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.

Linking dissipation, anisotropy, and intermittency in rotating stratified turbulence at the threshold of linear shear instabilities

“…These scaling laws extend smoothly to the purely stratified flows we have analyzed, where, for regime II, the reduced mixing efficiency was found in 4 to scale linearly with F r. These results are also compatible with other results obtained for that case (e.g. 13,14,[25][26][27] and references therein).…”

Turbulence in forced stratified shear flows

“…There is no significant evidence of an asymptote to a constant as proposed by Maffioli et al. (2016) and Garanaik & Venayagamoorthy (2019) for small Froude numbers. However, our simulations differ from those reported in Maffioli et al.…”

“…2016 b ; Venayagamoorthy & Koseff 2016; Zhou et al. 2017 b ; Garanaik & Venayagamoorthy 2019). Here, we diagnose the dependence of the time-dependent mixing efficiency in reference buoyancy coordinates (as defined in (5.2)) on several (also time-dependent) non-dimensional numbers to understand these relationships in a continuously forced system.…”

“…Specifically, Maffioli et al. (2016) proposed that and Garanaik & Venayagamoorthy (2019) proposed an additional intermediate scaling for , where in both cases we have written the scaling in terms of variables calculated in sorted buoyancy coordinates. Since the Reynolds number, buoyancy Reynolds number and Froude number are related through by definition (e.g.…”

The dynamics of stratified horizontal shear flows at low Péclet number

“…In addition to verifying (1.14 a , b ) and (1.15), we are also interested in testing the assumption that all energy dissipation is exclusively viscous. Although this assumption is superficially plausible, there is growing evidence (Maffioli, Brethouwer & Lindborg 2016; Garanaik & Venayagamoorthy 2019) for flows with that non-trivial irreversible mixing converting kinetic energy into potential energy continues to occur even in the limit of extremely strong stratification.…”