Long-time effects of bottom topography in shallow water

Abstract

We present and discuss new shallow water equations that provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity. We consider the regime where the Froude number is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained at first order in an asymptotic expansion of the solutions of the Euler equations for a shallow fluid by using the small parameter δ 2. The leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects. Both sets of equations conserve energy and circulation, convect potential vorticity and have a Hamiltonian formulation. The corresponding energy and enstrophy are quadratic integrals with which we can bound the cumulative influence of the nonhydrostatic effects.