On the inference of the state of turbulence and mixing efficiency in stably stratified flows

Abstract

Scaling arguments are presented to quantify the widely used diapycnal (irreversible) mixing coefficient $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D716}_{PE}/\unicode[STIX]{x1D716}$ in stratified flows as a function of the turbulent Froude number $Fr=\unicode[STIX]{x1D716}/Nk$. Here, $N$ is the buoyancy frequency, $k$ is the turbulent kinetic energy, $\unicode[STIX]{x1D716}$ is the rate of dissipation of turbulent kinetic energy and $\unicode[STIX]{x1D716}_{PE}$ is the rate of dissipation of turbulent potential energy. We show that for $Fr\gg 1$, $\unicode[STIX]{x1D6E4}\propto Fr^{-2}$, for $Fr\sim \mathit{O}(1)$, $\unicode[STIX]{x1D6E4}\propto Fr^{-1}$ and for $Fr\ll 1$, $\unicode[STIX]{x1D6E4}\propto Fr^{0}$. These scaling results are tested using high-resolution direct numerical simulation (DNS) data from three different studies and are found to hold reasonably well across a wide range of $Fr$ that encompasses weakly stratified to strongly stratified flow conditions. Given that the $Fr$ cannot be readily computed from direct field measurements, we propose a practical approach that can be used to infer the $Fr$ from readily measurable quantities in the field. Scaling analyses show that $Fr\propto (L_{T}/L_{O})^{-2}$ for $L_{T}/L_{O}>O(1)$, $Fr\propto (L_{T}/L_{O})^{-1}$ for $L_{T}/L_{O}\sim O(1)$, and $Fr\propto (L_{T}/L_{O})^{-2/3}$ for $L_{T}/L_{O}<O(1)$, where $L_{T}$ is the Thorpe length scale and $L_{O}$ is the Ozmidov length scale. These formulations are also tested with DNS data to highlight their validity. These novel findings could prove to be a significant breakthrough not only in providing a unifying (and practically useful) parameterization for the mixing efficiency in stably stratified turbulence but also for inferring the dynamic state of turbulence in geophysical flows.