Scaling Behavior of a Turbulent Kinetic Energy Closure Scheme for the Stably Stratified Atmosphere: A Steady-State Analysis

We present a turbulent kinetic energy (TKE) closure scheme for the stablystratified atmosphere in which the mixing lengths for momentum andheat are not parameterized in the same manner. The key difference is that,while the mixing length for heat tends towards the stability independent mixinglength for momentum in neutrally stratified conditions, it tends towardsone based on the Brunt–V¨ais¨al¨a time scale and square root of the TKE inthe limit of large stability. This enables a unique steady-state solution forTKE to be obtained, which we demonstrate would otherwise be impossibleif the mixing lengths were the same. Despite the model’s relative simplicity,it is shown to perform reasonably well with observational data from the1999 Cooperative Atmosphere-Surface Exchange Study (CASES-99) usingcommonly employed model constants. Analyzing the scaling behavior of thenon-dimensional velocity and potential temperature gradients, or of the stability(correction) functions, reveals that for large stability the present modelscales in the same manner as the first-order operational scheme of Viterboet al. (Quart. J. Roy. Meteor. Soc. 125, 2401–2426, 1999). Alternatively,it appears as a blend of two cases of the TKE closure scheme of Baas etal. (Bound.-Layer Meteor. 127, 17–36, 2008). Critically, because a uniquesteady-state TKE can be obtained, the present model avoids the non-physicalbehavior identified in one of the cases of Baas et al. (2008).