The formation of shear and density layers in stably stratified turbulent flows: linear processes

Abstract

The initial evolution of the momentum and buoyancy fluxes in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s. velocity u′0 and integral lengthscale l0 is calculated using a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) in order to study the growth of small temporal and spatial perturbations in the large-scale mean stratification N(z, t) and mean velocity profile ū(z, t) (here N is the local Brunt–Väisälä frequency and ū is the local velocity of the horizontal mean flow) when the ratio of buoyancy forces to inertial forces is large, i.e. Nl0/u′0[Gt ]1. The lengthscale L of the perturbations in the mean profiles of stratification and shear is assumed to be large compared to l0 and the presence of a uniform background mean shear can be taken into account in the model provided that the inertial shear forces are still weaker than the buoyancy forces, i.e. when the Richardson number Ri = (N/∂zū)2[Gt ]1 at each height.When a mean shear perturbation is introduced initially with no uniform background mean shear and uniform stratification, the analysis shows that the perturbations in the mean flow profile grow on a timescale of order N-1. When the mean density profile is perturbed initially in the absence of a background mean shear, layers with significant density gradient fluctuations grow on a timescale of order N−10 (where N0 is the order of magnitude of the initial Brunt–Väisälä frequency) without any associated mean velocity gradients in the layers. These results are in good agreement with the direct numerical simulations performed by Galmiche et al. (2002) and are consistent with the earlier physically based conjectures made by Phillips (1972) and Posmentier (1977). The model also shows that when there is a background mean shear in combination with perturbations in the mean stratification, negative shear stresses develop which cause the mean velocity gradient to grow in the density layers. The linear analysis for short times indicates that the scale on which the mean perturbations grow fastest is of order u′0/N0, which is consistent with the experiments of Park et al. (1994).We conclude that linear mechanisms are widely involved in the formation of shear and density layers in stratified flows as is observed in some laboratory experiments and geophysical flows, but note that the layers are also significantly influenced by nonlinear and dissipative processes at large times.