Countergradient Transport Revisited

Abstract

A modified diffusion equation for a passive scalar is proposed. It is similar (but not equal) to the ?telegraph? equation and describes also ?finite-velocity? diffusion. It is shown that earlier countergradient, or better, nonlocal diffusion theories for the quasi-steady convective boundary layer, are special cases of this diffusion equation. The Reynolds-averaged equations give support for the particular form of the equation that exhibits the nonlocal behavior. Solutions of the diffusion equation were compared with a large eddy simulation and the agreement is fair, both for the top-down and bottom-up diffusion. It appears that the general shape of the convective boundary layer (CBL) profiles for mean potential temperature or a passive scalar is largely determined by the vertical inhomogeneities in the CBL turbulence profiles of the variance and timescale and to a lesser extent by the skewness. The modified equation offers a simple framework to include more physics in the description of dispersion in a CBL. It leads generally to higher concentration predictions in the upper 30% of the CBL, and therefore, estimates of entrainment based on concentration differences between a (well mixed) boundary layer and inversion layer concentration may be too high and can be corrected using the nonlocal formulation, though corrections will generally be small.