Three-dimensional instability of shear flows with inflection-free velocity profiles in stratified media with a high Prandtl number

Abstract

In stably stratified media with a Prandtl number Pr ≫ 1, vertical scales of the density (l) and horizontal velocity variation (L) are quite different, l/L = O(Pr−1/2) ≪ 1, and this influences the flow stability. In particular, shear flows without inflection points on the velocity profile are unstable even in an ideal incompressible fluid. The maximum instability growth rate for sufficiently small l/L is of the same order as in homogeneous mixing layers, with mainly three-dimensional rather than two-dimensional oscillations increasing in a wide range of parameters. This paper focuses on the three-dimensional instability of such flows. It is shown that the spectrum of unstable oscillations is essentially anisotropic in the case of a relatively weak stratification when the bulk Richardson number J ≤ O[(l/L)3/2]. The results of the asymptotic analysis are illustrated by calculations for a model flow in a two-layer medium (l = 0) as well as for flows with values of l/L corresponding to a temperature or salinity stratification of the water.