The evolution of a quasi-steady critical layer in a stratified viscous shear layer

Abstract

In this paper we examine the evolution of the critical layer in a viscous stratified fluid when the Richardson number J = ¼ and the Reynolds number R is large. The basic flow consists of a hyperbolic tangent profile for both the velocity and the density variation, and on this is superimposed a free oscillation periodic in x . It is the determination of the nonlinear equation for the amplitude of this oscillation, which is characterized by a small parameter ε , that is our prime concern. This has been achieved analytically for a quasi-steady critical layer when the Prandtl number Pr is unity and when the order of magnitude of R bears a certain relation to that of ε . The thickness of the layer is O ( R -1/3 ) and the time scale on which the development takes place is large, specifically O ( R -2 ε -4 ). Discussion of the effects of the distortion of the mean flow is also included. When Pr ≠ 1 the appropriate time scale is shorter and O ( R -1 ε -2 ). The results of the analysis are confirmed by a numerical study for large but finite R which indicates that the form of the amplitude equation is different for Prandtl numbers other than unity. As R →∞ the basic shear is supercritically stable or unstable according as Pr ≷ 1.