Routes towards homogeneity in stratified turbulent flows

Quantifying the rate at which a stratified turbulent flow mixes a density field is of crucial importance for many environmental and industrial applications. In the absence of molecular diffusion $\kappa$ (i.e. in the absence of irreversible mixing), a stratified turbulent flow forced so as to have a constant kinetic energy will converge towards a statistical steady state whose density field geometric properties depend on the Richardson number $Ri$ (defined as the ratio of the kinetic energy in the flow to the amount of energy required to overturn the full water column). This statistical steady state is reached after vertically disturbed fluid parcels have explored the depth that is accessible energetically and have returned to their neutrally buoyant position, i.e. after a ‘resetting time’ $t_{R}$ . The magnitude of $t_{R}$ is controlled by stratification strength $N$ and the buoyancy Reynolds number $Re_{b}$ , quantifying the ratio between the Kolmogorov and Ozmidov scales, and hence the range of scales effectively unaffected by stratification. When $\kappa \neq 0$ , a second time scale needs to be considered: the mixing time scale $t_{M}$ . Within a mixing time, diffusion smooths the density field. We show that the ratio of the mixing and resetting times $t_{M}/t_{R}$ , as well as $Ri$ , control how fast stratified turbulent flows mix a density field into a fully homogeneous state and, hence, the history of mixing in such flows. In particular, we identify three regions in the $(Ri,t_{M}/t_{R})$ parameter space for which the time evolution of measures of mixing is controlled by different algebraic combinations of $t_{M}/t_{R}$ and $Ri$ . These scaling laws are compared with idealised direct numerical simulations. Using these findings, we propose a simple model for the time evolution of the density histogram in stratified turbulent flows.