40 45 12, 60 40 50 30 40 46 25 30, 180 40 40 17 The non-linear diffusion equation governing longwave propagation in a shallow inlet is (LeBlond, 1978) 3t 2|u+u| 3x 2 At slack water, since | u+U | = 0, this equation fails; however it holds best at the time of full ebb. At ebb time, (10) becomes (11) where K = (2 |u+U|) (12) Writing u = u Q . e at x = (13) where u is the dominant frequency, we get 3u 3t" K 3 2 u 3x 2 -1 K = (2 u+U | ) = u o- — iwt e at x = u(x,t) = u . exp i (w/2k)^ x - tat - v i (w/2k) 2 x - lot - (w/2k) 2 x (14) Thus the response is partly wave-like with a propagation speed V Q of J- J< (2 <(jj) 2 and partly damped with an e-folding length of (2k/oj) 2 . In dimensional terms, LeBlond ( 1978) showed that the propagation speed of the long gravity wave in shallow water is |g(H+n) V = u + U + g(H+n) (15) The travel paths from Cape Scott to the tide stations (Figure 3) are uncertain. Using several different travel paths, the travel time between Cape Scott to Johnson Point is bracketed between 185 and 210 minutes which compares reasonably well with the observed value of 200 minutes. 18 CONCLUSIONS For tsunami travel in shallow water, in certain situations, the standard long wave formula greatly underestimates the travel times. In those situations, the frictionally dominated diffusive flow regime may be a better approximation. ACKNOWLEDGEMENTS I thank Ms. Dorothy Wonnacott for typing the manuscript, and Ms Coralie Wallace for drafting the diagrams. 19 REFERENCES Abbott, M.R. (1956) A theory of the propagation of bores in channels and rivers, Proceedings of the Cambridge Philosophical Society, Vol. 52, 344-362. Dohler, G. (1964) Tides in Canadian Waters, Report of Canadian Hydrographic Service, Marine Sciences Branch, Ottawa, Canada, 14 pages. Dronkers, J.J. (1964) Tidal computations in rivers and coastal waters, John Wiley and Sons Inc., New York, 518 pages. Kreiss, H. (1957) Some remarks about nonlinear oscillations in tidal channels, Tellus, Vol. 9, 53-68. LeBlond, P.H. (1978) On tidal propagation in shallow rivers, Journal of Geophysical Research, Vol. 83, No C9 , 4717-4721. Longuet-Higgins , M.S. (1952) On the statistical distribution of the heights of sea waves, Journal of Marine Research, Vol. 11, No 3, 245-266. Longuet-Higgins, M.S. and R.W. Stewart (1960) Changes in the form of short gravity waves on long waves and tidal currents, Journal of Fluid Mechanics, Vol. 8, 565-583. Longuet-Higgins, M.S. and R.W. Stewart (1961) Short gravity waves on non- uniform currents, Journal of Fluid Mechanics, Vol. 10, 529-549. 20 Loucks , R.H. (1962) Investigation of the 1960 Chilean tsunami on the Pacific Coast of Canada, M.S. thesis, Department of Physics, University of British Columbia, Vancouver, Canada, 21 pages & 5 figures. Takahasi, R. and T. Hatori (1961) A summary report on the Chilean tsunami of May 1960. In "Report on the Chilean tsunami", Field Investigation Committee for Chilean tsunami, Tokyo, Japan, December 1961, 213-34 Whitham, G.B. (1974) Linear and nonlinear waves, John Wiley and Sons Inc., New York, 636 pages. 21 The Experiments on the Seiche Hisashi MIYOSHI 3-3-33 Minarai-Gizumi, Nerima-ku, Tokyo, 177, Japan The considerations on the seiche became, in Japan, very important from the viewpoint of tsunami research. Using a round pond, the shape of which is extremely geometrical and the diameter of which is as long as 13.23m, the artificial seiches of Bessel function types were generated. At the center of this pond, several jets of water are spouted. When the water surface at the center is elevated by the crest of the seiche, the strength of the jets is cut down, and the water surface is depressed, the jets jump up. This remarkable dance easily show us the periods of the seiches, the modes of them and the remarkable phenomena of the sudden shifts of the modes. The period of the simplest mode is 4.15 seconds, and those of the second simplest one, third and 4-th are 2.40, 1 .80 and 1.47 seconds, respectively. When the last mode whose period is 1.47 seconds appears, the shoreline is confirmed to be the 5th loop. While there is a close relation between the water temperature and the modes of the seiches, the area of the shocked surface plays an important role. 1 . Introduction The considerations on the seiche became, in Japan, very important from the viewpoint of tsunami research. In 1960, for example, the coast of Cfunato Bay in the Sanriku District was damaged most severely, by the seiche stimulated by the tsunami from Chile. J. 3. McNown (1952) conducted an elaborate experiment, in which the circular port installed in the canal was made of 15 concrete blocks each 20cm high and subtending an arc of 7C/8 radians. The entrance, of the same angle, was centrally placed on a diameter parallel to the axis of the canal so that the approaching wave crests were parallel to the entrance chord. The diameter of the port was 3.20m, and a constant water depth of 16cm ( or /I /10; was utilized in the experiment. This report is the result of the extensional one of this previous paper, in which the diameter of the utilized circular pond is as long as 13.23m. Through the entrance, McNown sent the wave train produced by a wave maker of the horizontal-displacement type, into this port, which resembles the phenomenon of the actual tsunami entering a bay. "It has been amply proved that the motion produced in a port can have an amplitude not only equal to but even a number of times greater than the amplitude of the wave that produces it. Furthermore, from theoretical considerations, this amplitude can occur equally well with an entrance width that is extremely small," 23 he wrote. No phenomenon of tsunami occurred however in his mind at that time, and he didn't, for example, report after how many incident waves the seiche was stimulated. Perry Byerly ( written communication, Feb. 1, 1954 ) called D. J. Miller's attention to the above-mentioned McNown's statement, when Miller's mind was torn among many possible causes of the giant waves on Oct. 27, 1936 in an uninhabited bay, Alaska afterwards. We are here only referring to the fact that even an American geologist noticed the relation between the tsunami and the seiche. Incidentally, it is a pity that Wilier once approached the truth when he described : "By analogy with the 1958 wave, a falling mass that caused the 1936 waves in Lituya Bay should have come from the southwest wall of Crillon Inlet", and then threw it away taking the words of a braggart eyewitness, F. H. FREDERICK SON, seriously. A pond, the shape of which is a much more accurate nearcircle surrounded by 90 sides ( concrete blocks ), was here utilized. Its diameter is 13.23m long, and the difference of the lengths of two diameters which meet at right angle, is negligible. At the center of this pond, several jets of water are spouted. Since such a pond seldom exists, experiments were conducted in detail. When the height of the water surface was adjusted, the dance of the jets began. Figures 1 and 2 show this dance, the period of which is 2.40 seconds. 2. The dance of the jets which expresses the seiche The water mass drops and gives impulse to water surface. Cauchy- Poisson wave is generated and is reflected almost completely along the shoreline of the pond. Here, stationary waves, the periods of which are various, appear. The stationary waves, the periods of which are the same as those of the seiches of the pond, predominate among many waves, and their wave heights reach several centimeters. 'when the water surface at the center of the pond is elevated, the water layer covers the spout holes and the strength of the jets is considerably cut down and the scene shown in Figure 1 appears. And when the water surface is depressed and the jets jump up (Figure 2) and beat the water surface when they drop. We can, therefore, estimate the period of the seiche by reading the period by which the scene shown in Figure 1 and that in Figure 2 alternate. A remarkable fact in this mechanism is that when a certain period seiche is established, water masses drop and give an impulse every period to the water surface. Impulses, the period of which is same as that of the seiche, act, therefore, on the water surface, and the pond itself stimulates the seiche eternally. If no frictions existed, the amplitude of the seiche would become larger and larger. This mechanism resembles the socially imp© rt ant phenomenon of the seiches of larger bays in the Sanriku District stimulated by the tsunami of 1460 from Chile. As a matter of fact, a certain stationary state is established, in which the balance between the motive power and the friction is attained. The above-mentioned inference was led by the thought that all the motive power comes from the impulse by the dropped water mass. 24 ■?& Figure 1. When the water surface at the center of the pond, the diameter of which is 13.23 m, is elevated, the strength of the jets is con- siderably cut down. ... mMI Figure 2. Approximately 1.2 seconds after the scene shown in Figure 1, the water surface is depressed. Alternation of Figure 1 and Figure 2 tells us that the period of the seiche is 2.40 seconds. 25 There is, however, another thought: It is illustrated by Figure 3. Stage a shows the moment at which the scene shown in Figure 1 "begins to shift to that in Figure 2. Stage b shows that at which the scene in Figure 2 almost ends and is going to return to the scene in Figure 1 . The jets are obviously giving energy to the rising crest of the wave in stage a. And they are taking it away from the falling crest in stage b. In a sense, the giving and the taking away may cancel each other. It is possible, however, that these two phenomena are asymmetric, which can become a kind of motive power. Since there are no basic data showing how many percent of the kinetic energy change into the wave energy, when the falling body beats the water surface, we are afraid it is not clear-cut to say, that in spite of the above-mentioned another thought, almost all' the motive power of the seiche may come from the impulse by the dropped water mass. We read the period of the seiche on Oct. 25, 1977, for the first time. Water temperature was 18.8°C from the beginning to the end of the experiment. This temperature was also uniform from bottom to water surface. The average water depth near the shoreline was 77.02cm. When this pond was emptied later, the fact that the depth became a little deeper toward the center of the pond was observed. Incidentally, several days before Oct. 25, a preliminary observation was done and the same period of 2.40 seconds was confirmed . 3. Considerations Introducing the polar co-ordinates, and assuming that «& ( elevation of the water surface ) is the function of V and 6 , the equation takes the form ya - .2 / ya .ill , Ltl_) (D where C is the wave velocity, and the wave is not a completely long one, so c =^\fyT" ( o(A) here occurred. It is a little hard to understand that the shift of the period from 2.40 seconds to 4.15 seconds occurred, under the condition that the impulse had been given every 2.40 seconds. Observations were continued and the modes of (C) and (D) in Figure 4 were confirmed. Their periods were 1 .80 seconds and 1.47 seconds, respectively. And the shifts of the periods were frequently observed, which are shown in Table 4. The (C) state in the third case in this table was maintained for only 10 minutes. And the (D) and (C) states in the fourth case were maintained for almost one hour and only 4 minutes, respectively. The (C) state is unstable, which is difficult to understand. 6. Considerations It is noticed in Table 4, that there occurs the shift from a complicated mode to a simpler one, but the reverse never occurs, and that the shift, for example, of (D)— »(C)->(B) is possible, but that at a bound, for example, of (D)->(B) is impossible. The next fact that attracts our attention is that the water temperature is the higher, the simpler mode appears. The simplest 29 Figure 4. The elevations of water surface in the cases of A, B, C and D-type seiches. The left-hand sides are the center of the pond and the right-hand, shoreline. These figures show that the surface is now elevated at the center. 30 Table 4. Shifts of the modes. Date Water Temperature ( °C) Shifts of the modes October 25, 1977 18.8 (B) May 16, 1978 23.8 (B)-»(A) October 25, 1978 18.0 (C)-*(B) December 26, 1978 7.7 (D)-*(C) -»(B) mode, (A) appeared only when the water temperature was 23.8 C. It is easy to understand from the viewpoint that the motive power of the seiche may be the impulse by the dropped water mass. When the water temperature varies, the physical properties of water, that is, surface tension and viscosity vary. And the shapes of the jets themselves and the dynamics of the collision of the water mass with the pond surface would vary. The confirmation that the seiche whose period is 1 .47 seconds corresponds to the mode (D) in figure 4, that is, the shoreline, to the 5th loop, is here necessary. When this confirmation will be obtained, the seiche whose period is 1.80 seconds is the (C) mode, and thus 2.40 seconds and 4.15 seconds are automatically (B) and (A) modes, respectively. This confirmation can be obtained as follows: When the seiche, the period of which is 1.47 seconds, appears, let us assume that the shoreline is the 72/th loop. Of course, 71^5. From the fact that the square of the velocity of the sinusoidal wave whose wave length is /_, , is given by Z7C U4/ "' rt/ L we get $r i i zzl Iqm% =1. (10) Let us here imagine an infinitely wide pond, and the Bessel function becomes a sinusoidal function at infinity, the wave length of which is given by [^ ^ =2 7tfl//X ( & is the radius of the pond ). Here A, is the solution of «£'(*)= -J,(x)=o. 00 The 4th and 5th solutions are given by A^=13.324 and ^,£ =16.471, respectively; thus b*t to =3.1 18m, C\2) L$oo =2. 552m. Substituting 1.47 seconds for ~]~ , and 80.0cm, for -ft, , and finally 3.118m and 2.552m, for /_, in equation (10), we get: The left side of (10)=0.998, when ^=3. 118m, /^n = 1 .286, when ^ = 2. 552m. It is here obvious that A.^=1 3.324 appears, and it is, therefore, confirmed that the mode in this case^Ls that of (D) , and the shoreline is the 5th loop. By the way, there was a slight discrepancy of the pond depth 31 at every experiment, corresponding to which there was also a slight discrepancy of the period of, for example, some 2.40 seconds of (B) mode. We call all of them, however, "2.40 seconds" for convenience' sake. 6. Energy of the seiche The seiche of mode (A) is the most convenient, by which we can estimate the amplitude Ho at the center of the pond. From Figure 4, we can easily find that the amplitude along the shoreline in the case of (A) is some C.4oHo» And in the caseS of (B), (C) and (D) , the amplitudes along the shoreline are C3Cno> C.25J~lo and 0.22 rfo> respectively. When the seiche of mode (D), the amplitude at the center of which is the same as that of mode (A), appears, the amplitude along the shoreline becomes, therefore, only some half as that of the latter. Besides this fact, it is easily supposed that the amplitude /-/ itself at the center may be apt to become larger in the case of a simpler mode. The reading the wave amplitude along the shoreline in the case of a complicated mode is, therefore, doubly difficult. At the same time, it can be suggested that it is possible that the shift from a complicated mode to a simpler one, means an aquisition of larger energy to the system, as the energy of impulse is constantly being given. As the examples of the natural seiches continuously stimulated by the external forces, one of the seiches in the Atlantic Ocean stimulated by the semidiurnal tide, those in a bay (Ofunato Bay, for example), by the tsunami ( especially by that of 1960 from Chile, which had a long duration ), and some seiches, by some meteorological factors— those factors are apt to continue for a while — can be observed. This phenomenon is, therefore, rather universal . In the case of the round pond, during the dancing of the jets whose period is 4.15 seconds, there exist three groups of the disturbances, that is, rather long period seiche, the disturbances the period of which are some 1 second ( since their motions are quite irregular, it is difficult to regard them as the seiches of J\ or Jx -type ), and the short crested waves which are more or less controlled by surface tension. After the jets are stopped, the last group disappears rapidly. The reading of the amplitude of the seiche is not, therefore, so difficult. Multiplying this read value by 5/2, we can estimate the fundamental amplitude }-\ at the center of the pond. In Figure 5, we show the double amplitudes, that is, the wave heights, one minute , two minutes, three minutes, four minutes and five minutes after the stop of the jets. The wave height after minute is lower than the expected one, which is due to the complicated superimposition of the waves whose periods are one second or so. In this semi-logarithmic diagram, we can easily read that the wave height is 2.2cm when "6=0. Multiplying this 32 cm 3-0 20- 10- 08- 06 5min Figure 5. Decay of the wave height of the seiche, the period of which is 4.15 seconds, after the stop of the jets. The wave heights were measured along the shoreline. 33 wave height along the shoreline by 5/2, we can get the fundamental wave height at the center of the pond, that is, 2H =5.5cm. Substituting 3.832 for 7U| in equation (8), we get C =0.958 Jffi. (14) Since the central part of the pond is a little deeper, the wave velocity is a little less than 95.8% of that of the long wave. This wave is, however, a long wave essentially. Then, let us estimate the energy of this seiche. Assuming that the pond water is the ideal fluid, and that when the water surface at the center of the pond takes its highest or lowest position, the kinetic energy of water particles disappears and we can substitute the potential energy for the total energy of the seiche. It is given as follows; ^ ^ n m s2 __hMHij\(x(^))ir = ^L(7 (A,))! Here, f is the density of water (=0.997 fy/orfl ) . Substituting —0.403 and 2.75cm for J~ (7\,) and y\ Q in (15), we can estimate the energy of the seiche as follows, £ =8.25 X 1C g ergs. (16) When we assume that the amplitude ( after we stopped the jets of water ) is given by a function of time t/ as follows, H=H t Xt ^ (H =2.75cm). (17) The energy of the seiche is, of course, expressed by t=E e ■ da) Now we can estimate the rate at which the energy of the seiche decays just after the stop of the jets. TxH =?lEA=4.09Xl0 6 ergs/sec. t-0 (19) This equation is considerably significant. If we had not stopped the jets of water, the state in which Jt =£To would have kept. Because the balance between the fall of the jets and the seiche was in a stationary state. As the equation (19) expresses the rate of energy decay owing to the stop of the jets, this value itself means, in other words, the rate at which the kinetic energy of falling water changes into the energy of the seiche. In spite of the fact that by equation (19), we cannot, of course, discuss all the phenomena, the quantity of energy change from kinetic one of falling body to that of wave, may be able to be estimated to be more than or even at a number of times greater than that given by (19). The remainder of the given kinetic energy might be spent to make the water temperature rise or might disappear changing into the sound energy. Or it might become the latent heat to vaporize the sprays. By the accurate measurements of the volume of the water jets and the maximum heights reached by them, this experiment could be utilized 34 to estimate the born wave energy component of the given energy "by a falling body. For this purpose, we should adopt a simpler model of experiment, when we should see where we stood, for Gauchy-Poisson waves and the short crested waves would appear at the same time. This consideration is a little important from the tsunami theory. Only in one case, it was suggested that two percent of the energy of the rockslide into Lituya Bay in Alaska ( 1958, it was near the borderline between rockslide and rockfall as defined in two classifications of landslides ) , had changed into the energy of the famous giant wave. The reporter of this giant wave, Don J. Miller (1960) didn't do any experiment which backed up this suggestion. This study may be our urgent business. It will be linked directly with important social problems. The question, for example, "what scale was the tanah longsor*"( landfall ) from the cliff on Lomblen Island, Indonesia, judging from the disastrous tsunami on July 18, 1979, which attacked the villages around this cliff?" would be solved by it. 7. The puzzle of the disappearance of shifts This section has a little meaning. It is curious enough that the period shifts disappeared when we stood the tubes of the spout holes made of alloy more vertically. It is a little reasonable that when the area shocked by the water drop decreased, long period wave components might hardly appear. Then, why the shift (B)-»(A), for example, occurred against the expectation that (A) type could appear from the beginning, when the shocked area was wide? And even the previous pattern that when the water temperature is high, the simpler mode is apt to appear and when the temperature is low, the complicated one is, this pattern was turned. As an example of a fine observation, we conducted an experiment on Nov. 6, 1980, when the water temperature was 14.3°C, when only (A) mode, the period of which was from 4.10 to 4.13 seconds continued for some 15 minutes, and since there was no expectation of the period shift, the observation was stopped. And on one day in summer, the shootings on the 8mm cinefilms v/ere tried, when only (B) mode whose period was 2.40 seconds continued and the observations were stopped after some two hours. That when water temperature was 14.3°C, (A) mode appeared, and when more than 25°0, (B) mode did, was here noticed. Under these circumstances, the descriptions till 6th section mean they are true under one condition that the impulse by the water mass is given over a large area. This report ends, suggesting that the surface area over which the impulse is given might cosiderably cotrol the phenomena, and expecting the next one on the following researches. 8. Conclusions Using a round spray pond, the shape of which is extremely 7V This word "tanah longsor" was mistranslated into "eruption", and some confusion resulted from this mistranslation. 35 geometrical and the diameter of which is more than 13m long, the artificial seiches of considerable scales were generated. Judging from their scales, the stir created by these phenomena might not be so small. Two natural phenomena occurred in our mind during the experiments. (1) The huge semidiurnal tide in the Atlantic Ocean is, to tell the truth, one of the seiches of the ocean whose period is some half a day, which is eternally stimulated by the semidiurnal tide itself. The range of the diurnal tide is, however, equal to or less than 38cm in general in the same ocean. (2) At the last stage of the tsunami invasion into Miyako Bay in the Sanriku District, Japan, the long wave whose period of which was as long as 80 minutes was observed. It can be suggested that in the case of our round pond, the amplitude of the seiche becomes sufficiently large comparing with the diameter of the pond, which wouldn't occur in the first case, but would, in the second case. But this is only suggestive. It can also be suggested that the period of shelf seiche used to be as long as some 80 minutes. Finally, itemizing the results of our experiments: (1) "when the water surface of a geometrically round pond is adjusted as high as the spout holes at the center of the pond, the seiches of beautiful Bessel function types appear, which are represented by the dances of the jets and can be easily read by our eyes. (2) In the cases of the wide shocked areas, when the water temperature is the higher, the simpler modes of seiches appear, and it is suggested that the surface tension etc. of water might play an important role. (3) In the above-mentioned case, the phenomenon shifts from a more complicated mode to a simpler one. This shift is completed in a short time. (4) Through the reading of the amplitude of the simplest mode, energy of the seiche and the rate at which energy is being supplied by the impulse are easily able to be estimated. (5) When the shocked area was narrow, the shifts of the periods disappeared, and the correspondences that when the water temperature was high, the modes were simple, and when it was low, the modes, complicated, this correspondences also disappeared, which suggested that the shocked area was an important factor, too. REFERENCES KODAIRA, Y. , Butsurisugaku (I), 336-340, 1931 (in Japanese). LAMB, H. , Hydrodynamics, Camb. Univ. Press, 6th edition, 1932. MCNOWN, J. S. , Waves and seiche in idealized ports, Proc. NBS Symp., Gravity waves, Nat. Bur. Stand. Circ, 521, 153-164, 1952. MILLER, D. J. , Giant waves in Lituya Bay, Alaska, Geol. Surv. Prof. Pap., 354-C, 51-86, 1960. WATSON, G. N. , A treatise on the theory of Bessel functions, Camb. Univ. Press, 2nd edition, 1948. 36 Seiches in Bays Forming a Coupled System ( Correction and Supplement ) Masito NAKAi\0* ana Naohiro FUJIMOTO** Faculty of Marine Science and Technology, Tokai University, Orido, Shimizu-shi, Shizuoka-ken, Japan ADstract*.- The seiches occurring in two adjacent bays, Koaziro Bay and Fioroiso Bay, situated in Miura Peninsula at the mouth of Tokyo 3ay, are very regular, and the phenomenon of oeat appears in their amplitudes of oscillation! Concerning this phenomenon, NAKAN0( 1932 a ) presented a theoretical explanation, considering two rectangular bays of the same size and form as well as of uniform depth, Dy assuming that a kind of coupling takes place oetween the two oays through a portion of water flowing across the mouth-line of each Day. In the present paper, the validity of the theory has Deen proved in a series of hydraulic model experiments. The experiments have shown that, in the case of the two adjacent rectangular Days mentioned aoove, there are two distinct modes of oscillation viz.(l) the co-phasic oscillation and (2) the contra- phasic oscillation, and since the frequencies of these oscillations are close to each other, when they come to interfere, the Deat phenome- non occurs. Moreover, in the former reports( References (3)-(6)), in the hydraulic model experiments, the beats occurring at the time of forced oscillations were regarded as those shown in the theory, Dut in the present investigation it has Deen shown that these two things are different phenomena, and correction has been made. * Present address: 4-8-14, Higashi, Kunitachi-shi , Tokyo, 186 Japan. ** Present affiliation: Nohmi Bosai Kogyo Co., Ltd. 7-3, Kudan-Minami 4-Chome, Chiyoda-ku, Tokyo, 102 Japan. 37 It has also oeen found that, when a tsunami wave enters the two bays, the mode of oscillation of the seiches occurring in them changes in the following mariner: the co-phasic oscillation ± the oscillation accompanied with the beat phenomenon >the contra-phasic oscillation. As regards this phenomenon, the authors have presented a possible explanation by considering a kind of " perturbation" and the dissipation of energy. 1 . Introduction The seiches occurring in two adjacent bays, Koaziro Bay and Moroiso Bay, situated in Miura Peninsula at the mouth of Tokyo Bay, are very regular, and the oeat phenomenon occurs in their amplitudes of oscil- lation( Fig.l ). Concerning this phenomenon, NAKANO(1932 a) presented a theoretical explanation, considering two adjacent rectangu- lar Days of the same size and form as well as of uniform depth, by assuming that a kind of coupling takes place between the two bays through a portion of water flowing across the mouth-line of each bay. FUJIMOTO and NAKAN0(1982) reported, from the result of hydraulic model experiments, that when a tsunami wave( a single wave) enters two adjacent bays, the beat phenomenon can occur as well(Fig.2). In the present paper, some hydraulic model experiments have oeen carried out to prove the theory. In addition, a further experimental investigation has been made on the oscillations in the two bays caused by a tsunami wave, and some discussions have been made on the transition of the mode of oscillation. 2 . T heoretical consideration s First, consider a rectangular bay of length L, breadth b and of uniform depth h, and suppose that in this bay there is a stationary oscillation which can oe expressed by t. =s 3 cos I A > A > ' * r J t 1 * A 2 * * s <■ A A ' A s A ' A t 4 / A ' A / * > A S A * * / • «- * / { ^ ■' * I ^ \ A 'ft, SSSSAAtA/ '„,, *x;j + M,x i0 . (5) Hence, the total kinetic and potential energies of the coupled system under consideration are as follows: Suppose that the present system is an isolated one, namely, that its total energy is constant. Then the Lagrangian equation fjr f — . ) — v — — — gr ( k = 1, 2 ) gives the following simultaneous equations, U + »x /fl +/VX. /"-K+M'K* vx /( .+/^X.-o. where H'«M+/L'=-gfcO+Q,)+/uf, (7) 4-eL *■.*+*-*&+*'& 42 The general solutions satisfying (6) are given by X /o = G)° C«s (n,t+ 6,)+ G}**Cos(n z t + e x ), y where * n ^nr^y /-/ M+/u.'j-f*>> ' \ (9) and £";, na)™' r C g z-rz aroitrary constants. Since U, and 7t^ are nearly equal, we can say that Doth X, anc ^ X are macie U P °f two oscillatory motions with nearly equal frequencies. The same can oe said for the case of the vertical motion of water surface which is connected with the horizontal motion by the equation of continuity. Therefore, we may say that a train of seiches exhibiting the appearance of beat can occur in a pair of bays lying side by side. ^e shall now consider some particular cases of (8). (l) Case wnen seicnes exist initially with the same phase in the two bays. If we taKe/Y )-/ , flfc) =r t ( X )=: / » (X ) - » then we have X,.~ C.sn t t,*7 ° '" ° W X^ fl =: CoS U, tj Namely, in this case the two oays oscillate with the same phase with period iH. the co-phasic oscillation. * n, (ii) Case when seiches exist initially with the opposite phase in the two oays. If we ta*e (x )=/ * (X ) = » (X )=-/$ (% ) =0 , then we have v „ '°t~° ' a t*o *•*=„ *•*=<» vo-cosnj, | ho = c °sn z t, Namely, the two Days oscillate with the opposite phase with period ±- 1 — the contra-phasic oscillation. (in) Case when seicnes exist initially in only one of the two bays. If we taKefx )-.£ , (x) =0 ,/)< ) = » fX ) = , then we have X /o = Jg gQ S (*i+*+)t. Cos ( n ^')t J I (12) Namely, both bays oscillate with period -rr, Tl s/o" anci amplitudes varying slowly with period- 7 — . In other words, the beat phenomenon occurs. 43 3. experiments In oraer to verify the aoove tneory, a series of hydraulic model experiments was carried out. fig. 4 shows the general configuration of tne experimental apparatus, and Fig. 5 shows the dimensions of each part of the experimental tank. At one end of the tank is a plunge r^P' 7 to generate waves, oy which the period ana height of the waves can oe altered. The model oays are made of wood, and near the head of each one is located a servo-limnimeter sensor "G". Taole 1 shows the dimensions of three kinds of bays used in the experiment and the maximum water-depth, h , r r max necessary to satisfy the long wave condition for each oay. Again, Taole 2 shows the minimum value of the period, I , of the incoming waves or the plunger ST motion, necessary to satisfy the long wave condition for a given value of the water-depth h (shallower than h ) for each oay. max 7 J (A) The oeat phenomenon l\iow, in order to see if seiches accompanying with oeats can occur in the two oays, an experiment was carried out in the following way in conformity with the initial condition stated aoove* * In their previous papers( References(2)-( 5) ) the present authors reported the result of a series of hydraulic model experiments made with an intention to verify the aoove-stated theory. In those experi- ments, an assumption was made that the whole of the two coupled Days forms an "isolated system", namely, the total energy of the system under consideration remains constant, in other words, the amount of energy dissipated from the system always balances with that supplied thereto. Thus, on Keeping moving the plunger with constant period and amplitude so as to satisfy the aoove condition, it was found that beats actually occurred in the seiches in Doth bays according to initial conditions. But the seiches in that case are forced oscillations oDviously, while the theory treats free oscillations, so we may say that the oeat phenomenon shown in the former experiment is different from that treated in the theory. In the experiments which follow in the present paper, the bay-water oscillations suosequent to stopping the plunger, viz. free oscillations(free damped oscillations) have oeen examined. 44 Figure 4. General view of the experimental apparatus Figure 5. Dimensions of each part of the experiment tank (unit: cm). P... plunger. F...wave filter. A. . .wave-absorbing device. I... iron frame. G. . . limnimeter . N,L. . .wave measuring points 45 Table 1. The maximum water-depth, h , for three kinds of Days used in the experiment. Length of Breadth of Mouth-correction Maximum water-depth tne Day tne bay factor (cm) b (cm) f (cm) u max 51.2 15.0 1.203 12.3 71.2 15.0 1.168 lb. 6 91.2 15.0 1.145 20.8 Table 2. The minimum period, T . , of the incoming waves for a given value of water-depth, h , for each Day. Length of Water- ■dep th Minimum period of the bay the incoming waves L (cm) h (cm) T (sec) p min 8.7 1.89 51.2 8.8 8.9 1.90 1.91 9.7 1.99 71.2 3^ 9.9 2.00 2.02 10.8 2.10 91.2 10.9 11.0 11.1 2.11 2.12 2.13 46 First, oy sending waves to Bay 2 oy operating the plunger, seiches were generated in Bay 2, while the mouth of Bay 1 remained closed. After the seiches in Bay 2 attained a steady state, the plunger was stopped, and when the seiches in Bay 2 oegan to decrease in amplitude, or when free oscillation oegan to occur, the mouth of Bay 1 was quickly opened. Fig. 6 shows an example of records of the seiches thus obtained. Looking at these records, we can see that the beat phenomenon actually occurs, and that the variations of amplitudes of the seiches in the two Days are opposite in phase. Thus, we may say that the theory and the experiment agree well with each other, except that the actual oscillations are damped oscillations owing to the dissipation of energy, while in the theory which assumes the constancy of energy the oscillations are undamped oscillations. Fig. 7 is an example of records showing the behaviour of Deats in the two bays when the period, T , of the incoming waves is changed. From this figure we can see that the oeat period in the time of free oscillation remains nearly constant, although the wave-height of the forced seiches in Bay 2 changes as the period of the incoming waves is changed. Fig. 8 is an example of records in the case when the distance, d , between the two Days is changed. Again, Table 3 shows the values of the distance, d , Detween the two Days, the period, T , of the incoming waves, and the Deat period Tu for the case of L = 51.2 cm as an example. As seen in this taDle, scarcely any relation exists between T-, and I , while some definite relation exists between T^ and d, namely T^ increases steadily as d increases. (B) The co-phasic and contra-phasic oscillations Now, from the aDove-stated theory it has Deen known that the beat phenomenon in question occurs as the result of composition of the two normal oscillations viz. the co-phasic oscillation of angular frequency n^ and the contra-phasic oscillation of angular frequency n ? . In order to see if each of these normal oscillations can occur separately, an experiment was made in the following way. First, Dy sending waves to the two Days Dy operating the plunger, seiches were generated in the bays. After the seiches attained a steady state, the plunger was stopped, and the free damped oscillations of the two bays were observed. Fig. 9 shows 47 E E o a> O S C\J CD CD CD >, u 03 ■H rH -o i—l d •H •H u w OJ o 5-1 3 T3 00 0) •H CX 4-1 e 03 OJ -o X! 0) eg d u ■H 4-1 3 LM o 0) TO E ■H OJ • +J x T3 H 0) OJ d x 0) *J a 0) o +J +J TO o CO z 03 Cfl 3 >> fl rH ,0 r-N s >> o u TO 3 CO *J o U-l eg lO O X 4-» II X m XJ 3 o O - S w S c u 0J o X >H O J-> fJ • 03 m X i— 1 i—i u H •H •H II X U 3 ta ■Q o +J ~ n3 4-i e o u d s-l cm CO d • 4J o i—i to ■H LO d > ■H CB II X! <3J Qj ^ X CQ w 4J vO OJ u 3 00 fe 48 o II CO CD CD CO s e 1 >. Qj u c CO X o ■H OJ X 4-> • X IT) Qj +J o d r-H rl 3 OJ 2 X •P 0/ II oo u 5 •H •H • OJ *J -Q 4h X T3 x 0/ 3 Qj 4J X *■• 01 d s X 4J OJ d ~ u +J 03 & •H 73 CN d 4-1 o [fl »■• i— i •H c w ■U 3 0) c ii o Cfl X CO M d i—i 4J ^ (-1 •rl M-l n RJ >. O ■H Qj CO -O T3 Qj X CG )-i QJ X 4^ 3 OJ Mh CH X d w o ■rl 4J CO OJ > X *J X 03 d u 0) (0 +J XI OJ 4-) u 3 Qj X w o •H O pq s ■H % -o a 00 QJ u 3 oo ■rl ^ CVJ CO CD 4-J 3 CO 00 O c ^ 4-1 -o w ■rl u d 0/ >> s II CO CO d CO O 4-> QJ X u -Q Qj W & ti X C O •rl *^ H ■rl 3 S W *J OJ u OJ CO X CN X 3 (U 4-> OJ 4-1 X r-H 4-> i— i 4-> 4-1 m o w o z 0) >i 3 n 4-) CO ■H *> CO CQ a ^ U W h s. • • — s >H 4-4 4-> E •a O CO « -o u d OJ -o e OJ > W u X X CO d U o 4J u X 0) > . •H OJ X CO m c X CQ 3 3 r ~' ■rl 3 r~~ QJ U 3 00 ■H to 49 Taole 3. delation among the distance, d , between the two Days, the period, T , of the incoming waves, and the beat period T b . Length of Distance oetween Period of the 3eat period the bay the two bays incoming waves L (cm) d (cm) T (sec) T (sec) "P x b 2.96 27.84 2.5 2.b9 27.96 2.53 28.14 3.00 30.74 5.0 2.71 30.66 2.55 30.64 3.01 36.41 51.2 10.0 2.75 36.45 2.61 36.53 2.96 37.82 15.0 2.80 38.00 2.64 37.88 2.96 39.54 20.0 2.83 39.50 2.64 39.32 50 [a) Case of T = 3.08 8ec Bay 2 Bay1 Bay 2 Bay 1 (b) Case of T = 2.57 8ec gjsec : Tc~ ■fe^-f- W(^WVW«««"~ " Figure 9. Sample records showing the behaviour of free damped oscillations cm .cm cm, in the two bays (L = 51.2 , b = 15.0 , d = 5.0 ). 51 an example of records thus obtained. Fig. 9(a) is the case where the period, T , of the incoming waves or plunger motion is 3.08 sec. In this case both Days perform damped oscillations with the same phase without oeing accompanied with beats. The oscillation period, To, say, in the time of damping in this case may De regarded as to correspond 2-rc to the period, ~y^~ , of the co-phasic oscillation without damping, derived from the theory. Fig. 9(b) is the case where the period, T , of the incoming waves is 2.57 sec. In this case, in both bays, simul- taneously with damping, oscillations accompanying with beats begin to occur, which gradually turn into contra-phasic oscillations and go to decay keeping that phase relation unaltered. The period, T , say, of the contra.-phasic oscillation in the time when beats can no longer J9-JT De seen, may be regarded as to correspond to the period, ~— , of the 'la- contra-phasic oscillation without damping, derived from the theory. Thus, from the aDove experiment it was found that the two kinds of normal oscillations mentioned above can actually occur. Table 4 shows the measured values of Tj and T . As seen in this table, T a is larger than T c , that is, n ± i s smaller than n 2 . Again, in the two rectangular bays of the same size and form, T, decreases, while T increases as d increases. This corresponds to the fact that the beat period T b increases as d increases( Table 3 ). For, as is derived from the theory, there is the relation - 1 - = -L f -L~ _ _L. \ (13) Moreover, oy the use of the data given in Table 4, Fig. 10 was construct- ed, which shows the relation among T., T and d. Furthermore, Table 5 shows the values of the Deat period T^*, say, measured directly on the limnimeter records and of the beat period T b calculated from the values of T, and T using the relation (13) . Looking at the table, we see that T b * and T, agree well with each other. (C) Experiment on a tsunami wave . Concerning what kinds of mode of oscillation may occur in case when a tsunami wave(a sigle wave) enters two adjacent Days, a model experi- ment was carried out. The experimental procedure is as follows. After removing the plunger P and the wave filter F from the said experimental tank( Fig. b) , a model tsunami was generated oy one stroke with a plate near the center of the tank, and the oscillations of water surface were 52 Taole 4. The co-phasic oscillation period T, and the contra-phasic oscillation period T . c Length of Distance Detween Water-depth Proper oscil- Co-phasic Contra- the oay the two bays lation period oscil- phasic of single bay lation oscil- period lation period L (cm) d (cm) h (cm) T (sec) T (sec) (sec) 2.5 8.7 2o67 3.10 2.54 5.0 8.7 2.67 3.08 2.56 51.2 10.0 8.7 2.67 3.04 2.61 15.0 8.8 2.65 3.03 2.61 20.0 8.8 2.65 3.03 2.64 2.5 9.7 3.41 3.86 3.22 5.0 9.8 3.39 3.82 3.24 71.2 10.0 9.8 3.39 3.79 3.30 15.0 9.9 3.38 5.76 3.30 20.0 9-8 3.39 3.75 3.35 30.0 9.8 3.39 3.63 3.37 2.5 10.8 4.06 4.31 3.84 5.0 11.0 4.02 4.25 3.83 91.2 10. 10.9 4.04 4.24 3.88 15.0 11.1 4.00 4.19 3.83 20.0 10.9 4.04 4.21 3.95 53 1.05 - \ l/D to 1.00- I 0.95- + % h Figure 10. Relation among T. , T and d. 54 TaDle 5. Comparison of the values of the Deat period. Length of Distance between Jeat period 6e at period calcu tne oay the two ( >ays measured di- la ted from the rectly on the va lues of T , and limnimeter record L (cm) d (cm y T *(sec) x b T (sec) b 2.5 27.98 28.01 5.0 30.68 30.49 51.2 10.0 3b. 46 36.10 15.0 37.90 38.02 20.0 39.45 40.73 55 measured with limmmeters placed at the heads of the bay models. Fig. 2, referred to oefore, shows an example of records thus obtained. As seen in this figure, the two bays oscillate, at first, with the same phase, out soon thereafter they begin to be out of step, together with the appearance of oeats. fhus the oscillations gradually turn into contra- phasic oscillations and go to decay keeping that phase relation unalter- ed. Table 6 shows the measured values of the period, T,, of the contra-phasic oscillation caused Dy the model tsunami ( a single wave ) ana of the period, T^ , of the contra-phasic normal oscillation mentioned before. From this table, we can see that T. is the same as T . c 4 . Qn tne transition of the mode of oscillation According lo the result of the above experiment( Fig. 2 ), when a tsunami wave( a single wave ) enters a pair of coupled bays, the mode of oscil- lation of the seiches occurring in them changes in the following manner: the co-phasic oscillation(i . e. the two bays oscillate with the same phase) *the oscillation accompanied with the beat phenomenon > the contra-phasic oscillation( i.e. the two Days oscillate with the opposite phase). And it is likely that when once the contra-phasic oscillation is attained, it does not return to the co-phasic oscillation any more. Concerning tnis phenomenon, some considerations will oe made in the following. As already mentioned, the co-phasic and contra-phasic oscillations are expressed oy Eqs.(lO) and (llj, respectively. We shall now investi- gate how these expressions will be altered when the initial condition in each case receives a slight change or a kind of "perturoation" . Case I . Case wnen the original motion is the co-phasic oscillation Let the original or unperturbed Y and X be written as % and f r , respectively, and those perturbed as Q -^ q and £ -f- ? » respectively. Then, we have X'^fa+t, = CoSM / + ^ l (14) Here, the values of £ , Q , 9 , 9 at t = are assumed to be small. And and Q must satisfy tne equation "k U K UK 56 Taole b. delation between the period, T t , of the contra-phasic oscillation caused oy the model tsunami (a single wave) and. the period, T , of the contra-pnasic normal oscillation. Length of Distance oetween Period of the period of the the Day (cm) 71.2 the two oays (cm) 2.5 5.0 10.0 15.0 20.0 30.0 contra-phasic oscillation caused by the model tsunami m (sec; _t 3.23 3.24 3.30 3.31 3.35 3.37 contra-phasic normal oscil- lation T c 3.22 3.24 3.30 3.30 3.35 3.37 ( sec) 5 7 where Tnus, we have the following simultaneous equations, cm +/c') \ t^-tc^m, + *% x ^°> i A £, + m+/^&+ vfyitt+mj^oj d 6 ) Tne general solutions for Q and v satisfying (16) are tr, = c, Cosn z t ±c z sCfin>t t c 3 co$n t t + c+s£n,?L / t, ) f /z = -c, Cosn z t - c^s^nft^t-f C » r^ are aroitrary constants. Hence, if we denote the values of <> , S , ^ , tf at t = by £ , 6 ' » £ » £' > respectively, ve i I * ' ' then we have X = Cosn to (18) These are the required solutions for the oscillations in the case when the "perturbation" is applied to the original co-phasic oscil- lation. Two special cases will be considered. (l) If we take £=£( = £), £'=£' = o, then we have X /o= r (it €) toS7\,£, y X^ = (/+e) Cos n,t. ) Namely, the two bays keep on oscillating with the same phase, (ii) If we take £=£,£ = -£, £*=£' - 0, then we have X Zo = Cosn,t- € Cos 7l z tJ Namely, in this case a contra-phasic oscillation is newly added to the original co-phasic oscillation in both bays. Case II . Case when the original motion is the contra-phasic oscillatio n . Tn this case, we put f>.- COS TV*, 1 (21) Tnen, in the same way as in the previous case(Case I), we shall obtain the following result, corresponding to (18;, (19) (20) 58 (22) Two special cases will oe considered. (1) Take £=£-( = € )» €' = £' = 0, then we have \= Cos n x t + € coSn,t, 7 (23) X Ao = - Co s 7fe,£ + € cos n, t, ) showing that, in ooth oayt>, a co-phasic oscillation is newly added to the original contra-phasic oscillation. (ii) TaKe £ = £,£ = -£,£' = £'= 0, then we have ^,o^(\-l-e)CoS?) z t, l (24) showing that the two Days keep on oscillating with the opposite phase. Thus far, we have assumed that the energy of the whole system under consideration is constant and there is no dissipation of energy.* If the dissipation of energy be taken into account, the oscillation in question will necessarily oecome a damped oscillation. And, as may also oe inferred from the configuration of streamlines shown in Fig. 11, the rate of dissipation of energy must be larger in the case of the co-phasic oscillation than in the case of the contra- phasic oscillation. Thus, when the dissipation of energy is considered, the terms involving n-, or n ? in the aoove equations should oe multiplied oy a factor such as £ or j£^ ( 0<$<°t- j, say, respect- ively. Hence, for example, Eqs. ( 2u ) and (23) will then be rewritten, respectively, as follows: X /0 = e~** Cos «,£ + £"£? C cvsr> t t, \ Xz6 = e" ut -Cos7i,t - -e~l f ecosnz t.j ( 20 ) ' * The dissipation of energy of the seiches in a Day is due to two causes Namely, the one is the effect of friction, and the other is that of "yielding". The latter phenomenon is analogous to the damping of oscil- lation occurring in a pendulum whose suspension point is loosely fixed. Of these two causes, the latter is, in general, far more effective. Concerning this proolem, NAKAN0(1932 b) once made a detailed discussion. According to his theory, for a given Day, the dissipation of energy of the seiches due to this "yielding" ooeys an exponential law such as e with respect to time. 59 Case of the co-phasic oscillation Case of the contra-phasic oscillation Figure 11. General aspects of streamlines in the co-phasic and contra-phasic oscillations of the two bays (schematic diagram) . 60 -ott -Bt / ~ * * -A In the case of Eg.. (20)', at t = 0, the amplitude of the co-phasic oscillation^ angular frequency, n-, ) is 1, and that of the contra- pna^ic oscillation( angular frequency, n^ ) is £ . Fig. 12(a) is a schematic diagram snowing the manner of decay of the amplitude of each normal oscillation with the lapse of time. At the beginning of the motion, the co-phasic oscillation prevails, out as time goes on, the amplitudes of the two normal oscillations draw near to each other and thus beats begin to occur. But, since the decay of the co-phasic oscillation is more rapid than that of the contra-phasic oscillation, Deats soon disappear, and at last the latter oscillation surpasses the former in amplitude. The transition of the mode of oscillation as seen in Fig. 2 is perhaps due to the above reason. Next, consider the case of tiq.(23)'. In this case, the amplitudes of the co-phasic and contra-phasic oscillations at t = are 6 and 1, respectively . Fig.l2(bJ shows schematically the manner of decay of the amplitudes of the respective oscillations with the lapse of time. As serin in this figure, the contra-phasic oscillation having smaller rate of decay prevails over the co-phasic one in amplitude from the very Qeginning. Thus, we may say that when once the contra-phasic oscillation is attained, it does not return to the co-phasic oscillation any more. 5 . Concluding remarks jased on the result of hydraulic model experiments with a pair of rectangular bays, the periods, Tj and T , of the two normal oscillations viz. the co-phasic oscillation and the contra-phasic oscillation, respectively, have been measured. The ueat period, Tv , of the seiches calculated from those values of T and T , agreed with its value d c measured directly from the experimental records. Thus, the theory developed has been proved adequate for explaining the phenome- non at least as & first approximation. 61 Tsunami Flood Control at the Opening of a Bay or Harbour SHIGEHISA NAKAMJRA Shirahama Oceanographic Observatory DPRI, Kyoto University Shirahama, Japan 649-23 ABSTRACT Tsunami flood control at the opening of a bay or harbour was studied in order to clarify the maximum current velocity which affects to hazards of navigation as well as of aquacultural plants and the other human activities. A simplified model of a bay with a single channel connecting the open ocean was considered. The depth of the bay was assumed to be constant. The channel's dimension is given by its width, its length and its water depth. A friction factor was considered in the model. An application of the model was considered to estimate the maximum current velocity at the opening of a bay or at the entrance of a harbour. The solution shows that the most important factors are the length of the channel, the area of the bay or harbour and the cross- section of the channel in relation to the frequency of the incident tsunami. With this, the author proposes a reasonable way to control tsunami- induced currents at the opening of a bay, i.e., to construct a long and narrow channel connecting the bay and the open ocean. Of cause, it should be evaluated in advance what is its effect to the environments in case of practice. 65 INTRODUCTION Tsunamis at the opening of a "bay or harbour are studied to evaluate how the tsunami- induced currents can be controlled by a channel connecting the ocean and the bay or harbour. In order to protect the coastal area where human activity is concentrated, many of the pertinent countermeasures have been introduced against the tsunamis caused by the undersea earthquakes and the storm surges accompanied by the typhoon, hurricanes or cyclones. Especially on the coast facing the western North Pacific Ocean, most part of the coast has suffered and threatened by the tsunamis and storm surges to have a hazard and damage. In addition, the typhoons, hurricanes or cyclones scratch several times every year in the monsoonal district of the mid lattitude to result a severe damage caused by the accompanied storm surges. In some of such district, frequent experience of storm surge suffer has lead to have an effort to establish a complete protection of the coastal area from any storm surges and they seems to believe that this protection is also enough against any possible tsunamis. On the other hand, the coast at higher lattitude on the circum-Pacif ic seismic zone has had severe damage frequently by the tsunamis in the past. As a countermeasure for the tsunamis, construction of the coastal structures has been promoted. However, it seems not yet well established how to manage the vessels, ships and boats around the harbours, coastal area and in navigation. In case of a storm surge, the vessels, ships and boats can escape from any severe damage in advance after receiving the meteorological predictions and warning. However, it is hard at present to predict when, where and how big tsunami occurs. Adding to that, we have only a limit ted time to escape from the tsunami at receiving any tsunami warning except in case of a distant tsunami which comes across the ocean. This is the essential difference of the warning for the tsunami from the storm surge warning. At present, it is only possible to estimate the next tsunami appearance at a certain coastal area statistically referring to the past documents, records and experiences. ALthough, we have only a limitted number of data about the tsunami accident experience of the vessels, ships and boats. It is generally said that the boats offshore never feel any tsunami passing by because 66 its wave length is very long and its wave height is very small at a certain distance from the wave source as well as from the coast. On the other hand, the effect of the tsunamis is more significant in the coastal area. Many of land-use damage on the coast caused by the tsunamis is well known and can he found in reports, notes or scientific publications. Some of them must be must be fading out or vanishing from our memory with the time elapse. One of a captain's talking about his experience of a tsunami was very interesting for the author. One day his vessel was just navigating neighbour the entrance of Osaka Harbour and he thought at that time it was unusual for him to feel it to be hard to manage his ship on the water. He has had never experienced such a case. His vessel was already controlled by a strong flooding stream at the entrance. It was the day of I960 Chilean Tsunami. At receiving this report from the captaion, the Japan Captain's Association considered it necessary to learn and evaluate what is the raximum or possible current velocity induced by the tsunami at the entrance of the harbour, or especially at the entrance of the important harbours facing the western North Pacific Ocean. The Kobe Shipping Casualty Research Commission proposed to promote a scientific research to evaluate such the current and to distribute the results for the navigator's use as >a manual. With this, a solution for practical use has been requested. In this study, a harbour model with a channel connecting to the open sea is considered and to analyze theoretically its response property to the incident tsunami. By an analysis, it is shown that the important factors controlling the current velocity in the channel are the area of the harbour, width and length of the channel and water depth of the harbour area as well as that of the channel. It is clarified that, especially, the ratio of the harbour's area and the channel's width is important to control the response property of the current velocity in the channel or of the water-level oscillation in the harbour. As an application of the model, a harbour on the coast of Osaka Bay and Kii Channel, Japan was considered as a parameterized proposal to control the current in the channel in a scope to ease 67 management of the vessels, ships and "boats and to know how to escape from any maritime accident even at any incident tsunami. MODELLING OP A BAT OR HARBOUR When a hay (or harbour) is connected to the open ocean "by a channel, response in the harbour must he affected "by the normal oscillation property of the channel. The combined system of the harbour and the channel consists the other response property different from that in the harbour and in the channel. If the period of an incident tsunami into the combined system coincides to the norm a l modes of the combined system, resonance should be appear to amplify the amplitude of the water level oscillation in the harbour and to magnify the induced current velocity in the channel. One way to study such a problem is to utilize a time-stepping numerical model of the harbour as an application of a finite difference method £f or example ? Nakamura, 1 981 J • The other technique must be an application of a boundary element method or a numerical contour integral method j[f or example, Lee, 197 2; Nakamura,1979 and 1980). This method is good for response of the water level in a harbour of a given coastline with a constant water depth at an incident sinusoidal wave. A numerical computation for normal modes of each harbour [f or example, Nakamura and Loomis ? 1980a and b; Nakamura, 1 980) is also useful to learn resonance if a dynamical consideration is given to an incident tsunami's condition. A simple model can be utilized as did Nakamura and Loomis£l980j, though only the initial stage of the first wave could be analyzed by the model. We can find many of theoretical analyses which have been published already farrier and Shaw, 1970; Carrier et al#? 1971; Hwang and Tuck,1970; L ee and Raichlen,1970j Mei and Petroni,1973; Miles and Munk,196l$ Shaw and Lai, 1974} • Anyway it is hard work to complete the analysis for all of the harbours which have improving and renewing under the designing plans, and the profiles of the harbours vary year to year so much that the exaustive analyses should be undertaken successively at each step of the renewal. Hence we have to be clever enough to get a pertinent solution by a simple analysis 68 of a harbour model which includes the essential factors,, With this consideration, the author has get to consider to develope a theoretical model of a simple harbour which has a channel connecting to the open ocean as follows. For a simplicity, a harbour model is considered as in Pig.l, where the bottom is flat in the harbour and in the channel. If the friction at the bottom is proportional to the current velocity, the equations of motion and of continuity are wriiten as follows in a simple form; ^z -rM t Itt. =c (l) where the current velocity u along the axis of the channel as the x axis is considered as a function of time t. The water level, the water depth, the gravitational accerelation and a friction factor are denoted by ^ , h, g and c< respectively. As for the boundary condition^, the following two conditions are assumed. That is, a wave from offshore is observed at the entrance of the channel (x = 0) first as expressed in form of and the disturbance propagated into the harbour through the channel spread at an instance in the harbour after passing through the channel (x = L). Then, ~rr - p U at X-^L ^4) where /3 = Bh/s(Bs width of the channel and Si area of the harbour). With the conditions (3) and (4)? the simulaneous differential equations (l) and (2) can be solved to obtain a solution for wave height in a channel as follows \ r-4(L-*) + (*/s)-+U-x) ,. ,. M :h [Auot) (5) \ r-ftt) i-(b/s)^(L) ex 69 9 Fig.l Schematic model of a harbour with a channel connecting to the sea Shigehisa Nakamura, Kyoto University, Japan Tsunami Control at the Opening 7 where B/S is the ratio of the width B of the channel and the area of the harbour S. In the solution (5)> the following functions and notations are used; ^fiiv^l-^ , f=£U^^\ oj = If we take the real part of the solution (5)> the wave height can he written as follows; 2 + -f- I j>^ 2|>L-^»hz ? LJ J (6) 71 and the maximum current velocity as an average in the cross-section of the channel is lf]« 1 v nl l y (L-x)r stiff [L-JC) { + ±\ } ^^(L~x)-r%^^(L-z)} fVf) {s^l^^L-r 5-.>"|L | t — j- s JmK v» L -t- c fc« --'- -: = lh ---"*-*-- — A r — ... -- v\ * 1 J/S:1 V A N 1 l/S =.o 5 7 0.1 1 10 T HOURS Pig. 3 Same as Pig. 2 except for the water depth of 12 m, Shigehisa Nakaraura, Kyoto University, Japetn Tsunami Control at the Opening 74 Pig. 4 Same as Pig. 2 except for the water depth of 15 m. Shigehisa Nakaraura, Kyoto University, Japan 'Tsunami Control at the Opening 75 For example, if we consider a sinusoidal incident wave of f\ = 1 m and T = 0.5 hour instead of an incident tsunami of realistic one to a model harbour of L = 10 m, h = 10 ro and B/S = 10~ , then, we have (u/^ o ) m 3.6 in Fig. 2 and u = 3.6 m/s at x = L. As for a sinusoidal wave of r } m 1 m and T = 12 hours instead of a semidiurnal tides, we have (u/ ^ ) ■ 0.14. That is to say, the solution suggests that, even in the channel of the harbour, the maximum current velocity u for a model tsunami could be as much as 26 times of it for a semidiurnal wave. Adding to the above, we can find that the maximum current velocity in a harbour of B/S ■ 10 '' is smaller as much as two decimal factors than in that of B/S « 10~ 4 in Fig. 2 in the range of 0.1 < T < 12 hours. And that, it should be remarked that the largest value of the maximum current velocity in a harbour of B/S *> 10 '' is almost same to the smallest value of the maximum current velocity in a harbour of B/S = 10 , Similar result can be found for h « 12 and 15 m in Figs. 3 and 4 respectively. As for the effect of L, it can be found in Figs. 2, 3 and 4 for a fixed values of the water depth and of the ratio B/S. In this case, the effect of L is more significant except neighbour the range of tides, A caption reported an anmarously strong flood flow caused by i960 Chilean Tsunami at the entrance of Osaka Harbour. If we refer to the nautical chart published by Japanese Martime Safety Agency (revised by 1982), order of the inner harbour of Osaka Harbour is h = 12 m, L = 10 m and B/S = 2.3 X lo"~ 5 (i.e., B = 275 m and S = 1.18 x 10 7 m 2 ) at the time of the lower low water. In this case, the value of (u/ n ) for a tsunami of 50 min(= O.83 hour) can be found as about 7.5 from Fig. 3. When 17 = 1.0 m, then, the maximum current velocity in the cross-section at the entrance of the Osaka Harbour could be u = 7.5 m/s. Although, the value of (u/>^ ) for the semidiurnal tides(T ■ 12 hours) is about 1.2 so that we should exect u = 1.2 m/s for a given value of rj = 1 m. This suggests that the tsunami- induced current at the entrance of Osaka Harbour must be about 6 times of the ordinal current caused by the semidiurnal astronomical tides. 76 By using a similar model, to this HiguchiQ.959a and b) had studied on the water-level response in a channel which connected a "brackish lake to the open sea. CONTROL OP TSUNJSMI FLOOD In order to make easy and safe to navigate at the entrance of a harbour, it is necessary to control the maximum current velocity induced by any tsunami. Of cause, there must he various method to control the current. One of the methods is to reconstruct the geometry of the harbour entrance with an utilization of the above results of the analyses. a) Primitive concept is to select the geometry and the area of the harbour not to coincide the period of the normal modes of the harbour and the significant period of the incident tsunami. Hence, we have to learn what is the resonant condition of the harbour for an incident wave. b) As stated in the above section, the value of (u/n ) at the end of the channel can be controlled by selecting a large value of (b/s). However, to select a large value of (B/s) is same to take a more wide width of the channel for a fixed area of the harbour or to consider a smaller area of the harbour area for a fixed width of the channel. Then, such the selection leads to make an easy condition of the incident tsunami into the harbour so that it is not appropriate in practice. c) If the water depth h is taken to be deeper, the larger be the cross-section of the channel and the smaller the current velocity in the section inversely. Recent design of harbours has a direction or tendency to take a deeper channel and to dredge the navigation course for bigger vessels, tankers and flighters, so that this increase of the water depth in the channel and harbour agrees well to have a pertinent current control at the entrance. However, we have to be aware that to increase the water depth is equivalent to make more susceptible for the incident tsunamis as well as to select a larger value of (B/S) . is so. 77 d) It is cleax that the larger the value of L, the smaller the maximum current velocity for a fixed wave height in Figs. 2, 3 and 4. If the values of L are taken as (i) 10 m, (ii) 100 m and (iii) 1000 m, the period of the incident tsunami is in a range of 30 min to 50 min, and the value of (B/s) is 10~5, the value of (u/ ^ ) is in the range of (i) 15 to 53, (ii) 5.1 to 5.2 and (iii) about 0.5, respectively. That is to say, we never have the hagher value of (u/ \ ) than about 0.5 if L = 1000 m for any value of (b/s). Hence the author tends to consider that the most effective method is to select a large value of L, for example r L = 1000 m. That is to construct a long channel to connect the harbour and the open sea. Fortunately, in some cases of the reclamation projects for promoting higher use of the coastal area resulted to form a long channel of an approach for navigation from the open ocean to a harbour, and these cases were unwillingly in a successful direction to control the tsunami flood at the entrance of a harbour. In many cases, a pair of breakwaters with about 10 m wide is constructed at the entrance. If so, the author wishes to recommend to construct a wider breakwater to form a longer channel at the entrance of the channel, or to construct a pair of breakwaters parallel to each other to form a long channel at the entrance of the harbour. With the above consideration, a recommendable length of the channel is L = 1000 m rather than 100 m. As stated above, we could find an appropriate method to control tsunami flood at the entrance of a harbour on the basis of a simple model. At the same time, we have to remind that the water-level response in the harbour is generally significant in a specific frequency band. In some other case as reported by Nakamura and All is on £l 981} » the mooring ropes of the fleighters or cargo— vessels at the pier of Dampier Harbour, which is on the coast of the northwestern Australia, were broken at the incident tsunami accompanied by the 1977 Sumbawa Earthquake. As for this problem, we should have an alternative review and promote another pertinent research as we did that on tsunami control at the entrance of a harbour. 78 ACOO WLEDGEMKNTS Professor Yoshito Tsuchya encouraged the author to complete this work and the author appreciates the support by Professor Tokujiro Inoue, Kobe University of Mercantile Marine and -^rs. Toshio Mihara and Shozo Sakai,. Kobe Maritime Casualty Commission. A part of this work was read at the IUGG Symposium on Assessment of Natural Hazards held in Hamburg in August 1983. The author thanks to consideration by Professors Hisashi Miyoshi and Harold G. Loomis. Lastly, the author wishes to dedicate this manuscript to the late Professor Haruo Higuchi, Shime University, 79 REFERENCES Carrier, G. P. and R.P.Shaw, Response of narrow-mouthed harbors to tsunamis, in "Tsunamis in the Pacific Ocean", W.M.Adams ed. , Honolulu, East West Center Press, 377-398, 1970. Carrier, G. P., R.P.Shaw and M.Miyata, The response of narrow-mouthed harbors in a straight coastline to periodic incident waves, Trans. ASME, Journal of Applied Mechanics, Ser.B, 38(2),. 335-344, 1971. Higuchi,H., Hydraulic model experiment on the oscillation of water level in Sakai Channel (i), Annuals Disas.Prev.Res.Inst., Kyoto Univ., No.3, 54-64, 1959. Higuchi,H., Hydraulic model experiment on the oscillation of water level in Sakai Channel (il), Annuals Disas.Prev.Res.Inst., Kyoto Univ., N0.4, 237-249, 1961a Higuchi,H., Hydraulic model experiment on the oscillation of water level in Sakai Channel, Coastal Engineering in Japan, 4, 35-45, 1961b. Hwang, L.S. and E.O.Tuck, On the oscillations of harbors of arbitrary shape, Jour. Fluid Mech., 42, 447-467, 1970. Lee, J. J., Wave induced oscillation in harbors with connected basins, Proc.ASCE, 98(WW3), 311-332, 1972. Lee, J.J. and P.Haichlcn, Resonance in harbors of arbitrary shape, Proc. 12th Conf. Coastal Eng., 2163-2180, 1970. Mei,C.C. and R.V.Petroni, Waves in harbors with protruding break- waters, Proc.ASCE, 99(WW2), 209-229, 1973. Miles, J. W. and W.Kunk, Harbor paradox, Proc.ASCE, 87(WW3), 111-130, 1961. Nakamura,S., Oscillations induced in Osaka Bay model, Proc. 26th Conf. Coastal Eng. in Japan, 139-142, 1979. Nakamura, S., A note on modes of oscillation induced in a Osaka Bay model, Proc. Internat. Conf . on Water Resources Development, Taipei, Taiwan, Republic of China, 835-843, 1980a. Nakamura, S. , Normal modes of oscillation in relation to storm surge and tsunami in Osaka Bay, Japan(3), La Mer, 18(4), 179-183, 1980b. 80 Nakamura, S., A numerical tsunami modeling in Osaka Bay and Kii Channel, La 5-'er, 19(3), I05-IIO, 1981. Nakamura, S. . and H.Allison, On long period waves on Western Australian coast, Proc.28th Conf. Coastal Eng. in Japan, 44-48, 1981. Nakamura, S. and H.G.Loomis, Normal modes of oscillation in relation to storm surge and tsunami in Osaka Bay, Japan(l), La Ker, 18(2), 69-75, 1980a. Nakamura, S. and H.G.Loomis, Normal modes of oscillation in relation to storm surge and tsunami in Osaka Bay, Japan(2 ; , La Wer, 18(2), 76-80, 1980b. Nakamura, S, and H.G.Loomis, Lowest mode of oscillations in a narrow- mouthed hay, Marine Geodesy, 4(3), 197-222, 1980c. Shaw, R. P. and C.K.Lai, Channel friction and slope effects on harbor resonance, Proc.ASCE, 100 (TO), 205-215, 1974. 81 Excitation Mechanisms of the ' Abiki' Phenomenon (a Kind of Seiche) in Nagasaki Bay TOSHIYUKI HIBIYA Earthquake -Research Institute University of Tokyo, Tokyo, Japan ABSTRACT Large seiches are often observed in several bays along the west coast of Kyushu in winter. Especially, those in Nagasaki Bay, called 'Abiki', are noted for their extraordinary amplitudes and frequent occurrences. The largest 'Abiki' recorded in the past 20 years at the tide station -t Nagasaki occurred on March 31, 1979. Simultaneously, a distinct atmospheric pressure disturbance of solitary type with an amplitude of about 3 mb was recorded at several neighbouring stations in Kyushu, which indicated the pressure disturbance travelled eastward over the East China Sea with an average speed of about 110 km/h. In the present study, the quantitative relation between this pressure disturbance and notable seiches observed in Nagasaki Bay is examined by means of numerical simulation and it is confirmed that the exceptionally large range of oscillations in the bay, which reached 278 cm at the tide station, was indeed produced by this travelling pressure disturbance. At the same time, several amplification mechanisms of water waves during the course of their propagation are clarified. 83 1. INTRODUCTION Large seiches in Nagasaki Bay called 'Abiki' are noted for their extraordinary amplitudes and frequent occurrences, particularly in winter (^Terada et al.,1953; Nakano & Unoki, 1962; Akamatsu, 1 9 78 J • Sometimes, strong currents induced by notable seiche cause severe damage to cargo and fishing boats by breaking their mooring lines. Partly from the necessity to avoid such accidents, detailed studies have been made of this phenomenon by Terada et al. (1953) and Ishiguro and Fujiki (1955). They clarified possible modes of oscillation in Naga- saki Bay by analysis of tide-gauge records together with some theoretical considerations and also by use of a hydraulic model experiment. On the other hand, some statistical studies on the occur- rence of 'Abiki' were made by Terada et al. (1953.) and Akamatsu (1978), who pointed out that the occurrence of 'Abiki' is always accompanied by a sudden change of atmospheric pressure recorded not only at Nagasaki but also at Tomie in the Goto Islands located to the west of mainland Kyushu. Therefore, it has been speculated that long-period oceanic waves induced by travelling pressure disturbances in Goto Nada (the Sea of Goto: see Fig.-8 ) excite large seiches in Nagasaki Bay. The problem of coupling between moving air-pressure disturbances and sea level variations was discussed long ago fproudman, 1952; Platzman, 1958] . Therefore, the basic under- 84 standing of the air-sea coupling of this type is now established and the problems to be clarified for • Abiki ' phenomenon are 1 ) To find a quantitative relation between the travelling atmospheric pressure disturbance and notable seiches in Nagasaki Bay, and 2) To understand the nature of observed atmospheric pressure disturbances. In the present study, the first problem is studied by taking the large oscillations in Nagasaki Bay observed on March 31, 1979 as an example. At the same time, the mechanisms of amplification of water waves which occur at various stages during the course of their propagation are clarified. 2. OBSERVATIONS Nagasaki Bay is a narrow and long bay, connected with the open ocean through its mouth which opens to the west (Fig. 1). The characteristic length and width of the bay are about 6 km and 1 km respectively and the average depth is 20 m. On the afternoon of March 31, 1979, a notable seiche was observed on the tide-gauge as shown in Fig. 2. After a few oscillations beginning at 12:10 JST, the maximum range of 278 cm was reached at about 13:30. Shortly after the first significant water wave was recorded at Nagasaki, a remarkable atmospheric pressure change of solitary type was observed at the weather station at Nagasaki although no particular changes of weather conditions were recognized. Barograph records obtained at Meshima Island, Fukue and 85 v^rwv^ \ \ \ ***^ ^r / \ "~\ • /■'.•'■ ( \ \ f.~. -\ \ \ a / j •aJ° 6hoto / \ »**v ; •;: '■_> Js* / \ ""^''■•'•ijr J*\ ■■; 1 s % * "*' \ *\ i \ 1 J W/ Kiboehl Vo^./i -/V ^Tld« Station j 1 /'■'■' J Jr • _ M _ \, t s' / /•'■'•' ■■'.:• Vy f .■■'/■ r^ s ' 1 * /•""' '/If v • /.'' j xJi /■' / V. 1 > . ' • - rMcgomi 1 ' ; C^l~'~ _\ 1 T -?.-,£?/ > v : ID ">* /J 1 ^ilp? 1 '-,">■.-»■ 1 1 \\: ^\J "\ ' - ^ ^ ■' r*-"^l~~' '" ' / j l8jlmo— lV - • / - •'P***^-' *f '} '•■ /*~&l \ltlond 1 ,"" !«"'/ '■•/" 'I'.'' ^ / \ ^«J\ /" jyJ -/'' / (*>. 21^11 '. Okl nojhimo ^■S ■ ,'"^-'" w -s / f^ W - 'V ^V** ilond t 1 ' ** *IN^ i \ '~~ V ^^^ '' W ^* \ &*lfff. ^v/^ *\ o\r \ v I 2 3 km \ _J Figure 1. Map of Nagasaki Bay. 86 m 3 Ki •-;:r| ::■; :rz:-:]-: ; ::: "MAR. 3 171979 -i . j— r~~ — "'. ..*LL«*_ .J.. z_i::: — ; — ' ' | .r+l ; j ...; :..]... i .j. . ■ !-:-| T |-- L f4--j • L -t-j--h|"M-j- H-| — j- -j- \ ■ I 1" i 1 -i-fl-H- ;jiH- :-;-!-— j— t- J i ; ' . . _ ' .' ■-H--J--J— : i i T : 1 ! INHOAdAM : :± p-i ■m. 1 i gf, I\ ' -•-I _. M "\ / \ | , 1 . 1 /I |A| : 1 * . j ~_\ 7 \ 1 1 ! i : 1 ii. 1 !/ \ 1 "| r • i - J--J. rU I \ /\ 1 fli ' ! ' : i , 1 ■ : p | /\ i 1 / .' ■ I : f\ ;- I i V Mi 1 I i l 1 ~T"" ! j 1 i \ ; "~T& / 1 ' — ; 4-- V V I ■ :r^ ■i -L.- J :-: -- ' "! / r~ 1 ■ l 1 TT ! 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" ■ •-! z! ! ■ "1 ■ :. : 1= "' "- .-:-* = ■-. ■- ■ - I . ; --■-■1 ■-■ "■ I '»' "1 1 "'■ •■i-:-! «rtBE . : ." -j. rj =^ -.- - T* . 1 [ - 1 lih ■: - 1 ;- :-:-!- i : i ' 2 n 4" : ! 1- : • ■ ;..}-,- ' r"-3jd fH-v- ^EA-T 11 ft- i _i -"Jr '• JlaIx£ ■mm « ■ - ! - . -—L-U-L P^P T'TT --»-)— : h^r- L ,1 !--!■ 1 ■ ! i i 1 - 1 - ; m ' i ! i I"-": I ii ■■i-NJij- 1 1 1 » i A i W IMi=i--lH ' i 1 : i :■ : 1 •'-: i i u | 1 : A 11 In A a ii :-^- -, • 1 ■ j i 1 ; . 1 ■ 1 -.■H-.'T:1 nil " Ti T "l 1 1 ill irrfPr y'T ' ; - ! • • 1 • - ' !t 1 F~ i . -'.; •■: ■i 1 l 3 V Mil a i/1 " V/r TF.-H : t" ; r ■- : | : : '. - I ' I'-i^ ' ' ! : {■;■: 1 , ■ 1 • V ' \ | Ij 1 Mtlfc T-: :: ■I r :i:: : | : | ■ | i , [•.■ j;j -, | \ ;.•■ =m-T4 i h fj MTT hl ; ' — "— .-, 'j'-l i i 1 : - ~t~T^ 1 m | • . , - \ i : Jii- i : ' ■ \/ : i !' j ; i ! -i : |-: ■-I ; ;i- : :r i- : ■ i- i ;■■ .:i T VI : ^ • -,..(. ,. ■ i-.r - - ! ■ -j !•] 1 ■■ •:■,.■ | 1 ;-!-l •■'•■ i ] 1 i! , ■ M , 4-tm 1 ! ■■■! ■ ■ : l ' ' : ! i ; ' ! I'l { j : j'| !■ !-T ■; 1 ! i ■ m : i 1 i ! ! -JJ 1 ' :l : ' i ' ! I'll : itl y ■ ; .;■ j-" _j| ! - ! ■ \ ] - i 1 •! J : • ! 1:1 ~j£ : ; Li. -j-H-vf:- 1 ■ L_ • i\l 1 ! I , ;-. ':- 1 1 , , | : • i i'l~ : ■ i . ; J ' LJ I r -• : i ; i i>|! h^i- 1 - : -i i ! i i ■ : 1 | _ : : i ; i ■; ! '-- : ! U 1 -1 H : i 1 ■ 1 ; ! i . : i ■ ■ ■ 1 ; i : I ■ : : ! 1 • ! i i 'I 1 ff- A ■ '"-- 1 1 j ; 1 : i =Ti i i i i V-- ,1 1: ■ i i ! : ; j f' ' / k: ;.- : p !;|-!i ! ! ' ' i i i ' i : 1 V ? U / > ■ j /( !> i ;// " 1 1 /V ! a> m WJ£,"! | 1 y t> 1 ' t ' . If'. \ 1 ! jW i .1'/ >'> ->» 7 hr. 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 MAR. 31. 1979 (J.S.T) Figure 2. The tide-gauge record at Nagasaki on March 31, 1979 87 Nagasaki are shown in Fig. 3 (for the locations of these stations, see Pig. 4). As is typically seen on the record at Meshima Island, the pressure change began with a rapid increase for about 15 min followed by a gradual decrease to the previous level in. about 90 min. The magnitude of the pressure jump was 2 to 6 mb depending on the station. If we assume the pressure disturbance to propagate in the form of a line front and the phases indicated by arrows in Fig. 3 to represent the same feature, the propagation velocity of the pressure wave is estimated to be 1.38 km/min in the direction of 5.6 degrees north of east. 3. NUMERICAL SIMULATION a) Formulation Let x, y denote horizontal coordinate directed to the east and south respectively and z the vertical coordinate directed upward with the origin at the undisturbed sea surface. In this coordinate system, shallow water equations of motion and continuity can be written as 3Qx - - Sh-ji- 11-1*) <0 and zTT = ~ & n '^ |$£ = - gh-^- (1 -.1*) (2) 14 --<¥!*¥£> (3) 88 (J.S.T) mb LV'. 8 / 9 i 10/ -\\J: \2[: 13/// 147- 157' -|6J' 17 h r. 1030 1020 1010 1000 990 HHm=t=^ .JD.0 . iuii £fr II /.///■// 8 9 10 ' NAGASAK 7"7 12 ' 5 / t4 / 15 / 16/ 17 hr. 1000 990 Figure 3. Barograras of an atmospheric pressure wave on March 31, 1979 89 where h is the depth of the sea, Qx and Qy are the depth- integrated transport components in the directions of x and y, *] is the water surface elevation and ^ = -p/C/'g) where p is the atmospheric pressure deviation at the sea surface, g is the acceleration of gravity and P is the mean density of sea water, Using square grids with spacing b s in horizontal space and applying the centered difference and leap-frog scheme with a time increment /it, (1) to (3) can be replaced by a finite difference scheme and the spatial distribution of 1 and those of Qx and Qy can be computed alternately with suitable initial and boundary conditions. The domain of the calculation is shown in Fig. 4. The grid interval as is 4 km in Region I and 2 km in Region II. Further- more, regions in the vicinity of tide stations of interest, Nagasaki and Fukue in the Goto Islands, are divided into finer meshes in order to calculate the wave forms accurately ^Aida, 1 974 J • Taking the stability condition of calculation into account, 4t is chosen to be 0.05 min. b) A model of the atmospheric pressure wave A model of the air-pressure disturbance applied here is shown in Fig. 5. Assuming the propagation velocity v of this pressure wave to be 1.88 km/min, the numerical values relevant to this model are determined as follows: The atmospheric pressure increases linearly by & p = 3 mb in a distance of Li (= 28 km) and returns to its previous level also linearly in a 90 100 200 Km Figure 4. Domain of the numerical computation. The grid interval is 4 km in Region I, and 2 km in Region II. The shaded area indicates the assumed initial location of the pressure wave. 28km Figure 5. Schematic view of the pressure wave model propagation is indicated by the arrow. The direction of 91 distance of L2(= 169 km) with no lateral gradient within its assumed width of 241 km. This width of 241 km was determined by examining the distribution of pressure jump intensities recorded at various weather stations in Kyushu. The pressure wave modelled above is assumed to occur instantaneously in the shaded region in Fig. 4 at t=0 and to propagate in the direction indicated by the arrow (5.6 degrees north of east) with a constant velocity of propagation of 1.88 km/min without change of shape, c) Initial and boundary conditions Initial conditions are that the ocean is at rest at t=0. The coast is assumed to be a rigid vertical wall with perfect reflection. An approximate radiation boundary condition is applied on open boundaries to allow waves to escape from the region of calculation. 4. RESULTS Calculated sea level variations at the tide stations at Nagasaki and Pukue are shown respectively in Fig. 6 a and b by the solid lines while the observed ones are indicated by the broken lines. The agreement between the calculated and observed oscillations at Nagasaki is very close. The maximum range of sea level variations takes place at the third oscillation in both the observed and simulated records. The agreement of the calculated pattern of sea level variations at Fukue with the 92 m (a) calculated observed 400 min m (b) i - -1 -2 L FUKUE A\ 250 i / calculated observed 300' 350 400 min Figure 6. Simulated sea level variations (solid lines) at the tide stations at Nagasaki (a) and Fukue (b) . Observed ones are shown by broken lines. 93 observed one is not so good, but general characteristics of the oscillations are simulated well. It is now clear that the conspicuous seiches with their maximum range of 273 cm at the Nagasaki tide station can be explained by the observed travelling pressure wave with an amplitude of 3 mb. Thus, it is interesting to explore how the long-period water wave, induced in the East China Sea by the pressure disturbance, is amplified as it propagates toward the coast of Kyushu and excites large seiches in Nagasaki Bay. 5. DISCUSSIONS a) Amplification of water waves on the continental shelf off China A shallow continental shelf in the East China Sea extends eastward for about 600 km from the Chinese mainland. Over most of the shelf, the depth is 50 to 150 m. The velocity of shallow water waves propagating in this region, therefore, is about 1.3 to 2.3 km/min and is comparable to that of the assumed atmospheric pressure wave which is 1.88 km/min. Thus, efficient coupling between the atmospheric pressure wave and the generated shallow water wave may be expected to occur. Results of the present simulation at Locations A and B (see Fig. 4) are shown in Fig. 7. The maximum elevation at Location A is only 2 cm but it increases to about 12 cm at Location B, which is located about 300 km from the initial location of the front of the assumed pressure wave. 94 China Elevation Pressure mb 2 cm - B c10 o Elevat o i i i ' / 1 1 " ■ " - -1-- J 100 L 200^X^n mm mb 5 w (A e 3 o d: Figure 7. Typical profile of the continental shelf off China in the direction of pressure wave propagation (left figure); and the calculated sea level time-variations at Locations A and B selected along the propagation direction of the pressure wave (right figure). 95 It is well known, in the case of one-dimensional propagation in the x-direction, that the forced wave *} «(x - vt) in water of constant depth, generated by a travelling atmospheric pressure disturbance with the velocity v of the form ^(x - vt) is given by *] „ = 1/(1- v /c ) where c = gh. Thus, *} f is in phase with i if v < c and out of phase if v > c. When c approaches v, reso- nance makes •],, formally infinite. However, in the vicinity of resonance condition v = c, we have to treat the problem as an initial value problem, in which the amplitude of the water wave increases linearly with time t. For the same atmospheric pres- sure model as in the simulation study, the maximum elevation of water level can be computed by the method of characteristics. The maximum appears in the front part of the forcing region, Xr. = vt, and is given by *1 = -( 47*/ Ll ). Xf /2 (4) where x f is the distance travelled by the front, 4'/ - - ^P/C/^-g) where ap is the pressure amplitude and L. is the distance from the pressure maximum to the front. Thus, if we take L. = 30 km and x f = 300 km, the amplification factor \^1 /d'f |is about 5 and if d'?*= -3 cm, as is assumed in the present case, ^7 would reach 15 cm. It is interesting to note that the rapid increase in atmospheric pressure is essential for a large amplification factor at a fixed distance x«. A more realistic situation is examined for the case of one- 96 dimensional propagation by taking the realistic depth change of the .Wast China Sea into account. For the same initial condition as in the case of numerical simulation, the maximum water level elevation at Location B is numerically computed for different propagation velocities of the atmospheric forcing. The result indicates that the assumed speed of 1.33 km/min is near the optimum for resonant amplification. b) Transformation in Goto Nada Between Locations C and I), shallow shelves are separated by a deep trough (see Fig. 4), so that the leading water wave moves ahead of the forcing region as a free gravity wave. As seen in Fig. 8» the direction of propagation of the first wave crest is gradually turned to the north by refractive effects of bottom topography and is almost normally incident in the shelf region (Goto Nada) between the mainland of Kyushu and the Goto Islands. The excitation of natural oscillations in Goto Nada can be noticed in the power spectra of simulated water level variations at several locations in Goto Nada. In Fig. 9, the eigenoscilla- tions with periods of 70, 36 and 24 mins are present. In the same figure, the power spectrum of incident waves at Location C is also shown, which exhibits no distinct peak at periods shorter than about 50 min. To determine the characteristics of natural oscillations in Goto Nada more definitely, the eigenvalue problem is solved numerically and it is found that the modes with periods of 64, 36 and 24 mins are lateral oscillations with nodes run- 97 t = 200 min t = 2 I min 5r»B £/ Nogosoki Bay t = 220 min Figure 8. Patterns of the propagation into Goto Nada of the first water wave crest (shaded area). Numerals on the contours represent the water surface elevation. 98 i a c E E 10 o S io 3 0) o (L 10 s 100 50 40 — I i i i — i 1 r Period (min) 30 20 Frequency (xiO'*cpm) Figure 9. Power spectra of simulated water level variations at Locations C, D, E, F (Fig. 4) and at the Nagasaki tide station. 99 ning in the north-south direction, suggesting that the waves reflected at the coasts of the Goto Islands and mainland Kyushu form an oscillating system, c) Amplification in Nagasaki Bay As is evident in Fig. 9» "the level of the power spectrum at Nagasaki relative to those in Goto Nada is very high and mean power amplification in the period range shorter than about 50 min amounts to 40. Superimposed on this mean amplification, the amplification due to resonance is noted at the periods of 36 and 23 mins, which are not only the eigenperiods of Goto Nada but also coincide with the periods of oscillations in Nagasaki Bay. To understand the mechanism of this large amplification, let us consider two factors: one is the amplification of progres- sive waves as they proceed toward the bay head due to topographic convergence effects and the other is the effect of dynamic reso- nance . The wave signature at Location G (see Fig. 11) located outside Nagasaki Bay is shown in Fig. 10, where the simulated first crest height is about 20 cm. However, if we formally separate the v/ater surface elevation into "incoming" and "outgoing" one- dimensional waves by taking information on time variations of transport into account, it is found that the first crest height A« of the "incoming" wave is about 16 cm (Fig. 10). The increase of the apparent first crest height as well as that of the "in- coming" waves as they approach the head of Nagasaki Bay is shown 100 200 Incoming Wave Outgoing Wave Figure 10. Calculated sea level variations (solid line) at Location G (see Fig. 11), along with the time histories of "incoming" (dashed line) and "outgoing" (dash and dot line) waves at the same location. 101 in Pig. 11, in which amplification obeying Green's law is also shown for reference. The total topographic amplification, A-rAV,, of the "incoming" wave from Locations G- to L is 33/16. Ju u This increase of the "incoming" wave crest height is dictated by the combined effects of topographic convergence inside the bay and partial reflection at the bay mouth in the vicinity of Location J and at the constriction near Location K. Now, the resonant effect of the fundamental mode in one- dimensional waves can be evaluated as follows. By the incidence of a uniform train of waves with the resonant frequency 60 o, wave energy is accumulated in the bay by total reflection at the head and partial reflection at the mouth of the bay. The final amplitude B at the head of the bay is max J W 2A > ■ 1 /< 1 - '*') <'5> where i\ is the "incoming" wave amplitude at the location of the bay head without the effect of resonance and |R\ is the absolute value of the reflection coefficient at the bay mouth. .It is easily shown that a nearly resonant amplitude (90 percent of the stationary amplitude) is attained only after, say, (-2*log|Rl) cycles from the start of oscillations. In terms of oL , the ratio of the admittance in the bay side to that in the outer channel side at the junction at Location J, the reflection coefficient R is expressed as 102 cm 50 • - - First Wove Crest — pj rs t Incoming Wove Crest * — Green's Low I J K L boy head Locations GV' i Ohoto Tide Station Figure 11. Calculated first crest heights of water waves at several locations selected along the bay-axis (locations are indicated in the map on the right). The first crest heights of "incoming" waves separated from "outgoing" waves are also plotted, along with those estimated by Green's formula, assuming both are equal in height outside the bay mouth at Location H. 103 R = (1 -«)/(1 +0C) (€) and the relative band width All) "between the half points of the power spectrum near resonance at the bay head is ALO/LOo r-> 4'CL/7l (7} The quality factor Q of the oscillating system with radiation damping is approximately given by tc/(4-oc). Since 4*0/ u> o £r0.7/2.75 in Fig. 9, it is reasonable to assume oL = 0.2 for the fundamental mode. It follows R = 0.667, and the Q factor is about 4. The amplification due to resonance is B max /(2A) ~ 3.0 (8) and this saturation amplitude is attained only after 2 to 3 wave cycles. Now since the topographic amplification from Locations G to L is 38/16, we may estimate the resonant amplitude at the bay head relative to the mean amplitude of the "incoming" wave train at Location G to be (38/16)x3*2 = 14.25. Returning to the time history of the incident waves at Loca- tion G (Fig. 10), a wave train with the period of about 35 min is recognized in the early stage. The mean amplitude of these first oscillations is estimated to be, say, 10 cm, so that the resonant amplitude at the bay head after 3 cycles of incident 104 waves is about 10 x 14.25 Cr 140 (cm). The total range would be about 280 cm which is very close to that actually observed. 6. CONCLUDING REMARKS Needless to say, the present study is concerned with only one typical case of the 'Abiki' phenomenon and there is the possibility that large seiches along the west coast of Kyushu might be produced by other types of pressure waves. Further investigation into the characteristics of atmospheric pressure waves is desirable to obtain a complete picture of the 'Abiki' phenomenon, the prediction of which will be possible in the future through the combination of observations of water waves and atmospheric pressure waves preferably on the continental shelf region in the East China Sea. REFERENCES Aida, I., Numerical computation of a tsunami based on a fault origin model of an earthquake (in Japanese), J. Seismol. Soc. Japan, Ser. II, 27, 141-154, 1974. Akamatsu, H. , Abiki phenomenon in Nagasaki Harbour (in Japanese), in 100th Anniversary Volume of the Nagasaki Marine Observa- tory, edited by the Nagasaki Marine Observatory, pp. 154-162, Nagasaki, 1978. Ishiguro, S., and A. Fujiki, An analytical method for the 105 oscillation of water in a bay or lake, using an electric network and electronic analogue computer, J. Oceanogr. Soc. Japan, 11, 191-197, 1955. Nakano, M. , and S. Unoki, On the seiches (the secondary undula- tions of tides) along the coasts of Japan, Records Oceanogr. Works Japan, Special No. (6), 169-214, 1962. Flatzman, G-. W. , A numerical computation of the surge of 26 June 1954 on Lake Michigan, Geophysica, 6, 407-438, 1958. Proudman, J., Dynamical Oceanography, 409 pp., Methuen, London, 1952 B Terada, K., Z. Yasui, and S. Ishiguro, On the secondary undula- tions in Nagasaki Harbour (in Japanese), The Report of the Nagasaki Marine Observatory, (4), 1-73, 1953. 106 Reconsiderations on the Huge Tsunamis — Utility of the Seawall » Hisashi MIYOSHI 3-3-33 Minami-Oizumi, Nerima-ku, Tokyo, 177, Japan We mean principally the tsunami of 1896 and that of 1771 by "the huge tsunamis". Yamashita (1982), a nonprofessional researcher, who came of a Ryori Village family, issued an important book, in which he described that the tsunami of 1896 killed approximately 22,000 people. There are sufficient evidences that he is correct, and the previous thought that some 27,000 people had been killed was given up. He also suggested that the survivors (male; 65, female; 14) at Hongo, Toni Village (male;394, female;4C9, before the tsunami) were only those who were fishing in boats offshore or visiting other villages as June 15, 1896 was a festival day. It means that this was an actual annihilation, which could be compared only with the perfect annihilation at Makayome Village (male; 140, female; 143), Ishigaki Island in 1771. How do these annihilations occur? We have usually exlained them as the results of clashes between the floating people and the wooden pieces. But this explanation is insufficient. We should assume the death resulting from a kind of bombing or from the grinding action of the turbulent water itself. The tsunami of May 26, 1983 might happen to demonstrate that this assumption was right. The differences between the dynamics of the huge wave attack on a victim and those of the famous diving at Acapulco etc. are that the victim is then adjacent to a solid body and that there is a violent turbulence in the water, the former of which looks the more essential. 1 . Introduction We have, for a long time, considered on the two representative huge tsunamis (1771 and 1896) in Japan, especially on the perfect annihilation at Nakayome Village, Ishigaki Island in 1771. Even a qualitative consideration is the first step to discuss the utility of the seawall. In spite of the fatal fault of the seawall, which has seldom been discussed formally, that the huge tsunami which overflows it cannot return to the sea, and the village sinks in water, the plan to build the seawall to cut off the huge energy of the tsunami which causes the annihilation will, prove to be right. With the progress of our considerations, many facts were found and we encountered the most instructive tsunami of May 26, 1983. 107 2. The Tsunami of 1896 In our previous paper (1983), we reported the results of our surveys at Tanohata Village and revised the runup height of the 1896 tsunami at Raga, Tanohata from 22.9m to 29.0m. We noticed, at the same time, that the previous measurer (Igi, 1896) didn't bring a handlevel during his on-site inspection. We suppose that he stretched his hand horizontally instead of using the handlevel. He might have a habit to stretch his hand a little high, which led to his underestimations of the runup heights. Matsuo (1933) revised the runup height of the same tsunami at Okubo, Ryori Village, a saddle point which is situated right behind the head of a large V-shaped bay. In spite of the f orgetfulness of the inhabitants, we believe in this revised figure, judging from the size of Ryori Bay compared with that of Raga Inlet, and from the above-mentioned habit of the previous measurer. Ckubo point is a saddle point, the altitude of which is 32m above mean sea level, which is an adequate height to check the runup height. The combination of Ryori Ray and this saddle point means, in a sense, a huge energy accumulating apparatus and its measuring instrument for the future tsunamis. Fortunately, as this bay is rather a part of ocean, less people dwell along its shoreline compared with other bays. While revising the number of people who were killed by the tsunami of 1896 from 27,OOC to 22,OCO, and confirming the revised runup height at Okubo point judging from his feel of the place, Yamashita suggested one important fact. The survivors (male; 65, female; 14) at Hongo, Toni Village (male; 394, female ;4C9, before the tsunami) were only those who were fishing in boats offshore or visiting other villages. This actual annihilation reminded us the perfect annihilation at Nakayome Village in 1771 . The dynamics of the huge wave attack on a victim resembles that of the diving at Acapulco, which is repeatedly demonstrated without any accidents. And on July 9, 1960, a boy of 7 years of age survived a drop over Niagara Falls without elaborate protection, only with a life jacket. This fact suggests that we must not attribute the dynamics of Acapulco to the diver's diathesis. Then, why do the annihilations occur in case of the huge tsunamis? We considered that the dynamics of the huge tsunami differed in the situation that the human body was then adjacent to a solid body, and concluded that the death resulting from a kind of bombing was possible, when the tsunami of May 26, 1983 attacked the Japanese coast of the Japan Sea. And we found a clue to solve this puzzle on a caisson at Moshiro City, city which was most badly hit by this tsunami. It is, however, a little difficult to look through the true nature of this tsunami. 3. The Japan Sea Tsunami of 1983 When we consider on this tsunami of May 26, 1983, the consideration on the monsoon from Siberia is indispensable. This monsoon is 108 , over flowed Figure 1 . Ckubo point is a saddle one which the torrent of the 1896 tsunami but that of the 1933 one might not. In 1896, the measurer, T. IG1 didn't bring here a handlevel. Rut in 1933, H. FiATSUC did. The latter also confirmed that the runup height tsunami had been as high as 38.2m, can believe. This saddle point prove to be the measuring instrument of the 1896 in which we itself will for the future huge tsunamis. 109 V very strong and blows during al^the winter, and the width of the Japan Sea is considerably large. Since both the duration and the fetch of this strong wind are large, the wind wave is extremely violent in winter along the Japanese coast of the Japan Sea. Figure 2 shows a typical town along the coast of the Japan Sea. This tsunami was some 1m or less lower than the top of this seawall. Its runup height was 4.4m here. A man stands on the road, the altitude of which is some 4.4m. Except one house, this town was utterly unhurt. The wind waves in winter are higher than this tsunami, and invade into this town several times a year. In such cases, this seawall used to change into a breakwater. This town looks, in a word, like a castle surrounded by a castle wall. The school children on a trip from a primary school among the mountains were descending towards this beach. When the earthquake occurred, they were in a bus. And Japan Meteorological Agency failed to issue a quick tsunami warning. And it meant that the children went out of the castle wall, which they couldn't climb anymore. And 13 children were killed. In spite of the fact that many ships were destroyed by this tsunami, we can say that their deaths were rather exceptional comparing with this unhurt town. And in spite of the fact that approximately 100 lives were lost in this way, we can say that this tsunami occurred under three fortunate circumstances and left cities, towns and villages rather unhurt. (1) V-shaped bay does not exist. (2) The cities, towns and villages stand rather high. (3) This tsunami occurred in early summer and was not accompanied by the storm surge. Figure 2 shows the village located on the top of Oga Peninsula, from where the tsunami runup height became higher and higher towards Noshiro City, and reached as high as 13.6m at Menagata (lida et al, 1983), which is 12km north of Noshiro City. Even at Noshiro Harbor, the scene shown by Figure 3 was witnessed. The tsunami is breaking against a breakwater. Its wave height was, we inferred, as high as 8m. In spite of the fact that the violence of this tsunami was hidden under the above-mentioned fortunate circumstances, we should recognize that this tsunami was a huge one. Not only we should recognize this, but also we should here recognize that the tsunami of August 28, 1741, which occurred at 41.5°N, 139.4°E and killed 1,467 persons principally along the southwestern coast of Hokkaidc&iight be huger than the Sanriku tsunami of 1896, judging from the fact that the Japan Sea might be calm on this date and very few inhabitants lived along this coast. The next earthquake, to which we should refer, is the Niigata Earthquake, the magnitude of which was only 7.5 or less, caused a considerably large tsunami for the earthquake's magnitude. At Nezugasaki, the runup height reached 5.5m, which the Japanese Society easily has forgotten, being dizzied by the rare phenomenon of quick sand. This f orgetfulness was partially due to the fortunate season of early summer in which the Japan Sea was calm. no Figure 2. The town on the top of Gga Peninsula surrounded by the construction like a castle wall, where the tsunami runup height measured 4.4m. This construction is essentially a breakwater against the violent wind waves in winter. The school children had been on this beach. 13 of them were killed on May 26, 1983. Figure 3. Even at Noshiro Harbor, this tsunami, the accurate height of which was difficult to measure, was huge. We inferred it was here as high as 8m. According to K. IIDA et al, at Menagata, 12km north of Noshiro City, the runup was much higher than at Noshiro City and reached as high as 13.6m. The true scale of this tsunami was apt to be hidden under the complicated circumstances. ill Figure 4. ± : Noshiro Harbor, Q Uenagata, <§> : ga Peninsula, X: Ckubo point, Q. Tanohata Village, respectively. 112 Among us, the researchers of the tsunami, the tsunami in the Japan Sea had been a source of anxiety. We tried to explain that the efficiency of the tsunami was high in the Japan Sea, because the angle of the fault was some 50°, comparing with that off the Sanriku District, which might be some 20 . Another explanation has appeared, which suggested that a reverse fault, the angle of which was from 20 to 30°, occurred. We must notice that, in this case, we should assume a broad bottom deformation And the parameter ^VW|/ suggests the period of the tsunami. Here d , 9" and IV are the effective diameter of the bottom deformation, the acceleration of gravity and the sea depth at the origin, respectively. - In the case of the Japan Sea Tsunami of 1933, ru is some 2,900m, which is equivalent to that of the typical tsunamis off the Sanriku District. If we assumed a large value of (Z, , the short period of some 6 minutes of this tsunami would become difficult to be explained. The fact itself that the efficiency of the tsunami -in the Japan Sea is high, is much more important. And the meaning of the word of "efficiency" is quite different from that comes inevitably from an elementary dynamics (Miyoshi, 1954 and 1977). 4. The New Phenomena on a Row of Caissons at Noshiro City Generally speaking, when a man is pursued by the tsunami, he tries to run away more 1m or half a meter. And when he falls down, he gets up to run again. Are these attitudes right? Figure 3 shows the tsunami, the wave height of which is some 8m, attacking the breakwater at Noshiro City. Besides these breakwaters, there is a row of caissons near Noshiro Harbor. This row embraces a large rectangular water pool, one of the longer sides of which is the shoreline itself. Another longer side is parallel to the shoreline and some 1km apart from the latter. This is a thermal power plant construction site. More than 30 workmen were engulfed by the tsunami and killed along the breakwaters and the row of caissons. We must notice that a unique situation appeared along these breakwaters or the row of caissons. In the case of the workmen who were working along them, all their hopes for escapes were gone. One workman prefered spontaneous falling down onto the caisson just in front of the tsunami, than being smashed against concrete. He survived. Another workman prefered jumping into the water pool in a life jacket, and survived. The former testimony seems to be more important, because this situation is more universal. Of course, we are afraid that the data are yet too few. But we think that the above-mentioned new experience might prove to be a clue to solve the puzzle about the inhabitants' annihilations in 1771 or in 1896. Since this paper consists of a long-term consideration including the results of the studies by a nonprofessional researcher, we 113 decided to present this rather prosaic paper, under a condition that we must gather the basic data concerning this problem from rlow on. The basic data here may include those of postmortem examinations of the human bodies. 5. Other Points at Issue Raised by the Tsunami of 1983 Repeatedly saying, the runup height was not large along the coast of Oga Peninsula. It was as high as only 3.3m around the coastal aquarium. One Swiss woman was, however, killed here and her husband was also swept away but managed to reach safety. One Japanese testified that the Japanese language of these Swiss was fluent. But there are three stages of fluency. The first stage means that they can easily do their shopping. The second stage means that they know the word "tsunami". The third stage means that when the microphone announces the tsunami warning, they can understand it. We are afraid that these Swiss were in the first stage, judging from the fact that their country is that of mountains and the length of their stay in Japan hadn't been so long. This accident suggests us that the tsunami warning should be issued at least in two languages. Next point is as follows: When an earthquake occurs, JMA (Japan Meteorological Agency) is in duty to issue the tsunami warning 20 minutes or less after the earthquake. This figure, 20 minutes, was decided when there were few conputers in Japanese Society, and proved to be the main cause that many persons were killed by this tsunami. JMA is now trying to shorten this interval, in which they will easily succeed. The last point is that, in a vast area, the quick sand phenomena occurred in the case of this earthquake. And one is apt to overestimate the energy or the magnitude of this earthquake, being dizzied by this phenomenon. We observed, however, many phenomena that might suggest that the energy of this earthquake had been rather small. We hope that there might not be a relation between the temporarily decided magnitude and the finally decided one. 6. Conclusions (1) According to Yamashita and us, the tsunami of 1896 killed some 22,000 persons along the coast of the Sanriku District, and reached as high as 38.2m at Okubo point, Ryori Village. (2) Yamashita 1 s more important pointing out is that an actual annihilation of the inhabitants of Hongo, Toni Village, which could be compared with the perfect annihilation at Nakayome Village, Ishigaki Island in 1771, occurred in 1896. (3) The most remarkable difference between the dynamics of the huge wave attack on a victim and those of the famous diving at Acapulco etc. is that the victim is then adjacent to a solid body. (4) The Japan Sea Tsunami of 1983 was a huge one, which was hidden under three fortunate circumstances. Generally speaking, the efficiency of the tsunami in the Japan Sea is high. Here, the meaning of the word of "efficiency" is quite different from that comes inevitably from an elementary dynamics. 114 (5) One workman prefered spontaneous falling down onto the caisson just in front of the tsunami of May 26, 1983, than being smashed against concrete. He survived. This new experience is very, very enlightening, which suggests the mechanism of the inhabitants' annihilations by the past huge tsunamis. (6) In spite of the fatal fault of the seawall, the plan to build the seawall to cut off the huge energy of the tsunami proves to be right. REFERENCES IGI, T. , A report of the field investigation of the tsunami of 1896 in the Sanriku District, Shinsai Yobo Chosakai Hokoku, 11, 5-33, 1897 (in Japanese). IIDA, K. , et al., Investigation of the Nippon-Kai Chubu Earthquake of May 26, 1983, special issue from Aichi Inst. Tech. , Toyota, Japan, 1-29, 1983. MATSUO, H. , A report of the field investigation of Sanriku Tsunami, Doboku Shikenjyo Hokoku, 24, 83-112, 1933 (in Japanese). MIYOSHI, H., K. IIDA, H. SUZUKI and Y. OSAWA, The largest tsunami in the Sanriku District, Tsunamis - Their Science and Engineering, TERRAPUB, Tokyo, 205-221, 1983. YAMASHITA, F. , Aishi Sanriku otsunami , 1-413, 1982 (in Japanese). 115 REGIONAL TSUNAMI WARNINGS USING SATELLITES Eddie N. Bernard 1 Gerald T. Hebenstreit 2 James F. Lander 3 Paul F. Krumpe 4 ABSTRACT This study shows that satellite technology has potential applications to the problem of providing early tsunami warning information in developing nations of the Pacific not having their own regional warning network. A simple conceptual model is developed that shows how these technologies could be integrated into an early warning "system". The basic elements are existing instrumentation connected to satellite communication devices for rapid data acquisition and analysis and quick information dissemination. 1 Pacific Marine Environmental Laboratory Seattle, WA 2 Science Applications, Inc. McLean, VA 3 National Geophysical Data Center Boulder, CO 4 Office of U.S. Foreign Disaster Assistance, AID Washington, D.C. 117 1 . INTRODUCTION Tsunami are very long ocean surface waves generated by the rapid dis- placement of large volumes of sea water. The most common generators of tsunami are sea floor motions caused by large (Ms magnitude 7.0 or higher) submarine earthquakes occurring in shallow coastal areas. As Figure 1 shows, most tsunami-generating earthquakes occur in the coastal margins of the Pacific. Tsunami in the open ocean are generally low amplitude, long waves in- distinguishable from normal surface motion. When they enter shallow water, however, they can grow to great amplitudes and destructive power. Although really destructive tsunami do not occur often, their effects can be quite devastating. Thompson (1982) estimates that during the period 1947-1981 a total of 10 tsunami caused over 8500 deaths. The economic cost of damage is difficult to estimate, especially in developing nations, where post-event surveys are rare. 2. THE PACIFIC TSUNAMI WARNING SYSTEM On 1 April 1946 an earthquake in the Aleutian Islands generated a tsunami which caused approximately 250 deaths and many millions of dollars in property damage in the United States (Iida et al . , 1967). This single event precipitated the formation of the forerunner of what is now the Pacific Tsunami Warning 118 minim Figure 1. Locations of earthquakes that generated tsunami between 1876 and 1976 are denoted by small squares. Produced by World Data Center A, National Geophysical Data Center, NOAA/NESDIS, Boulder, Colorado. Shaded area represents areal coverage by GOES West Satellite. System, which is centered at the Pacific Tsunami Warning Center (PTWC), Honolulu, Hawaii. The present warning system, operated by the National Oceanic and Atmosperic Administration (NOAA) , is composed of a seismic network in the Hawaiian Islands and approximately 60 tide gauges located in 22 nations throughout the Pacific (Fig. 2). Regional networks are in operation in Alaska, Japan, and the USSR. PTWC detects major earthquakes in the Pacific region, evaluates the earthquake tsunami potential in terms of epicenter and Richter scale magnitude, determines if a tsunami has been generated, and issues appropriate warnings and information to minimize the hazards of tsunami. The international warning system employs teletypewriter and voice communication links to acquire data and disseminate tsunami information to twenty-one nations. Transmission times 119 Figure 2. Pacific Tsunami Warning System. can take from 10 minutes to 1 hour, depending on the efficiency of communication relay points. The warnings delivered by PTWC include earthquake locations (±50 km), earthquake Richter scale magnitude (±.3), tsunami arrival time (±20 min) , and reports of tsunami wave heights as recorded by tide gauges. The earthquake parameters and tsunami arrival times are usually disseminated to the 54 international warning points within one hour after the occurrence of an earthquake. The time of receipt of tsunami wave reports varies with the travel time of the tsunami from its origin to the tide gauges, the dependability of observers, and the communication links. 120 3. REGIONAL WARNING SYSTEMS A 45-60 minute warning time is quite adequate for warning coastal nations a long distance from the generating area. For example, a tsunami originating off the southern Chile coast would reach Hawaii in approximately 15 hours and Japan in 22 hours (Dept. of Commerce, 1971). Thus even a delay of several hours is acceptable for warning distant areas. Unfortunately, tsunami threaten more than just distant locations. In actuality, three levels of tsunami threat can be delineated by establishing three zones based on tsunami travel time. These are depicted in Figure 3 for a hypothetical earthquake occurring just south of Valparaiso, Chile. Zone 1 includes areas within 10 minutes travel time of the source. This is close enough to the source so that some ground motion will probably be felt. Indeed, this motion is the only reliable warning of a possible tsunami that people in Zone 1 are likely to get. Zone 2 covers areas from 10 to 60 minutes travel time away. These areas are much less likely to experience ground motion and yet are too close to be warned by the existing tsunami warning system. Zone 3 includes all areas beyond 60 minutes travel time. These are reasonably well covered by warnings from PTWC. The hazard Zones 1 and 2, which represent a regional area, are essentially excluded in the coverage by the existing Pacific wide tsunami warning network. It is apparent that some sort of regional coverage is necessary to fill these gaps. Some regions of the Pacific, specifically Alaska, the USSR, and Japan do have operational regional warning systems. Most other regions do not. None of the so-called developing nations have such systems, even though a glance at Figure 1 shows that many of these area are sources of tsunamigenic earthquakes. These areas are especially vulnerable if large populations live in low-lying, unprotected coastal locations. 121 >60 MIN ZONE 3 VALPARAISO • SANTIAGO CONCEPCION 200km Figure 3. The three tsunami threat zones described in the text are indicated for a hypothetical tsunami originating in a source area (cross hatching) south of Valparaiso, Chile. 4 . THRUST The Office of U.S. Foreign Disaster Assistance (OFDA) of the Agency for International Development has commissioned a pilot study known as THRUST (Tsunami Hazard Reduction Using System Technology). The THRUST project team is made up of personnel from OFDA, three elements of NOAA, and two scientific research firms. The goal is to demonstrate that regional systems can be assembled, using existing technology, and integrated into established disaster warning and relief infrastructures in developing nations. The program is based on an earlier study (Bernard et al . , 1982) showing that such systems are technically feasible. 122 The THRUST effort is primarily focussed on the tsunami threat to Zones 1 and 2, although a successful program will enhance the protection of Zone 3 as well. The immediacy of the threat to Zone 1 requires a somewhat different approach than is needed for Zone 2. In Zone 1, there is insufficient time to activate and disseminate warnings. The population at risk must be trained so that if they are on the shore and feel earth tremors they immediately head for high ground as a precaution. This seemingly simple operating principle must be ingrained in the threatened population by a public education program. In Zone 2, the need for education is coupled with a need for the technical capability to issue warnings rapidly. People in this zone will not have earth tremors to alert them to put their educated response into action. Instead, they must rely on an external warning mechanism. It should be apparent that the location and extent of Zones 1 and 2 will vary with the tsunamigenic epicenter. Thus both the education and the technology aspects of THRUST must be applied throughout the entire area covered by the warning system. Both technology and education are necessary to build an effective system. 5. THE THRUST PROJECT The structure of the THRUST study is best described in matrix form, as indicated in Table 1. Two time frames are specified: Pre-event time is the period, however long, before the next disaster. During this time historical data can be compiled and studied, plans can be laid and evaluated, and educational programs can be conducted. Real-time is the time which commences with the earthquake. During real time, data are collected, warnings are 123 issued, and appropriate responses are taken. During each time frame, three activities are conducted— data collection , data analysis , and dissemination . 5 . 1 Pre-event Data Collection Data collection during this phase consists of a thorough compilation of existing data to archive and file all available tsunami-related information for the threatened area. The purpose is to establish a comprehensive, readily accessible data base to be used for planning, education and rapid response techniques . 5.2 Pre-event Data Analysis Since tsunami are rare events, the data base created by scanning historical records will be very limited. In fact, it may be so limited that we must apply analytical techniques to extend the base. Two techniques that are used are statistical evaluations and hydrodynamic numerical simulations. Statistical evaluations focus on time considerations such as frequency of occurrence, rates of stress release, and seismic gap theory. Hydrodynamical numerical simulations enable scientists to interpolate maximum run-up heights adding to areal considerations . From such analyses, time and space parameters can be estimated in areas of high tsunami risk. Application of these parameters define the specific risk for a particular location and serve as a guide for emergency planning. 5.3 Pre-event Dissemination This is essentially an educational phase — education of both disaster control and planning personnel and of threatened populations. The data base 124 H H O Hi rH M ■a H to < H s a w i H W 1 5S o t— I H < H s w CO CO M CO < < < 2 O M H CJ W hJ (-5 o u < < Q H fa l CO CM TO % O ft CJ u o 3 fa i-h jh d U V o n e -h O 00 W d d •h o (0 3 4-> TO T3 rH W 3 TO S N -H £0 C/D S TO d 3 Q 01 a I fa 00 3 O u 3 6 0J CD TO 2 4J 00 4-1 Si u u w 00 >> -H ^ i) h > s m 4J H D W d to q M fa 0) O -P •H CJ TO T3 O H CU 2 QJ a u o w d OJ 00 3 •H CO CO 0) + CJ o S-l Oh 3 rH O OJ •H > CO u h CO CO CO H CO 3 4J CO 4J 3 CO 5H -u c •H o PL, T3 3 CO CO 3 o CO /—s -o 4-) c 3 CO •H r-t o CO Cm M T3 3 3 •H CO 00 CO CO N_^ a 3 cu x: s CO o 3 O •u cu > !-i CO CU 5 4-1 CD o s ■g o a co co •H •H 0) 0) to co CO 4-> •H CO i .-I (0 Cm S-i CU S r-H CO Cm c ■a !m cu 4J CD S o S co ■r-t CU o co cu >-i > CO cfl ^H s CO 4-> g co CU co u H CO cu CU 4-) (0 4J 3 CO 1 o CN CO M CO CO H CO 3 ^> o CO 4-1 c •H o Cm 3 CO co CO c CO cu X) CO c co 4_> CO CO 3 4-> 0) •H CO cn <*s ^^ c 3 ■fl-fl o o 4-1 c •H o Cu 3 CO •g-g o o -cf O CO M •— \ CO ^ CO CO .-H •H CO T3 c O 13 3 c 'SI o o co vO X cu c ^"«N co G ■^s 4-1 CU bO >» 3 CO CO rH bO CU -3 bO C ■a •H > CO •H c > 3 >-l co 3 4-> co cu CNJ ^ 4^ CU >-l T3 3 c bO CO •H PH co a. CU CO cu bO CO 3 cu CO 0) cu bO u 3 .3 •H 4-> 3 J-l M-l CO o 3 cu u •g 3 3 4-> 3 Q, 3 2 co H cu 4-1 -C 3 4-> co c CU cu > cu 3 s 3 4-> cu J3 1 3 CO 3 CU 3 4-> CO 3 4-> C •g -3 3 3 3 •H O CO •g >, CO CO •H rH CU CU CO T3 u-i ■3 o CU 4-> ^i O 3 •H > T3 •H CU U U i-i Cm 3 160 Epicentral Sea-Level Monitoring Data from the epicentral region of a future earthquake are impor- tant for several reasons other than the acquisition of timely tsunami amplitude data. Using tide-gauge long-term vertical deformation in the region can be monitored to an accuracy of a few cm per year to provide an indication of pre-seismic strain accumulation [Kato, 1983]. Such studies enable refinements in the forecast of the timing of earthquake rupture [Wyss, 1976]. Several eye witness accounts of coastal uplift and depression immediately prior to major earthquakes have been reported in Japan [Imamura, 1930] but there are few instru- mental records that have recorded these forms of precursor. The study of earthquake mechanism in large historical events has been made poss- ible, in part, by the availablity of sea-level data obtained before, during and after the event [Fitch and Scholz, 1971]. Data on tsunami run-up in the epicentral region is seldom available in the detail that coastal engineers would like. Spatial measurements of tsunami run-up would, moreover, enable the inversion of these data with suitable corrections for near-shore effects, to estimate the mechanism of tsunami generation and the time history of sea floor movement. For these applications it is desireable to install several sea- level monitors within and close to the epicentral region of potential future plate-boundary earthquakes. Since a major part of the useful- ness of a multipurpose sea-level monitoring array is centered near the time of the event where precursory deformation, co-seismic movements and tsunami generation occur it is vital that epicentral sea-level 161 monitors are "hardened" against destruction by the earthquake or resulting tsunami. An Indestructable Sea-Level Monitor A number of features of existing tide-gauges and tsunami -gauges contribute to a poor data survival rate during major earthquakes. Many such gauges are installed on harbour structures that are damaged co-seismically and data can be lost due to power interruption, tele- metry failure or the destruction and loss of the entire recording system. A totally indestructable gauge may not be feasible but there are several design improvements that can contribute to surviveability. The instrument should be totally submersible, should use an uninter- ruptable power source, should avoid man-made harbour structures and should not use land telemetry. Engineering an instrument to these specifications is technically realistic and can take many forms. Three solutions to the problem are discussed. The first of these is a proposed modification to remote sea-level monitors operated in the Shumagin Islands, Alaska, to improve their surviveability during a major earthquake. The second solution is a possible design for an inexpensive tsunami run-up gauge that memorises data from a few hours before to several hours after an event but includes no telemetry. The third solution is the possibility of monitoring the spatial loading of the tsunami using sensitive extensometers or tiltmeters buried inland. 162 Tsunami -Hardening the Alaskan Gap Gauges The sea-level monitors in the Shumagin Islands seismic gap, Alaska, consist of ceramic pressure gauges bolted to a rock foreshore slightly above lower-low-water level (Figure 3) [Bilham et al. , 1981]. An amoured submarine cable bolted at frequent intervals to the shore conveys data to an electronic package 2-20m above high water. Sea-water pressure, atmospheric pressure and sea-water temperature are converted to frequency and gated every 12 minutes for transmission at 3 hour intervals as 14 bit digital words via the GOES satellite. The existing sea-level system is designed to monitor long-term, tectonically-induced, vertical deformation at the cm level although it is evident that long-period waves with amplitudes of the order of 2 mm are detectable at 30 minute periods (Figure 3). To convert the gauge to record tsunami periods requires an addi- tional electronic package to store and/or transmit the higher data rates encountered co-seismically. A block diagram of the proposed modifications are shown in Figure 4. The tsunami data are passed to an additional counter that samples data averaged over 2-30s inter- vals. These data are transmitted or are stored in a fail-safe storage device analagous to a flight recorder. The satellite telemetry could use either the GOES emergency channel with phased 30 minute telemetry or an interrogateable platform [Clarke, 1976]. The storage device would be buried deeply for field retrieval after the event should the antenna be destroyed. Weak links in the proposed system are the possible destruction of the armoured cable from the pressure gauge to the electronic package, 163 GOES TRANSMITTER ELECTRONICS, AIR PRESSURE + BATTERIES CABLE BOLTED TO ROCKY FORESHORE BENCH MARK HH_W DEEP WATER EN1 LLW UNDERWATER PRESSURE TRANSDUCER 0-20psi L0G1I CrOUER 5PECTRRL DENSITY IPRC - PRS Ht SEA SLOPE (40 km separation) ?.•»-- i. is - 8.18 !.••■ ;.••-■ 3.M J- semi-diurnal tide frequency in cycles per day > 24 30 36 48 — »- 52 — t- 50 —I 30 minutes Figure 3 . Sea level monitoring system in Shumagin Islands seismic gap. Gauges (top) are installed on islands approximately 40 km apart. In the power spectrum (bottom) of the difference in sea height between two adjacent gauges can be observed a spectral peak corresponding to waves of 28 minute period and peak- to-peak amplitude 2 mm. Non-linear shelf-tides and seiches are evident at longer periods . 164 SEA LEVEL DATA ATMOSPHERIC PRESSURE- SEA TEMPERATURE— SEA WATER PRESSURE- GOES SELF-TIMED MODE 12 min 1 mm RESOLUTION V 3 hours TSUNAMI DATA INERTIAL TRIGGER WITH N-MINUTE DELAY GOES EMERGENCY MODE 3s 1 cm RESOLUTION V 30min V MEMORY -4 hrs +16 FAIL-SAFE STORAGE Figure 4 . Proposed recording and telemetry systems to be added to the existing Lamont-Doherty sea level monitors that transmit data every 3 hours. A 30 minute emergency channel would telemeter data to the Alaska Tsunami Warning service and a fail-safe memory would store data for post-event retrieval. 165 the difficulty in securing an offshore pressure gauge (the present scheme monitors low-water levels only if an entry tube anchored off- shore is attached to the onshore gauge) and the difficulty in ensuring that the satellite antenna will not be destroyed by strong ground motion or by excessive tsunami run-up. A Tsunami Surge Gauge This device is proposed as an inexpensive solution to providing tsunami data, pre-seismic and co-seismic deformation data and engi- neering run-up data, but without the benefit of real-time or inter- rogateable telemetry. A 5 cm diameter, 20 to 30 cm long sealed can- nister is proposed that contains power for a year or more and has the ability to record sea-level data at 2-4s samples for about 20 hours about the time of a major earthquake. The package is inserted into a diamond core-hole on a rocky beach at low tide or bolted to coastal structures below low-water-level. In a core-hole installation the surge gauge is essentially invisible to the dynamic and destructive effects of a tsunami but can measure wave action with great fidelity. A schematic diagram of a possible configuration for a tsunami surge gauge is shown in Figure 5. Two sensors are provided — a low-power pressure gauge senses sea-level to 4 cm resolution over a 10 m range (1 cm resolution using a 10 bit A-D converter) and a hori- zontal strong motion switch activates the solid-state memory. A 16K CMOS RAM provides up to 20 hours capacity (with 8 bit samples every 4s) and a clock provides 1 minute absolute time resolution (0.1s rela- tive time resolution during recording). By operating the transducer 166 SHORELINE CORE-HOLE SEA-CLIFF PROBE 5cm- REMOVABLE UNIT OIL FILLED TRANSDUCER ELECTRONICS LITHIUM BATTERIES INSTALLATION CASE CEMENT 4CM RESOLUTION (8 BIT) 1CM RESOLUTION (10 BIT) LITHIUM POWER r i % PRESSURE SENSOR lOM RANGE FILTER (10S) 8 BIT A/D (4S) | 1 16 K CMOS RAM (20 HOUR) MEMORY CLOCK 1M/YR LOGIC STRONG MOTION SWITCH ' ALIljo Figure 5 . Installation and schematic operation of proposed tsunami surge gauge 167 and CMOS memory continuously it is possible to provide several hours of pre-event memory. The component costs for a tsunami surge gauge appear to be less than $600 (1983 prices) making the device suitable for dense deploy- ment in a number of tsunamigenic epicentral regions. The shortcomings of the tsunami surge gauge are that data are unavailable until retrieved and the installation method provides no data on negative excursions of sea-level unless the gauge is installed on submarine probes attached to man-made structures or near-vertical sea cliffs. Indirect Methods for Monitoring Tsunami Run-Up The weight of the sea causes elastic deformation of the earth's crust beneath the sea floor and for a large distance inland. These load tides have been studied extensively using gravimeters, tiltmeters and extensometers [e.g., Melchior, 1978]. The latter two classes of instrument are particularly sensitive to local loading and it is poss- ible in principle, to obtain a transfer function relating observed tilt or strain to the instantaneous spatial distribution of sea-level [cf Marthelot et al. , 1980]. Figure 6 illustrates two examples of the monitoring of sea-level variations by instruments installed 100 m inland. A difficulty of the method is that strains and tilts caused by co-seismic crustal movements will be superimposed on the tsunami-load signal. Strain and tilt data are tensor and vector quantities respec- tively requiring more data recording capacity than direct (scalar) 168 ICY BAY STRAIN N8°W CAPE YAKATAGA SEA LEVEL Figure 6 . Two examples of inland surface deformation resulting from sea loading effects. Above: Sea level and the strain-signal normal to the coast 100 m inland at a depth of 2 m measured by a 17 m long carbon-fiber extensometer . Below: Record of the Tonga earthquake and following tsunami in the New Hebrides (19 Dec. 1982) recorded by a 100 m long water-tube tiltmeter. The tsunami wave (bandpassed 20-1500s) arrives 200 minutes after the seismic wave. Courtesy of George Hade, Cornell University. 169 measurements of sea height, and the inversion of these data may pro- vide ambiguous information on the spatial distribution of transient post-seismic sea-level surges unless several sites are monitored. Local tsunami run-up signals could be distinguished relatively easily from more distant loads using small arrays of tiltmeters, for example, disposed perpendicular to the shore. The advantages of monitoring sea-level transients from their resulting strain and tilt fields are that it is possible to map the offshore development of tsunami waves, the sensors can monitor a nega- tive tsunami and finally since the measurement is "non-contacting" the sensors are inherently tsunami surviveable. Further investigation of these methods appears to be worth persuing. Discussion The acquisition of epicentral data to satisfy the needs of long- term tectonic studies (secular, co-seismic and post-seismic trends) and transient signals (precursory uplift, tsunami run-up and tsunami modelling) are not easily satisfied by a single instrument if it is to be sufficiently inexpensive for widespread deployment. A diverse instrumental approach may be necessary to yield optimum data return given a limited budget. For example, the pressure transducers required for secular studies require high resolution (mm) and good long-term stability (cm/year) and are relatively costly whereas those required in tsunami studies demand cm resolution and stability for a limited period only. Similarly the data sampling rates needed for secular studies are one or two orders of magnitude lower than those required for transient studies. 170 The serious weakness in the two shore-based measurement systems discussed is the inadequate sampling of sea-level below lower-low water. Ideally, the sensor would be in deep water off-shore bolted to bedrock. In Alaskan waters this is a costly undertaking. The use of passive offshore sensors attached to massive sea-bottom anchors is not a reliable solution for secular studies (settlement uncertainties) or for transient studies (tethered buoys were moved bodily during the 1964 Alaskan tsunami). A pressure tube attached to an onshore gauge with an orifice tethered offshore has been used in the Shumagin gauges to provide sea-level data below lower-low water but this is suscept- ible to storm or tsunami damage. The error in mean sea-level esti- mation arising from the absence of the pressure tube at times of lower-low water can be eliminated by predictive-filtering methods but the occurrence of a major earthquake at a time of low tide may lead to partial loss of data if the offshore entry tube is destroyed, especially should uplift occur. These criticisms apply to the tsunami surge recorder but are less important if the gauge is installed primarily to monitor positive run-up. It may be possible to deploy the surge gauge in a larger number of environments including a sampling of harbour facilities where a deeply submerged, retrievable package can be installed at low cost. The measurement of the deformation field of the tsunami through the use of an inland or onshore strain or tilt array may provide a useful form of short-term warning (e.g. , 10-30 minutes) of the arrival of a tsunami for large urban areas away from the epicentral region particularly for tsunamis that are not generated by seismic disturb- 171 ances (e.g., submarine slumps). Under these circumstances the tsunami -induced deformation field will not be masked by the near-field or propagating strains of a major earthquake and it may be possible to implement a detection algorithm to identify an incoming offshore wave. The detection of storm surges in the North Sea using tiltmeters has been discussed by Rumpel [1983]. The cost of a sufficiently low-noise tilt or strain array may be prohibitive compared to a more direct form of warning system [Zielinski and Saxena, 1983] but may be feasible where such arrays already exist for crustal strain measure- ments, e.g., Japan and Peru. Conclusions Sea-level data from the epicentral regions of future major earth- quakes are important for several reasons: the monitoring of long-term tectonic trends to refine earthquake hazard forecasts, the acquisition of short term precursory deformation to provide evidence for seismic imminence, the monitoring of tsunami generation and amplitude for the Tsunami Warning Service, the acquisition of tsunami run-up data for coastal engineering design, and for the study of tsunami generation processes and the modelling of earthquake mechanisms. Despite the importance of epicentral sea-level data, relatively few instruments presently exist that are either optimally located or able to survive earthquake shaking or tsunami damage. The locations of at least the next few circumpacific tsunamigenic seismic events are known with reasonable certainty and the technology to improve the reliability of sea-level monitors to resist damage is within present 172 capabilities. Several of the seismic gaps in the Pacific are at an advanced stage in the seismic cycle [e.g., Jacob, 1983]. Twenty or more tsunami run-up gauges in each of these seismic gaps are likely to provide a wealth of new data even should only one of the regions rup- ture in the next decade. It is important that the scientific com- munity .together with those concerned with tsunami warnings, act promptly if we are to monitor the epicentral coastal effects of the next great cirumpacific earthquake. Acknowledgements . The monitoring of sea-level in Alaskan seismic gaps is supported by the U.S. Geological Survey under contract USGS- 14-08-0001-21246 and NASA contract NA-5-27232. The article follows stimulating discussions with John Beavan, Klaus Jacob, Douglas Johnson, Tom Pyle and Lynn Sykes. Lamont-Doherty Geological Observatory Contribution Number 3577. 173 References Beavan, J., E. Hauksson, S. R. McNutt, K. Jacob, and R. Bilham, Tilt and seismicity changes in the Shumagin seismic gap, Science , 222 , 322-325, 1983. Bernard, E. N., J. F. Lander, and G. T. Hebenstreit, Feasibility study on mitigating tsunami hazards in the Pacific, NOAA Technical Memorandum ERL PMEL-37, 1982. Bilham, R. , J. Beavan, and K. Hurst, Installation of sea level moni- tors to detect vertical motion and tilt in Alaskan seismic gaps, EOS, Trans. AGU , 62, 1053, 1981. Clarke, H. E., Jr., Tsunami tide system, U.S. Geol. Surv. Open File Rept. 76-735 , 44 pp. Fitch, T. J., and C. H. Scholz, Mechanism of under thrusting in SW Japan: A model of convergent plate interactions, J. Geophys. Res . , 76, 7260-7292, 1971. Imamura, A. , Topographic changes accompanying earthquakes or volcanic eruptions, Publ. of the Earthquake Investigation Committee in Foreign Languages , 25 , Tokyo, 1930. Jacob, K. , Alaskan seismic gaps quantified, EOS, Trans. AGU , 64 , 258 1983. Kato, T. , Secular and earthquake related vertical crustal movements in Japan as deduced from tidal records 1951-1981, Tectonophysics , 183-200, 1983. Kumpel, H. H. , The possible use of tiltmeters in forecasting storm gauges in the German Bay, North Sea, in Proc. 9th Int. Symp. Earth Tides , 1981, J. T. Kuo, ed. , E. Schweizerbart 'sche Verlagsbuch- handlung Stuttgart, pp. 359-371, 1983. 174 Nishenko, S. P., and W. R. McCann, Seismic potential for the World's major plate boundaries: 1981, in Earthquake Prediction - An Inter- national Review , Maurice Ewing Series, Vol. 4, D. W. Simpson, and P. G. Richards, eds., AGU, Washington, D.C., pp. 20-28, 1981. Marthelot, J. M. , E.Coudert, and B. L. Isacks, Tidal tilt from localized ocean loading in the New Hebrides arc, Bull. Seismol. Soc. Am., _70, 283-292, 1980. Melchior, P. J., The Tides of Planet Earth , Pergamon Press, New York, 609 pp., 1978. Spaeth, M. G., and S. C. Berkman, The tsunamis as recorded at tide stations and the seismic sea wave warning system in The great Alaska Earthquake of 1964 , Oceanography and Coastal Engineering, Nat. Acad. Sci., pp. 38-110, 1972. Van Dorn, W. G. , Tsunamis, in Advances in Hydroscience , V, T. Chow, ed., New York Academic, 2, pp. 319-366, 1965. Wyss, M. , Local sea level changes before and after the Hyaganada, Japan earthquakes of 1961 and 1968, J. Geophys. Res ., 81 , 5315, 1976. Zielinski, A., and N. K. Saxena, Tsunami detactability using open- ocean bottom pressure fluctuations, IEEE J. Oceanic Engineering , OE-8, 4, 272-280, 1983. 175 THE NONLINEAR RESPONSE OF A TIDE GAGE TO A TSUNAMI Harold G. Loomis Department of Ocean Engineering University of Hawaii Honolulu, Hawaii 96822 ABSTRACT We calculate the response of a tide gage to a tsunami by creating a pseudo-tsunami by running white noise through a filter with a response func- tion like that of a tsunami spectrum. The resultant time series has the appearance of a tsunami. This time series is used as the driving function of a differential equation corresponding to the mechanical arrangements of a tide gage. The resulting time series differs most from the input series in the cases where the amplitude is high and/or the spectrum is broad. 1 . Introducti on The typical tide gage in the United States for the last hundred years has looked like Fig. 1. The first consideration was to have a rugged, dependable instrument that is easily maintained. It has been known for a long time (Cross 1968) that the response of the tide gage is both nonlinear and frequency dependent. A further annoyance is the lack of up and down symmetry that results from the placing of the orifice at the end of a funnel, stilling well , area A 2 orifice, area A-, Fig. 1 Typical Tide Gage Stilling Well 177 The purpose of the stilling well is to dampen the high frequency wind waves and swell. The casing of the stilling well is normally 12 in. diameter and the orifice size is chosen for the location. The five National Ocean Service tide gages in the Hawaiian Islands have orifices of .75", 1.0", and 1.5". The orifice fills with marine growth and approximately once a year each gage is serviced. The only major change in recent times has been to replace the strip chart recorder with a digital record. The rotating arm is clamped once every six minutes and that reading is punched into a paper tape. Obviously a sampling interval of six minutes is useless for tsunami work. Where a tsunami gage is needed, a bubbler gage is installed. The hose of the gage is still placed in the stilling well so that alteration of the record is now combined with the response of the bubbles mechanism. That the frequency response is important in tsunami work is clear from looking at Fig. 2. The sharp bend in the response curve occurs right at the higher tsunami frequencies. The fact that different response curves are given for different amplitude waves points up one aspect of the nonlinearity of the tide gage. We are going to demonstrate that the mixture of frequencies creates another nonlinear response problem. IOOO 2000 Sec. j L 10 20 30 Min. WAVE PERIOD, T Fig. 2 Tide Gage Response Versus Wave Period, 3/4- inch Diameter Orifice (From Cross, 1968) 178 2. Theory Assuming streamline flow through the orifice, we can apply Bernoulli's equation along a streamline. Then where v is the velocity at vertical coordinate z and p is the pressure at that point, is the velocity potential. Assume that for some short time the flow is approximately constant so that 3/3t = 0. Then the terms gz + p/p are combined and their value is constant along the length of the orifice and is, in fact, equal to the difference in pressure due to the head inside and outside the stilling well. This is expressed in the equations v = c /2g(y-x) , y £ x or (2) v = - c/2g(x-y) , x I y , where c is an orifice coefficient which has been measured in laboratory tests at around .8 for inflow and 1.0 for outflow. The inflow (outflow) of water raises (lowers) the water level in the stilling well according to dt A 2 v or which is sometimes written H . (jL . c • **g) sign (y-x) /abs (y-x) (3) where A, is the area of the orifice and A ? is the area of the stilling well If one sets then y = a sin(27Tt/T) R = a_/x mav o max is the amplitude response of the tide gage. Curves such as those shown in Fig. 2 result. 179 We, instead, create a pseudo- tsunami y(t) for an input. The spectrum of y(t) is chosen to be the same as tsunami spectra. The output x(t) also has a spectrum and one is tempted to find the response function from the cross spectrum. Because the system is nonlinear, this would be incorrect. There are two parts to the problem. The first is to create the pseudo- tsunami. We take white gaussian noise z(t) and convolute it with a kernel k(t) M y(t) = I z(t) k(t-i) T=-M 3. Results In addition to the filtering of white noise to create a pseudo-tsunami, various envelopes were used to vary the amplitude slowly. Figure 4 shows a tsunami with periods centered around 12 min. At small half-amplitudes there is little error. As the half-amplitude gets to the order of 5 ft, the error is significant as is seen in the right of the figure. Figure 5 has approx- imately the same frequencies but larger amplitude and with a wave of 14 ft above mean sea level, the gage under records by 37%. Figure 6 shows a broader 180 so that the spectrum of y, S (f ) is the same as one of the typical tsunami spectra. If k(t) has transform K(f), then Sy(f) = S z (f) * K 2 (f) . The procedure is first to specify k (f), then find the cosine transform, k(t), if K(f) (which assumes K(f) is a symmetric function). Because k(t) would have values for t in (-«>, °°) it is necessary to truncate which is done by fading with a sin t/t function with zero crossings at -M and M. For an infinite sample y(t) would have the desired spectrum. For a finite sample, y(t) approximates the desired spectrum and this is calculated and compared. The second part of the problem is to solve differential equation (3). We tried various differencing schemes and they could be made to agree well. The one used is x(t+At) - x(t) = ± (A 1 c/2g~/A 2 ) • At /±(y(t) - X(t)) (6) Schematically, Fig. 3 shows the procedure. white noise actual spectrum x(t) dif. eqn. Fig. 3 Schematic of Procedure for Creating Pseudo-Tsunami and Passing It Through a Tide Gage spectrum and in this case a wave of 14 ft is under-recorded by 50%. In Fig. 6, the spectrum is broad and even at small amplitudes, there is an under-recording of wave heights. Finally, Fig. 7 shows the case of a double of the wave motion causes a severe under-recording of wave heights with errors of 67% in the right part of the record. 4. Conclusions The main conclusion is that a large tsunami with a broad spectrum may cause excursions on the tide gage of only 2/3 of its actual range. The larger the tsunami, the greater the error and the broader the spectrum, the greater the error. 5. Acknowledgements This work was supported by a grant from the National Science Foundation, CEE 8120160. The manuscript was typed by the staff of the Center for Engineering Research. 181 o o o o o o o CO to •1— CD o c o •r— • _J CJ r- -o *r— r— o to o 00 •r- CU i— DC Q. E CD o •a: en o (O cu o o +j in fD CU S- "O 0) T- •o 1— o S 00 o o o < o 3* E • TO CD C > 3 fO oo 3 r> 1— o i— to o (0 C U S- OJ •i- CD Q.+-> >> x h- UJ o 00 -1 — CD c o •1 — o _J o -0 CO •1 — 1 — C/~) o E (D re ro CO 5 cd -0 i~ T- o s- l— o • A3 (13 ,o C 3 3" 3 00 1— 1— (D C CD S- O cn cd o J- +-> (T3 X .0 _J UJ (T5 u- er> 00*S2 00*02 OO'Sl 00*01 OD'S 00*0 dWO 183 00*02 00*SI 00*01 OO'S OO'O dWU oo*s- 00*01- OO'Sl- 184 00 o CD ■ c o •1 — r» 1 -o •r— I o o o o Oi U3 • 00 E c 3 o J- Q. +-> co U CD O 0) DC O CL a o en a; o .^ fo -o o O- •!- o h- • ai o ■ — oo ^ -O -i- 3 o < Q o *o • o o +-> ft3 en •r- 3 s ^— •I- A3 E c o ZS 4-> o 00 X cu t— LU I s *. o en O -r- 00"S2 00*02 OO'Sl OO'Ot OO'S OD'O dWU OO'S- 00*01- OO'Sl- 185 TSUNAMI OF MAY 11, 1981 ON THE COAST OF SOUTH AFRICA S.O. Wigen T.S. Murty D.G. Philip Institute of Ocean Sciences Sidney, B.C. Canada ABSTRACT On May 11, 1981, tide stations on the coast of South Africa registered unusual tsunami -like waves. They are clearly apparent on the analogue records from Saldanha Bay to East London, 1000 km eastward along the coast, and at intermediate tidal stations. Largest maximum wave was registered at Mosselbaai with a trough-to-crest height of 58 cm after removal of tide. Source of these tsunami-like waves has not been absolutely determined. No earthquake approaching tsunamigenic potential preceded the arrival of the waves. Atmospheric conditions were not abnormal, nor were the characteristics of the initial waves indicative of a meteorological origin. Submarine landslide may have initiated the disturbance, and this possibility is investigated. The source position has been estimated by inverse calculations of wave velocities from the observed arrival times at coastal positions. It appears to have originated on Agulhas Bank, 90 km southward of Cape Agulhas, in a charted depth of 140 m. Analogue records of the waves at tide stations have been digitized, and spectral analyses are used to make further inferences about the generation of this event. 187 1. INTRODUCTION On May 11, 1981, tide stations on the coast of South Africa registered tsunami-like waves (I.T.I.C. 1981). The tsunami waves are clearly apparent on the analogue records from Saldanha Bay in the west to East London (Figure 1), some 1000 km eastward along the coast. Six of the South African tide stations show the tsunami, and on three others, the waves are either concealed within normal seiche action, or are not present. These nine stations are listed in Table 1 along with their geographical positions. The source of these tsunami-like waves has not been absolutely determined. No earthquake approaching tsunamigenic potential, off South Africa or elsewhere, preceded the arrival of these waves. Commander Stokes (personal communication) reported: "Meteorological conditions along the coast were normal, and no unusual weather was reported at sea, in the area". The only observer report came from the tide gauge custodian at Knysna, who reported: "I have never experienced such a pronounced rise and fall in so short a period". He is a reliable custodian who has looked after the Knysna tide gauge for twenty years. Wave arrival times in Table 1 suggested a source southward of the South African coast, on Agulhas Bank. However oil drill rigs operating there experienced no unusual sea conditions or waves on that day. Since seismic or meteorological source can be ruled out, by inference, a submarine landslide appears to have initiated the disturbance. 2. DETERMINATION OF THE SOURCE AREA FOR THE DISTURBANCE Using the stations for which the time of wave arrival could be most clearly defined (ie. Simonstown, Mosselbaai and Knysna) the source position was estimated by inverse calculations of wave velocities from the observed arrival times at these coastal positions (Figure 2). This determination gives a source at 35°37'S, 20°16'E, and the time of occurrence of the submarine slide around 0710 G.M.T. on May 11, 1981. This location is on the Agulhas Bank, 90 km south of Cape Agulhas, in a water depth of 140 m. Agulhas Bank is roughly triangular, with the apex about 250 km southward of Cape Agulhas. From the apparent source position, the nearest edge of the bank (200 m contour) lies about 40 km to the southwest. Agulhas Bank is noted as a region of fairly strong currents and occasional very rough seas. Buildup of submarine banks to a point of instability is a reasonable possibility. However an overall increase in depth from 140 m to 200 m over a distance of 40 km to the edge of the bank 188 \ J J I °*' f ® o Oo _ ra N <1 n n 1-5 J$ '■'?'*% 5^r>^_ kT> >- < •««»«■$ r *4t a "< I u IS Z < co a 3 t v Q c ~ n n ~" ^w K -r 4 '%K Z o c 4- ' *~%\ o a "■w ' K Z o. »v O D a> ^A < •A X t— UJ CD < o o o o o o CD _ p> _j |v — ci UJ II CM < IV LU U 2 fl ct ce '1 o Q_ < u. ' A L/l < 7 __ C 1^ X tf) o V LU J Q£ U- »i 3 o < i o z A I— CO -« u- >■ « _ CO < o u UJ X Z i— < if A cd ;y uj >HL O #k £ V «., J l-M *~ - '] _ff. °o> _ .g''':''.^*/ .» ^ z •Uts*^ £ > < CD o I— to ° K f-'l^^ < z •^ "" - Isl , , ^ X o - " »— X'- i»' I z 5 a. UJ gap o < a lO Q — ■ — < 3 ■''ft* —J o z < ii£jr~* j ' "to 1— "o "r- •, /« 1 O CL 1 1 1 189 u X < _0 co □ co CO u □ co o- C_J CO ■ — co co ro i*> A ct -1- CM CO A A a ' — A CM □ co r*> A '— c O o o o o o C a A r^ r^ CO C 1 ^, ■j-- r- cm ~~ r ~ ^~ — CM CM CM csi r^i UJ CO < co □ ■s. co LA — CM , r^ CSJ CO I — — LOO cc: >- LO >- 1 — - co- 0" C_i _) CO - c O — h» §o- CO / V' — r^ / \/ cEg to e LO t < 7/ > \ P A \ // ^L V / / ^.E c 3 Em CO UO II CO g c o w (- TJOO .. < = cr>.c z LU > LU z LU a LU §5 roc f£ m t- LU -J >P z 2 Z _! * LU O o CO jl GO < LU *" GO o - DC _- UJ CO D O .. CM C < / \ / ~ *tT ( '\ \ en 1 - s V ' — z O O) ' \ ' \ >- LU c CD ^ UJ V 1/) ^^ > ^--' \ < t- o z — LU cr o — - \ ] / LL — > "~~"^"X s __!_«o / / < o- ^\ ^^-~^^ / / LU X ~~^ \ f~ v \_./i1q/ 1- LU / /^\| r ~~-^?-V. " ( / ( D -I *"\/ l/> V ( Jj f ~~ ; ^ s \ / \ O CO < / m x V^ l re I ^^\ 1 \ 7 ) OV S n IN \ CO < ^^v °~ — • V / ) •'^ ' \ J \\ \ CD O LX r \ u o Q Q_ y-^^^ X -0- \^ / /\y ^^'* v -- — ' j^^^^ ^-- — ^__ ooT CN z J flsy > • V ^' ">^. LU 3 i/ ^ ^ — \ Q r> / / i \ / ■ > 191 from the source is an extremely small average gradient. Summerhayes (1979) recognizes a slump off the west coast of Africa involving 250 km^ of material on a gradient of less than 1°. Hence submarine slides on an even smaller gradient is a possibility. There are limitations on the accuracy of determination of the source. If the submarine slide was not instantaneous, but occurred progressively, wave arrival times inferred would be less accurate. An even more probable limitation on accuracy lies in the question of how the waves reacted to the edge of the bank, and whether the assumed path of travel in the inverse calculations is significantly in error. Until some geophysical evidence of the existence of the submarine slide is found, this remains an open question. Figure 3 to 6 respectively show the tide gauge records for Simonstown, Mosselbaai, Knysna and Port Elizabeth. In each diagram the upper part shows the recorded water level, and the lower part shows the curve after the tide is removed. Since the tsunami was recognizable at East London, about 800 km eastward of the assumed source, and at Saldanha Bay, about 400 km northwestward, ocean wide propagation was investigated. No tsunami was detectable at Monti video and La Paloma, Uruguay. Nor was it detectable on seven tide stations in Australia, Carnarvon, Geraldton, Bunbury, Alhany, Esperance, Point Lonsdale, or Williamstown. No tide records were available from British administered islands in the South Atlantic or French administered islands in the Indian Ocean. 3. ESTIMATION OF THE SIZE OF THE SUBMARINE SLIDE An approximate estimation of the energy involved in the suhmarine slide was made using a technique described by Slingerland and Voight (1982). From dimensional arguments they write for the wave characteristic a: IT, w h. JL A ^i n d ft$ i t /s -i, p iLiaa , Jr, t A (l) where: V A \6 l d' d' /-J P y ' d' / d d = average depth offshore from slide site; E|< = maximum dimensionless slide kinetic energy; f|< = kinetic friction coefficient g = gravitational acceleration; h = slide thickness; i = angle of slope; k = constant ; l = slide length r = radial horizontal distance from slide; s = downslope distance of travel of slide; t = time; V = maximum velocity of slide; w = slide width; 192 "2 -■O Oco o-' <.- 3< o (SS313W) 13A31 «3iVM ooooooooooooo I I I I I ( SH3HW I 13A31 K31VM i i i i i ISM313WI T3A31 K3iVM ISK313WI 13A31 «3iVM 193 6 6 I I I I I ( S83J.3W) 13A31 H31VM I I I I I I I II I I I I I *. q> *, ^ v I I I I I I I ' SS313 W ) 13A31 M31VM TTJ1 T 1 A . ^ OOOO OOOO O Q I I I I I '; * i 1 i * 1 13A31 a 3iVM { sasiaw i i3Aii S31VM 194 n m A P Ps maximum wave amplitude at site for particular slide event as measured from mean water level; angle of slide entry with respect to horizontal; water molecular viscosity; dimensionless characteristic of wave; density of water; slide bulk density; and function. Hydraulic model studies have shown that one can ignore the Reynolds Number by assuming that the viscous drag of the slide is much smaller than the pressure drag. Also ignoring 8, the dimensionless maximum wave amplitude can be expressed as: r/d-4 (l Mi -} vl) ^ 2 d 3 p gd' >(E k ) (2) Here the dimensionless maximum wave amplitude was distance r = 4d where d is the average water depth, above nonlinear relationship gives: taken at a horizontal A log-log plot of the log -j a + b log (E, ) (3) From hydraulic experiments, Equation (3) can be written as -1.25 + 0.71 log (E k ) (4: Using the relationships given in Camfield (1980) between coastal amplitudes and deep water amplitudes of long gravity waves, for each of the tide stations considered in this study, knowing the maximum coastal amplitude (from Figures 3 to 6), the deep water amplitude at r/d = 4 was deduced. Then using Equation (4), the maximum submarine slide kinetic energy E|< was estimated. From the estimates for the different stations, an average value was deduced, and from this, the volume of the material involved in the slide was estimated to be at least 4 million cubic metres. In reality, this figure has to he increased by at least an order of magnitude. Because, the above estimate assumes that, all the energy of the submarine slide goes into generating a tsunami which is totally radiated towards each tide station. Also in this estimation, energy losses due to spreading, scattering, diffraction, refraction and bottom friction are not taken into consideration. 195 4. SPECTRAL ANALYSIS OF THE TSUNAMI RECORDS Fourier analysis techniques were employed to identify any frequency bands dominant in energy content. Analysis was performed only on tide gauge records from Simonstown, Mosselhaai, Knysna, and Port Elizabeth, since these four ports were subject to clearly defined tsunami activity. At Saldanha Bay and East London the tsunami activity was partially obscured by seiche action, and at Luderitz, Port Nolloth, and Richards Bay, completely or almost completely so. The frequency spectra derived from these analogue records would be of uncertain interpretation. Applying routines described by Wigen (1981), the analogue records from the four tide stations were first digitized, then interpolated at a sampling interval of one minute. Mathematically continuous series of cubic polynomial splines were calculated to approximate the digital record; a least squares error criterion was used to establish the best possible fit. This approximating curve was then subtracted from the tide record to yield a time series without semidiurnal and diurnal tidal fluctuations. This preliminary extraction step was required to prevent the tide waves from appearing in the frequency domain and blurring the tsunami spectrum. Figures 3 to 6 illustrate the digitized tide charts and the tsunami waveform following tide removal for the four ports being considered. A cosine bell taper was applied to the first and last ten percent of each time series prior to the implementation of a Fast Fourier Transform computer algorithm. Compared to a rectangular data window, the cosine bell has smaller side lobes in the spectral domain and as a result spectral estimates calculated following the application of the cosine bell are of greater accuracy. The Fourier coefficients produced by the Fourier Transform algorithm were used to compute the variance at each frequency and variance estimates were then band averaged to increase statistical stability. Finally, the variance was divided by bandwidth to produce the spectral density estimates, which were then plotted. The confidence limits of the spectral estimates approximately folow a Chi Square probability distribution, with the degrees of freedom dependent upon the number of variance estimates per band multiplied by the number of points involved in the cosine taper, divided by the total number of points in the time series. Plots of spectral density versus frequency have been prepared for Simonstown (Figure 7), Mosselhaai (Figure 8), Knysna (Figure 9) and Port Elizabeth (Figs. 10, 11). In each case the time series length was twenty hours (1200 minutes) and the maximum number of spectral estimates calculable for a time series of such length is 600. The frequency ranges from cycles/hour to the Nyquist frequency (30 cycles/hour) but spectral estimates above 6 cycles/hour were deemed insignificant and were truncated from each plot. The four plots from Port Elizabeth are presented to illustrate the dependence of the spectral density estimates on bandwidth; wider bands provide greater statistical accuracy but poorer frequency resolution. The following inferences are made from these spectra. Both Port Elizabeth and Simonstown exhibit a large peak at a period of about 145 minutes, and in fact most of their energy is consumed in this band. In contrast, the spectra for Knysna and Mosselhaai do not have this peak and the tsunami energy seems well distributed across the frequency spectrum. 196 a ) 300 BANDS -to- rn z 111 u ' o- b) 120 BANDS r. 1 I3 °-: "e U 1 l ' l I I I ll I I I i'i I I I i ' i I I I i'i I I I I'l I I I I ' l I I I I'l I I I I'l I I I I I I I I I I I I 5 2 JO FREOUENCY ICYCIES/HR) I 1-| | i i | i i | " ii|i i ■ ■ | i i i i — | 1 1 r • lao 120 «0 J 5 30 23 20 13 PERIOD (MINI 6 1 I I I l I I I I | I I II | II I I) I I ll |l II l|) I II |ll I l|l I II | | 2 3 FREQUENCY ICYCIES/hRI I Tl I'M 1 • 110 120 40 13 l""l M " I ' ' ' 30 2 5 20 PERIOD (MINI FIGURE 7 SPECTRAL ANALYSIS OF TSUNAMIGRAM FOR SIMONSTOWN 1200 MAY 11 TO 800 MAY 12. 1981 a ) 300 BANDS I II o -. ~E Z »o VI Z < i Z '0-. b ) 120 BANDS „ z '2 0-; ~E i i | i i'i i t i r i i | i r 00 10 20 30 40 FREOUENCY (CYCIES/HR) i — n I'M' ' — r ™ I ' ' ' ' I ' ' I., ' ,,!, 1 , ,!,, ! ,,[ » H0 120 60 13 -■ — r- 30 23 20 13 PERIOD (MINI lll|lll l|llll|l i — n i"i' " 180 120 60 13 2 3 FREOUENCY ICYCIES/HRI -* — r lllllll l lll l""l ' ' ' ' I ' 30 23 20 PERIOD (MINI I 5 FIGURE 8 SPECTRAL ANALYSIS OF TSUNAMIGRAM FOR MOSSELBAAI 1400 MAY 11 TO 1200 MAY 12, 1981 197 □ ) 300 BANDS r, 1 ' 2 ° o o -p 44 b) 120 BANDS p. 1 ' 2 ° "e i i I i i i i | i A i | i i'i i | i i' i i | i i i i | i i i i | i i i i | i I i i I ' I ' ' | ' i ' ' I ' ' ' ' | 00 10 JO JO 4 JO 60 FREQUENCY ICYCIES/HRI | 1— ) |i ■ | i — I | 'i n | l l ■ ■ | i ■ ■ 1 1 1 ■ ■ l | • 110 120 60 45 JO 23 30 13 '0 PERIOD iMINl i — n i"i ' • 190 130 60 4 J I I I | I II I 1 I I I I | I I I I | I I M | 3 JO 4 FREQUENCY ICYCIES/HRI t i — | i i I I I""! ''"'I JO 33 30 PERIOD IMINl "1 i o FIGURE 9 SPECTRAL ANALYSIS OF TSUNAMIGRAM FOR KNYSNA 1250 MAY II TO 2250 MAY 11. 1981 o ) 200 BANDS ~E u 10- b) 120 BANDS X 13 "e i — n I' M 1 « ia0 120 60 4 5 I I I I I I I I i'i I I I i ' i I I I i'i I | I I I I | I I I T I I I I I I I I I I ] 30 3 40 SO 60 FREQUENCY (CYCIES/HRI I ' ' ' ' 1 ' ' ' ' 1 l""l' 30 33 PERIOD I M IN ) i — n l"l ■ HO 120 60 4 3 i ' i ) i ' i I |' l I I I I I'I I I | I I I I | '■■ ■ | 20 30 40 50 60 FREQUENCY ICYCIES/HRI "I l""l""l ' ' ' 30 23 3 PERIOD IMINl I FIGURE 10 SPECTRAL ANALYSIS OF TSUNAMIGRAM FOR PORT ELIZABETH 1600 MAY II TO 1200 MAY 12. 1981 198 a) 600 BANDS I !} o- E o — 10 0-_ _ »o-: Mi z ui < \J * 0-. t/1 20- b) 300 BANDS R I 12 0- ~E 'O z iu ° 60-3 \j *0-. 2 3 4 FREQUENCY (CYCIES/HR) i I i i | i i I i | i i i i i i i i i | I n I' M 1 ' l "" l ' ■ ' ' I ' ' ' ■> 1)0 120 00 4 3 30 23 20 PERIOD (MINI -| , 1 r- I 5 I 2 3 4 FREQUENCY (CYC LES/HR) n r'-'-T ■ -> — r rrr 7 I ' l' p 'l' l ' pi I'l 1 ' I ' » ' ' I ' ■ I ' I * 190 120 60 4 3 - | — i — l — l 1 1 1 1- 2 3 2 13 PERIOD (MINI ~1 I o FIGURE 11 SPECTRAL ANALYSIS OF TSUNAMIGRAM FOR PORT ELIZABETH 1600 MAY 11 TO 1200 MAY 12. 1981 199 The path of the tsunami from the probable epicenter to each of the four ports can not be established with certainty. However, it is reasonable to suppose that the first waves to reach Simonstown passed off the continental shelf and travelled more rapidly through deep water. Conversely, the likely course of the first waves to Mosselbaai and Knysna would be across the top of the continental shelf; the path to Port Elizabeth was apparently across the tip of the Agulhas shelf and then into deep water. In the case of Mosselbaai and Knysna, interaction between tsunami waves and the shallow sea floor of the shelf caused a dispersion of wave energy across a wide spectrum, whereas the tsunami arriving at Simonstown was nearly unaffected by the sea floor. In Port Elizabeth, some energy dispersion did occur, but the characteristic 145 minute period is still evident. The spectral plots demonstrate the differences in the amplitudes of the tsunami waves arriving in each port; Port Elizabeth and Mosselbaai show more activity than do Simonstown and Knysna. For this there are several possible reasons: 1) Tsunami energy at the source may not have been distributed equally in all directions; depending on the impulsive nature of the submarine slide, and the position and geometry of the moving material, and the direction of the slide, the waves may have been generated asymmetrically. 2) The effects of submarine topography on tsunami propagation may have accounted for differences in wave amplitudes. For instance, wave propagation between the epicenter and Simonstown followed almost a direct course through deep water, yet tsunami energy was smaller than that recorded in Port Elizabeth, where the wave travelled much greater distances on top of the continental shelf. Submarine topographical features may have channeled energy towards Port Elizabeth, or, the shelf edge may have reflected more energy back to the ocean at Simonstown than at Port Elizabeth. 3) Coastal topography has had a definite influence on tsunami amplitudes because the geographical configuration of each port determines the resonant frequencies of that port. Knysna is located on a relatively straight section of coastline, and therefore it is not as susceptible to resonance oscillations as are Mosselbaai and Port Elizabeth. This may partially account for the low tsunami energy evident in Knysna 's frquency spectrum. Simonstown is located on the bank of False Bay, which is about 31 km in width and has a mean depth of about 45 m. In contrast to other ports, there is not the apparently indiscriminant shifting of energy to higher frequencies in Simonstown, rather there is apparent energy shift into discrete pockets at about 2.7 and 4.6 cycles/hour, which may be the second and third harmonic frequencies of the bay. The basic resonance period across the bay is about 1.5 cycles per hour. 4) It is possible that tsunami waves arriving at a single port from two different directions may interact to generate a new waveform. In the Knysna tide record, the initial three hours of intense tsunami activity are followed by four hours of waves of lesser intensity, and then another few hours of greater excitation. This oscillation may represent a 'beat' phenomonen ; perhaps one train of waves travelled directly from the epicenter 200 across the continental shelf to Knysna, while another train passed off the shelf at some point, and travelled on a curved course to Knysna through a stretch of deep water then hack up onto the shelf. Both wave trains would have had distinct characteristic frequencies since they travelled different courses; if these frequencies were slightly different, then beats would have occurred and there would have been a cyclic pattern of constructive versus destructive interference. The frequency spectrum from Knysna shows predominant peaks at .7 and .9 cycles/hour and therefore wave interaction causing beats is a possibility. 5) Seiche activity before the arrival of the tsunami was more pronounced in Port Elizabeth and Mosselbaai. This may be a result of meteorological conditions (possible low atmospheric pressure or wind conditions) or it may represent the predisposition of these harbours to seiche and/or energy influx from the swells of the southern ocean. 5. CONCLUSIONS The tsunami -like waves recorded by tide gauges on the coast of South Africa on May 11, 1981, were neither preceded by an earthquake nor a meteorological disturbance. By assuming the source of these waves as a submarine slide, using inverse techniques, a possible source areas was determined. If indeed, this event were due to a submarine slide, it presents one of the few known cases on record when a tsunami from a submarine slide has been registered by several coastal tide gauges. 6. ACKNOWLEDGEMENTS We thank Commander C.F. Stokes of the South African Hydrographic Service, the Uruguay Hydrographic Service and Dr. G.W. Lennon, Flinders University of South Australia, for supplying us with tidal records from their respective countries, and supporting information. 7. REFERENCES Camfield, F.E. 1980. Tsunami Engineering, Special Report No. 6, February 1980, U.S. Army Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Virginia, U.S.A. 222 pages. International Tsunami Information Center. 1981. Tsunami Newsletter, Vol. XIV, No. 2: 11. Slingerland, R. and B. Voight. 1982. Evaluating hazard of landslide-induced water waves. J. Waterway, Port, Coastal and ocean Div., Proc. Am. Soc. Civ. Eng., 108: WW4, 504-512. 201 Summerhayes, C.P., B.D. Bornhold and R.W. Embley. 1979. Surficial slides and slumps on the continental slope and rise of South West Africa: a reconnaissance study, mar. Geo!. 31: 265-277. ! Wigen, S.O. 1981. Digitization of tsunamigrams. Proc. 1981 Int. Tsunami Symposium, 213-224 (in press). 202 RUNUP OF A TRANSIENT WAVE ON A SLOPING BEACH by K. Kajiura Earthquake Research Institute, University of Tokyo, JAPAN Experimental data of solitary type wave runup on a sloping beach published in the past by various investigators are reanalyzed and expressed by a single relation between R/A and a/s, where R is the runup height, A is the crest height of a solitary wave at the toe of the sloping bottom, s is the bottom slope, and a is a representative non-dimensional time scale of a solitary wave. This relation is somewhat analogous to the case of periodic wave runup but simpler because A and a are related in a solitary wave . The maximum of R/A is about 3 to 3.5 which occurs at a value of a/s of about 4 to 5 . Roughly speaking, the critical value of a/s separates the non-breaking and breaking wave regimes. For a/s larger than this critical value, the wave always breaks and R/A decreases with the increase -3/4 of a/s (proportional to (a/s) for a/s larger than about 10). This empirical relation betweeen R/A and a/s is compared with numerical solutions of the nonlinear shallow water wave equations (Euler equations). In particular, the effect of bottom friction, quadratic in velocity, upon the runup height is examined. Numerical integration is made by means of a simplified two-step Lax-Wendroff method and runup on dry bed is computed by extending grid points on the dry bed. The precise position of the wave front is computed at every time step by extrapolation. 203 -3 It is found that the friction coefficient of about 2 X 10 gives reasonable fit to the experimental data. In numerical solutions, R/A has slight dependency on the relative height A/H (H, the constant depth of a channel) but overall tendency of experimental runup with respect to a/s can be reproduced by ignoring this slight deviation. For a/s less than about 2, the effect of friction is minor. On the other hand, for a/s > 4, the frictional effect is very large because the solitary wave breaks before reaching the original shoreline and the thickness of the water layer running up the slope becomes very thin. Since most tsunamis are considered to be non-breaking, the effect of -3 friction would be minor if the friction coefficient is 2 x 10 . However, in realistic simulation of runup, it seems necessary to adopt a large friction coefficient of the order of 0.1 to explain the extent of observed inundation, because of the presence of various obstacles on land. 204 RUN-UP OF LONG WATER WAVES by G. Pedersen and B. Gjevik Department of Mechanics, University of Oslo P.O. Box 1053, Blindern, Oslo 3, NORWAY A numerical model based on a Lagrangian description has been developed for studying run-up of long water waves. The performance of the numerical scheme has been tested by comparing with analytical solutions and experimental data. Simulations of the run-up of solitary waves on relatively steep planes (inclination angle >15°) show surface displacements and run-up heights in good agreement with experiments . For waves with relatively large amplitude the simulations reveal the development of a breaking bore during the back-wash. Results of run-up heights in converging and diverging channels are also presented. 205 OPEN OCEAN SIGNATURE OF TSUNAMI ON THE SEA FLOOR: OBSERVATION AND USEFULNESS by J. H. Filloux Scripps Institution of Oceanography La Jolla, CA 92093 USA Recording of tsunamis over the open ocean would permit (1) investiga- tion of the shoaling transformation and associated amplitude magnification of tsunamis impinging on a coastline, (2) implementation of a tsunami warning system by rapid digital simulation of the propagation of a threatening tsunami observed over the open ocean at an early time, and (3) acquisition of indirect, though extremely valuable information on tsunami generation processes by retro step modeling of the open ocean waves back to the earthquake faulting area. However, tsunamis are not only rare events, but they are also unpredictable. For this reason the recording on the seafloor of the pressure signal of a small and unthreatening tsunami 1000 km from its originating area is of importance. The referred earthquake occurred on March 14, 1979 along the Pacific coast of Mexico, with a magnitude Ms = 7.6. The resulting wave was observed roughly 1.5 hours later in deep water off the Gulf of California with an initial amplitude of .5 cm. Five successive waves are clearly recorded, the first three with nearly the same period (~45 mn) , the next two considerably shorter. Thereafter the tsunami signal is overwhelmed by the background pressure noise of very long ocean surface waves and 207 internal waves (~.l cm rms in significant band). Tsunami travel time is consistent with a propagation velocity v = (gh) , h = oceanic depth along straight ray path. Shortly after the main earthquake shock a large pressure disturbance is recorded on the seafloor, resulting from the vertical acceleration of the oceanic water mass by the seismic perturbation. The seismic travel time is consistent with the velocity of propagation of the seismic mantle waves over oceanic areas while the amplitude of the pressure signal is consistent with the estimated earthquake moment and current theories of seismic energy attenuation with distance. The pressure signal on the seafloor of an earthquake with tsunamigenic capabilities is distinctive enough to constitute a possible means to initiate the recording of an incoming tsunami. The use of such a triggering scheme would greatly reduce the data logging capacity required from a long endurance tsunami recorder. 208 LONG WAVE OBSERVATIONS NEAR THE GALAPAGOS ISLANDS by E. N. Bernard, H. 0. Mofjeld, H. B. Milburn, and E. G. Wood Pacific Marine Environmental Laboratory Seattle, Washington 98105 USA Two pressure gauges were deployed from April 1982 to April 1983 at (2.5°S, 95.0°W) in 3571 m water depth and at (0.5°S, 89.5°W) in 20 m water depth to ascertain the background level of energy of waves over a broad frequency range. Descriptions of the instruments, method of data collection, and relative accuracy are presented. Analysis of data reveals the wave "climate" of this region of the Pacific Ocean. Observations of the tsunami generated by the December 19, 1982 earthquake south of Tonga Islands (24°S, 176°W) are contained in this data set. 209 DIFFRACTED LONG WAVES ALONG CONTINENTAL SHELF EDGES T.S. MURTY INSTITUTE OF OCEAN SCIENCES DEPARTMENT OF FISHERIES AND OCEANS SIDNEY, B.C. , CANADA AND H.G. LOOMIS DEPARTMENT OF OCEAN ENGINEERING UNIVERSITY OF HAWAII HONOLULU, HAWAII, U.S.A. ABSTRACT The Lateral wave at a depth discontinuity in the ocean, in the context of tsunami propagation has been studied by King and LeBlond (1982) making use of geometrical optics techniques as well as laboratory experiments. This paper supplements the above work in three ways: (i) instead of ray techniques, a numerical finite-difference model for the hydrodynamic equations was used, (ii) in addition to a depth discon- tinuity between a shelf of uniform depth and the deep ocean, a sloping shelf situation also is considered, (iii) some attempts were made to identify the Lateral waves with tsunami forerunners. 211 1. INTRODUCTION King and LeBlond (1982) introduced the term "Lateral wave" to describe a wave that arises at a depth discontinuity in the ocean such as at the boundary between a continental shelf and the deep ocean. They pointed out that the identification of a Lateral wave in coastal tsunami records will help in tsunami warning due to the fact that, at sufficiently great distances from the tsunami source, the Lateral wave arrives faster than the direct wave. Figure 1 shows schematically a continental shelf sloping to a deep ocean. The blackened area represents the region of tsunami generation due to a submarine quake on the shelf. The direct wave travels entirely in shallow water, whereas part of the energy from the source is radiated into the deep water. From the deep water wave travelling along the edge of the shelf (on the deep water side) energy is continuously diffracted back onto the shelf and the coast, and this wave motion constitutes the Lateral wave. The path taken by the direct wave is entirely in shallow water, whereas for the Lateral wave, a substantial part of the travel occurs in deep water. Since for long waves the speed of travel is proportional to the square root of water depth (ignoring other effects) the Lateral wave can arrive at a given coastal location even earlier than the direct wave. The difference between the arrival times at a coastal location between the Lateral wave and the direct wave increases with the distance of the observation point from the tsunami source. The present study supplements the work of King and LeBlond (1982) in three ways. Firstly, instead of using ray techniques, in this study the 212 REFLECTING m m o o I *» — Isource SHELF hi ?//////// .slope 7 /////// n DEEP OCEAN h 2 i TRANSPARENT 33 > c/> > m Figure 1. Schematic representation of a continental shelf, slope and deep ocean. 213 Lateral wave is simulated numerically using hydro-dynamic equations. Secondly, the earlier case of a shelf of uniform depth meeting the deep ocean through a depth discontinuity is extended for the case of a sloping shelf. Thirdly, some attempts were made to identify the Lateral wave with tsunami forerunners. 2. THEORETICAL CONSIDERATIONS We will briefly summarize the relevant points from the work of King and LeBlond that are useful in developing the numerical model and inter- preting the results. Let hi and ti2 respectively be the depths of the shelf and the deep ocean. Assume a depth discontinuity between h\ and ti2 , and also that the coastline acts as a completely absorbing boundary. From the tsunami source, a circular wavefront travels with a speed Cj= Wghi. At the depth discontinuity, the incident waves are partly reflected and partly refracted into the deep ocean. With reference to the top part of Figure 2, the reflected waves propagate on the shelf with speed C^ and one can imagine an image source for them across the discontinuity. The refracted rays get deflected from their original paths according to Snell's law sin 62 /h2 (-jn sin B\ ^f hj Since h2 > hi , total reflection occurs provided 81 > 9 (2) 1 - c e P = sin 1 / 111 Vh 2 214 Deep Water Shelf •»- x ^& * P^y) Figure 2. (Top) Wavefronts propagating from an impulsive symmetric source situated at S on a shallow shelf adjacent to a deep ocean. The direct wavefront (I) is centred on S; the reflected wave (II) seems to emanate from an image source at S'. A deep-water refracted wavefront (IV) and the lateral wavefront (III) complete the picture. Only selected rays are shown; one subcritical direct ray (0 X < ) and its refracted continuation (0 2 < VO are represented by the dot-dash line. The solid-line rays all contribute to the lateral wave. (Bottom) Travel paths to a remote observation point P from the source S. The direct wave travels a distance R in a straight line between the two points. The path of the Lateral wave consists of the three segments L , L 2 and L x . The path of the specularly reflected ray is not shown. (From King and LeBlond, 1982.) 215 If the angle of incidence is greater than the critical angle , no energy can enter the deep ocean from the source, according to geometric optics. However, a more rigorous analysis reveals that, for 0i > C a wave (whose amplitude decays exponentially away from the discontinuity) travels along the depth discontinuity on the deep water side with speed C2 = \l g^2- With reference to the top part of Figure 2 , at a point A along the depth discontinuity for which 0i > C , wave energy first arrives with speed C2 in the form of a critically refracted wave propagating along the boundary (wave front IV) before any direct wave travelling at speed Ci (this direct wave arrives at point B, when the refracted wave arrives at A) . For all points between B and A along the depth discontinuity, the wave disturbance first arrives from the deep water side and this can act as a source of radiated energy for the shelf; and this energy is the source of the Lateral wave. Because the Lateral wave travels with speed C\ on the shelf, its wavefront (denoted by III in the top part of Figure 2) must have the angle C with the discontinuity to be able to keep up with the deep ocean wave. King and LeBlond (1982) summarize thus; waves arising from the original impulse include a direct (I) and a reflected (II) wave spreading radially on the shelf at speed C\ from centers S and S 1 respectively, a deep ocean wave (IV) travelling at speed C2 , and a Lateral wave (III) diffracted back onto the shelf by the deep ocean wave, but travelling at speed Cj at an angle to the discontinuity. With reference to the bottom part of Figure 2 , at a point P on the shelf, the direct wave arrives at time 216 where R = J (x 2 - xi) 2 + y 2 (5) The Lateral wave arrives at ? at time T L = L ° + Ll + ^ (6) Ci C 2 The direct wave and Lateral wave arrive at the same time at a distance R c given by (xi + x 2 )(Ci - Ci) 2 (y) R = c C 2 - Ci sin

R c it is the Lateral wave that arrives first. 3. THE NUMERICAL MODEL To keep the mathematical problem simple, linearized shallow water equations are used. With reference to a Cartesian coordinate system, the equations of motion and continuity are at " " 8 ax - (8) 9v . m (9) 3t g 3y K } ft - - i (hu > - i (hv) (10> where x, y are horizontal coordinates, t is time, g is gravity, h (x,y) is the water depth, n is the deviation of the water level from its equilibrium position, and U and V respectively are the x and y components of the water transport. Here the coriolis forces were ignored because the period of the tsunami waves we are interested in are, at most, of the order of several 217 minutes and are small compared to the period of earth's rotation (24 hours). We have also ignored bottom friction, mainly because in this preliminary study we are not so much interested in the dissipation of the Lateral wave as in proving its existence through numerical simulation. However it is a relatively easy matter to include coriolis forces and bottom friction. A left-right symmetry was assumed at the coastline with a symmetric wave source. The following boundary conditions are used. At the reflecting boundaries, U = at the shoreline and V = at the axis of symmetry. The transparent boundaries (Figure 1) are accomplished by extrapolating the wave motion to points just outside of the boundary and using the values at these points in the finite difference equations. Tests showed that this procedure was quite satisfactory. The dependent variables, U, V, n are prescribed on a staggered grid in space and are computed at alternate time steps (i.e.- U, V at even time steps and ri at odd steps) using standard techniques (e.g. Henry, 1982). The calculations are performed for the four topographic configurations shown in Figure 3. Figures 4, 5 and 6 respectively show the results of the computations corresponding to the topographies shown in Figures 3a, 3b and 3d. In each of these diagrams, the abcissa is the time in minutes that elapsed after the impulsive generation of the tsunami. The computed water levels are shown as a function of time at various distances from the source. Along the ordinate the location marked by zero is the source of the tsunami. The disturbance at the source as a function of time is also shown. The scale for the amplitude of the disturbance at the source is also shown. Although the 5m amplitude for the disturbance is somewhat higher than is likely, since the mathematical problem considered here is linear, results 218 0_ UJ o cr UJ I 100 160 240 180 1000 5000 - 1000 DISTANCE FROM SHORE KM ► Figure 3. Various topographies used in the computations. a) island chain, b) continental shelf, c) Japan trench, d) idealized shelf. 219 for different initial amplitudes can easily be deduced. In Figures 4, 5 and 6 the crests identified by Vj is the Lateral wave. V2 is the first significant crest of the tsunami and V3 is the crest of the wave of maximum amplitude (energy) . In the Figure 5 V3 is off the area shown in the diagram. 4. DISCUSSION OF RESULTS Figure 4 shows the results of the computations for the island chain. In this case, the slope is more important than the shelf in the resulting wave motion. The average speed of the lateral works out to be 687 kph whereas the average speed of travel of the first significant crest is about 467 kph. The average velocity of the cluster of trapped waves (which appears to have the maximum energy) is about 570 kph. Note that, in practical tsunami prediction, the travel-time curves correspond to the arrival of the first significant tsunami wave (identified by V2) and the wave with the maximum energy (identified by V3) is referred to as the highest wave. Figure 5 shows the results for the case of a continental shelf (topog- raphy shown in Figure 3b) and Figure 6 shows the results for the idealized shelf (topography shown in Figure 3d) . In the case corresponding to Figure 3b, the shelf is more important than the slope in determining the wave motion. Table 1 summarizes the values of Vj , V2 , V3 for the cases shown in Figures 4, 5, 6. 220 INITIAL AMPLITUDE = 5m j i i i j i i i i i i i 20 40 60 80 MINUTES Figure 4. Results for the island chain. The meaning of V ly V 2 , V 3 is explained in the text. 221 60 90 MINUTES Figure 5. Results for the continental shelf case. 222 Figure 6. Results for the idealized shelf case. 223 TABLE 1 Values of the speeds (kph) of different waves identified in Figures 4, 5, 6, Vj: Lateral wave V2: First significant wave of the tsunami V3: Tsunami wave with the maximum energy Topography as shown in Figure Vl V 2 v 3 3a 3 b 3d 687 634 779 467 187 356 570 356 Since we have not included bottom friction we cannot comment on the dissipation of the Lateral wave as it progresses. King and LeBlond mention that the amplitude of the Lateral wave decays as -3/2 n L R (see Figure 2 for the meaning of these symbols) , n L -R •3/2 (11) By comparison, the amplitude of the direct wave decays as -k n D -R (12) Thus, in spite of its earlier arrival, the amplitude of the Lateral wave is considerably smaller than that of the direct wave. 224 Figure 7 shows tsunami forerunners in two cases. Although it is tempting to identify these as Lateral waves, until a proper numerical model is used with the actual topography, such an identification cannot be done. 5. CONCLUSIONS The so-called Lateral wave which arises mainly from diffraction of wave energy at a continental shelf edge has been simulated through a numerical model for different topographies. Actual topographies with realistic coastlines have to be used in the numerical model before identifying the observed tsunami forerunners with Lateral waves. It appears that Lateral waves might prove to be useful in practical tsunami prediction. 225 J I I I I I I MAY 23-24, I960 J i i I L J L 1 J I J I I L 12 14 16 18 20 22 APPROX. HOURS G.M.T. 10 MAY 23-24, I960 l i I L 12 14 16 18 20 APPROX. HOURS G.M.T. 22 Figure 7 . a) bj Water level record at Tofino, B.C., Canada associated with the Chilean earthquake of May 1960. Arrow shows a forerunner. Water level record at Crescent City, California, U.S.A., for the same tsunami. 226 ACKNOWLEDGEMENTS We thank Mrs. Dorothy Wonnacott for typing the manuscript and Ms. Coralie Wallace for drafting the diagrams. REFERENCES Henry, R.F. (1982). Automated programming of explicit shallow water models: part 1: linearized models with linear or quadratic friction, Canadian Technical Report of Hydrography and Ocean Sciences, No. 3, Institute of Ocean Sciences, Department of Fisheries and Oceans, Sidney, B.C., Canada, 70 pages. King, D.R. and P.H. LeBlond (1982). The Lateral wave at a depth dis- continuity in the ocean and its relevance to tsunami propagation, Journal of Fluid Mechanics, Vol. 117, 269-282. 227 : NUMERICAL SIMULATION OF A TSUNAMI ON A TRIANGULAR MESH Harold G. Loomis Department of Ocean Engineering University of Hawaii Honolulu, Hawaii 96822 ABSTRACT A triangular mesh for numerical time-stepping of the long wave equations has some advantages in studying tsunami wave propagation. The mesh can be refined easily in shallower water so that for each triangle, area/(g*depth) is approximately constant. This allows for approximately the same number of mesh points in each wave length thereby making effi- cient use of computer storage and computing time. Also, the triangular mesh can follow a coastline more exactly than a rectangular mesh. Formulas are written for the partial derivative of u,vand n in terms of the values at neighboring points and these are used in the set of three first order, linearized hydrodynamic equations. Initially, this scheme was tested on simple geometric region to test the stability and fidelity of the scheme as an approximation to the con- tinuous fluid motions. Then the scheme was applied to idealized island bathymetries that are imbedded in and ocean of constant depth. These can be compared to analytical solutions for steady state cases with sinusoidal waves. The most troublesome problem, in which we have had some success is in matching the numerical boundary to an ocean of constant depth, treated analytically, so that the radiated portion of the solution passes the boundary without reflection. Some success has been gained on this problem. 1 . Introduction The linear, shallow water, long wave equations that we solve are u t - - g d h x . v t = - g D H y , (l ) H t " " U x - V y > where U and V are the integrated vertical velocities, H is the water level measured from the undisturbed level and D is the depth of the water D is variable. 229 « introduced by Thacker (1977, 1978). The The method of solute was in tr odu ce deriy . mai n idea is that a triangular gn - £ ^ ^ from the ativ es are calculated at any grid point by ^ ^ triangular ;r:;,rrr,rr ::;:.-— -- Wake Island Grid North Fig. 1 ;scr ffiK&1 sr- 230 Figure 2 demonstrates the approximations to H and H , x y Fig. 2 Organization of Triangular Grid About Point j. The neighboring points of point j are temporarily labeled j,, j 2 , — , j 5 and the neighboring triangles have areas A-,, A 2 , — , Ar. The sum of the areas is taken to be A. Then the partial derivative at.j is approximated by < H x>j = I (\h ' V A (2) , th where (H ). is the value of H on the i triangle and i runs through the indices of the neighbors of j in a counterclockwise direction. The bar over (H ). indicates an average. Equation (2) turns out to be relatively simple in terms of the coordinates of the surrounding points and the neighboring values of H. Namely, < H x>j = H*i + i-*i-i> V< 2A > (3) Similarly (" y )j " I ( x i+ i " Vi) H i /(2A) (4) 231 and A is found conveniently by 2A = I(y 1+l -y^Jx, ■ (5) The time derivatives are leap-frogged. With H°, U , V given, we calculate, in order, U 1/2 , V 1/2 , H 1 , U 3/2 , V 3/2 , H 2 , etc., by jn+ l/2 u ^-1/2 _ D At (H )n v n+l/2 = v n-l/2 . g D A t (H ) n (6) H n+1 = H n - At (U x ) - At (V y ) . 1/2 1/2 In the first calculation of U ' , V ' , a half time step is used. In order to take a time step, we need to know for each point, whether it is an interior point or a point on a reflecting or transparent boundary. We need to know how many neighbors it has and the indices of the neighbors in a counterclockwise direction. We need to know the c's and d's that are used to effect the partial derivatives. We need the bathymetry at that point. In order to plot the output and contour it, we need to know the x,y coordinates of each point and we have to know how the points are arranged into triangles. In spite of the complexity of this, it is really very simple and it is suited to large problems. The dynamic variables U, V and H are kept in core for the entire region. The constants can be stored in blocks. They are read in and the dynamic variables in that block updated and then the same is done for the next block and so forth. Even when there is plenty of core, it is possible to arrange the paging so that the computa- tion is fast. Figure 3 shows a typical organization of blocks. We can stretch and shrink the mesh so that the mesh is optimized for the bathymetry, i.e., so (area of triangle/gD) is approximately uniform. The mesh can follow curved boundaries easily as is shown in Fig. 4. 232 region 2 Fig. 3 Organization of Computing by Blocks Fig. 4 Mesh Points Can be Slid Over to be Exactly on a Boundary Contouring is particularly simple. We go through all of the triangles drawing the appropriate contours in each triangle and when finished the contours will be connected. This is shown in Fig. 5 where the values at the vertices are H, , H~ and H~. If a value H. lies between H, and H 2 ,'then a point on the side is located by linear interpolation. Then it must be so that H. lies between either H, and H 3 or hL and hL and a second point is located and a line segment is drawn between them. Similarly for all other values that are to be contoured. This work is interesting because a whole new set of programming con- siderations comes into play. For example, it is possible to write computer programs to do most of the operations automatically, such as - select grid, interpolate bathymetry - optimize grid for uniform area/gD - organize points into triangles 233 Fig. 5 Diagram for Contouring the Field H - identify neighbors of points - classify points as to interior, reflecting boundary, transparent boundary - find direction cosines of normal to boundary - contour any field 2. Calculations A set of calculations was done with reflecting boundaries to test the stability and accuracy of this method. We took a closed circular basin and an initial displacement to be a normal mode of the basin. The equations were time-stepped for several hundred time steps and the periods were calculated from the number of time steps necessary for an integer number of complete cycles. There was agreement to within a few percent after several cycles. The actual accuracy depended on the wave lengths of the normal modes. When wave lengths were short the accuracy was worse. Also, the total energy in the basin was relatively constant. These tests indicated stability and reasonable agreement between numerical celerity and theoretical celerity. A second set of tests was done with a transparent boundary. In the case mentioned above, the boundary was changed to transparent and it was observed that the outward wave passed through with little (but observable) reflection. 234 Then another problem was done with an island was sitting on a parabolic mound as shown in Fig. 6. The mesh shown is optimal in that the travel time along each leg of a triangle is approximately the same, A mound of water is released on the flank of the island and the sub- sequent wave records are recorded at each shoreline point. The outer boundary is (hopefully) transparent. The subsequent wave records are shown in Fig. 7. Fig. 6 Tsunami Generated on the Flank of an Island with Parabolic Bathymetry. Shaded Portion is 10 m. Uplift 235 Water Level At Source Water Level At 90° From Source TIME Water Level At 180° From Source Fig. 7 Wave Records at the Shoreline of Parabolic Island Due to 10 m. Displacement on the Flank In this case, some of the energy is trapped on the flanks of the island and goes round and round as is seen from Fig. 7. The result looked reasonable but no quantitative comparison could be made because an analytical solution to this problem is not known. For the above reason it was decided to solve the above problem for a wave incident from the deep ocean. This is a solved problem. At this point we discovered that our boundary conditions were not good enough. 236 It is necessary at the boundary to separate the incident and the radiated wave and to carry the radiated wave outward by extrapolation. About a year's time was consumed in this effort and finally it was suc- cessful. Basically we calculated, by formula, the incident wave on the outer two boundary rows. Equations (3) were used to calculate the total wave height at all inner grid points. The reflected wave heights at the inner boundary points (one row inside the outer boundary) were found by subtracting the incident wave from the calculated wave. This was then propagated directly outwards to the outer row and added to the incident wave height then for a total wave height to be used in the next time step for the inner points. It took a great deal of thinking to get the time and spacial interpolations and extrapolations correct. The final results are shown in Fig. 8 where the numerical solution is compared with the theoretical solutions by Homma (1950). pa LU Q PARABOLICAL ISLAND 16 -MINUTE INCIDENT PLANE HRVE RNRLTTIC SOLUTION A NUMERICAL MODEL ^.00 20.00 40.00 60.00 80.00 100 00 AZIMUTH 120.00 140.00 160.00 180.00 Fig. 8 Test of Boundary Conditions with Steady State 16 min Incident Wave. Wave Amplitude is Plotted Around the Island. 180° is Bow End of Island 237 3. Acknowledgements This work was supported by a grant from the Pacific Marine Environmental Laboratory, National Oceanographic and Atmospheric Administration. Ramon Cabrera was wery skillful in programming the last and hardest portion of the work and the staff of the Center for Engineering Research was helpful in producing the finished paper. 4. References Thacker, W.C. (1977): Irregular grid finite-difference techniques: simulations of oscillations in shallow water circular basin, J. Phys. Oceanography, Vol. 7, No. 2, pp. 284-292. Thacker, W.C. (1978): Comparison of finite-element and finite difference schemes. Part I: One-dimensional gravity wave motion; Part II: Two-dimensional gravity wave motion, J. Phys. Ocean, Vol. 8, No. 4, pp. 676-689. Homma, S. (1950): On the behavior of seismic sea waves around circular island, Geophysical magazine, Tokyo Central Observatory. 238 PROPAGATION OF TSUNAMI OVER THREE-DIMENSIONAL SHELVES by Theodore Y. Wu California Institute of Technology Pasadena, California, USA Helene Schember TRW Defense and Space Systems Group Redondo Beach, California, USA A slowly modulating nonlinear long-wave model of the Boussinesq class is applied using a finite difference numerical scheme to investigate wave focusing, reflection and transmission, converging and diverging of tsunami waves during their propagation over shelves of typical three-dimensional topography. These results are exploited to examine the range of validity of some approximations such as linear, nondispersive long-wave model and geometric wave theory. This study has been supported jointly by NSF Grants PFR-77-16085 , MEA-8118429, and ONR Contract N00014-82-K-0443 . 239 Energy of the Tsunami Converging onto an Island Hisashi MIYOSHI 3-3-33 Minami-Oizumi, Nerima-ku, Tokyo, 177, Japan Munk et al (1947) drew two curves, one of which was computed according to the solitary wave theory, and another of which was computed according to Airy's theory. The latter gave better agreement with observations at various positions along the beach adjacent to Scripps Institution. This result can be utilized when we study the tsunami approaching an island. Absorption of the tsunami energy by the insular slope and shelf enhances the deceptive sharpness of directivity of tsunami, which can be studied by means of calculus of variation ( in accordance with Fermat's principle ) with the predominance of wave refraction suggested by fiunk in mind. 1 . Introduction The tsunami of April 1, 1946 looked to have a sharp directivity. Only the Hawaiian Islands were hit badly, and, for example, Kidway was safe from this tsunami. The more remarkable fact is that the islands lying along the extended part of the line from the origin of this tsunami to the Hawaiian Islands, were safe from this tsunami. This suggests that the sharp directivity of this tsunami is only deceptive. And the tsunami of May 24, 1960 from Chile, attacked from the whole coasts of Japan including Okinawa, to Attu Island in the Aleutian Islands, which suggested that the directivity of this tsunami was blunt. On the other hand, the analyses of the directivity both from the viewpoint of the refraction of tsunami and from that of the elliptic shape of the origin, suggest that the directivity of tsunami is rather blunt. We considered that the deceptively sharp directivity of the tsunami of April 1, 1946 was due to the absorbing effect of the broad insular slope and shelf around the Hawaiian Islands, which consisted of basalt. This is the background of this study. 2. Analyses of the refraction of tsunami The curves shown in Figure 1 are considerably famous in oceanography. But they are apt to be unrecognized from the viewpoint of tsunami research. 241 80 160 140 £3 120 en w 100 > "S- b0 4 ?0 AIRY 1 WAVE 1 THEC IT )RY \ K '", 7 6 O iVEBAG £ o y. ° O c 1 LEGEND O OBSERVATIONS < 1 FACTION DIAGRAM WNW I3 ! SOL TARY WAVE T HE0RY r c > f s ° 1 s o AVERAI ;e o y_ o ■v^O^, o SOU! ORTH 4000 3000 DISTANCE FROM 2000 1000 1000 SCRIPPS PIER IN FEET Figure 1 . The circles indicate observed wave heights at various positions along the beach adjacent to Scripps Institution pier. The solid lines give computed changes in wave height for a special wave train. The upper curve is computed according to Airy's theory, and the lower, to the solitary wave theory. 242 These curves were introduced twice. At first, Munk and Traylor published them in 1947, when they measured the wave heights along the coast around Scripps Institution. on the abscissa means the spot of famous pier. The bay off this institution is one of those, the depth distributions of which 3cf6 most well known in the world. They could easily draw a refraction diagram in the case, for example, when a special wave train from the WNW of 13 second period comes, and estimate the energy distribution along this coast. At these estimations, the upper curve appears when the Airy wave theory is assumed, and the lower curve, the solitary wave theory. The lower curve gives better agreement with observations than the upper curve. They concluded, therefore, every wave crest near the coast is a solitary one. Then, they refered very briefly to the fact that the predominance of wave refraction existed as compared with diffraction etc. And as the wave period is 13 second, its wave length on the deep sea is given by which means, so to speak, it is a tsunami of a small size near the coast. And corresponding to the existence of the trench in the case of tsunami, there exists the sharp submarine canyon off Scripps Institution in the case of this wave train. We can, therefore, utilize this fact of the predominance of wave refraction, when we discuss the energy covergence of the tsunami approaching an island, by means of calculus of variation ( in accordance with Fermat's principle ). When the circle shown in Figure 2 represents the cross section of the earth, AQB means the path of the seismic wave, which is well known. The wave goes into the earth the more deeply, its velocity becomes the larger. And when the same circle r epr esents the bird's-eye view of the outer edge of insular slope, APB is the expected path of the tsunami wave, which will be studied here, the point P being expressed as that at which dT / cLB =0. Let us assume that the depth distribution around a round island is given by (2) Here (h is the radius of the island, b is the radius of the outer edge of the insular slope, J-/ (= 4,0C0m ) is the depth of the deep-sea floor, and T is the horizontal distance from the center of the island. And when T>b > -&( Y* )= H . The ray of the tsunami will draw a curve, along which r ,,___ ^ v ^r- w (3) fPTr 5 yj will take its minimum value. According to the theory of calculus of variation, and putting 243 Figure 2. The circle shown in this figure has two meanings. When this circle is the cross section of the earth, A"Q]5 means the path of the seismic wave. And when this circle is the outer edge of the insular slope, a"p~5 is the expected path of the tsunami wave, which is looked for in this study. 244 V r *+ r '_ . = J>(r,r') (4) we get the differential equation fC^.l^r'+c . (5) by which we can draw the curve of ray. It becomes Q is given by c= J^m£ . ( 7) Finally, we get T a bsino( (8) Putting Y* / = 0, when V" = Y^, , we get To bs£*dL (I); when 7^=2, (9) becomes To - a- fa - 0/ ' which becomes \ i? does not, therefore, exist. (II); when IV =1, (9) becomes To \)SWick Since , rf, / ^ \ Yi-£4, (9) ^l ] [r^TJ 2(t t -&y/ z ' do) (11) the left side of equation (10) decreases from To = &"f" to 7^=2^/ , and increases from T = 2&. And it takes the value of i ' 245 when 7" = b . when (j<2fl, , oi which satisfies equation (10), does not exist. And when b-2& , such o( exists, which will he reported in detail in the following paper. The cross sections of three important islands have been completed and can he presented here. These cross sections are instructive. Figure 3. The vertical cross section and the map of Oahu Island. The cross section is drawn on a scale of 1 to 2,000,000 horizontally, and of 1 to 200,000 vertically. The map is drawn on a scale of 1 to 7,000,000. In the map, the direction in which tsunami came and the line along which the cross section was made, are shown by arrow and chain line, respectively. Figure 4. The vertical cross section and the map of Midway. The cross section is drawn on the same scales as those in Figure 3. The map is drawn on a scale of 1 to 360,000. In the map, arrow and chain line are the same as those in Figure 3. 246 Figure 5. The vertical cross section and the map of Hachijo Island. The cross section is drawn on the same scales as those in Figure 3. The map is drawn on a scale of 1 to 100,000. In the map, the meaning of chain line is the same as that in Figure 3. The tsunami of 1605 came approximately perpendicularly to this line. 3. Conclusions Taking further recognition of another consequence of the famous figure drawn by Munk et al (1947) , the predominance of wave refraction as compared with diffraction, friction and reflection was observed. It was applied to the case of tsunami approaching an island. After some trials and errors ( for example, "Energy of the Tsunami which Converges into an Island" ( H. MIY0SHI, in Japanese, 1978 )), the fact that the angle cl shown in Figure 2 existed, was confirmed. A quantitative discussion will be tried in the next paper. The cross sections of three islands were completed, which are instructive . REFERENCES HATORI, T. , Rekishi tsunami, 1-125, 1977 (in Japanese). LOOMIS, H. G. , Tsunami wave runup heights in Hawaii, NOAA-JTRE-161, HIG-76-5, 1-95, 1976. MIYOSHI, H. , Directivity and efficiency of tsunamis, NOAA-JTRE-190, HIG-77-4, 1-32, 1977. MUNK, W. H. and M. A. TRAYLOR, Refraction of ocean waves: A process linking underwater topography to beach erosion, J. Geol., 15, 1-26, 1947. MUNK, W. H. , Solitary wave theory, Annal. New York Acad. Sci., 51, 376-424, 1949. SHEPARD, F. P., G. A. MACDONALD and D. C. COX, The tsunami of April 1, 1946, Bull. Scripps Inst. Oceanogr., Univ. Calif., 5, 391-528, 1950. 247 NUMERICAL SIMULATION OF THE 1975 SHIKOTAN TSUNAMI by A. S. Alekseev, V. K. Gusiakov Computing Center, Novosibirsk, USSR L. B. Chubarov, Yu. I. Shokin Institute of Pure and Applied Mechanics Novosibirsk, USSR The submarine earthquake of June 10, 1975 which happened near the Shikotan Island caused appreciable tsunami in the southern Kurile region and the northern part of Hokkaido. The main feature of this earthquake was the generation of relatively large tsunami waves of intensity 1=2 according to the Soloviev-Iida scale for a moderate source magnitude (M = 7.0). The paper presents the results of numerical simulation of this tsunami generation and propagation. The total scheme of the numerical experiment is divided into two stages. At the first stage, static bottom deformations in the epicentral region are computed for the dimensional dislocation model of a seismic source with the parameters obtained from seismological observations. At the second stage the computed bottom deformations are used as the initial conditions for the tsunami propagation problem. This problem is regarded within the framework of linear theory of shallow water and solved numerically by the finite-difference method on a rectangular 117 x 62 grid, whose mesh size is 4.76 km. This grid covers 550 x 300 km region in the part of the ocean under consideration. 249 • Several variants of seismic sources with somewhat different fault parameters are regarded. The total volume of the bottom displacements and the initial tsunami energy are evaluated. Theoretical mareograms are computed at a number of points along the coastline. The comparison of the computed mareograms for several sources shows that in spite of the difference in the form of the initial elevation, the oscillations of the water level near the coast have a similar form. Thus one can make a conclusion that the resonance characteristics of the bottom relief, even for open and half-open regions like those considered in this numerical experiment, essentially affect the wave form of the tsunami. At six points, where mareograph records of this tsunami were made, the mareograms computed are compared with those observed. It turns out that the model of the earthquake source inferred from seismological data (a subvertical rupture, branched upwards from the lithospheric interface) reproduces the main features of this tsunami including the time and the sign of arrivals, amplitudes and periods of several first waves. 250 Seiche on a Parabolic Sea s helf SHIGEHISA NAKAMURA Shirahama Oceanographic Observatory, DPRI Kyoto University Shirahama, Japan 649-23 ABSTRACT An analytical model was developed to study seiche on aparabolic sea shelf formed along a straight coastline facing a semi- infinite open ocean by using a lineal ized equations of motion. Oscillation on the shelf was assumed to he normal to the coastline and any propaga- tion of waves along the coast was not considered in the model. The profile of the paracolic sea shelf was assumed so as that the water depth was proportional to square of the distance from the coastline. With the assumed "boundary condition, analytical solutions were obtained to find the envelope of the oscillation normal to the coast. These solutions show that an exponential envelope for an oscillation of a frequency tu can be found if u> ga. 2 Especially, in case of a) = ga, the solution can he expressed by a sum of a constant and a term inversely proportional to the distance from the coastline. An application to an observed oscillations was discussed to learn what is the meaning of the critical condition. 251 INTRODUCTION A simple model of shelf- seiches on a parabolic profile of the sea bed off a coast facing an open ocean is developed to have a dynamical understanding of resonant sea-level variations. Such a resonant oscillation could he induced by any external disturbances, for example, local variations of atmospheric pressure, wind stress acting on the sea surface, local variation of currents on the shelf which was formed off the coast or any impact at the sea bed. This model must be also useful to learn a tsunami- induced local oscillation which has a specific period. Bottom profile off Susami, facing the Pacific, can be approximated by a parabola so that some part of the significant oscillation observed at the Susami tide stationCNakamura and Serizawa, 1983; Current Obser- vation Group, I983JI must be understood as that formed by a quite similar mechanism as considered in the model. And the author believes that this work could give a suggestion to a, tsunami- induced local oscilla- tion. In these years, many of works has concerned to the waves propagating along a coast on a continental shelfffor example, Buchwald and de Szoeke, 1973; Webb, 1976; le Blond and Mysak,1978; Warren and Wunsch, 1981]. However, the author feels that these theories are not necessarily applicable or appropriate to have a dynamical understanding of tsunami- induced oscillations observed on the coast, especially on the coast of the Japanese Islands facing the Pacific. There must be a possible modes of oscillation normal to the coast rather than a wave propagating along the coast. As for the classic linear theory on seiche in a narrow rectangular bay, we have a Merian's formula to determine the period of the seiche [for example, Def ant, 1961; Proudman,1953]| . Seiches were studied first for oscillations of the water-level in the Lake of Jeneva, and later it has been shown theoretically that seiches can be observed not only in lakes but also in bays, channels and banks ^see a reviewal note by Hakamura, 19813. Seiche on a sea-shelf is sometimes called as the secondary undulation which has been taken essentially to be on an extention of the theory of the seiches in a lake, and an adjustment for the Merian's period of the seiche in a bay or a shelf was introduced 252 to get a reasonable agreement between the observed result and the theory [Def ant, 1961} Norn it su, 1937] . Hidaka[l935] developed a theory of shelf- seiches as an application of the Fathieu's functions. However, each of the theories by Defant, Nomitsu and Hidaka, has left an uncertain factor to determine the dimension of the sea-shelf, i.e., a reference water depth or length of the sea-shelf. Allison and Glassia[l979j have worked a statistical study on the sporadic oscillations of the sea-level on the Western Australian coast. Successively, Tuck[l980] developed a mathematical theory of oscillations induced in a shelf-sea with a reef facing an open ocean. Tuck et al. [1980] have applied Tuck's theory to find mechanism of the sporadic oscillations on the Western Australian coast. A theory developed by Buchwald and Miles [l98l] is an extensive theory of Tuck's one to evaluate the effect of the reef's width. However, the author could not consider that these theories were appropriate to understand oscillations of the sea-level on the coast of the Japanese Islands facing the Pacific vrhere the reef is not so significant or is trivial and the^shelf is rather steep than the shelf off the Western Australian coast. With the above consideration, the author developed a linearized theory of seiches on a parabolic shelf. The assumption is that the shelf is less than several ten kilometers in dimension. The solution will show some different properties from those obtained previously for the waves found along the coasts [Nakamur a, 1976a,b; 1979;198l}. A MATHEMATICAL FORMULATION Now consider a straight coastline which bounds a semi- infinite sea. As for the co-ordinates, let us take a cartesian co-ordinates so as to be the sea surface of still state is on the x-y plane where the y axis is along the coastline and the x axis is normal to the coast- line. The z axis is taken to be upward positive on the coastline. The disturbance £ of the sea-surface is considered to be small for a convenience of the analysis. No wave propagating along the coast are considered in this analysis. Simply, oscillations normal to the coastline is the author's interest in this work, ^hen the author wishes to start the analysis refering to the following linearized 253 equations of motion and continuity, i.e., (1) and -ft ** (2) ! where u is the x component of the current velocity, P a is an external forcing factor expressed in form of a potential, T is a stress acting the sea water. The notation t, g and h denote time, gravitational accerelation and water depth from the sea surface to the bottom respectively. If the coastline is parallel to a meridian, the effect of the earth's rotation vanishes apparently in the equations of motion to result a same equation as written in (l). Rven if the effect of the earth's rotation is considered, the equation (l) can he utilized for problems just neighbour the equator. So that, there must be a solution which is essentially same in mathematical expression for each of the problems. Introducing the volume transport instead of>the current velocity and write as follow T7= M k. (3) V=vL (4) and putting F-/f.A ( 5 ) -t, then, an approximated alternative expression of the equations (l) and (2) can be obtained as ^=-j(^)¥ ^ + (T S -T b ) ( 6) and 254 (?) In the equation (6), the stresses at the sea surface and "bottom are expressed hy and b respectively. When and h are functions of x respectively, we have the following equation from (6) and (7) after eliminating U. - x z ( S r^¥ j-t-r -r\] , wll ■ ?^.'\^ (8) If the terms of the second or higher order are smallenough to he taken negligible, the above equation is rewritten as lie* J *** ^ r^F . ,„ ~r , i , „^ ">r a St" iiMVVlJ M 3a a (9) for the conditions of t" « h and (^/"bx ) <1, Assuming a periodical wave ?> r e «*p (fo*0 (10) and substituting (10) into (9) with rewriting £ instead of £ for a convenience, then 4 Jx ^x (ii) In this equation (11 ), the effects of the external actions are included in the term £i;£ + <■*--*)] (12) When this term equals to zero, the equation (11 ) can be expressed as T-. + "J" -r~ -r— + ^7~ C^ 1 77 77 + js: ^ " ° (13) even if F / 0, Then, the equation (13) cincides to the classic equation of seiches for a constant water depth because the second term in the equation (13) vanishes when (dh/dx) = 0. In case of an uniform slope 255 of the sea bed, irtst and the water depth is proportional to the distance from the coast, the equation (13) is a Bessel's differential equation. Now, if we write (13 ) in form of (*a* r ^ ( '^ + ) * " =0 (15) with ^ +V H kit (16) X. * t'% t U7) then, the conditions that the equation (15) is equivalent to the equation (13) should be \\)~iiil*H%T-*f^ as) Hence, to solve the equation (9) is equivalent to solve the equation (l5)« That is, in other words, to solve simultaneous equation i ji + 4 ) i » ° (19) (fi <""*) V = I (20) Solving the equations (19) and (20), we have £«**(, [~j f J* ] (21) and 256 ?=^[-Jt^J'{ i ex r[ j(- /ft fe fc -«£ (2i) using the expression of (l8). BOUNDARY CONDITION In this study, the bottom profile is assumed and expressed by a parabola, i.e., it. * ax. (24) for the water depth h at a distance x from the coastline, where a is a constant of real number. Using the expression (24)> the solution (22) is rewritten as follows ^H^/TjgjAx] { j«r[iyi^-Ax]«fc t/ } (25) where, of cause, the value of x should be positive. The form of the solution (25) is quite similar to that of a trapped mode of edge wave travelling along a coast, though the author never had any willing to get the solution in relation to an edge wave. If we introduce an expression Y= exb [ <* hx") (26) as an equivalent form to the expression of J„X~- Li*) (27) 257 then, the solution (25) can "be rewritten ?=x ' ** . J x ; fy* +.4 (28) ; or where the value of X is 1 or 3 corresponding to the sign of plus or minus in the solution. Prom the solution (29)> we can find an amplitude distribution from the coastline to offshore along the x axis stretched normal to the coastline on the given parabolic bottom profile. All terms in the solution (29) is real for (l - z ) ^ so that we have a linear combination of the two monotonous function with the variable x. On the other hand, for (l - jr ) < ; the solution (29) is consisted by a term which is an oscillatory solution and a term which is for an damping oscillation. Especially in case of (l - -~r ) = 0, the solution (29) is written simply as follows; r * i + A* 1 (30) Although the solution (30) is the zeroth mode. OSCILLATIONS OBSERVED AT SUSAMI As already noted£Nakamura and Serizawa, 19&33 » a significant oscilla- tion with 5 cpH(l2 min in period) is observed through a year at Susami on the coast facing the Pacific. The bottom profile off Susami can be expressed by a parabola approximately, i.e., the water depth is proportional to the square of the distance from the coastline. I ! ?• r=\ x + A x (29) 258 Prom the "bathymetry read from a nautical chart around Susami, the profile is given "by an empirical formula d - £v,c~ 6 • ^ (31) 6 l 2 Then, with a= 6/ 10" (m ) and g ■ 9.8 m /s for > ^"pc)z ° (32) we have a corresponding expression for frequency , i.e., «* 7-7* the general solution for the equation (ll) can he written as T s - T- h ) J h ^.{I'^'gHT.-T.yjJx (35) with ^ - £ i^ - ^^ (36) where A]_ and Ag are real constants determined to satisfy the given boundary conditions respectively. Refering to the solution (35) > "we have to consider that essential is the initial existence of of the oscillations expressed by T- and <£_ a* the two free modes and at the input of a disturbance the main factors governing the oscillation on a parabolic shelf profile are (l) a variation of the atmospheric pressure, or an equivalent pressure disturbance in the sea or on the sea surface or at the bottom, (2) wind stress acting on the sea surface and (3) bottom stress caused by the effect of friction against currents. Any effect of the disturbance in a limitted area of the bottom will be accepted as a tsunami on the mareogram obtained on the coast. Another possible cause must be a variation in the general circulation, for example, meander of Kuroshio south of the Japan Islands in the western Pacific or that of Gulf Stream east of the United States in the western Atlantic. 260 One of the examples for the effects is a "barometric disturbance as studied by Hibiya and Kajiura 0-9^2]. They considered that a signifi- cant seiche could be amplified by a combined effect of the resonant bathymetric condition on the shelf and the local geometry of the almost- closed bay at an incident sea-level variation induced by a barometric disturbance. However, the author feels that it is hard to consider that the oscillation observed at Susami cannot be always amplified simply by such the barometric effect. The author could not find any barometric disturbance neighbour Susami, even though the specific oscillation is observed at Susami. At present, it must be appropriate to consider that the oscillation at Susami is governed by the local wind action to the water on the parabolic sea-shelf. If any earthquake under sea generates a tsunami, the tsunami- induced oscillation can be easily distinguished from the others. A detailed discussion of the mechanism should be given after refering to the observed data of the sea-level variations and of the currents on the shelf, though we have no data available for promoting our dynamical understanding of the shelf seiche at present. ACOO WLPIDG1MENTS This work was presented at the IAPSO Symposium on tsunami prpegation of the XVIIIth General Assembly of the International Union of Geodesy and Geophysics held in Hamburg in August 1982. The author thanks to Professor T. Tsuchiya's permission for the presentation as well as to consideration by Dr. K. C. La Pond and Professor S. L. Soloviev and to Professor T. Y. Wu's support. Professors H.Miyoshi, P. H. Le Blond and H. G. Loomis are appreciated for their valuable discussion. 261 REFERENCES Allison, H. and A.^lassia, Sporadic sea-level oscillations along a Western Australian coastline, Aust. Jour. marine and freshwater Res., 30, 723-730, 1979. Buchwald,V.T. and R.A.de Szoeke, The response of a continental shelf to travelling pressure disturbances, Aust. Jour. marine and fresh- water Res., 24, 143-158, 1973. Buchwald, V.T. and J.W.Miles, On resonance of off-shore channels bounded by a reef (unpublished), 1981. Current Observation Group, Long continuous observation of currents in Tanabe Bay, Ann.Disaster Prevention Res. Inst., Kyoto Univ., 26 (to be published), 1983*. Defant,A. , Physical Oceanography, Vol.2, Pergamon Press, N.T., 1961. Hibiya,T. and K.Kajiura, Origin of the Abiki phenomenon(a kind of seiche) in Nagasaki Bay, Jour.Oceanogr.Soc. Japan, 38(3), 172-182, 1982. Hidaka,K. , Study on bank-seiches and shelf-seiches, Umi to Sora, 15 (7), 223-229, 1935. Le Blond, P. H. and L.A.Mysak, Waves in the cean> Elsevier Oceano- graphy Ser., 20, Amsterdam, 1978. Nakamura, S. , Edge waves as linear solutions, La Mer, 14(1), 1-6, 1976a. Nakamura, S. , A linear edge wave excited by an external action, La Mer, 14(3-4), 139-143, 1976b. Nakamura, S., Sur ondulation lineaire le long de la cote circulaire, La Mer, 17(1), 11-15, 1979. Nakamura, S. , A note on classic theory of seiche, CSIRO Division of Land Resources Management Perth, Tech. Memo. 8l/7, 198la. Nakamura, S., Ondes lineaire et leir instability dans la voisinage de la cote de forme elliptique, La Mer, 19(1 ), 1-5, 198lb. Nakamura, S. and S.Serizawa, Shelf-seiches off Susami, south of Japan, La Mer, 21, 119-124, 1983. 262 Nomitsu,T., Conditions to form shelf- seiches in lakes or seas and adjustment for Merian's period, Chikyo Butsuri (Geophysics in Kyoto Univ.), 4(1), 38-46. Proudman,J. , Dynamic Oceanography, Methuen, London, 1953. Tuck, 3.0. , The effect of a submerged harrier on the natural frequencies and radiation damping of a shallow-basin connected to open water, Jour. Aust. Math. Soc. , 22(Ser.B), 104-128, 1953. Tuck,K.O., H.Allison, S.R.Field and J.W.Smith, The effect of a submerged reef on periods of sea-level oscillation in Western Australia, Aust. J our. Marine and freshwater Res., 31, 719-728, 1980. Warren,B.A. and C.Wunsch, Evolution of physical Oceanography, MIT Press, Cambridge, 1981. Webh,D.J., A model of continental shelf resonances, Deep-Sea Res., 23, 1-15, 1976. 263 ! A HYBRID FEM-MODEL FOR TSUNAMI AMPLIFICATION IN NEARSHORE REGIONS. Lars Behrendt 1 Ivar G. Jonsson 1 Ove Skovgaard 2 institute of Hydrodynamics and Hydraulic Engineering, (ISVA). laboratory of Applied Mathematical Physics, (LAMF). The Technical University of Denmark, DK-2800, Lyngby, Denmark. A two-dimensional hybrid finite element model for small water waves over a variable bottom has been developed for diffraction and harbour resonance studies. The basis is the linear mild- slope wave equation and the model includes absorbing boundary conditions. In the present contribution the model is used for long wave response in an idealized bay. The influence of the incidence angle and of the presence of an absorbing boundary are shown for one specific situation. Introduction. When dealing with linear water waves over a slovly varying ba- thymetry the govering wave equation is the well known mild-slope wave equation. This equation which reduces a three-dimensional problem to a two-dimensional one was first derived by Berkhoff (1972). It has been discussed in detail by several authors, see e.g. Jonsson and Brink-Kjar (1973). A hybrid finite element model based on the shallow water ver- sion of the mild-slope wave equation was developed by Chen and Mei (197*0. Bettes and Zienkiewicz (1977) introduced a finite element model based on the mild slope wave equation using infinite elements along the open boundaries. Houston (1978) used a shallow water hybrid finite element model for calculation of the interaction of tsunamis with the Hawaiian Islands. It was demonstrated that the results of the finite ele- ment model was in good agreement with what was actually recorded on several locations. In these proceedings a hybrid finite element model based on the mild-slope wave equation is presented. The functional which in- cludes an absorbing boundary condition is in principle the same 265 as the one used by Bettes and Zienkiewicz (1977) and Houston (1981) and it is a generalized version of the functional ori- ginally developed by Chen and Mei (197*0. The finite element model. The mild slope wave equation reads : 2 to 7- (cc v*) + - = (1) where c is the phase velocity, c the group velocity, 00 the an- gular frequency and <}) = = along B' (3) where k is the wave number and a is a real absorption coeffici- ent. When defining the reflection coefficient b as the ratio of the reflected wave amplitude to the incident wave amplitude this quantity is gived by : b - 1-q 1+a < a < 1 (4) By assuming constant water depth in the infinite outer region R (see Pig. 1.) the mild-slope wave equation becomes the Helm- holtz equation for which semianalytical solutions are known. Denoting these d) D and demanding the potential d) to be a smooth function everywhere, the functional -which is required to be stationary with respect to the first variation in ) = j| [cc g (v V 2+: OJ •2] dA ucv^-tV^^V^ ds -1* ( V^-glnT dS A 'A £cc ika A ds B a r (5) Here 4> A is the potential function in area A and cf> is the poten- tial of the undisturbed incident waves of constant period. n„ is an outward normal to the area A. By subdividing the area A into simple three-node triangular fi- nite elements and by extremization of the functional, one yields a set of linear equations which can be solved numerically by the Gaussian elemination technique. 267 Results . The simple geometry chosen for this occation is shown in Pig. 2 Ocean / t 777777777/ 2500 m 1500 m Fig. 2. Horizontal sketch of idealized bay with plane sloping bottom. For y>0 the water depth is 60 m. At y=-2,500 m the wa- ter depth is 10 m. A wave period T = 5 min. was chosen corresponding to wave lengths of approximately 7.3 km in the ocean and 2.8 km at the inner boundary of the bay. In Fig. 3. the effect of an absorbing boundary is shown. Fig. 3a shows the amplitudes in the bay and in the ocean for waves com- ming in at a right angle to the coastline, all boundaries being fully reflecting. Fig. 3b. shows the same thing but now with the inner boundary of the bay being fully absorbing. What is seen in Fig. 3a. are the amplitudes of a standing wave and in Fig. 3b. the amplitudes of a propagating wave. In Fig. 4. the /effect of a varying "angle of incidence" 0. (see Fig. 2.) is shown. One notices that even though the amplitudes in the ocean change rather drastically from Fig. 3a. to 4a. to 4b,, the amplitudes inside the bay do not change very much. As one would expect the amplitudes inside the bay are decreasing when 9. is decreasing. 268 2^ -0.75 ■0.50 ■0.35 a) b) Fig. 3. Relative wave amplitude in the bay and in the ocean out- side it for right angle incidence. 3a, all boundaries fully re- flecting. 3b j the inner boundary of the bay fully absorbing. /777/7777777777777777777A / -0.25 -0.28 -0.80 -0.78 -1.00 -1.28 V\\\\\\V^\N\\\\\\N\\N\\\\ /77777777777777777777777. tS — 1.28 — 1.S0 W.VVVAWVA'AVA'A'A'AV S c SSSSS! '/ ////////////;////>/;/ < b) Fig. 4. Relative wave amplitude in the bay and in the ocean out- side it for two different angles 9 4a 6. =240 l 4b e.=2io i 269 The figures 3 and 4 do not show the phase angles and thus the visualization of the water surface motion is poor. Therefore in Fig. 5- a three-dimensional plot of the relative surface eleva- tions at a chosen time are shown corresponding to the amplitude distributions from Fig. 3. The damping effect from the absorbing inner boundary of the bay is noticeable. a) b) Fig. 5. Relative surface elevations at time t/T=0.125. The am- plitudes are shown in Fig. 3. 5a, all boundaries fully reflec- ting. 5b, the inner boundary of the bay fully absorbing. In Fig. 6. the relative surface elevation is shown at four dif- ferent times for the situation 6. =210 . This corresponds to the amplitude field shown in Fig. 4b. One notices that the wave mo- tion in the inner half of the bay seems to be almost one-dimen- sional . Finally in Fig. 7. an example of a net of finite elements in the bay is shown. The grid is unnessacaryly fine for the wave lengths chosen but the grid illustrates the principle. 270 c) d) Fig. 6. Relative surface elevations at four different times The amplitudes are shown in Fig. 4b. 6a, t/T=0.000 6b, t/T=0.125 . 6c, t/T=0.250 . 6d, t/T=0.375 . e.=2io 1 271 Fig. - 7. An example of a net of finite elements. Conclusion. The finite element method is a powerful method for diffraction calculations for small water waves in water areas of arbitrary shape. The method has previously proved to give good results for tsu- nami calculations (Houston, 1978). However, the results are on- ly good as long as the amplitudes remain small. When the ampli-^ tudes become large i.e. near the shore line, the non-linear ef- fects in nature become dominant and our results loose their re- liability. It has been shown how an absorbing boundary condition can be used in the model. Tsunamis, however, will very seldom be fully absorbed but the model is able to deal with partially absorption also . 272 References. Berkhoff J.C.W. (1972): Computation of combined refraction-dif- fraction. Proc 13'th Coastal Engng. Conf. Vancouver. ASCE, New York, 1 , chap. 24, 471-^90. Bettes P. and Zienkiewicz O.C. (1977): Diffraction and refracti- on of surface waves using finite and infinite elements. Int. J. Num. Eng., 11 , no 8 , 1271-1290. Chen H.S. and Mei C.C. (197*0: Oscillations and wave forces in an offshore harbor. Massachusetts Institute of Techno- logy. Parsons Laboratory, Report no 190. Houston J.R. (1978): Interaction of tsunamis with the Hawaiian Islands calculated by a finite-element numerical model. J. Phys. Oceanography, 8 , 93-102. Houston J.R. (1981): Combined refraction and diffraction of short waves using the finite element method. App . Oce- an Res., 3 , no 4, 163-170. Jonsson I.G. and Brink-Kjaer 0. (1973): A comparison between two reduced wave equations for gradually varying depth. Inst. Hydrodyn. and Hydraulic Engrg. (ISVA), Techn. Univ. Denmark, Prog. Rep. no 31, 13-18. 273 •A U.S. Government Printing Office 1984 - 776-001/4 105 Reg. 8 ^.f™ 7 * UNIVERSITY LIBRARIES A00007CHMSS0fl