Monin–Obukhov Similarity and Local-Free-Convection Scaling in the Atmospheric Boundary Layer Using Matched Asymptotic Expansions

Abstract

The Monin–Obukhov similarity theory (MOST) is the foundation for understanding the atmospheric surface layer. It hypothesizes that nondimensional surface-layer statistics are functions of [Formula: see text] only, where z and L are the distance from the ground and the Obukhov length, respectively. In particular, it predicts that in the convective surface layer, local free convection (LFC) occurs at heights [Formula: see text] and [Formula: see text], where [Formula: see text] is the inversion height. However, as a hypothesis, MOST is based on phenomenology. In this work we derive MOST and the LFC scaling from the equations for the velocity and potential temperature variances using the method of matched asymptotic expansions. Our analysis shows that the dominance of the buoyancy and shear production in the outer ([Formula: see text]) and inner ([Formula: see text]) layers, respectively, results in a nonuniformly valid solution and a singular perturbation problem and that [Formula: see text] is the thickness of the inner layer. The inner solutions are found to be functions of [Formula: see text] only, providing a proof of MOST for the vertical velocity and potential temperature variances. Matching between the inner and outer solutions results in the LFC scaling. We then obtain the corrections to the LFC scaling near the edges of the LFC region ([Formula: see text] and [Formula: see text]). The nondimensional coefficients in the expansions are determined using measurements. The resulting composite expansions provide unified expressions for the variance profiles in the convective atmospheric surface layer and show very good agreement with the data. This work provides strong analytical support for MOST.