The viscous Holmboe instability for smooth shear and density profiles

TL;DR

This paper investigates the viscous Holmboe instability in smooth stratified shear flows at finite Reynolds numbers, revealing new unstable regimes and challenging traditional stability criteria, with implications for understanding wave resonance mechanisms.

Contribution

It introduces the first linear stability analysis of the Hazel model at finite Re, identifying new instability regions even when the Richardson number exceeds 1/4.

Findings

Unstable modes exist at finite Re with Richardson number > 1/4.

Maximum growth rate occurs at large Re despite viscosity.

Holmboe instability characterized by propagating vortices, not sharp interfaces.

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.