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

TL;DR

This paper introduces a TKE closure scheme for the stably stratified atmosphere that allows for a unique steady-state solution by differentiating heat and momentum mixing lengths, and demonstrates its reasonable performance against observational data.

Contribution

It presents a novel TKE closure scheme with distinct mixing length parameterizations for heat and momentum, enabling steady-state solutions and improved modeling of stable atmospheric conditions.

Findings

Model scales similarly to the first-order scheme of Viterbo et al. (1999).

The scheme aligns with two cases of Baas et al. (2008) but avoids non-physical behaviors.

Performs reasonably well with observational data from CASES-99.

Abstract

We present a turbulent kinetic energy (TKE) closure scheme for the stably stratified atmosphere in which the mixing lengths for momentum and heat are not parameterized in the same manner. The key difference is that, while the mixing length for heat tends towards the stability independent mixing length for momentum in neutrally stratified conditions, it tends towards one based on the Brunt-V\"ais\"al\"a time scale and square root of the TKE in the limit of large stability. This enables a unique steady-state solution for TKE to be obtained, which we demonstrate would otherwise be impossible if the mixing lengths were the same. Despite the model's relative simplicity, it is shown to perform reasonably well with observational data from the 1999 Cooperative Atmosphere-Surface Exchange Study (CASES-99) using commonly employed model constants. Analyzing the scaling behavior of the non-dimensional velocity and potential temperature gradients, or of the stability (correction) functions, reveals that for large stability the present model scales in the same manner as the first-order operational scheme of Viterbo et 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 et al. (Bound.-Layer Meteor. 127, 17-36, 2008). Critically, because a unique steady-state TKE can be obtained, the present model avoids the non-physical behavior identified in one of the cases of Baas et al. (2008).