Asymptotic analysis of mixing in stratified turbulent flows, and the conditions for an inertial sub-range

Abstract

In an important study, Maffioli et al. (J. Fluid Mech., Vol. 794 , 2016) used a scaling analysis to predict that in the weakly stratified flow regime $F{r}_h\gg 1$ ($F{r}_h$ is the horizontal Froude number), the mixing coefficient $\Gamma$ (defined as the ratio of the dissipation rates of potential to kinetic energy) scales as $\Gamma \sim O(F{r}_h^{-2})$. Direct numerical simulations confirmed this result, and also indicated that for the strongly stratified regime $F{r}_h\ll 1$, $\Gamma \sim O(1)$. Furthermore, the study argued that $\Gamma$ does not depend on the buoyancy Reynolds number $R{e}_b$, but only on $F{r}_h$. We present an asymptotic analysis to predict theoretically how $\Gamma$ should behave for $F{r}_h\ll 1$ and $F{r}_h\gg 1$ in the limit $R{e}_b\to \infty$. To correctly handle the singular limit $R{e}_b\to \infty$ we perform the asymptotic analysis on the filtered Boussinesq-Navier-Stokes equations, and demonstrate the precise sense in which the inviscid scaling analysis of Billant \& Chomaz (Phys. Fluids, vol. 13, 1645-1651, 2001) applies to viscous flows with $R{e}_b\to \infty$. The analysis yields $\Gamma \sim O(F{r}_h^{-2}(1+F{r}_h^{-2}))$ for $F{r}_h\gg 1$ and $\Gamma \sim O(1+F{r}_h^2)$ for $F{r}_h\ll 1$, providing a theoretical basis for the numerical observation made by Maffioli et al, as well as predicting the sub-leading behavior. Our analysis also shows that the Ozmidov scale $L_O$ does not describe the scale below which buoyancy forces are sub-leading, which is instead given by $O(F{r}_h^{1/2}{L}_O)$, and that the condition for there to be an inertial sub-range when $F{r}_h\ll 1$ is not $R{e}_b\gg 1$, but the more restrictive condition $R{e}_b\gg F{r}_h^{-4/3}$.