Statistical Theory on the Functional Form of Cloud Particle Size Distributions
Abstract
Abstract 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 processlevel 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 fourparameter generalized gamma distribution is proposed as the analytical form of a PSD using the principle of maximum (relative) entropy with assumptions on powerlaw relations among state variables, scale invariance, and amore »
 Authors:

 Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois, and National Center for Atmospheric Research, Boulder, Colorado
 Cooperative Institute for Mesoscale Meteorological Studies, and School of Meteorology, University of Oklahoma, Norman, Oklahoma
 Publication Date:
 Research Org.:
 Univ. of Illinois at UrbanaChampaign, IL (United States); University Corporation for Atmospheric Research, Boulder, CO (United States); Univ. of Oklahoma, Norman, OK (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Biological and Environmental Research (BER)
 OSTI Identifier:
 1462291
 Alternate Identifier(s):
 OSTI ID: 1541823; OSTI ID: 1593792
 Grant/Contract Number:
 SC0014065; SC0016476 through subcontract SUBAWD000397; SC0016476
 Resource Type:
 Published Article
 Journal Name:
 Journal of the Atmospheric Sciences
 Additional Journal Information:
 Journal Name: Journal of the Atmospheric Sciences Journal Volume: 75 Journal Issue: 8; Journal ID: ISSN 00224928
 Publisher:
 American Meteorological Society
 Country of Publication:
 United States
 Language:
 English
 Subject:
 54 ENVIRONMENTAL SCIENCES; Meteorology & Atmospheric Sciences
Citation Formats
Wu, Wei, and McFarquhar, Greg M. Statistical Theory on the Functional Form of Cloud Particle Size Distributions. United States: N. p., 2018.
Web. doi:10.1175/JASD170164.1.
Wu, Wei, & McFarquhar, Greg M. Statistical Theory on the Functional Form of Cloud Particle Size Distributions. United States. doi:10.1175/JASD170164.1.
Wu, Wei, and McFarquhar, Greg M. Tue .
"Statistical Theory on the Functional Form of Cloud Particle Size Distributions". United States. doi:10.1175/JASD170164.1.
@article{osti_1462291,
title = {Statistical Theory on the Functional Form of Cloud Particle Size Distributions},
author = {Wu, Wei and McFarquhar, Greg M.},
abstractNote = {Abstract 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 processlevel 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 fourparameter generalized gamma distribution is proposed as the analytical form of a PSD using the principle of maximum (relative) entropy with assumptions on powerlaw relations among state variables, scale invariance, and a further constraint on the expectation of one state variable (e.g., bulk water mass). The fourparameter generalized gamma distribution is very flexible to accommodate various type of constraints that could be assumed for cloud PSDs.},
doi = {10.1175/JASD170164.1},
journal = {Journal of the Atmospheric Sciences},
number = 8,
volume = 75,
place = {United States},
year = {2018},
month = {7}
}
DOI: 10.1175/JASD170164.1
Web of Science