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

Chenning Tong Department of Mechanical Engineering, Clemson University, Clemson, South Carolina

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Mengjie Ding Department of Mechanical Engineering, Clemson University, Clemson, South Carolina

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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 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 and , where 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 () and inner () layers, respectively, results in a nonuniformly valid solution and a singular perturbation problem and that is the thickness of the inner layer. The inner solutions are found to be functions of 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 ( and ). 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.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Prof. Chenning Tong, ctong@clemson.edu

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 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 and , where 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 () and inner () layers, respectively, results in a nonuniformly valid solution and a singular perturbation problem and that is the thickness of the inner layer. The inner solutions are found to be functions of 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 ( and ). 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.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Prof. Chenning Tong, ctong@clemson.edu
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