The Transition between Kelvin–Helmholtz and Holmboe Instability: An Investigation of the Overreflection Hypothesis

Abstract

We consider the evolution of small disturbances on an inviscid, Boussinesq, stably stratified free shear layer. This flow may deliver either Kelvin-Helmholtz or Holmboe instability, depending on the details of the background stratification. Conventional one-dimensional linear analysis is employed to study the temporal and spatial structures of these instabilities and the physical mechanisms which govern their evolution. Attention is focussed upon the manner in which Kelvin-Helmboltz instability is replaced by Holmboe instability for a sequence of background flows with successively larger values of the bulk Richardson number. Unstable normal modes that exist in the transition region between the Kelvin-Heimboltz and Holmboe regimes exhibit a distinctive spatial structure, are characterized by relatively low growth rates, and are shown to occur under conditions for which overreflection of neutrally propagating internal waves apparently cannot occur because the gradient Richardson number at the steering level exceeds ¼. Detailed calculations of the propagation characteristics of internal waves in stratified shear layers the extent to which resonant overreflection theory, based upon the reflection properties of temporally neutral waves, may fail to yield physical insight into the stability characteristics of such flows.