Mixing efficiency dependence on overturning and turbulence intensity in stratified shear layers

It is well-known that both the total amount of irreversible mixing and its efficiency in stratified shear flows are strongly time-dependent. We consider shear layers that are susceptible to primary Kelvin-Helmholtz instabilities, developing relatively large billow overturnings that in turn are subject to various secondary instabilities which trigger turbulence transition. Valuable insights can be gained by considering the time-dependence of three characteristic length scales of the flows: the overturning Thorpe scale $L_T$; the largest turbulence scale unaffected by stratification known as the Ozmidov scale $L_O=\sqrt{\epsilon/N^3}$; and the Kolmogorov scale $L_K=(\nu^3/\epsilon)^{1/4}$, where $\epsilon$ is the kinetic energy dissipation rate, $\nu$ is the kinematic viscosity, and $N$ is the buoyancy frequency. Provided $L_O/L_K$ is sufficiently large, we show that $L_T$ first grows as the primary billow develops, but then falls rapidly as the turbulence onsets and $L_O$ increases in turn and then decays more slowly, leading to a typical monotonic increase in the ratio $L_O/L_T$ with time. Both the most efficient and the most vigorous mixing occurs when $L_T \simeq L_O$, which has important implications for the interpretation and modelling of real oceanic mixing events.