Stability of downslope flows to two-dimensional perturbations

Abstract

We consider the stability problem for wide, uniform stationary open flows down a slope with constant inclination under gravity. Depth-averaged equations are used with arbitrary bottom friction as a function of the flow depth and depth-averaged velocity. The stability conditions for perturbations propagating along the flow are widely known. In this paper, we focus on the effect of oblique perturbations that propagate at an arbitrary angle to the velocity of the undisturbed flow. We show that under certain conditions, oblique perturbations can grow even when the perturbations propagating along the flow are damped. This means that if oblique perturbations exist, the stability conditions found in the investigation of the one-dimensional problem are insufficient for the stability of the flow. New stability criteria are formulated as explicit relations between the slope and the flow parameters. The ranges of the growing disturbances propagation angles are indicated for unstable flows.