Instability criterion for oblique modes in stratified circular Couette flow

TL;DR

This paper derives an analytical inviscid stability criterion for stratified Taylor-Couette flow, explaining the conditions under which oblique modes become unstable, and aligning with recent experimental and numerical findings.

Contribution

It introduces a new instability criterion considering finite gap effects and non-axisymmetric perturbations, extending the understanding of stratified flow stability beyond classical Rayleigh conditions.

Findings

Instability occurs for rotation ratios $<^*$, with $^*$ depending on the gap size.

For narrow gaps, $^* o 1$; for wide gaps, $^* o 2\u03b4^2(1+)$.

Instability is predicted for $>_c=^2$, consistent with centrifugal instability.

Abstract

An analytical approach is carried out that provides an inviscid stability criterion for the strato-rotational instability (in short SRI) occurring in a Taylor-Couette system. The control parameters of the problem are the rotation ratio $\mu$ and the radius ratio $\eta$. The study is motivated by recent experimental \cite{legal} and numerical \cite{rudi, rudi2} results reporting the existence of unstable modes beyond the Rayleigh line for centrifugal instability ($\mu =\eta^2$). The modified Rayleigh criterion for stably stratified flows provides the instability condition, $\mu<1$, while in experiments unstable modes were never found beyond the line $\mu=\eta$. Taking into account finite gap effects, we consider non axisymmetric perturbations with azimuthal wavenumber $l$ in the limit $l Fr<<1$, where $Fr$ is the Froude number. We derive a necessary condition for instability : $\mu < \mu^*$ where $\mu^*$, a function of $\eta$, takes the asymptotic values, $\mu^* \to 1$ in the narrow gap limit, and $\mu^* \to 2\eta^2(1+\eta)$ in the wide gap limit, in agreement with recent numerical findings. A stronger condition, $\mu<\eta$, is found when $\eta>0.38$, in agreement with experimental results obtained for $\eta=0.8$. Whatever the gap size, instability is predicted for values of $\mu$ larger than the critical one, $\mu_c=\eta^2$, corresponding to centrifugal instability.