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

A direct method for analyzing diapycnal mixing in a stably stratified fluid (Winters et al., 1995) has been applied to the stably stratified shear lay er. The diapycnal flux and mixing efficiency are computed as functions of t ime, whatever the turbulent activity in the fluid. The mixing properties of two- and three-dimensional numerical simulations of the Boussinesq equatio ns 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 ov erall mixing that occurs as the computed flows evolve from unstable initial conditions. In three dimensions, the flow dynamics exhibit three successiv e stages, each with different mixing properties. During the first stage, a primarily two-dimensional Kelvin-Helmholtz instability develops and the mix ing efficiency is high (the flux Richardson number Rf(b) ranges between 0.3 7 and 0.68, decreasing as J increases). The second stage is characterized b y the development of small-scale three-dimensional instabilities. These mot ions result in significantly higher diapycnal flux than during the first st age but in only moderate mixing efficiency (Rf(b) similar or equal to 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 rel aminarizes. During this final stage, Rf(b) approaches an asymptotic value c lose to 0.25 and the diapycnal diffusivity displays a clear functional depe ndence on a gradient Richardson number Ri(b) of the form Ri(b)(-2.) As expe cted, the two-dimensional flows are unable to reproduce the mixing properti es of the flow, except during the first stage. During the subsequent turbul ent regime, both the diapycnal flux and the dissipation rate of kinetic ene rgy are too small (because, for the latter quantity, of the nonlinear enstr ophy conservation constraint). The final stage consists in a quasi-stationa ry weakly turbulent regime, for which the diapycnal diffusivity behaves as Ri(b)(-1). It should be noted that, despite these differences, Rf(b) relaxe s toward the 0.25 value found in three dimensions. (C) 2000 The Japan Socie ty of Fluid Mechanics and Elsevier Science B.V. All rights reserved.