1、The Importance of the Shape Parameter in a Bulk ParameterizationScheme to the Evolution of the Cloud Droplet Spectrumduring CondensationJun ZHANG1,3,Jiming SUN*1,2,3,Wei DENG1,3,Wenhao HU1,3,and Yongqing WANG1,41Laboratory of Cloud-Precipitation Physics and Severe Storms,Institute of Atmospheric Phy
2、sics,Chinese Academy of Sciences,Beijing 100029,China2Nanjing University of Information and Technology,Nanjing 210044,China3University of Chinese Academy of Sciences,Beijing 100049,China4Beijing Key Laboratory of Cloud,Precipitation and Atmospheric Water Resources,Beijing 100089,China(Received 15 Ma
3、rch 2022;revised 1 June 2022;accepted 22 June 2022)ABSTRACTThe shape parameter of the Gamma size distribution plays a key role in the evolution of the cloud droplet spectrum inthe bulk parameterization schemes.However,due to the inaccurate specification of the shape parameter in the commonlyused bul
4、k double-moment schemes,the cloud droplet spectra cannot reasonably be described during the condensationprocess.Therefore,a newly-developed triple-parameter condensation scheme with the shape parameter diagnosed throughthe number concentration,cloud water content,and reflectivity factor of cloud dro
5、plets can be applied to improve theevolution of the cloud droplet spectrum.The simulation with the new parameterization scheme was compared to those witha high-resolution Lagrangian bin scheme,the double-moment schemes in a parcel model,and the observation in a 1.5DEulerian model that consists of tw
6、o cylinders.The new scheme with the shape parameter varying with time and space canaccurately simulate the evolution of the cloud droplet spectrum.Furthermore,the volume-mean radius and cloud watercontent simulated with the new scheme match the Lagrangian analytical solutions well,and the errors are
7、 steady,withinapproximately 0.2%.Key words:cloud microphysics parameterization,cloud droplet spectrum,double-moment scheme,shape parameterCitation:Zhang,J.,J.M.Sun,W.Deng,W.H.Hu,and Y.Q.Wang,2023:The importance of the shape parameter in abulk parameterization scheme to the evolution of the cloud dro
8、plet spectrum during condensation.Adv.Atmos.Sci.,40(1),155167,https:/doi.org/10.1007/s00376-022-2065-7.Article Highlights:The shape parameter of the Gamma size distribution is calculated accurately in the new scheme.The new scheme improves the simulation of the cloud droplet spectrum.1.IntroductionO
9、ne of the important processes for cloud formation anddevelopment is water vapor condensation.The evolution ofthe cloud droplet spectrum (CDS)during condensationhighly impacts the other microphysical processes,such asthe formation of rain,the nucleation of ice,and so on(Mil-brandt and Yau,2005a;Pette
10、rs et al.,2006;Milbrandt et al.,2021).The numerical simulation for the condensationalgrowth of a population of cloud droplets remains a challenge(Khain et al.,2015).The evolution of the CDS can be described by the twomathematical approaches.The first is by bin schemes inwhich the number concentratio
11、n in each bin is determinedindividually for all microphysical processes(Tzivion et al.,1994;Khain et al.,2000).Another method is by bulk parame-terization microphysics schemes in which the size distributionis prescribed to fit a function form(e.g.,a gamma distribu-tion),and the parameters vary accor
12、ding to changes in prog-nostic moments.In most bulk parameterization schemes,theCDS is expressed by a Gamma distribution function withthree parameters,the intercept parameter,the shape parame-ter,and the slope parameter.The bulk parameterizationschemes can be classified into three kinds of schemes b
13、y thenumber of parameters considered:single-moment schemes,*Corresponding author:Jiming SUNEmail:ADVANCES IN ATMOSPHERIC SCIENCES,VOL.40,JANUARY 2023,155167 Notes&Letters Institute of Atmospheric Physics/Chinese Academy of Sciences,and Science Press and Springer-Verlag GmbH Germany,part of Springer
14、Nature 2023double-moment schemes,and triple-moment schemes.The hydrometeor mass mixing ratio is predicted,andthe slope parameter is diagnosed in single-moment schemes(Kessler,1969;Lin et al.,1983;Cotton et al.,1986;Tao andSimpson,1993;Kong and Yau,1997;Bae et al.,2018).Indouble-moment schemes,the in
15、tercept parameter and slopeparameter are diagnosed by the number concentration andthe mass mixing ratio(Ziegler,1985;Levkov et al.,1992;Harrington et al.,1995;Cohard and Pinty,2000;Seifert andBeheng,2001;Morrison et al.,2009).However,the shapeparameter is either fixed or diagnosed by empirical formu
16、laswithout adding a new degree of freedom (Milbrandt andYau,2005a;Morrison and Grabowski,2007;Xie et al.,2013,2017).For the double-moment schemes with a constant shapeparameter,the simulated cloud spectra for pseudo-adiabati-cally ascending cloud parcels may not obey their condensa-tional growth behavior.If there are not the new clouddroplets from nucleation,the breadths of the spectra willbecome narrower and narrower because the growth rates ofsmaller droplets are higher than those of bigger on