Lateral mixing in shallow convection: In theory and in practice

Analytical expressions, in principle also valid for an ensemble of clouds, are derived for $\eps$ and $\delta$ from first principles and with a minimum of assumptions. A good correspondence is found when these expressions are validated against $\eps$ and $\delta$ as usually diagnosed from LES (here considered as best possible estimates). Further, the expressions give us insight into the behavior of the mixing coefficients. For example, analysis of LES for the non-steady ARM case revealed that the $\pdd{a_c}{z}$ term dominates the detrainment coefficient and is mainly responsible for the larger variation in $\delta$ than in $\eps$. As this term is strongly linked to the mass flux, via $M=a_cw_c$, it therefore seems plausible to let the variability of the mass flux profile be controlled by the detrainment only (as argued by \cite{Rooy08}). Besides giving insight, the expressions can help us to judge existing parameterizations as well as be a source of inspiration for future parameterization developments.