Abstract
The Holmboe wave instability is one of the classic examples of a stratified shear instability, usually explained as the result of a resonance between a gravity wave and a vorticity wave. Historically, it has been studied by linear stability analyses at infinite Reynolds number, $Re$, and by direct numerical simulations at relatively low $Re$ in the regions known to be unstable from the inviscid linear stability results. In this paper, we perform linear stability analyses of the classical `Hazel model' of a stratified shear layer (where the background velocity and density distributions are assumed to take the functional form of hyperbolic tangents with different characteristic vertical scales) over a range of different parameters at finite $Re$, finding new unstable regions of parameter space. In particular, we find instability when the Richardson number is everywhere greater than $1/4$, where the flow would be stable at infinite $Re$ by the Miles-Howard theorem. We find unstable modes with no critical layer, and show that despite the necessity of viscosity for the new instability, the growth rate relative to diffusion of the background profile is maximised at large $Re$. We use these results to shed new light on the wave-resonance and over-reflection interpretations of stratified shear instability. We argue for a definition of Holmboe instability as being characterised by propagating vortices above or below the shear layer, as opposed to any reference to sharp density interfaces.
Highlights
Stratified shear flows are ubiquitous in the oceans and atmosphere
We have described a new, inherently viscous instability and have demonstrated that it shares many of the characteristic features of the classic, inviscid Holmboe wave instability, namely manifesting as a propagating vortex on either side of the mixing layer and appearing to be caused by the interaction of internal gravity waves on a shear interface
Since it exists in regions of parameter space where no instability is predicted in the inviscid limit, we term it the viscous Holmboe instability, or VHI
Summary
Stratified shear flows are ubiquitous in the oceans and atmosphere. Their instabilities are believed to be relevant to a variety of geophysical processes, and understanding them is important, for example, in the irreversible mixing of fluid of different densities in the abyssal ocean to close ocean energy budgets. A different perspective, developed by R.S. Lindzen and coauthors, and reviewed in Lindzen (1988), is based on the idea that when the local Richardson number is less than one quarter, the critical layer of a normal-mode wave incident on a stratified shear layer will ‘over-reflect’, and in the correct configuration, this may lead to exponential growth. Reviewed in Lindzen (1988), is based on the idea that when the local Richardson number is less than one quarter, the critical layer of a normal-mode wave incident on a stratified shear layer will ‘over-reflect’, and in the correct configuration, this may lead to exponential growth This theory, harder to understand intuitively than the wave-resonance picture, is attractive as it explicitly includes the Miles–Howard criterion.
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