Determination of similarity functions of the resistance laws for the planetary boundary layer using surface-layer similarity functions

Abstract

Functional forms of the universal similarity functions A, B (for wind components parallel and normal to the surface stress), and C (for potential temperature difference) are determined based on the generalized theory of the resistance laws for the Planetary Boundary Layer (PBL). The similarity-profile functions for the surface layer are matched with the velocity and temperature-defect profiles that are assumed to have shapes modified by certain powers of nondimensional height z/h, where h is the PBL height. The powers of the outer-layer profile functions are determined, so that the functions become negligible in the surface layer. To close the temperature defect law, an assumption that the temperature gradient across the top of the PBL is continuous with the stratification of the overlying atmosphere is used. The result of this assumption is that nondimensional momentum and temperature profiles in the PBL can be described in terms of four basic ratios: (1) roughness ratio η= ζ/h (2) scale-height ratio λ=|f|h/u_*, (3) ambient stratification parameter κ=γh/θ_*, and (4) stability parameter μ=h/L, where L is the Monin-Obukhov length, z₀ is the surface roughness, γ is the upper-air stratification, u _* is the friction velocity, and θ _* is the temperature scale at the surface. For stable conditions, the scale-height ratio can be related to the atmospheric stability and the upperair stratification, and the generalized similarity and Rossby number similarity theories become identical. Under appropriate boundary conditions, function A is explicitly dependent on the stability parameter γ, while B is a function of scale-height ratio λ, which in turn depends on the stability. Function C is shown to be dependent on the stability and the upper-air stratification, due to the closure assumption used for the temperature profile.

The suggested functional forms are compared with other empirical approximations by several authors. The general framework used to determine the functional forms needs to be tested against good boundary-layer measurements.