Mixing efficiency for forced, stationary, stratified turbulence: when is it constant?

Due to the large range of spatial and temporal scales of turbulent, environmental flows, larger-domain simulations (e.g., global and regional simulations) have often used an eddy diffusivity to account for the irreversible mixing of density gradients, occurring at small length scales. To estimate this eddy diffusivity, one needs to know the mixing efficiency, which involves the ratio of the buoyancy flux to the sum of the mechanical sources of turbulent kinetic energy (Ivey and Imberger (1991)). Recently, a set of forced, sheared, stably-stratified DNS runs by Portwood et al. (2019) showed that the stationary value of the mixing efficiency remained around 0.2 for buoyancy Reynolds numbers between 36 and 900. These results are ostensibly contrary to the values expected at buoyancy Reynolds numbers above ~O(100) (e.g., Monismith et al. (2018)).

In an attempt to reconcile this discrepancy, we consider the effects of forcing on the stationary value of mixing efficiency over a range of flow conditions. We consider a set of DNS runs at different turbulent Froude numbers (defined as one over the product of the large-eddy time scale and buoyancy frequency) while keeping the Prandtl number and desired turbulence Reynolds numbers fixed. Stationary flow states are achieved through time-varying forcing based on methods suggested by Bassenne et al. (2016). In addition, new expressions, derived to calculate the mixing efficiency for stationary, homogeneous flows explicitly accounting for the time-varying forcing, are evaluated and analyzed.