Direct numerical simulation of turbulence–mean field interactions in a stably stratified fluid

Abstract

Freely decaying turbulent flows in a stably stratified fluid are simulated with a pseudo-spectral numerical code solving the fully nonlinear Navier–Stokes equations under the Boussinesq approximation with periodic boundary conditions. The flow is decomposed into a turbulent field and a horizontal mean flow ū(z, t) defined as the average of the horizontal velocity component in a horizontal plane at height z and time t. Similarly, the density field is decomposed into a turbulent field and a (stable) mean density profile ρ(z, t) defined as the average of the density field in a horizontal plane at height z and time t. Attention is paid to the effect of the turbulent velocity field on an initial z-periodic horizontal mean flow (Simulation A) or an initial z-periodic perturbation of the mean density profile (Simulation B). Both A and B are performed under conditions of moderate and strong stratification and are compared to the non-stratified simulations.

Simulation A shows that the turbulence–mean flow interaction is strongly affected by the buoyancy forces. In the absence of a stratification, the mean flow perturbation decays rapidly due to the turbulent diffusion of momentum. When a moderate stratification is applied, the mean flow perturbation decays much more slowly whereas it oscillates and grows with time when the stratification is strong. These results are interpreted by defining a time-dependent eddy viscosity. Whereas the eddy viscosity coefficient has positive values in the non-stratified simulation, it is affected by the buoyancy forces and decreases after a period of order N⁻¹. For large times, the eddy diffusivity oscillates and its time-averaged value over a few turnover timescales is positive but small when the stratification is moderate, and roughly zero when the stratification is strong. These results are interpreted by defining a time-dependent eddy viscosity. Whereas the eddy viscosity coefficient has positive values in the non-stratified simulation, it is affected by the buoyancy forces and decreases after a period of order N⁻¹ in the stratified simulations (where N is the Brunt–Väisälä frequency associated with the background linear stratification). At large time, we find that the eddy viscosity remains roughly zero when the stratification is moderate, whereas it oscillates but remains persistently negative in the strongly stratified case, which causes the horizontal mean flow to accelerate.

We conclude that the presence of a stable stratification strongly affects the temporal behaviour of the mean quantities ū and ρ in turbulent flows and partly explains the formation of horizontal layers in stratified geofluids such as oceans and atmospheres.