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.
VERTICAL VARIATION OP UNDERTOW IN THE SURF ZONE