Statistical Thermodynamics and the Size Distributions of Tropical Convective Clouds

Parameterizations for sub-grid cloud dynamics are commonly developed by using fine scale modeling or measurements to explicitly resolve the mechanistic details of clouds to the best extent possible, and then to formulating these behaviors cloud state for use within a coarser grid. A second is to invoke physical intuition and some very general theoretical principles from equilibrium statistical thermodynamics. This second approach is quite widely used elsewhere in the atmospheric sciences: for example to explain the heat capacity of air, blackbody radiation, or even the density profile or air in the atmosphere. Here we describe how entrainment and detrainment across cloud perimeters is limited by the amount of available air and the range of moist static energy in the atmosphere, and that constrains cloud perimeter distributions to a power law with a -1 exponent along isentropes and to a Boltzmann distribution across isentropes. Further, the total cloud perimeter density in a cloud field is directly tied to the buoyancy frequency of the column. These simple results are shown to be reproduced within a complex dynamic simulation of a tropical convective cloud field and in passive satellite observations of cloud 3D structures. The implication is that equilibrium tropical cloud structures can be inferred from the bulk thermodynamic structure of the atmosphere without having to analyze computationally expensive dynamic simulations.