Modelling boundary-layer clouds with a statistical cloud scheme and a second-order turbulence closure

We present a second-order turbulence model for the cloudy planetary boundary layer (PBL), which includes a statistical scheme of the sub-grid scale condensation. The model contains prognostic equations for the turbulent kinetic energy, total water, and liquid water temperature, the latter two being assumed to be conservative variables. Using these conservative thermodynamic variables the condensation process is formulated as a function of the departure of the total water from saturation and its variance. The computation of the variance requires second moment correlations which are modelled through the parameterization of the third-order moments using a convective mass-flux formulation. The inclusion of these third moments and new assumptions on heat flux transport lead to a nonlocal turbulence scheme with counter-gradient effects. The final form for the heat flux turns out to be a linearized version of a previously established result. For the statistical cloud formulation, a linear combination of a Gaussian and a positively skewed distribution function is used with a modified liquid water flux expression to account for non-Gaussian behaviour. The effect of the turbulence scheme on the boundary-layer cloud structure is discussed and the performance of the model is tested by comparing it against the large eddy simulation (LES) of the undisturbed period of the Atlantic Stratocumulus Transition Experiment (ASTEX). The model is able to produce both mean and turbulent quantities that are in reasonable agreement with the LES output of ASTEX.