The multilayer shallow water system in the limit of small density contrast

Abstract

We study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow and a smallness assumption on the initial surface deformation, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that there exists a small initial layer in time for ill-prepared initial data, accounting for rapidly propagating "acoustic" waves (here, the so-called barotropic mode) which interact only weakly with the "incompressible" component (here, baroclinic).1 Because we assume that the layers are all of comparable depth and the vertical stratification is balanced, in the sense that we fix m > 0 such that sup i∈{1,...,N } {δ i , δ −1 i , r i , r −1 i } ≤ m, then c 0 measures the typical velocity of propagation of the baroclinic modes; see Appendix B.

An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification

“…These stability conditions are rigorously characterized in [40], where a general criterion of hyperbolicity and local well-posedness is given, under a particular asymptotic regime and weak stratification assumptions of the densities and the velocities. A similar study has been realized in [25] in the limit of small density contrast. It is shown that, under reasonable conditions on the flow, the system is well-posed on a large time interval.…”

“…Even if the maximum admissible CFL is reduced, it is a remarkable result to find that taking one of the two stabilization constants to zero can be sufficient to obtain linear stability. These results may be set in relation with the optimized stability criteria issuing from the non-linear study, that is the one-dimensional relaxed condition (25) discussed in Remark 3.6. As the non linear study requires both α and γ to be strictly positive, only the case α = γ is explored in Fig.…”

“…α = γ (linear) α = γ (non linear) One-dimensional linear stability analysis: (CFL, α + γ) sampling in the particular cases α = γ (red) and αγ = 0 (blue) at first-order in space and time (left). Comparison with the relaxed condition issuing from(25) in the case α = γ (right). Two-dimensional linear stability analysis involves a rescaling of the CFL numbers dividing them by √ 2.…”