Generalizing ‘z‐less’ mixing length for stable boundary layers

Abstract

AbstractCurrent research shows, recent and former research suggests, that the nature and evolution of the stable atmospheric boundary layer(s) (SABL) is still understood and modelled inadequately. The ‘classical’ SABL, almost always stratified weakly (i.e. gradient Richardson number Ri ≪ ∞, typically 0 < Ri ≤ 1), has been modelled reasonably well during the last few decades or so, but the very stable case, i.e. the VSABL (Ri ≫ 1), is generally not well understood. Excessively diffusive and much too deep VSABL flows, as often appearing in numerical models, were recently addressed; the over‐diffusion was alleviated by improving the local turbulent mixing length. This demands an explicit inclusion of the vertical shear of horizontal wind, S, in the mixing length, besides the previously known role of buoyancy frequency, N. A generalization of this recent work is given here by a simplified turbulent kinetic energy (TKE) equation and a set of subsequent parametrizations for the eddy diffusivity and conductivity, i.e. K‐parametrizations, in terms of a generalized ‘z‐less’ mixing length, Λ. The aim is to produce a parametrization that is uniformly valid for all Ri ≥ 0. It is shown that Λ ∼ (TKE)1/2/ | S | ·f(Ri, Pr), uniformly valid for 0 ≤ Ri, regardless of the other parametrization details (the details appear as corrections); Pr is the turbulent Prandtl number and f(Ri, Pr) is a simple set of derived functions depending on the parametrization properties.A couple of important shortcomings of the current turbulence parametrizations for the SABL, as modelled in numerical weather prediction, air‐chemistry and climate models, will be remedied by using this new generalized ‘z‐less’ mixing length. This approach also recommends that it should be better if various mixing length scales were derived from simplified main principles, instead of only being guessed from plausible reasoning or dimensional analysis. In particular, it has often been assumed that K and Λ profiles could be chosen for modelling purposes more or less independently from each other. It is shown, based on a simple renormalization for Λ, that this is not the case if one wishes to use a more consistent parametrization scheme. Copyright © 2010 Royal Meteorological Society