Francois N. Frenkiel Award Talk: Relevance of the Thorpe length scale in stably stratified turbulence*

A relatively simple and objective measure of large-scale vertical overturns in turbulent oceanic flows is the Thorpe length scale, $L_{T}$. Reliance on common scaling between the Ozmidov length scale $L_{O}$ (which is a measure of the size of largest eddy unaffected by buoyancy in stratified turbulence) and $L_{T}$ is commonplace in the field of oceanography to infer the dissipation rate of turbulent kinetic energy $\varepsilon $. In this study, we use direct numerical simulations (DNS) of stably stratified turbulence to compare the Thorpe overturn length scale, $L_{T}$, with other length scales of the flow that can be constructed from large-scale quantities fundamental to shear-free, stratified turbulence. Quantities considered are the turbulent kinetic energy, $k$, its dissipation rate, $\varepsilon $, and the buoyancy frequency, $N$. Fundamental length scales are then the Ozmidov length scale, $L_{O}$, the isotropic large scale, $L_{k\varepsilon }$, and a kinetic energy length scale, $L_{kN}$. Behavior of all three fundamental scales, relative to $L_{T}$, is shown to be a function of the buoyancy strength parameter $NT_{L}$, where $T_{L} = k$/$\varepsilon$ is the turbulence time scale. When buoyancy effects are dominant (i.e., for $NT_{L} > 1$), $L_{T} $is shown to be linearly correlated with $L_{kN}$ and not with $L_{O}$ as is commonly assumed for oceanic flows. Agreement between $L_{O}$ and $L_{T}$ is only observed when the buoyancy and turbulence time scales are approximately equal (i.e., for the critical case when $NT_{L} \approx 1$). The relative lack of agreement between $L_{T}$ and $L_{O}$ in strongly stratified flows is likely due to anisotropy at the outer scales of the flow where the energy transfer rate differs from $\varepsilon $. The key finding of this work is that observable overturns in strongly stratified flows are more reflective of $k$ than $\varepsilon $. In the context of oceanic observations, this implies that inference of $k$, rather than $\varepsilon $, from measurements of $L_{T}$ is fundamentally correct when $NT_{L} \approx 1$ and most appropriate when $NT_{L} > 1$. Furthermore, we show that for $NT_{L} < 1$, $L_{T}$ is linearly correlated with $L_{k\varepsilon }$, \textit{when} mean shear is absent.