2. Numerical Scheme to Model the Tsunami
2.1. Statement of the Problem
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Figure 4
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[7] The aim of this work is a numerical simulation of such a process in the tsunami
source which would be a most relevant to possible motions occurring in the vicinity
of earthquake source during the first minutes after starting of the earthquake. The
process was modelled by vertical displacements of rectangular keyboard blocks
with height B (Figure 4). The keyboard block size varied from 50 km to 150 km long
and of 50 km wide. The number of keyboard blocks was varied from 3 to 8. In this
work, the simplified keyboard model with vertical displacements of keyboard
blocks is analyzed, horizontal movements of keyboard blocks were not considered
because of technical complexity to realize numerically the horizontal
displacements of the order of 10 m for keyboard blocks, 50 km wide. It is
considered a model problem of tsunami wave generation by dynamical source,
its propagation and run-up on the sloping beach. The source is modelled by seafloor
vertical displacements of keyboard blocks with given vertical velocity for each
block. For the first set of computations, the source was located at the distance of
100 km from a beach being parallel to the coastline. The water height above the
source was taken to be equal to
H = 1 km. There was considered the wave
propagation along even bottom in approximative geometry and run-up on a sloping
beach (Figure 4).
2.2. Governing Equations
[8] To describe the process of wave generation and propagation in
correspondence with above assumptions it was used the nonlinear system of
shallow water equations (1),
 | (1) |
where
h is the surface elevation,
u, v are
the horizontal ( x, y ) particle velocity components of the wave motion,
H is the undisturbed depth of the water,
B(t) is the seafloor deformation (seafloor
displacement relative to initial position),
g is the acceleration due to gravity
[cf. with Garagash et al., 2003].
The equations (1) were approximated by difference scheme according to
[Sielecki and Wurtele, 1970].
It was used computation grid with space intervals
Dx, Dy and with time step
Dt. The tsunami wavefield
was computed on a Cartesian coordinate system. The size of the calculated region
corresponds to 2400 km
2400 km. The size of computation grid was equal to
6.6 km. The computation was made every 10 s satisfying the Courant stability
condition.
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