  The rigid-lid approximation is a commonly used simplification in the study of
density-stratified fluids in oceanography. Roughly speaking, one assumes that
the displacements of the surface are negligible compared with interface
displacements. In this paper, we offer a rigorous justification of this
approximation in the case of two shallow layers of immiscible fluids with
constant and quasi-equal mass density. More precisely, we control the
difference between the solutions of the Cauchy problem predicted by the
shallow-water (Saint-Venant) system in the rigid-lid and free-surface
configuration. We show that in the limit of small density contrast, the flow
may be accurately described as the superposition of a baroclinic (or slow)
mode, which is well predicted by the rigid-lid approximation; and a barotropic
(or fast) mode, whose initial smallness persists for large time. We also
describe explicitly the first-order behavior of the deformation of the surface,
and discuss the case of non-small initial barotropic mode.
