The limiting form of inertial instability in geophysical flows

Abstract

The instability of a rotating, stratified flow with arbitrary horizontal cross-stream shear is studied, in the context of linear normal modes with along-stream wavenumberkand vertical wavenumberm. A class of solutions are developed which are highly localized in the horizontal cross-stream direction around a particular streamline. A Rayleigh–Schrödinger perturbation analysis is performed, yielding asymptotic series for the frequency and structure of these solutions in terms ofkandm. The accuracy of the approximation improves as the vertical wavenumber increases, and typically also as the along-stream wavenumber decreases. This is shown to correspond to a near-inertial limit, in which the solutions are localized around the global minimum offQ, wherefis the Coriolis parameter andQis the vertical component of the absolute vorticity. The limiting solutions are near-inertial waves or inertial instabilities, according to whether the minimum value offQis positive or negative.We focus on the latter case, and investigate how the growth rate and structure of the solutions changes withmandk. Moving away from the inertial limit, we show that the growth rate always decreases, as the inertial balance is broken by a stabilizing cross-stream pressure gradient. We argue that these solutions should be described as non-symmetric inertial instabilities, even though their spatial structure is quite different to that of the symmetric inertial instabilities obtained whenkis equal to zero.We use the analytical results to predict the growth rates and phase speeds for the inertial instability of some simple shear flows. By comparing with results obtained numerically, it is shown that accurate predictions are obtained by using the first two or three terms of the perturbation expansion, even for relatively small values of the vertical wavenumber. Limiting expressions for the growth rate and phase speed are given explicitly for non-zerok, for both a hyperbolic-tangent velocity profile on anf-plane, and a uniform shear flow on an equatorial β-plane.