Symmetry Analysis of Mean Velocity Distribution in Stratified Atmospheric Surface Layers

Abstract

The mean velocity distributions of unstably and stably stratified atmospheric surface layers (ASLs) are investigated here using the symmetry approach. Symmetry groups for the mean momentum and the Reynolds stress equations of ASL are searched under random dilation transformations, which, with different leading order balances in different flow regions, lead to a set of specific scalings for the characteristic length ℓ13 (defined by Reynolds shear stress and mean shear). In particular, symmetry analysis shows that in the shear-dominated region, ℓ13 scales linearly with the surface height z, which corresponds to the classical log law of mean velocity. In the buoyancy-dominated region, ℓ13/L∼z/L4/3 for unstably stratified ASL and ℓ13/L∼const for stably stratified ASL, where L is the Obukhov length. The specific formula of the celebrated Monin–Obukhov similarity function is obtained, and hence an algebraic model of mean velocity profiles in ASL is derived, showing good agreement with the datum from the QingTu Lake observation array (QLOA) in China.

Keywords

• atmospheric surface layer
• mean velocity profile
• symmetry analysis