Modification Of The Standard ∈-Equation For The Stable Abl Through Enforced Consistency With Monin–Obukhov Similarity Theory

A condition is derived for consistency of the standard∈-equation with Monin–Obukhov (MO) similarity theory of thestably-stratified surface layer. The condition is derivedby extending the procedure used to derive the analogous condition forneutral theory to stable stratification. It is shown that consistencywith MO theory requires a function of flux Richardson number, Rif, to be absorbed into either of two closure parameters, c∈ 1 or c∈ 2.Inconsistency, on the other hand, results if constant values of these are maintained for all Rif, as is done in standardapplication of the equation, and the large overpredictions ofturbulence found in such application to the one-dimensionalstable atmospheric boundary layer (1D-SBL) are traced to thisinconsistency. Guided by this, we formulate a MO-consistent∈-equation by absorbing the aforementioned function intoc∈ 1, and combine this with a Level-2.5 second-orderclosure model for vertical eddy viscosity and diffusivities.Numerical predictions of the 1D-SBL by the modified model converge to a quasi-steady state, rectifying the predictive failure of the standard∈ -equation for the case.Quasi-steady predictions of non-dimensional variables agree stronglywith Nieuwstadt's theory. Qualitative accuracy of predictionsis inferred from comparisons to field data, large-eddy simulationresults and Rossby-number similarity relationships.