PARAMETERIZING TURBULENT-DIFFUSION THROUGH THE JOINT PROBABILITY DENSITY

The "convective mass flux" parameterization often used in meteorological modeling expresses the vertical flux of a transported scalar as proportional to the product of the difference in mean values of the scalar in updrafts and downdrafts and their characteristic velocity. The proportionality factor is a constant to be specified. We show that this proportionality factor also appears in the "relaxed eddy accumulation" technique of Businger and Oncley. That associates the surface-layer flux of a scalar with the product of the standard deviation of vertical velocity and the mean concentration difference between updrafts and downdrafts. We show that this constant (b) is determined uniquely by the joint probability density (jpd) of vertical velocity and the scalar. Using large-eddy simulation, we generate this jpd for a conservative scalar diffusing through a convective boundary layer. It has quite different forms in "top-down" and "bottom-up" diffusion geometries. The bottom-up jpd is fairly well represented by a jointly Gaussian form and implies b approximately 0.6, in good agreement with the surface-layer value reported by Businger and Oncley. The top-down jpd is strikingly non-Gaussian and gives b approximately 0.47. Updrafts carry the bulk of the scalar flux - 70% in the bottom-up case, 60% in the top-down case.