Analytical expressions for entrainment and detrainment in cumulus convection

Analytical expressions for entrainment and detrainment are derived based on general total water specific humidity and mass budget equations. From these expressions, containing a small-scale turbulent and a larger-scale organized term, a physical picture emerges for a shallow cumulus cloud ensemble in which the individual clouds have a massive entrainment at the bottom, lateral turbulent mixing with constant mass flux between bottom and top, and massive detrainment at the top. Combining these results with the general budget equation for vertical velocity, new formulae for entrainment and detrainment rates can be expressed in terms of buoyancy, vertical velocity and cloud fraction. For a variety of shallow convection cases, results from large-eddy simulations show a good correspondence of these new formulae with more traditional methods to diagnose entrainment and detrainment rates. Moreover, the formulae give insight into the behaviour and the physical nature of the mixing coefficients. They explain the observed large variability of the detrainment. The formulae cannot be directly applied as a parametrization. However, it is demonstrated how they can be used to evaluate existing parametrization approaches and as a sound physical base for future parametrization developments. Copyright (C) 2010 Royal Meteorological Society