LONG-TIME EFFECTS OF BOTTOM TOPOGRAPHY IN SHALLOW-WATER

We present and discuss new shallow water equations that provide an est imate of the long-time asymptotic effects of slowly varying bottom top ography and weak hydrostatic imbalance on the vertically averaged hori zontal velocity of an incompressible fluid with a free surface which i s moving under the force of gravity. We consider the regime where the Froude number is much smaller than the aspect ratio delta of the shall ow domain. The new equations are obtained at first order in an asympto tic expansion of the solutions of the Euler equations for a shallow fl uid by using the small parameter delta(2). The leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects, Both sets of equat ions 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 influe nce of the nonhydrostatic effects.