RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES6003, doi:10.2205/2006ES000212, 2006

Numerical Modeling of Generation and Propagation of Tsunami Waves

Equations of Motion

[11]  We shall consider generation of tsunami waves in the shallow water approximation (see, e.g. [Gusyakov V.K., 1978; Pelinovsky, 1982]. We consider nonlinear equations (1), where function B(x, y, t) describes the vertical motion of key-blocks in the seismic source. The law of the motion of key-blocks changes depending on the version of the calculation. This dynamic model allows us to model various variations of the sea bottom without significant complication of the algorithm.

eq001.gif(1)

where x, y are spatial coordinates, t is time, u(x, t), v(y, t) are horizontal components of velocity, h(x, y, t) is perturbation of the free surface with respect to its level at rest, H is maximal depth of the basin, B is variation in the sea bottom (the account for the characteristics of dynamic seismic source), g is acceleration due to gravity.

Initial and Boundary Conditions

[12]  Boundary conditions in the problems of tsunami wave propagation differ depending on the peculiarity of the problems considered in the study. For example, if propagation of tsunami waves is considered in the basin of the sea, it is possible to specify the condition of complete reflection (specifying zero normal components of velocity un, vn, and derivative hn ) or to specify "free" boundary conditions. In the latter case, the wave passes through the boundary without distortions. However, a more complete description of the interaction between the wave and coast is needed in the solution of the problems of tsunami wave propagation in the shelf zone [Pelinovsky, 1982; Pelinovsky and Mazova, 1992].

[13]  We consider that before the generation, the fluid that fills the region with initial form of the bottom is at rest, i.e. the velocity and perturbation of the free surface are zero:

h (x, y, t) = u (x, 0) = v (y, 0) = 0.

Boundary conditions (full reflection) at 10-meter isobath are written as:

eq002.gif(2)

The conditions of free passing wave are written as

eq003.gif(3)

where n+1 denotes the time layer following layer n, and c = (g (H + h))1/2.

[14]  This model makes possible to correlate the wave with the generating source taking into account the characteristics of displacements and velocities at each point of variable sea bottom. The method allows us to combine the processes of numerical modeling of the generation and propagation of tsunami waves taking into account the effects of real basin bathymetry residual variations in bottom topography and superposition of subsequent perturbations on the wave propagating in the ocean. The seismic characteristics of the earthquake source influence the wave only by means of function B(x, y, t).

[15]  The numerical scheme described in and adjusted for the solution of the problems with dynamic source [Garagash et al., 2003] was used to calculate the generation and propagation of tsunami waves.

[16]  Bathymetry of the Black Sea with a resolution of 1.5 km was used in the modeling. The number of nodes in the numerical scheme is equal to 505 times 781=394,405. Modeling was carried out with a time step of 1 s (taking into account the tests for convergence and stability, see e.g. [Garagash et al., 2003]). Condition of full reflection (vertical wall) was specified at the last point of the sea region at a depth of 10 m.


RJES

Citation: Mazova, R. Kh., and E. A. Tresvyatskaya (2006), Numerical modeling of the generation of long waves by a dynamic seismic source and their propagation in the Black Sea, Russ. J. Earth Sci., 8, ES6003, doi:10.2205/2006ES000212.

Copyright 2006 by the Russian Journal of Earth Sciences

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