RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES5004, doi:10.2205/2006ES000213, 2006


Main Equations

[3]  To describe the generation and propagation of a wave, we use the nonlinear system of equations of shallow water

eq001.gif(1)

Here grad(s) = sxi + syj, div(s) = (s1)x + (s2)y, i and j are unit vectors directed along the x and y axes,

eq002.gif

eq003.gif   f = 2W cos(q) is the Coriolis parameter, W is the velocity of the Earth, q is the latitude, g is the gravity acceleration (9.8 m s -2 ), Ch = (H + h - B)0.4/sh is the Shesi coefficient, sh is the asperity factor, and B(x, y, t) is the law of motion of the basin bottom. If a point lies in the bottom motion region, the wave height is converted to the increment value with the corresponding sign. The found values of the wave height are then used for updating velocities:

eq004.gif

eq005.gif

eq006.gif

eq007.gif

[4]  The inferred values are then corrected for the Earth's curvature. The time step was chosen from the stability condition of the difference scheme in use.

[5]  For an adequate implementation, we introduced a model grid with spatial intervals (in degrees) Dx and Dy and with a time step Dt. The spatial intervals were corrected for the Earth's curvature: Dxp = (Dxcdot pcdot R Earth)/180, where R Earth is the radius of the Earth, and the meridian intervals (in meters) between neighboring nodes were found from the relation Dypj = (Dxp cdot pcdot cos (yn + jcdot Dy))/180; as mentioned above, the time step (in seconds) was calculated from the stability condition of the difference scheme.

[6]  To describe free boundaries in the problem of tsunami wave propagation, we chose the well-known Sommerfeld condition, according to which the part of the wave field beyond the boundary is transferred along the outer normal with a constant velocity defined by the basin depth near the boundary and without changing the waveform. The normal component of the velocity at the free boundary is calculated from the known relation un = c h/(H + h), and the tangential velocity component vanishes, ut = 0.


RJES

Citation: Kataeva, L. Yu., A. V. Romanov, R. Kh. Mazova, and I. V. Kozhevnikov (2006), A software complex for the numerical simulation of generation and propagation of tsunami waves in various marine areas from dynamic seismic sources, Russ. J. Earth Sci., 8, ES5004, doi:10.2205/2006ES000213.

Copyright 2006 by the Russian Journal of Earth Sciences

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