  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).
