Statistical Theory on the Analytical Form of Cloud Particle Size Distribution

Several analytical forms of cloud particle size distributions (PSDs) are used empirically in modeling and remote sensing retrievals, including exponential, gamma, lognormal, and Weibull distributions. However, the physical basis for the use of particular analytic forms of PSDs has not been well established. In this study, the principle of maximum entropy (MaxEnt) is proposed as a method for determining the analytical form of PSDs. MaxEnt theory states that for a group of probability density functions (PDFs) that satisfy given properties of a variable, the PDF with the largest information entropy should be chosen. Thus, a uniform distribution (most uncertain) is most probable if no additional information is provided. But, with extra information, the most probable PDF will be changed. By using MaxEnt, the generalized gamma distribution is derived theoretically for describing cloud PSDs in this study. Ramifications for both cloud modeling and remote sensing retrievals will be discussed. For cloud resolving models, where a limited number of moments are predicted (e.g., mass and number concentration), other moments need to be calculated using additional assumptions about the form of PSDs that are now known from MaxEnt. For radar retrievals, only radar reflectivity and other polarization variables are measured with other moments retrieved again using the form of the PSD from MaxEnt. The generalized gamma distribution is also tested by an idealized warm rain simulation with bin microphysics including diffusional growth and collision-coalesce growth.