Multiple linear instability of layered stratified shear flow

Abstract

We develop a simple model for the behaviour of an inviscid stratified shear flow with a thin mixed layer of intermediate fluid. We find that the flow is simultaneously unstable to oscillatory disturbances that are a generalization of those discussed by Holmboe (1962), purely unstable modes analogous to those considered by Taylor (1931), and a new type of oscillatory disturbance at large wavelength. The relative significance of these different types of instability depends on the ratio R of the depth of the intermediate layer to the depth of the shear layer. For small values of R, the new type of oscillatory wave has both the largest growthrate for given bulk Richardson number Ri0, and is also primarily unstable to disturbances propagating at an angle to the mean flow, i.e. such modes violate the conditions of Squire's theorem (1933), and thus the assumption of initial two dimensionality of such flows is invalid. For intermediate values of R, the Holmboe-type modes and the Taylor-types modes may have wavelengths and phase speeds conducive to the formation of a resonant triad over a wide range of Ri0. Thus the presence of an intermediate layer in a stratified shear flow markedly changes its stability properties.