Strato-rotational instability without resonance

Abstract

Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between normal modes of the internal or Kelvin variety in three-dimensional settings in which the stratification and rotation are orthogonal to both the background flow and its shear. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by certain singular critical levels. These levels are not classical critical levels, where the phase speed $c$ of a normal mode matches the mean flow speed $U$, but are a different type of singularity where $(c-U)$ matches a characteristic gravity-wave speed $\pm N/k$, based on the buoyancy frequency $N$ and streamwise horizontal wavenumber $k$. Instead, it is shown that a variant of SRI can occur due to the coupling of a Kelvin or internal wave to such ‘baroclinic’ critical levels. Two characteristic situations are identified and explored, and the conservation law for pseudo-momentum is used to rationalize the physical mechanism of instability. The critical level coupling removes the requirement for resonance near specific wavenumbers $k$, resulting in an extensive continuous band of unstable modes.