Justification of the shallow-water limit for a rigid-lid flow with bottom topography

The so-called lake equations arise as the shallow-water limit of the rigid-lid equations-three-dimensional Euler equations with a rigid-lid upper boundary condition-in a horizontally periodic basin with bottom topography. We prove an a priori estimate in the Sobolev space H-m for m greater than or equal to 3 which shows that a solution to the rigid-lid equations can be approximated by a solution of the lake equations for an interval of time which can be estimated in terms of the initial deviation from a columnar configuration and the magnitude of the initial data in H-m, the gradient of the bottom topography in Hm+1, and the aspect ratio of the basin. In particular, any solution to the lake equations remains close to some solution of the rigid-lid equations for an interval of time that can be made arbitrarily large by choosing the aspect ratio of the basin small.