The role of vertical shear in the slow return to isotropy in rotating stratified turbulent flows

Abstract:

We examine some of the characteristics of a return to isotropy for rotating stratified turbulence through the analysis of a large number of direct numerical simulations performed in a cubic box in the absence of forcing, with spectral accuracy and mostly with 1024³ data points. Initial conditions are either random for the velocity and no temperature fluctuations, or else in quasi-geostrophic (QG) equilibrium. Froude numbers, Fr, measuring how fast the waves are with respect to nonlinear eddies, vary from 0.001 to a few. Reynolds numbers, Re, vary from 1600 to 55,000, and rotation is moderate to weak, with Rossby numbers as low as 0.1. Such flows are known to display three regimes of dissipation in terms of Fr, and they also show a remarkable correlation between the Richardson number Ri (measuring vertical shear of horizontal winds) and the buoyancy Reynolds number R_IB (measuring the efficacy of dissipation) throughout the whole range of parameters and covering the three regimes. This emphasizes the importance of vertically sheared horizontal winds in the dynamics of such flows and their achieving localized balance with dissipation, making the fluid prone to instabilities close to a critical threshold of (local, gradient) Richardson number Ri_g~ 0, even for low Froude numbers when the fluid is strongly stratified. We concurrently find a slow return to isotropy for the velocity, as well as for its gradients, for temperature gradients and even more so for the vorticity. A clear transition occurs for Richardson and Froude numbers of order unity, i.e. when the flow globally reaches the fully turbulent regime. The strong vorticity anisotropy can be linked to the persistence of intense localized small-scale vortices, for example in the lanes separating large-scale eddies which are themselves close to a QG state.