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We study the inviscid multilayer Saint-Venant (or shallow-water) system inthe limit of small density contrast. We show that, under reasonablehyperbolicity conditions on the flow and a smallness assumption on the initialsurface deformation, the system is well-posed on a large time interval, despitethe singular limit. By studying the asymptotic limit, we provide a rigorousjustification of the widely used rigid-lid and Boussinesq approximations formultilayered shallow water flows. The asymptotic behaviour is similar to thatof the incompressible limit for Euler equations, in the sense that there existsa small initial layer in time for ill-prepared initial data, accounting forrapidly propagating "acoustic" waves (here, the so-called barotropic mode)which interact only weakly with the "incompressible" component (here,baroclinic).

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