Some Characteristics of the Kelvin-Helmholtz and Resonant Overreflection Modes of Shear Flow Instability and of Their Interaction through Vortex Pairing

Abstract

A stably stratified parallel shear flow near the ground is able to support two distinct modes of instability, these being the well known Kelvin-Helmholtz disturbance and a secondary wave whose growth in time derives from the process of critical level overreflection. Here we first consider the two modes of instability in circumstances in which they exist in isolation from one another and present some new results concerning their finite-amplitude characteristics. In particular, we comment on the likely mechanism of secondary instability which marks the transition to turbulence in a Kelvin-Helmholtz billow. Further, simulations of the finite-amplitude evolution of an isolated resonant mode shows that nonlinearity, of itself, is not able to compensate for a small linear growth rate in making the resonant mode a prominent feature of the flow. We suggest that such may be accomplished, however, by the interaction between the two modes through the process of vortex pairing since the first subharmonic of the fastest growing Kelvin-Helmholtz mode may be a resonant mode. We provide a simple, weakly nonlinear calculation which supports this notion.