Geoscience Reference
In-Depth Information
A more complex version of the description of tsunami interaction with the coast
implies numerical simulation of waves running up the coast. We shall dwell upon
methods for resolving this problem in Sect. 5.3.
In those cases, when detailed simulation of the tsunami dynamics within a re-
stricted region is required, the necessity often arises to make use of boundaries that
freely transmit incident waves. In other words, the amplitude of a wave, reflected
from such a boundary, should be reduced to the minimum. The physical principle
for realization of such a 'non-reflecting' boundary condition is quite simple. At each
moment of time a boundary point is assigned that value, which should be brought to
it by the wave incident upon the boundary. However, technical realization of the con-
dition of free transmission turns out to be elementary only in the one-dimensional
case. If one considers wave propagation along the 0
x
axis, then the condition of free
transmission will be of the form
∂
u
c
∂
u
=
±
x
,
(5.45)
∂
t
∂
where
c
=
√
g
H
is the velocity of long waves. The quantity
u
in formula (5.45)
is understood to be any of the sought functions (the free-surface displacement or
the flow velocity component).
A condition of the same form as (5.45) is also applicable in resolving two-
dimensional problems, but it will no longer provide for ideal free transmission
through the boundary
x
= const of waves, travelling at a certain angle to the 0
x
axis.
Regretfully, no success has been achieved in totally avoiding the reflected wave,
when resolving the problem on a plane. It is possible to reduce the amplitude of
waves, reflected by the boundary, by enhancing the order of the boundary condition
approximation [Marchuk et al. (1983)],
2
u
2
u
c
2
2
2
u
c
∂
=
∂
∂
t
2
−
y
2
,
∂
x
∂
t
∂
∂
2
u
c
3
4
3
u
3
u
3
c
2
4
2
u
c
∂
∂
x
=
∂
∂
t
2
−
t
3
−
t
.
∂
∂
y
2
∂
∂
∂
y
2
∂
There also exists another approach to realizing nonreflecting boundaries. It con-
sists in the introduction of an absorbing layer in the vicinity of the boundary [Israeli,
Orzag (1981); Kosloff (1986)].
The third type of boundary conditions is the most simple to realize. If a certain
perturbation approaches the boundary from outside of the calculation region, then
at all points of the boundary one must set the velocity components and the surface
displacement to correspond to this perturbation. Depending on the concrete problem
these quantities can either be determined from the solution of another numerical
problem or be given by certain functions.
The key information, upon the reliability of which the precision of numerical
tsunami calculations depends, comprises data on the ocean bottom bathymetry
and on the topography of the coastal area. At present, free access is provided to a
2-min global database for the Earth's relief (ETOPO2, http://www.ngdc.noaa.gov/)
