Linear Instability of Stably Stratified Down-Slope Flows

Abstract

Fluid instabilities in the Prandtl model for down-slope flows are studied using linear modal analysis as well as direct numerical simulations. Given Prandtl’s analytical solution for uniformly cooled down-slope flows, we determine the point of instability initiation and the corresponding unstable flow modes. We show that down-slope flows are susceptible to transverse and longitudinal instability modes. The transverse mode consists of stationary longitudinal rolls whose axes are aligned parallel to the base flow direction, whereas the longitudinal mode emerges as transverse waves travelling along the streamwise direction. The emergence of these instabilities are controlled by the Prandtl number, the slope angle, and the stratification perturbation parameter, which is a measure of the strength of the surface buoyancy flux relative to the background stratification. When the other two dimensionless parameters are held constant, the stratification perturbation parameters determines whether the imposed surface buoyancy flux can overcome the stabilizing effect of the background stratification and give rise to dynamically unstable flow. Beyond the linear stability thresholds, these two type of instabilities coexist to form complex flow structures. The absence of strong non-normality of the operator is shown by calculating the pseudospectra for both types of instabilities.