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Atmosphere 2014, 5(3), 484-517; doi:10.3390/atmos5030484
Article

A Comparative Study of B-, Γ- and Log-Normal Distributions in a Three-Moment Parameterization for Drop Sedimentation

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Received: 14 April 2014; in revised form: 23 June 2014 / Accepted: 3 July 2014 / Published: 24 July 2014
(This article belongs to the Special Issue Cloud and Precipitation)
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Abstract: This paper examines different distribution functions used in a three-moment parameterization scheme with regard to their influence on the implementation and the results of the parameterization scheme. In parameterizations with moment methods, the prognostic variables are interpreted as statistical moments of a drop size distribution, for which a functional form has to be assumed. In cloud microphysics, parameterizations are frequently based on gamma- and log-normal distributions, while for particle-laden flows in engineering, the beta-distribution is sometimes used. In this study, the three-moment schemes with beta-, gamma- and log-normal distributions are tested in a 1D framework for drop sedimentation, and their results are compared with those of a spectral reference model. The gamma-distribution performs best. The results of the parameterization with the beta- and the log-normal distribution have less similarity to the reference solution, particularly with regard to number density and rain rate. Theoretical considerations reveal that (depending on the type of the distribution function) only selected combinations of moments can be predicted together. Among these is the important combination of “number density, liquid water content, radar reflectivity” for all three distributions. Advection or source/sink terms can only be calculated under certain conditions on the moment values (positivity of the Hankel–Hadamard determinant). These are derived from mathematical theory (“moment problem”) and are more restrictive for three-moment than for two-moment schemes.
Keywords: cloud microphysics; drop sedimentation; parameterization; method of moments; Beta distribution; Gamma distribution; Log-Normal distribution; Hankel–Hadamard; moment problem cloud microphysics; drop sedimentation; parameterization; method of moments; Beta distribution; Gamma distribution; Log-Normal distribution; Hankel–Hadamard; moment problem
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Ziemer, C.; Wacker, U. A Comparative Study of B-, Γ- and Log-Normal Distributions in a Three-Moment Parameterization for Drop Sedimentation. Atmosphere 2014, 5, 484-517.

AMA Style

Ziemer C, Wacker U. A Comparative Study of B-, Γ- and Log-Normal Distributions in a Three-Moment Parameterization for Drop Sedimentation. Atmosphere. 2014; 5(3):484-517.

Chicago/Turabian Style

Ziemer, Corinna; Wacker, Ulrike. 2014. "A Comparative Study of B-, Γ- and Log-Normal Distributions in a Three-Moment Parameterization for Drop Sedimentation." Atmosphere 5, no. 3: 484-517.


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