Long-time shallow-water equations with a varying bottom

Abstract

We present and discuss new shallow-water equations that model the long-time effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid possessing a free surface and moving under the force of gravity. We consider the regime where the Froude number ε is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained from the ε→0 limit of the Euler equations (the rigid-lid approximation) at the first order of an asymptotic expansion in δ². These equations possess local conservation laws of energy and vorticity which reflect exact layer mean conservation laws of the three-dimensional Euler equations. In addition, they convect potential vorticity and have a Hamilton's principle formulation. We contrast them with the Green–Naghdi equations.