The interaction of large amplitude barotropic waves with an ambient shear flow: critical flows

Abstract

This study discusses the evolution of long gravity waves on shear flows. Although the paper is concerned mainly with finite amplitude neutrally stable flows which contain a critical level, a new representation is given for the unstable mode solutions of the linearized equations. From these solutions it appears that focusing instabilities, usually associated with nonlinear viscous effects, can occur even in linear inviscid theory. For finite amplitude disturbances the analysis is restricted to polygonal shear profiles and only the neutrally stable solutions are considered. The theory is presented in detail for a simple two-layer profile which can support a critical mode. At small Froude numbers the critical mode is essentially an internal wave. This limiting solution also describes critical flows between parallel rigid boundaries when there is no body force. The finite amplitude solutions are generalizations of the classical simple wave solutions for unsheared flows. As in the classical case, those waves can break but it is found that the conditions under which they break can be markedly different for shear flows. Calculations for the particle trajectories are also presented. These trajectories differ from the usual Kelvin’s cat’s eye pattern in that they are, in general, no longer closed. Finally, it is observed that there are many other barotropic flows for which the governing equations can be reduced to a form equivalent to the shallow water equations discussed here. A list of such flows is given.