Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments

Abstract

A direct method for analyzing diapycnal mixing in a stably stratified fluid (Winters et al., 1995) has been applied to the stably stratified shear layer. The diapycnal flux and mixing efficiency are computed as functions of time, whatever the turbulent activity in the fluid. The mixing properties of two- and three-dimensional numerical simulations of the Boussinesq equations are analyzed and compared. The interest of the former simulations is to emphasize the fundamental role of three-dimensional effects in fluid mixing and to quantify it. We focus on the influence of stratification (measured by the minimum Richardson number J) and changes in Prandtl number on the overall mixing that occurs as the computed flows evolve from unstable initial conditions.In three dimensions, the flow dynamics exhibit three successive stages, each with different mixing properties. During the first stage, a primarily two-dimensional Kelvin–Helmholtz instability develops and the mixing efficiency is high (the flux Richardson number Rfb ranges between 0.37 and 0.68, decreasing as J increases). The second stage is characterized by the development of small-scale three-dimensional instabilities. These motions result in significantly higher diapycnal flux than during the first stage but in only moderate mixing efficiency (Rfb≃0.32), as the rate of kinetic energy dissipation is also high during this stage. Finally, the turbulent activity is progressively expulsed toward the outer regions of the shear layer and decays in time while the central region relaminarizes. During this final stage, Rfb approaches an asymptotic value close to 0.25 and the diapycnal diffusivity displays a clear functional dependence on a gradient Richardson number Rib of the form Rib−2.As expected, the two-dimensional flows are unable to reproduce the mixing properties of the flow, except during the first stage. During the subsequent turbulent regime, both the diapycnal flux and the dissipation rate of kinetic energy are too small (because, for the latter quantity, of the nonlinear enstrophy conservation constraint). The final stage consists in a quasi-stationary weakly turbulent regime, for which the diapycnal diffusivity behaves as Rib−1. It should be noted that, despite these differences, Rfb relaxes toward the 0.25 value found in three dimensions.