Treatment of buoyancy production at a cloudy-clear interface in a TKE model

Turbulence models that include a prognostic equation for the turbulent kinetic energy (TKE) must account for differences in the buoyant production of TKE in saturated and non-saturated conditions. As grid resolution increases the occurrence of grid saturation is more frequent and correct treatment of turbulent mixing in saturated conditions becomes even more important. The formulation for the buoyant production of TKE in fully saturated layers was derived by Deardorff in the 70's. Applying this formulation in a vertical one-dimensional (1-D) model, however, we found some problems dealing with adjacent layers, one of which is saturated and the other is not. This paper examines some assumptions that can be made for handling these interfacial layers and the effect that they have in the model results. We hypothesize that the way this interface is treated can affect the rate of entrainment and the stability of cloudy layers predicted by TKE models.

The paper first shows results obtained with a simple two-layer model using liquid water potential temperature and total water mixing ratio as conservative variables. An additional equation for the evolution of the TKE responsible for the mixing across the two layers is added. Four different basic ways of destabilizing this model are studied: a) heating the lower layer, b) moistening the lower layer, c) cooling the upper layer, and d) drying the upper layer. Options b) and c) allow for saturation to occur, and results with this simple model illustrate the different effects that the onset of saturation produces in the turbulence level of the model. Next, different ways of treating buoyancy production of TKE in the cloudy-clear interface are assessed: 1) ignoring any saturation effect, 2) ignoring saturation effects until both layers become saturated, 3) considering both layers to be saturated as soon as one becomes saturated, 4) performing a weighted average of the saturated and non-saturated buoyancy production rates. The last approach seems most reasonable but it still requires a method to define the weighted average. Several options are investigated. We also show multi-layer results obtained with the PSU-TKE model in a 1-D version of the MM5 modeling system. Results for idealized cases are shown and the effects of different treatments of the cloudy-clear interface layer are discussed.