Rossby solitary waves in the presence of a critical layer

Abstract

This study considers the evolution of long nonlinear Rossby waves in a sheared zonal current in the régime where a competition sets in between weak nonlinearity and weak dispersion. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean-flow velocity at a certain latitude, due to the appearance of a singularity in the leading order equation, which strongly modifies the flow in the critical layer. Here, nonlinear effects are invoked to resolve this singularity, since the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear critical-layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg-de Vries equation, modified by new nonlinear terms, depending on the critical-layer shape. These lead to depression or elevation solitary waves. © 2008 Springer.