High order models for the nonlinear shallow water wave equations on the equatorial beta-plane with application to Kelvin wave frontogenesis

Abstract

Abstract We describe two new numerical models for solving the nonlinear shallow water wave equations on the equatorial beta-plane. The finite difference code is eighth order in space and 4th order in time. The second model is a four-mode Hermite–Galerkin scheme in latitude combined with a Fourier pseudospectral algorithm in longitude and 4th order time-marching. As a first application, we test the Boyd–Ripa theory for Kelvin wave frontogenesis. The zeroth order strained coordinates approximation is good even for waves of nondimensional unit amplitude. The first order corrections grow. Even so, for small amplitude, the first order analytical approximations encapsulate most of the difference between the zeroth order approximation and the numerical solution if the longitudinal derivatives are interpreted as x -derivatives rather than as derivatives with respect to the strained coordinate ξ . The main discrepancy is that for moderate and large amplitude, the front curves westward with increasing latitude.