Instability and wave over-reflection in stably stratified shear flow

Abstract

We reexamine the related problems of instability of parallel shear flows and over-reflection of internal waves at a critical level, concentrating on the stratified case. Our primary aim is to delineate the specific aspects of a flow that permit overreflection and instability. A related and partly realized aim is to develop a mechanistic ‘picture’ of how over-reflection and instability work. In the course of this study we have also uncovered some new results concerning the instability of stratified shear flows – showing how regions of enhanced static stability and enhanced damping can destabilize otherwise stable flows.For the scattering of steady plane waves, we show that, of the conditions found by Lindzen & Tung (1978), in the unstratified case, only the existence of wave-propagation regions above and below the critical level is always necessary for over-reflection (at least in the absence of damping), although a trapping region around the critical level and a reflecting surface bounding the upper wave region may play crucial roles in some cases. Our results suggest that the role of the upper wave region may be to allow a wave flux through the critical level. Moreover, we show numerically that the effect of an upper wave region can be mimicked by a region of localized damping which leads to over-reflection as well.We also consider an initial-value problem, using numerical methods. When a wave is incident on the incident level, the reflection and transmission coefficients grow smoothly to their final values. The rate of growth depends on the flow parameters, but there is some evidence to suggest there is a characteristic timescale involved that depends only on the shear (and not on wave travel time). This fits a mechanistic picture of over-reflection and instability that we describe, in which the essential part is a kinematic interaction between wave and mean flow at the critical level, depending only on shear.