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

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.Atmosphere 2014, 5 485

Three‐Moment Representation of Rain in a Bulk Microphysics Model

“…In line with observed rain properties, a popular choice is the three‐parameter gamma distribution, where D is the diameter, N 0 is the intercept parameter, μ is the shape parameter, and λ represents the slope of the distribution. In a direct comparison with beta and log‐normal distributions, Ziemer and Wacker () concluded that the gamma distribution performs best in simulations of pure sedimentation.…”

Characteristics of Raindrop Size Distributions in Chongqing Observed by a Dense Network of Disdrometers

“…The combination of M 0 , M 3 and M 6 is selected. The calculation of the three gamma distribution parameters using this configuration is also described in Milbrandt and Yau (2005a) and Ziemer and Wacker (2014). Defining X as $$X=\frac{{M}_{0}{M}_{6}}{{M}_{3}^{2}}$$ and combining Equation with Equation , the shape parameter μ can be computed by solving the equation numerically: $$$X-\frac{(\mu +6)(\mu +5)(\mu +4)}{(\mu +3)(\mu +2)(\mu +1)}=0,\hspace*{.5em}\mu \in \mathbf{R}.$$$ …”

“…The combination of M 0 , M 3 and M 6 is selected. The calculation of the three gamma distribution parameters using this configuration is also described in Milbrandt and Yau (2005a) and Ziemer and Wacker (2014). Defining X as…”

“…More recently, the optical disdrometers became the most widely used instruments for RSD observations and research (Kathiravelu et al., 2016). To describe the observed RSD, many studies (e.g., Ulbrich, 1983; Costa et al., 2000; Ziemer & Wacker, 2014) have used the gamma distribution and found it a good fit: $$$N(D)={N}_{0}{D}^{\mu }{e}^{-\lambda D},$$$ where D is the raindrop diameter (mm), N ( D ) (m −3 mm −1 ) is the number distribution function, N 0 is the intercept parameter (m −3 mm −1− μ ), μ is the shape parameter, and λ is the slope parameter (mm −1 ). When μ equals to 0, Equation is also referred to as the Marshall‐Palmer distribution (Marshall & Palmer, 1948) in the form of exponential distribution.…”

Precipitation Sensitivity to the Mean Radius of Drop Spectra: Comparison of Single- and Double-Moment Bulk Microphysical Schemes

“…proportional or equal to the corresponding moment of the size distribution function. Therefore, depending on the number of moments-single-moment [12][13][14][15][16][17], double-moment [18][19][20][21][22] and triple-moment [23][24][25][26]-bulk microphysical schemes can be distinguished.…”