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Propagation of Tsunami on a Linear Slope Between Two Flat Regions

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Propagation of Tsunami on a Linear Slope Between Two Flat Regions J. Phys. Earth, 28, 531-541, 1980 PROPAGATION OF TSUNAMI ON A LINEAR SLOPE BETWEEN TWO FLAT REGIONS. PART I EDGE WAVE Hiroshi ISHII* and Kuniaki ABE** * Observation Center for Earthquake Prediction , Faculty of Science, Tohoku University, Sendai, Japan **Niigata Branch , Faculty of Dentistry, Nippon Dental University, Niigata, Japan (Received August 11, 1980; Revised December 18, 1980) An edge wave traveling along a continental shelf with a sloping region sand- witched by a flat shelf and an ocean bottom is investigated theoretically by esti- mating the dispersion relation and the amplitude distribution versus the distance from the coast. The results are discussed in comparison with a step shelf model. It is demonstrated that dispersion curves shifted toward a long period range with decrease of a dip angle of a sloping region and that the minimum point of a group velocity curve becomes less clearer in the case of a sloping model. It is found that the maximum amplitude of the Kamchatka tsunami observed in Japan can be accounted for by an edge wave propagated along the shelf. The result is useful for predicting arrival time of the maximum amplitude for a distant tsunami with an oblique incidence to the continental shelf. 1. Introduction Edge wave has so far been studied by many investigators. For examples, SEZAWAand KANAI(1939) pointed out an analogy between edge wave on a shelf with a step and Love wave in seismology. REID (1958) estimated the effect of Coriolis force on an edge wave on a sloping shelf. NAKAMURA(1962) theo- retically investigated generation of edge wave on a continental shelf for a cylindrical source and obtained a theoretical marigram. HATORIand TAKAHASI(1964) applied the dispersion relation by NAKAMURA(1962) to the Iturup tsunami of 1963. AIDA et al. (1968) investigated an edge wave traveling along a curved shelf experimentally. MUNK et al. (1964) discussed trapped and leaky modes on the continental shelf off California using the shelf profile approximated by a series of steps. Generally a typical continental shelf is approximated by a linear slope and a fairly uniform profile over a long distance. Thus a linear sloping model is considered to be ap- propriate for a typical shelf like the Pacific shelf of north east Japan. Therefore, it is of significance to obtain analytical solutions for a linear sloping model between two flat regions. Such analytical solutions make it easy to investigate the property of the model and to compare theoretical results to observed ones. 531 532 H. ISHII and K. ABE 2. Theory Let us express a continental shelf with a sloping bed as shown in Fig. 1 in which Cartesian coordinates (x,y,z) are used. The y-axis is taken parallel to the coast line, while the x-axis is directed off shore with the origin at the margin of the shelf. The z-axis is taken vertically upward. The depths of the shelf, sloping bed and outer ocean are denoted by h1, h2, and h3, respectively. The depths h1 and h3 are assumed to be constant and h2 is assumed to be expressed by: Fig. 1. Coordinate system and a model of continental shelf. h2=h1+xtanθ, in which θ is the dip angle of a sloping bed. The widths of the shelf and the sloping bed are denoted by l and a, respectively. It is known in the theory of long wave that the surface elevations ζ1, 2, 3 above the mean sea level satisfies. (1) where t and g represent the time and the acceleration of gravity, respectively. In these equations suffixes 1, 2 and 3 indicate the quantities for the shelf, sloping bed and outer ocean, respectively. After eliminating time factor eiωt, which is introduced by the assumption of purely periodic motion, a set of solutions are assumed in the form of plane waves as (2) (3) (4) where Propagation of Tsunami on a Linear Slope between Two Flat Regions 533 In these expressions c, ω, k, Ln(2λ) and F(-n:1:2λ) are the longshore phase veloc- ity, angular frequency and wave number, Laguerre polynomial and confluent hypergeometric function, respectively. Elevation ζ1 represents the waves reflected at the coast and the margin of the shelf, propagating toward the positive y direction while ζ2 is considered to be a kind of the stationary wave on the linearly sloping bed. The first term in ζ2 is equivalent to the solution derived by REID (1958) et al. and the second term is the one diverging when the depth is 0. Elevation ζ3 corresponds to the wave decaying in the outer ocean. Amplitudes A1, A2, B1, B2 and C are the constants to be determined from the boundary conditions. The boundary conditions are derived from the continuities of elevation and the flow normal to the y-axis at x=0 and a. At the coast the flow normal to the coast line is zero because a total reflection is assumed. These are written as (5) (6) (7) When formulae (2), (3) and (4) are substituted into (5), (6) and (7), five simul- taneous equations are obtained including five unknown parameters A1, A2, B1, B2 and C. A dispersion equation is derived from the existence condition of a non- trivial solution of the equation. The equation is written as follows: (8) where (9) (10) (11) (12) in which we use the Laguerre polynomial and the confluent hypergeometric function are 534 H. ISHII and K. ABE (13) (14) respectively, their derivatives with dash and λ1=2kh1/tanθ, (15) λ2=2K(a+h1/tanθ). (16) Transcendental equation (8) including polynomials (13) and (14) is solved numerically, so that an ω versus c relation can be determined. The c-ω curve thus obtained is called the dispersion curve for the phase velocity. The longshore group velocity U can then obtained by making use of the following formula: (17) The U-ω curve is called the dispersion curve for the group velocity. The dispersion curves represent the propagation property of the edge wave excited. The ampli- tude distribution against the distance from the coast can also be obtained by solving simultaneous equations (5), (6) and (7) for A1/C, A2/C, B1/C and B2/C, and substituting c-ω values obtained by the dispersion relation into them. 3. Dispersion Relation The dispersion curves are illustrated for the various sloping angles in Fig. 2. The curve for dip angle 90° was obtained by NAKAMURA (1962). Parameters h1, h3 and l are fixed to be 0.6, 6.0 and 50km, respectively. Only the fundamental mode is plotted and the other higher modes are not considered in this paper, because only waves having a long period are to be dealt with here. In the figure, c and ω are normalized as c/c1 and ωl/c1(=γ) respectively. It is seen that the phase velocity curve takes on a value for γ=0 and approaches asymp- totically to 1 for short-wave limit γ→ ∞. From c2/c1 to 1 the curves decrease monotonically and the point of inflection moves toward γ=0 along with a decrease of the sloping angle. It is found that the result for 90° agrees with the limiting case when the sloping angle is increased. The differences between the limiting case and the sloping one are obtained as -3% for 15° -10% for 5 .7° -37% for 1 .8°, Propagation of Tsunami on a Linear Slope between Two Flat Regions 535 Fig. 2. Dispersion curves of the fundamental mode edge wave for various dip angles. Phase (c) and group (U) velocities are shown by the solid and broken lines, respec- tively. when γ=1 is assumed. It is seen that the group velocity curve takes on a value for γ=0, and approaches asymptotically to 1 for short-wave limit γ=∞, A striking feature is the occurrence of a minimum value. It is found that the position of the minimum shifts towards a small value of γ with decrease of a dip angle. It is also shown that the case for 90° is the limiting case when the dipping angle is increased. The relative differences between the limiting case and the sloping ones for γ=1 are expressed as follows: -12% for 15° -35% for 5 .7° -61% for 1 .8°. The relative difference is larger than that of the phase velocity. 4. Comparison with Step Models It is interesting to investigate the role of the width of a flat shelf on the edge wave propagation. The dispersion curves for step models with three different 536 H. ISHII and K. ABE Fig. 3. Comparison of sloping models with step models of various widths. Phase (c) and group (U) velocities are shown with the solid and broken lines, respectively. For both the models the depths h1 and h3 are fixed to be 0.6 and 6.0km, respectively. widths are compared with those of the sloping models having a gentle slope amounting to only 1.8°, such a slope model corresponding to the case off north east Japan so called the Sanriku District. The sloping case is characterized by the constants of 0.6km, 6.0km, 1.8° and 50km for h1, h3, θ and l, respectively. The dispersion curves for various cases are shown in Fig. 3. Les us take the follow- ing three cases for step model. Case II, for which the shelf takes the shortest width amounting to 50km, is the limit of the above sloping model when the slope angle is increased. For Case IV, the shelf width which takes on a value amounting to 220km, the largest value among the models considered, is equal to the horizontal width of sloping shelf plus that of flat shelf which amounts to 50km.
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