Statistical Theory on the Functional Form of Cloud Particle Size Distributions

Wu, Wei; McFarquhar, Greg M.

doi:10.1175/JAS-D-17-0164.1

Title: Statistical Theory on the Functional Form of Cloud Particle Size Distributions

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 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 furthermore »« less

Authors:

Wu, Wei ^[1]; McFarquhar, Greg M. ^[2]

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:: Tue Jul 31 00:00:00 EDT 2018

Research Org.:: Univ. of Illinois at Urbana-Champaign, 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 0022-4928

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/JAS-D-17-0164.1.

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                    Wu, Wei, & McFarquhar, Greg M. Statistical Theory on the Functional Form of Cloud Particle Size Distributions.  United States.  https://doi.org/10.1175/JAS-D-17-0164.1

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                    Wu, Wei, and McFarquhar, Greg M. Tue .  
"Statistical Theory on the Functional Form of Cloud Particle Size Distributions".  United States.  https://doi.org/10.1175/JAS-D-17-0164.1.

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@article{osti_1462291,

  title        = {Statistical Theory on the Functional Form of Cloud Particle Size Distributions},

  author       = {Wu, Wei and McFarquhar, Greg M.},

  abstractNote = {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.},

  doi          = {10.1175/JAS-D-17-0164.1},

  journal      = {Journal of the Atmospheric Sciences},

  number       = 8,

  volume       = 75,

  place        = {United States},

  year         = {Tue Jul 31 00:00:00 EDT 2018},

  month        = {Tue Jul 31 00:00:00 EDT 2018}

}

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Works referencing / citing this record:

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Atmospheric Measurement Techniques, Vol. 12, Issue 11
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