Mixing efficiency in stratified turbulence

We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number Fr=εₖ/(Nu²) , where εₖ is the kinetic energy dissipation, u is a turbulent horizontal velocity scale and N is the Brunt–Väisälä frequency. For Fr 1 , in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as Γ∝ Fr⁻² , where Γ=εₚ/εₖ and εₚ is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with Fr 1 , we argue that Γ should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of $Fr$ and confirm that at high $Fr$ , Γ∝ Fr⁻² , while at low $Fr$ it approaches a constant value close to Γ=0.33 . The parametrization of Γ based on Reb due to Shih et al. ( J. Fluid Mech. , vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of Γ in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value $Fr=0.3$ found in our simulations.