Parameterization of turbulent diffusivity in stratified flows using microstructure observations and DNS

In oceanic flows, the eddy diffusivity of density, $K_{d}$, is commonly approximated using the Osborn-Cox model with a constant mixing efficiency, $\Gamma$. Many have sought to improve upon the accuracy of this approach by parameterizing the variability in $\Gamma$ using the buoyancy Reynolds number (\textit{Re}$_{B}$=$\varepsilon$/$\nu N^{2})$. In this study, we point out that \textit{Re}$_{B}$=$Fr^{2}$\textit{Re}$_{L}$ (where \textit{Fr=$\varepsilon$/ kN} and \textit{Re}$_{L}=k^{2}$/$\varepsilon \nu$) and is, thus, a mixed parameter that obscures explicit dependencies on the more fundamental parameters involving turbulent kinetic energy, $k$. Using microstructure observations, we demonstrate this non-uniqueness of \textit{Re}$_{B}$ and explore the independent effects of \textit{Fr} and \textit{Re}$_{L}$. Because $k$ is not readily available from microstructure measurements, however, we investigate alternative methods to infer its value from measured Thorpe scales, $L_{T}$. Through physical reasoning, we argue that $L_{T}$ should scale with a length scale dependent on $k$ and not solely on dissipation $\varepsilon$. We test this reasoning using DNS of decaying grid turbulence.