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Bounds on mixing efficiency and Richardson number in stably stratified turbulent shear flow

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The Miles-Howard theorem is a classical result, which has been extremely influential in the study of stratified shear flow. The theorem states that if the local Richardson number (i.e. the ratio of the buoyancy frequency to the square of the velocity shear) throughout a laminar inviscid stratified shear flow is everywhere greater than a quarter, the flow is stable to two-dimensional infinitesimal normal mode perturbations. Though heuristic energy arguments are commonly presented, and similar criteria based around bulk Richardson numbers (i.e. the ratio of the overall reduced gravity times the layer depth to the square of the velocity difference) are widely used to parameterize the mixing behaviour in fully nonlinear turbulent flows, rigorous theoretical results for flow stabilization by strong stratification have been elusive. We derive such a nonlinear result for a model flow (stratified Couette flow, where the top and bottom boundaries are set at constant relative velocity, and constant, statically stable densities) by generating rigorous bounds on the long-time average of the buoyancy flux, subject to the requirement that the ratio between the buoyancy flux and the forcing (i.e. the “mixing efficiency”) is an (arbitrary) constant, demonstrating that a statistically steady state is only possible for sufficiently small values of the bulk Richardson number. Conversely, for a given (sufficiently small) Richardson number, we show that the mixing efficiency has a strict lower bound within this model flow.

This talk is part of the Institute for Energy and Environmental Flows (IEEF) series.

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