Boundary-value problems in stratified shear flows with a nonlinear critical layer

Abstract

Approximate solutions of nonlinear boundary-value problems arising in inviscid stratified shear fluid flows over relatively small obstacles are considered. The presence of a critical layer in the flow field leads to a failure of the linear theory, necessitating the derivation of a nonlinear critical layer equation. The solution of this second-order differential equation for the horizontal velocity is found to be discontinuous along the streamline which passes through the upstream critical point. In the analysis, three constants of motion along a streamline are used to study flows involving slip and split streamlines emanating from this point. The matching of the critical layer (inner) and the linearized outer solutions is discussed and some examples of steady flows over an obstacle and waves propagating with constant speed are given.