A “Balanced” Approach to Stable Boundary Layer Dynamics

Abstract

Semiempirical arguments from Roach and Nieuwstadt concerning Richardson numbers Ri, Rf in stably stratified shear layers and boundary layers appear to bypass uncertainties of turbulence closures. Here it is shown that these arguments can be expressed systematically as a Richardson number ?balance? condition, derived formally from standard closure models implementing critical Richardson numbers. The range of validity of this procedure, and hence of the semiempirical arguments, is examined for both slow and rapid modes, and their stability analyzed. The rapid mode is stable under any reasonable assumptions, but the slow or balanced mode could be destabilized if a generalized form of the z/L stability parameter decreased with increasing Ri. Particular measured profiles are shown for illustration, and examples are given of how a Richardson number balance generates previously unsuspected quantitative predictions, which have been validated with models and simulations. Balance, where valid, is a radical simplifying approximation that generates a mathematical structure parallel in several respects to, for example, quasigeostrophic theory. A numerical implementation of balance is defined and may be a useful tool for further investigations.