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Size Distributions of Hydrometeors: Analysis with the Maximum Entropy Principle

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  • 1 GAME/CNRM, Météo-France, and CNRS, Toulouse, France
  • | 2 National Center for Atmospheric Research,* Boulder, Colorado
  • | 3 Department of Physical Geography, University of Lund, Lund, Sweden
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Abstract

This paper proposes that the maximum entropy principle can be used for determining the drop size distribution of hydrometeors. The maximum entropy principle can be applied to any physical systems with many degrees of freedom in order to determine a distribution of a variable when the following are known: 1) the restriction variable that leads to a homogeneous distribution without constraint and 2) a set of integrals weighted by the distribution, such as mean and variance, that constrain the system. The principle simply seeks a distribution that gives the maximum possible number of partitions among all the possible states. A continuous limit can be taken by assuming a constant bin size for the restriction variable.

This paper suggests that the drop mass is the most likely restriction variable, and the laws of conservation of total bulk mass and of total vertical drop mass flux are two of the most likely physical constraints to a hydrometeor drop size distribution. Under this consideration, the distribution is most likely constrained by the total bulk mass when an ensemble of drops under the coalescence–breakup process is confined inside a closed box. Alternatively, for an artificial rain produced from the top of a high ceiling under a constant mass flux of water fall, the total drop mass flux is the most likely constraint to the drop size distribution. Preliminary analysis of already-published data is not inconsistent with the above hypotheses, although the results are rather inconclusive. Data in the large drop size limit are required in order to reach a more definite conclusion.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Jun-Ichi Yano, GAME/CNRM, Météo-France, 42 av. Coriolis, 31057 Toulouse CEDEX, France. E-mail: jiy.gfder@gmail.com

Abstract

This paper proposes that the maximum entropy principle can be used for determining the drop size distribution of hydrometeors. The maximum entropy principle can be applied to any physical systems with many degrees of freedom in order to determine a distribution of a variable when the following are known: 1) the restriction variable that leads to a homogeneous distribution without constraint and 2) a set of integrals weighted by the distribution, such as mean and variance, that constrain the system. The principle simply seeks a distribution that gives the maximum possible number of partitions among all the possible states. A continuous limit can be taken by assuming a constant bin size for the restriction variable.

This paper suggests that the drop mass is the most likely restriction variable, and the laws of conservation of total bulk mass and of total vertical drop mass flux are two of the most likely physical constraints to a hydrometeor drop size distribution. Under this consideration, the distribution is most likely constrained by the total bulk mass when an ensemble of drops under the coalescence–breakup process is confined inside a closed box. Alternatively, for an artificial rain produced from the top of a high ceiling under a constant mass flux of water fall, the total drop mass flux is the most likely constraint to the drop size distribution. Preliminary analysis of already-published data is not inconsistent with the above hypotheses, although the results are rather inconclusive. Data in the large drop size limit are required in order to reach a more definite conclusion.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Jun-Ichi Yano, GAME/CNRM, Météo-France, 42 av. Coriolis, 31057 Toulouse CEDEX, France. E-mail: jiy.gfder@gmail.com
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