The instability of stratified flows at large Richardson numbers

Abstract

In contrast to conventional expectations based on the stability of steady shear flows, elementary time-periodic stratified flows that are unstable at arbitrarily large Richardson numbers are presented here. The fundamental instability is a parametric one with twice the period of the basic state. This instability spontaneously generates local shears on buoyancy time scales near a specific angle of inclination that saturates into a localized regime of strong mixing with density overturning. We speculate that such instabilities may contribute significantly to the step-like microstructure often observed in buoyancy measurements in the ocean.One of the basic analytical results for stably stratified fluid flows is the celebrated Miles-Howard theorem (1, 2). This theorem states that steady shear flows V = v z 0 0 in an inviscid stably stratified fluid are linearly stable for all Richardson numbers, i, satisfyingwith N 2 = −g ∂ρ ∂z /ρ b , the square of the buoyancy or BruntVaisala frequency. This criterion for stability is often interpreted and applied literally for time-dependent flow fields in both theoretical and numerical modeling for the atmosphere or ocean. For example, a popular turbulent eddy diffusivity used in numerical simulations in the atmosphere/ocean community is the Lilly-Smagorinsky eddy diffusivity (3, 4), where the turbulent eddy diffusivity is completely switched off and set to zero for i > i > 1 4 with i of order unity. Here, we present elementary examples, with firm mathematical underpinnings, which demonstrate that such reasoning can be violated in dramatic fashion for time-dependent strongly stratified flows. With appropriate nondimensional units explained below, we consider solutions of the inviscid two-dimensional Boussinessq equations in vorticity-stream form,whereρ is fluid density,is the vorticity, ψ is the stream function, and the field velocity v is given byNext, we build elementary time-periodic mean states for the equations in 2, and then we demonstrate their linearized and nonlinear instability for large Richardson numbers. Time-Periodic Mean States and the Nonlinear PendulumWe construct elementary time-periodic exact solutions of the Boussinessq equations with constant spatial gradients withThe publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. $1734 solely to indicate this fact.© 1998 by The National Academy of Sciences 0027-8424/98/957850-4$2.00/0 PNAS is available online at http://www.pnas.org.the formρDirect calculation yields the fact that special solutions of 2 have the structure in 4 provided that the phase function, θ t , satisfies the nonlinear pendulum equationwith the initial dataImplicit in both 2 and 6 is a nondimensionalization where we use the ambient density, ρ b , the stably stratified initial vertical density gradient, and gravity, g, to set the relevant scales. Thus, the unit of time in 2 is determined by the constant buoyancy frequency, N, w...

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