Numerical modeling of waves for a tsunami early warning system

Numerical modeling of waves for a tsunami early warning system
everywhere except in the area where the semi-elliptic landslide moves. At
the boundaries (x =0andx =10m) the waves are allowed to exit the
domain by imposing the radiation condition specified by the equation (3.34).
The simulations here presented reproduce a time series of 100 s, usingaΔt
of 0.1 s. This results in a total of 1000 time steps, that in the frequency
domain corresponds to the same number of angular frequencies ω, inthe
range between 2π · 10 −2 and 2π · 10. In the practice the MSE (4.8) is solved
just for the frequencies in the range of 2π · 10 −2 ≤ ω ≤ 2π · 2, where there is
a significant content of wave energy. The higher frequency component has,
for the considered water depth, a wave length of 0.4 m, thusthemaximum
distance between the computational nodes is of 0.04 m, since a resolution of
about 10 nodes per wave length is required. The resulting number of Degrees
of Freedom (DOF) is 201, and the elliptic equations for the 200 frequencies
are solved on a 2 GHz CPU computer equipped with 4 GB of RAM (as all
the other simulations presented here), with a total computational time of
about 7 s.
In the simulations carried out with the three dimensional model, the
Laplace equation (4.2) is solved in a two dimensional domain, again 10 m long
and 1 m high (as the water depth). At the bottom boundary the condition
of wave generation (4.4) is imposed, by letting the landslide to move along
the bottom for 2 s. In order to compare the results with those of the depth
integrated model, the same angular frequencies ω are solved (2π ·10 −2 ≤ ω ≤
2π · 2). The finite element grid is composed of linear triangular elements,
with the maximum distances between the nodes of 0.04 m, asforthedepth
integrated model. The number of DOF for these simulations is 8725 and the
computational time is about 53 s.
In figure 4.11 the results of the numerical simulations with the two models
are shown, in terms of the free surface elevation in time at the points
A (left panel) and B (right panel) of figure 4.10. The landslide for the
presented simulation is 4 m long and moves between t =10s and t =12s.
When the landslide starts moving, going in the direction of increasing x
values, it generates a first crest-trough wave propagating in front of it,
and a first trough-crest wave at its back. The results computed with the
depth integrated model (solid black line) almost perfectly superimpose to
the ones obtained from the three dimensional one (dashed red line). The
good agreement is both in the generation (left panel) and in the propagation
areas (right panel).
Simulations with the depth integrated model have been carried out also
Università degli Studi di Roma Tre - DSIC 55