Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

Abstract

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

Stratified shear flow instabilities in the non-Boussinesq regime

“…Waves that propagate vorticity anomalies may become phase-locked with each other via the advection by the background flow and action-at-a-distance of the nonlocal velocity field induced by local vorticity anomalies. With phase-locking, depending on the phase shifts, these waves may amplify each other and lead to instability 7,18,21,23 .…”

“…Sometimes it is useful to show the locations where the resonance condition is satisfied 7 . These are the locations where the counter-propagating edge waves have matching phase speeds, taking into account advection by the background flow.…”

“…Such an assumption is useful for the study of the onset of instabilities for several reasons: the resulting dispersion relation often reduces to a low order algebraic equation, for which analytical as well as asymptotic solutions exist; such solutions are often the leading asymptotic solution for general smooth profiles in the long-wave limit 6,8 ; there is a mechanistic interpretation for the instability, seen as the constructive interference of vorticity propagating waves travelling counter to the background flow 7 . The use of defects has life beyond linear theory, allowing the derivation of reduced models via matched asymptotic methods to investigate the nonlinear development and saturation of shear instabilities 9,10 .…”

“…This assumption of small density deviation is well satisfied in the ocean and remains useful for studying certain atmospheric flows. The Boussinesq equation has and still remains a useful model for investigating a variety of fluid dynamical phenomena in geophysical systems, such as convection 3,4 , wave-mean flow interaction 5 and shear instabilities 6,7 , the last of which will be our principal focus here.…”

“…In Section IV, we consider a more complex example where waves are now supported on defects, and rationalise also the changes induced by the non-Boussinesq term. In Section V, a slightly simpler version of the Taylor-Caulfield instability 7,[15][16][17][18] in the non-Boussinesq regime is investigated and analysed accordingly. This ties together the modification to the wave dynamics and the instability properties resulting from the action-at-a-distance interaction between non-Buossinesq interfacial waves.…”