Instability criterion for oblique modes in stratified circular Couette flow
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 $μ$ and the radius ratio $η$. 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 ($μ=η^2$). The modified Rayleigh criterion for stably stratified flows provides the instability condition, $μ<1$, while in experiments unstable modes were never found beyond the line $μ=η$. 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 : $μ< μ^*$ where $μ^*$, a function of $η$, takes the asymptotic values, $μ^* \to 1$ in the narrow gap limit, and $μ^* \to 2η^2(1+η)$ in the wide gap limit, in agreement with recent numerical findings. A stronger condition, $μ<η$, is found when $η>0.38$, in agreement with experimental results obtained for $η=0.8$. Whatever the gap size, instability is predicted for values of $μ$ larger than the critical one, $μ_c=η^2$, corresponding to centrifugal instability.