Rapid distortion theory for mixing efficiency of a flow stratified by one or two scalars

The mixing efficiency of unsheared homogeneous turbulence in flows stratified by one or two active scalars was calculated with rapid distortion theory (RDT). For one scalar the mixing efficiency $\eta $depends on the Schmidt number and the Grashof number. For two scalars the efficiency also depends on the density ratio $R_{\rho }$, which compares the density differences caused by temperature and salt. In the one scalar case when \textit{Gr }is large, $\eta $ decreases as \textit{Sc} increases. The mixing efficiency increases with \textit{Gr} up to a maximum value, as in numerical simulations and experiments. The maximum of approximately 30{\%} for low \textit{Sc} is consistent with simulations, while the maximum of 6{\%} for heated water is consistent with laboratory measurements. However, RDT underpredicts the maximum for saltwater and the value of \textit{Gr} at which the efficiency becomes constant. For two active scalars, $\eta $ decreases as $R_{\rho }$ decreases, as in experiments. Results from simulations with low \textit{Sc} likely overestimate the efficiency of turbulence in strongly stratified flows in lakes and oceans.