On the turbulent Prandtl number in homogeneous stably stratified turbulence

In this paper, we derive a general relationship for the turbulent Prandtl number Pr-t for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Pr-t is developed in terms of a mixing length scale L-M and an overturning length scale L-E, the ratio of the mechanical (turbulent kinetic energy) decay time scale T-L to scalar decay time scale T-rho and the gradient Richardson number Ri. We show that Our formulation for Pr-t is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio L-M/L-E approximate to 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, T-L/T-rho. We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1-20; J. Fluid Mech., vol. 525, 2005, pp. 193-214) to propose a new parameterization for Pr-t in terms of the gradient Richardson number Ri. The formulation presented here provides a general framework for calculating Pr, that will be useful for turbulence closure schemes in numerical models.