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 the case with one scalar the mixing efficiency η depends on the Schmidt number Sc = ν/D and the Grashof number Gr = NL2/ν, where ν is the kinematic viscosity, D is the molecular diffusivity, N is the buoyancy frequency, and L is a length scale representative of the large eddies. For the case with two scalars the efficiency also depends on the density ratio Rρ, which compares the density difference caused by temperature and the density difference caused by salt. In the one scalar case when Gr is large, η decreases as Sc increases. The mixing efficiency increases with Gr up to a maximum value, as in numerical simulations and experiments. The maximum mixing efficiency of approximately 30% for low Sc is consistent with simulations, while the maximum efficiency of 6% for heated water is consistent with laboratory measurements. However, RDT underpredicts the maximum efficiency for saltwater and also the value of Gr at which the efficiency becomes constant. The predicted behavior of the mixing efficiency for two active scalars is similar to that for one scalar, and the efficiency decreases as Rρ decreases, as in experiments and semi-empirical models. These calculations show that results from simulations with low Sc likely overestimate the efficiency of turbulence in strongly stratified flows in lakes and oceans.