Statistical theory on the functional form of cloud particle size distributions

Several functional forms of cloud particle size distributions (PSDs) have been used in numerical modeling and remote sensing retrieval studies of clouds and precipitation, including exponential, gamma, lognormal, and Weibull distributions. However, there is no satisfying theoretical explanation as to why certain distribution forms preferentially occur instead of others. Intuitively, the analytical form of a PSD can be derived by directly solving the general dynamic equation, but no analytical solutions have been found yet. Instead of a process-level approach, the use of the principle of maximum entropy (MaxEnt) for determining the theoretical form of PSDs from the perspective of system is examined here. MaxEnt theory states that the probability density function with the largest information entropy among a group satisfying the given properties of the variable should be chosen. Here, the issue of variability under coordinate transformations that arises using the Gibbs-Shannon definition of entropy is identified, and the use of the concept of relative entropy to avoid these problems is discussed. Focusing on cloud physics, the four-parameter generalized gamma distribution is proposed as the analytical form of a PSD using the principle of maximum (relative) entropy with assumptions on power-law relations among state variables, scale invariance, and a further constraint on the expectation of one state variable (e.g., bulk water mass). The four-parameter generalized gamma distribution is very flexible to accommodate various type of constraints that could be assumed for cloud PSDs.