Long-time shallow-water equations with a varying bottom

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 epsilon is much smaller than the aspect ratio delta of the shallow domain. The new equations are obtained from the epsilon --> 0 limit of the Euler equations (the rigid-lid approximation) at the first order of an asymptotic expansion in delta(2). 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.