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To describe the heat and scalar fluxes in the convective boundary layer, we propose expressions for eddy diffusivities and countergradient terms. The latter expressions can be used in a modified flux-gradient approach, which takes account for nonlocal convective vertical exchange. The results for heat are based on a derivation similar to that of Deardorff by utilizing the turbulent heat-flux equation, but the closure assumptions applied to the heat-flux budgets are different. As a result, the physical interpretation for the countergradient term differs; our countergradient term results from the third-moment transport effect, while Deardorff's results from the buoyancy production term. On the basis of our analysis, we are able to calculate an eddy diffusivity for heat, using large-eddy simulation results. The results are presented in the form of a similarity profile, using the convective velocity scale w* and the inversion height zi. It is shown that the latter profile is well behaved and that it matches the results of surface-layer theory. Using the top-down and bottom-up decomposition, we have generalized our findings for any scalar, such as the moisture field or an air pollution contaminant. We show that the eddy diffusivity profile for scalar C is sensitive to the entrainment–surface flux ratio of C. Therefore, a different scalar field should have a different eddy-diffusivity profile. The proposed expressions for the eddy diffusivities and the countergradient terms are easy to apply in (large-scale) atmospheric and diffusion models.
Holtslag et al. (Mon,) studied this question.