Numerical modeling of waves for a tsunami early warning system

Numerical modeling of waves for a tsunami early warning system
the a priori heavy computations with a unitary source term are solved, it
is shown how the model predicts in few seconds the water surface elevation
at one target point, as the data become available at a point close to the
generation area.
3.3 Application of the parabolic
approximation of the MSE
The MSE of elliptic type defines a problem which is in general properly
posed only if boundary conditions are specified along the boundaries of
the computational domain. In order to obtain a numerical solution over
a large area in the horizontal plane, a great amount of computing time
and storage is required, because the elliptic equation needs simultaneous
solution over the whole area. Therefore there was a natural incentive for
developing an approximation of the MSE which bypasses these numerical
difficulties. The parabolic equation method served that objective. It consists
in approximating the elliptic wave equation to a parabolic equation, which is
easier to solve numerically because allows a solution scheme which proceed
along the predefined wave direction. The standard parabolic wave equation
has the disadvantage that the direction of waves must be substantially similar
to the predefined one. As the waves refract and diffract, they change
direction and the accuracy of the approximation decrease with increasing
angle between waves and the initial predefined direction.
An application of the model implemented in this work regards a matching
between the elliptic MSE, solved in a closed domain where diffraction and
reflection are important, and the parabolic MSE, solved in a larger domain
where waves propagate undisturbed. The idea behind this matching is to
reduce the computational costs using a faster numerical technique when
solving the waves propagation over large oceanic areas, so that the model
can be more efficient and results of practical use.
A further point of interest is that the tsunamogenic source can be typically
viewed as a small region with respect to the full domain where the tsunamis
can propagate, this is specially true in the case of landslide generated
tsunamis. It is therefore clear that the waves will travel radially away from
the source and thus the use of cylindrical coordinates apperas to be very
appropriate. In the following the parabolic MSE is derived in a cartesian
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