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

Chapter 4
Numerical results
4.1 Hierarchy of one-dimensional models
For numerical implementations, we first normalize the shallow water velocity c 2 0 =
(
ρ2
ρ 1
− 1 ) of system (2.27) by settingη=η ∗ , u 1 = c 0 u ∗ 1 , t= t∗
c 0
. Dropping the
asterisks, the following Strongly Nonlinear Corrugated Bottom Model (SNCM) is
obtained,
⎧
η t − 1 [ ] (1−η)u1
ξ ⎪⎨ M(ξ)
= 0,
⎪⎩
u 1t + 1
M(ξ) u 1 u 1ξ − 1
M(ξ) η ξ= √ β ρ 2
ρ 1
1
M(ξ) T [0,2l]
[ (1−η)u1
]
ξt . (4.1)
We will work on the periodic domainξ ∈ Π[0, 2l], so that instead of the
operatorT its periodic versionT [0,2l] appears above. See Appendix C for the
definition ofT [0,2l] . The choice of a computational periodic domain was made to
be able to use spectral methods. To avoid the influence of the boundaries on the
evolution of the perturbationηinteracting with the bottom profile we added two
52