The Scaling Behaviour of a Turbulent Kinetic Energy Closure Model for Stably Stratified Conditions

Turbulent kinetic energy (TKE) closure is more and more applied in research models, but most large scale operational models still apply first-order closure. One of the reasons is that uncertainties exist in the scaling behaviour of TKE models. It is also questioned whether this behaviour can be controlled.

We investigate the scaling behaviour of a TKE closure model for stably stratified conditions. The mixing length scale for stable stratification is proportional to the ratio of the square root of the TKE and the local Brunt–Väisälä frequency, which is a commonly applied formulation. We analyze the scaling behaviour of our model in terms of traditional Monin–Obukov Similarity Theory and local scaling.

From the model equations, expressions are derived for the stable limit behaviour of the flux-gradient relations and other scaling quantities. It turns out that the scaling behaviour depends on only a few model parameters and that the results follow local scaling theory. The analytical findings are illustrated with model simulations for the second GABLS intercomparison study. We also investigate solutions for the case in which an empirical correction function is used to express the eddy diffusivity for momentum as a function of the Richardson number (i.e. an increasing turbulent Prandtl number with stability). In this case, it seems that for certain parameter combinations the model cannot generate a steady-state solution. At the same time, its scaling behaviour becomes unrealistic. This shows that the inclusion of empirical correction functions may have large and undesired consequences for the model behaviour.