The different 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 absence of molecular diffusion (i.e. in the absence of mixing), a stratified turbulent flow forced so as to have a constant kinetic energy will converge towards a statistical steady state (or state of maximum 'disorder') whose density field geometric properties depends on the Richardson number Ri (defined as the ratio of the kinetic energy to the amount of energy required to overturn the full water column) [Venaille et al. 2017]. We hypothesise that this statistical steady state is reached within a 'resetting time' [Petropoulos et al. 2025], the magnitude of which is controlled by the dissipation rate of kinetic energy and stratification strength. When molecular diffusion is not zero, a second timescale needs to be taken into account—the mixing timescale; within a mixing time, diffusion smooths the density field disordered state. We hypothesise that the ratio r of the resetting and mixing times, as well as Ri, control how fast a stratified turbulent stirring field fully mixes a density field into a fully homogeneous state. In particular we identify three region in the Ri-r parameter space ('fast' mixing, 'slow' but energetic mixing, 'slow' and 'lazy' mixing) for which the time-evolution of measures of mixing (such as the instantaneous mixing efficiency, destruction rate of density variance or more generally the density field p.d.f., for instance) is controlled by different algebraic combinations of 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 field p.d.f. in stratified turbulent flows.