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

In this study, we evaluate four different parameterizations of the turbulent Prandtl (Schmidt) number Prt = νt/Γt where νt is the eddy viscosity and Γt is the scalar eddy diffusivity, for stably stratified flows. All four formulations of Prt are strictly functions of the gradient Richardson number Ri, which provides a measure of the strength of the stratification. A zero-equation (i.e. no extra transport equations are required) turbulence model for νt in a one-dimensional, turbulent channel flow is considered to evaluate the behavior of the different formulations of Prt. Both uni-directional and oscillatory flows are considered to simulate conditions representative of practical flow problems such as atmospheric boundary layer flows and tidally driven estuarine flows, to quantify the behavior of each of the four formulations of Prt. We perform model-to-model comparisons to highlight which of the models of Prt allow for a higher rate of turbulent mixing and which models significantly inhibit turbulent mixing in the presence of buoyancy forces resulting from linear (continuous) stratification as well as two-layer stratification. The basis underlying the formulation of each model in conjunction with the simulation results are used to emphasize the considerable variability in the different formulations and the importance of choosing an appropriate parameterization of Prt given a model for νt in stably stratified flows.

► We evaluate four formulations of the turbulent Prandtl for stably stratified flows.
► We use relative model comparisons to show the differences between the formulations.
► We demonstrate the use of a simple 1-D model to study the effects of stable stratification.