Quasi-Steady Analysis of a PBL Model with an Eddy-Diffusivity Profile and Nonlocal Fluxes

Abstract

Analytic solutions to a planetary boundary layer (PBL) model with an eddy-diffusivity profile (i.e., a K profile) and nonlocal fluxes are presented for the quasi-steady regime. The solutions demonstrate how different processes contribute to the quasi-steady profiles of heat and/or other scalars in the convective boundary layer. It is shown that for a standard cubic form of the K profile, and flux scales based on the surface fluxes, the nondimensional nonlocal term should be less than six; larger values can cause scalar profiles of water vapor to increase with height in the upper portion of the PBL and can produce weakly superadiabatic layers in the upper PBL temperature profiles. Solutions are also shown to be sensitive to the choice of flux scale: fluxes scaled by their vertically averaged values imply that nondimensional profiles of top-down scalars will have a neutral point somewhere in the PBL, a result in conflict with previous work on the subject, and the predictions of the same model with fluxes scaled by their surface values. The analysis also shows that allowing K to go to zero with the square of the distance from the PBL top results in nonconvergent profiles; in general K should reduce to some positive value at the top of the PBL, or go to zero less rapidly. It is further shown that the class of models investigated here may be physically interpreted as relaxation models, that is, they tend to relax profiles of scalars in the PBL to implicitly defined similarity profiles on a convective timescale. Finally, analysis of a 1-yr integration of a climate model, interpreted in light of the author’s analytic results, suggests that a dynamically important aspect of the nonlocal term is its role in ventilating the surface layer, and thereby indirectly affecting the diagnoses of PBL depth in many models.