In contrast to the Miles-Howard theorem (Miles, 1961 JFM; Howard, 1961 JFM) for inviscid steady shear flow in stably stratified fluids, in an earlier work the authors constructed explicit elementary time periodic solutions of the Boussinesq equations that are unstable at arbitrarily large Richardson numbers. These solutions are parameterized by the Richardson number and the effective shear of the flow at the period time. Elementary flows with zero effective shear are purely vortical and are mathematically equivalent to solutions of a nonlinear pendulum equation; flows with maximum effective shear are simple shear flows. All other flows present mixed behaviour with the features of the both types above.
Through direct numerical simulation the authors showed that instability of purely vortical flows spontaneously generates local shears on buoyancy time scales near a specific angle of inclination which saturates into a localized regime of strong mixing with density overturning. The phase of these instabilities does not agree with that for internal gravity waves. We speculate that such instabilities may contribute significantly to the step-like micro-structure often observed in buoyancy measurements in the ocean. On the other hand, direct numerical simulations demonstrate that small perturbations of shear flows present transient amplification, but never generate overturning for large enough Richardson numbers. In this context the effective shear plays a crucial role. Here we review these recent results and also show that elementary flows with large effective shear are nonlinearly stable and shear-like, whereas flows with smaller effective shear develop instabilities and are vortical-like with strong overturning and mixing. We show that for each Richardson there exists a threshold value of the effective shear that separates stable and unstable flows.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics