Theoretical and Statistical Analysis of Deep Convection Based on Canonical Ensemble Framework

Abstract:

The collective characteristics of cumulus convection in statistical equilibrium vary in response to environmental large-scale forcing. Here we view the population of weakly interacting deep cumulus clouds under ideal conditions as a Gibbs canonical ensemble with convective kinetic energy as a reference quantity. With this as a starting point, we suggest a theoretical framework based on equilibrium statistical mechanics that relates the macroscopic state of the system to the probability distribution of individual cumulus convection. From these considerations, the form of the parameter representing the macroscopic state, which corresponds to the inverse temperature in statistical mechanics, can be inferred from the bulk thermodynamic quantities for the troposphere. The analysis of the cumulus convection ensemble generated by cloud-resolving simulations in radiative-convective equilibrium shows that the frequency distribution of convective cores calculated from the updraft kinetic energy follows a negative exponential (figure 1). Furthermore, both the average and the fluctuation of convective kinetic energy tends to be larger at higher spatially homogeneous sea surface temperature. This is in general agreement with the theory of the present work. Comparison between the vertical profiles of simulated thermodynamic quantities and the scaling exponent of cumulus probability distribution is also discussed.