Evaluation of energy consistent entrainment rate closure for cloudy updrafts

In the Earth’s atmosphere, Cumulus clouds are often modeled as an ensemble of statistically stationary moist plumes in a stratified environment. The entrainment rate coefficient for a plume relates the rate of change of volume flow rate along the plume axis to the center-line velocity and plume width. Its value determines the rate at which each plume gets diluted by the environment, and, consequently, the height of each plume. General Circulation Models typically assume a constant, empirically determined value of entrainment rate coefficient along cloud depth, which is the same for all plumes. In this work, data from Large Eddy Simulation of non-precipitating shallow Cumulus clouds is used to obtain an energy consistent closure for entrainment rate coefficient along the cloud depth in terms of the source terms in the mean momentum and vertical energy equation. The relative humidity of the environment is systematically varied over a set of three simulations, in which the control case corresponds to the Barbados oceanographic and meteorological experiment. A non-linear dependence of the local entrainment rate in terms of the local Richardson number is postulated for updraft patches, and the model constants are dynamically extracted from the relevant source terms in momentum and energy equation using the energy consistent approach. This model allows updraft patches to have a positive local entrainment rate even when the average entrainment rate for the ensemble of updrafts is negative. The resulting entrainment model for individual updraft patches may be used to obtain the fractional entrainment and detrainment rates of the cloud ensemble. The model yields reasonably low a posteriori error, especially for positively buoyant updrafts.