Wave packet critical layers in stratified shear flows

Abstract

Abstract In the inviscid theory of shear flow stability, the eigenvalue problem for a neutral or weakly amplified mode revolves around possible discontinuities in the eigenfunction as the singular critical point is crossed. Extensions of the linear normal mode approach to include nonlinearity and/or wave packets lead to amplitude evolution equations whose coefficients generally involve singular integrals. In the past, viscosity, nonlinearity or time dependence has been introduced in a critical layer centered upon the singular point to resolve these integrals. The form of the amplitude evolution equation is greatly influenced by which choice is made. In this paper, a new approach is proposed in which wave packet effects are dominant in the critical layer, and it is argued that in many applications this is the appropriate choice. The theory is applied here to two-dimensional wave propagation in stratified shear flows.