Breakdown of the shallow water equations due to growth of the horizontal vorticity

Abstract

In an oceanographic setting, the shallow water equations are an asymptotic approximation to the full Euler equations, in the limit ϵ =h0/L→ 0, withh0being the vertical length scale andLa horizontal length scale associated with the fluid layer. However, in arriving at the shallow water equations an additional key step in the derivation is the condition that at some reference time (e.g.t= 0) the thin-layer horizontal vorticity field is identically zero, which corresponds to the horizontal fluid velocity field being independent of the vertical coordinate,z, att= 0. With this condition in place, the ‘thin-layer equations’ reduce exactly to the shallow water equations. In this paper, we show that this exact condition may be unstable: small, even infinitesimal, perturbations of the thin-layer horizontal vorticity field can grow without bound. When the thin-layer horizontal vorticity grows to be of order 1, the shallow water equations are no longer asymptotically valid as a model for shallow water hydrodynamics, and the ‘thin-layer equations’ must be adopted in their place.