Simple PDF models for convectively driven vertical diffusion

Abstract The mode of vertical velocity in convective boundary layers (CBLs) is usually negative and the probability distribution function (PDF) of w, Pw, is rarely symmetric except near the top and bottom of CBLs. Consequently, vertical diffusion from elevated sources is usually asymmetric and exhibits a descending mode of concentration, causing higher peak surface concentrations than predicted by Gaussian models. The main concentration (χ) effects, we argue, can be modeled using the simplest of PDF diffusion models, with tracers responding to Pw at the source height with straight line trajectories and simple reflection at the surface and zi, the mixing depth. The critical element is the choice of Pw. Two Pw models are offered, a bi-Gaussian (BG) and a Gaussian-ramp (GR) formulation. Both have some observational support, and the resulting PDF models are mathematically tractable. Analytical solutions for key variables are given; these show some surprising contrasts between the BG and GR models, but both can approximate laboratory and numerical modeling results for ∝χdy patterns. A diverse selection of atmospheric turbulence measurements is presented; for measures that reflect asymmetry in Pw, the data show wide ranges and do not lend support to any particular form of Pw. Recent lidar measurements of oil fog plumes are presented that show a large variability in ∝χdy patterns, even with substantial averaging periods. The only concurrent turbulence measurement that strongly correlates with the observed vertical diffusion of oil fog is the mode of wind elevation angle. A simple adaptation of the BG model is recommended that fits the average peak ∝χdy and distance of occurrence as observed so far.