Unbalanced instabilities of rapidly rotating stratified shear flows

The linear stability of a rotating, stratified, inviscid horizontal plane Couette flow in a channel is studied in the limit of strong rotation and stratification. Two dimensionless parameters characterize the flow: the Rossby number ǫ, defined as the ratio of the shear to the Coriolis frequency and assumed small, and the ratio s of the Coriolis frequency to the buoyancy frequency, assumed to satisfy s ≤ 1. An energy argument is used to show that unstable perturbations must have large, O(ǫ) wavenumbers. This motivates the use of a WKB-approach which, in the first instance, provides an approximation for the dispersion relation of the various waves that can propagate in the flow. These are Kelvin waves, trapped near the channel walls, and inertia-gravity waves with or without turning points. Although, the wave phase speeds are found to be real to all algebraic orders in ǫ, we establish that the flow is unconditionally unstable. This is the result of linear resonances between waves with oppositely signed wave momenta. Three modes of instabilities are identified, corresponding to the resonance between (i) a pair of Kelvin waves, (ii) a Kelvin wave and an inertia-gravity wave, and (iii) a pair of inertia-gravity waves. Whilst all three modes of instability are active when the Couette flow is anticyclonic, mode (iii) is the only possible instability mechanism when the flow is cyclonic. We derive asymptotic estimates for the instability growth rates. These are exponentially small in ǫ, of the form Imω = a exp(−Ψ/ǫ) for some positive constants a and Ψ. For the Kelvin-wave instabilities (i), we obtain analytic expressions for a and Ψ; the maximum growth rate, in particular, corresponds to Ψ = 2. For the other types of instabilities, we make the simplifying assumption s ≪ 1 and find that Ψ = 2.80 for (ii) and Ψ = π for (iii). The asymptotic results are confirmed by numerical computations. These reveal, in particular, that the instabilities (iii) have much smaller growth rates in cyclonic flows than in anticyclonic flows, in spite of having both Ψ = π. Our results, which extend those of Kushner et al. (1998) and Yavneh et al. (2001), highlight the limitations of the so-called balanced models, widely used in geophysical fluid dynamics, which filter out Kelvin and inertia-gravity waves and hence predict the stability of the Couette flow. They are also relevant to the stability of Taylor–Couette flows and of astrophysical accretion discs.