Evaluation of turbulent Prandtl (Schmidt) number parameterizations for stably stratified turbulent flows

In this study, we evaluate four different formulations of the turbulent Prandtl (Schmidt) number $Pr_t=\nu_t/\Gamma_t$ where $\nu_t$ is the eddy viscosity and $\Gamma_t$ is the scalar eddy diffusivity, for stably stratified flows. All four formulations of $Pr_t$ are strictly functions of the gradient Richardson number $Ri$ which is a measure of the strength of the stratification. A zero equation turbulence model for the eddy viscosity $\nu_t$ in a one-dimensional, turbulent channel flow is considered to evaluate the behavior of the different formulations of $Pr_t$. Both uni-directional and oscillatory flows are considered to simulate conditions representative of practical flow problems such as atmospheric flows and tidally-driven estuarine flows, to quantify the behavior of each of the four formulations of $Pr_t$. We discuss which of the models of $Pr_t$ allow for a higher rate of turbulent mixing and which models significantly inhibit turbulent mixing in the presence of density stratification. The basis underlying the formulation of each model in conjunction with the simulation results are used to highlight the importance of choosing the appropriate parameterization of $Pr_t$, given a model for $\nu_t$ in for stably stratified flows.