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George C. Craig and Brenda G. Cohen


To provide a theoretical basis for stochastic parameterization of cumulus convection, the equilibrium fluctuations of a field of cumulus clouds under homogeneous large-scale forcing are derived statistically, using the Gibbs canonical ensemble from statistical mechanics. In the limit of noninteracting convective cells, the statistics of these convective fluctuations can be written in terms of the large-scale, externally constrained properties of the system. Using this framework, the probability density function of individual cloud mass fluxes is shown to be exponential. An analytical expression for the distribution function of total mass flux over a region of given size is also derived, and the variance of this distribution is found to be inversely related to the mean number of clouds in the ensemble. In a companion paper, these theoretical predictions are tested against cloud resolving model data.

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Brenda G. Cohen and George C. Craig


The theoretical predictions derived in Part I of this study for the equilibrium fluctuations of an idealized ensemble of noninteracting, pointlike cumulus clouds are tested against three-dimensional cloud resolving model (CRM) simulations of radiative–convective equilibrium. Simulations with different radiative cooling rates are used to give a range of cloud densities, while imposed vertical wind shear of different strengths is used to produce different degrees of convective organization. The distribution of mass flux of individual clouds is found to be exponential in all simulations, in agreement with the theory. The distribution of total mass flux over a finite region also agrees well (to within around 10%) with the theoretical prediction for all simulations, but only after a correction to the modeled variance to take account of the finite size of clouds has been made. In the absence of imposed vertical wind shear, some spatial clustering of convective cells is observed at lower forcings (−2 and −4 K day−1) on a scale of 10–20 km, while at higher forcings (−8, −12, and −16 K day−1), there is a tendency toward spatial regularity on the same scale. These localized cloud interactions, however, appear to have little effect on the magnitude of the mass flux variability. Surprisingly, the convective organization obtained in the simulations with vertical wind shear has only a small effect on the mass flux statistics, even though it shows clearly in the location of the clouds.

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