Turbulent mixing in a stratified shear layer with time-dependent forcing

We use direct numerical simulations to study the dynamics of the turbulence produced by stratified shear instability in tilted coordinates where the tilt angle continuously oscillates in time. Such a setup might model, for example, a layer of strong density stratification in the ocean undergoing shear oscillations due to a horizontally propagating gravity wave. Previous work by Inoue and Smyth (2008, J. Phys. Oceanogr., 39) has demonstrated that the mixing properties may be modified by the timing of the deceleration of the shear induced by the tilt oscillation. Here, we investigate in detail how the time dependence of the forcing relative to the development of growing Kelvin-Helmholtz instabilities modifies the energetic pathways that lead to turbulence at a high Reynolds number. We look at the quantification of mixing in terms of the change in the background potential energy (BPE), and in terms of the mixing efficiency, here defined as the ratio of the rate of increase in BPE to the total energy expended in producing the mixing. We discuss the importance of transient flow behaviour for the parameterization of mixing and outline the implications of our results for the development of practical parameterizations used to infer mixing rates from observational data.