a reduced model for internal waves interacting with submarine ...

4.4 Periodic topography experiments
When the bottom is flat, the topography dependent coefficient M(ξ) is identically
one. For the time being, we avoid the computation of M(ξ) from the variable depth
bottom, which can be costly even using Driscoll’s package [10]. Let us assume
that it is a function of the form M(ξ)=1+n(ξ) where n(ξ) describes periodic fluctuations.
This choice is not far from the real coefficient that comes from mapping
a periodic piecewise linear topography, see [21, 24, 22]. This strategy will prove
useful for testing the models and observing the phenomena we are interested in.
Example 4.4. As a first example of a rough bottom let us consider a periodic
slowly-varying coefficient M(ξ) defined on the domain [0, 16π] as
⎧
⎪⎨ 1+0.5 sin(5ξ), for 6π≤ξ≤12π,
M(ξ)=
⎪⎩ 1, elsewhere.
The bottom irregularities are located in the region 6π≤ξ≤12π. There are 15
oscillations. The period of the bottom irregularities is l=1.2566. The initial
perturbation of the interface is the Gaussian function
η 0 (ξ)=0.5e −a(ξ−π)2 /64
with a = 200, therefore its effective width is L = 2.4 and the ratio inhomogeneities/wavelength
is about 0.5236. For the mean velocity u 1 we choose the
corresponding initial condition that gives one propagation direction for the LFM
withβ=0.0001,α=0, as done in Section 4.3. See Fig. 4.9 where the numerical
solution for t=36.3247 is depicted together with the solution for the flat bottom
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