Parameterizations of different hydrometeor spectral relative dispersion in the convective clouds

Qinyo Zou , Lei Zhu , Chunsong Lu , , Gung J.Zhng , Xioqi Xu , Qin Chen , Dn Li

a Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, and Key Laboratory for Aerosol-Cloud-Precipitation of China Meteorological Administration, Nanjing University of Information Science and Technology, Nanjing, China

b Scripps Institution of Oceanography University of California, La Jolla, San Diego, CA, United States

Keywords:Hydrometeors Relative dispersion Cloud Volume-mean diameter

ABSTRACT Spectral relative dispersion of different hydrometeors is vital to accurately describe sedimentation.Here, the Weather Research and Forecasting model with spectral bin microphysics is used to simulate convective clouds in Shouxian of Anhui province in China to study the spectral relative dispersion of different hydrometeors.Firstly,regardless of clean or polluted conditions, the relative dispersion of ice crystal spectra and its volume-mean diameter are negatively correlated, while the relative dispersion of other hydrometeor spectra is positively related to their respective volume-mean diameter.The correlations for cloud droplets and raindrops are affected by the process of collision–coalescence; the correlations for ice crystals, graupel particles, and snow particles could be affected by the deposition, riming, and aggregation processes, respectively.Secondly, relative dispersion parameterizations are developed based on a comprehensive consideration of the relationships between the relative dispersion and volume-mean diameter under both polluted and clean conditions.Finally, the relative dispersion parameterizations are applied to terminal velocity parameterizations.The results show that for cloud droplets, ice crystals, graupel particles, and snow particles, assuming the shape parameter in the Gamma distribution is equal to 0 underestimates the shape parameter and overestimates the relative dispersion; and for raindrops, assuming the shape parameter is equal to 0 is close to the relative dispersion parameterizations.The most appropriate constant shape parameters are recommended for different hydrometeors.The relative dispersion parameterizations developed here shed new light for further optimizing the terminal velocity parameterizations in models.

1.Introduction

Clouds cover approximately two-thirds of the Earth’s surface and are essential for the balance of heat, moisture, and momentum of the Earth’s atmosphere ( Liu et al., 2011 ; Zhao et al., 2019 ).The physical processes of clouds and precipitation are important components of the atmospheric water cycle, in which the influence of cloud microphysical processes cannot be ignored ( Xie and Liu, 2015 ; Fu and Lei, 2017 ).Hence, it is necessary to accurately describe the microphysical processes ( Huang et al., 2016 ).Parameterizations of microphysical processes are mainly used to describe microphysical processes in models, and the terminal velocity during the sedimentation process is an important parameter ( Lord and Lord, 1988 ).The gamma distribution is often used in models to represent the hydrometeor spectra,and the terminal velocity is a function of the spectral shape parameter( Milbrandt and Yau, 2005a ).The shape parameter is often assumed to be 0 in models ( Morrison and Gettelman, 2008 ; Morrison et al., 2009 ).Also, Milbrandt and Yau (2005b) assumed the cloud shape parameter as 1 and the rain shape parameter as 2.However,in-situobservations also show that the shape parameter is not a fixed constant ( Yin et al., 2011 ),and it is unable to reasonably calculate the terminal velocity when the shape parameter is set as a fixed constant ( McTaggart-Cowan and Milbrandt, 2010 ).Milbrandt and Yau (2005a) proposed a diagnosis equation for the shape parameter and further analyzed the relationship between the shape parameters of precipitation particles and their meanmass diameters with model simulations.Seifert (2008) found that the shape parameter of raindrops and its volume-mean diameter are not simply positively correlated.Because of the limitations in current research on the terminal velocity using the shape parameter, it is necessary to parameterize the shape parameter and optimize the parameterization of terminal velocity.

Since the square of relative dispersion is inversely related to the shape parameter in the gamma distribution plus 1 ( Tas et al., 2012 ),it is easy to calculate the relative dispersion from the shape parameter.Using relative dispersion is equivalent to using the shape parameter, and many studies on the spectral width of droplets have used it( Liu and Daum, 2000 ; Liu et al., 2017 ).Existing studies of relative dispersion versus microphysical properties focus mainly on cloud droplet spectra ( Zhao et al., 2006 ) and raindrop spectra ( Seifert, 2008 ).However, even for the most studied relative dispersion of cloud droplet spectra, there are still many uncertainties ( Liu et al., 2006 , 2014 ).For example, some studies found that the relative dispersion of cloud droplet spectra is negatively correlated with the volume-mean radius( Liu et al., 2008 ), while others found positive correlations ( Tas et al.,2012 ).Lu et al.(2020) found that the relationship between relative dispersion and the volume-mean radius of cloud droplets changes from positive to negative as the volume-mean radius increases under different cloud microphysical processes.However, in-depth observational analyses on the relationship between the relative dispersion of ice-phase particle spectra and their microphysical properties are lacking.

To overcome the insufficiency of observational research on relative dispersion, especially for ice-phase particles, the Weather Research and Forecasting (WRF) model with the spectral bin microphysics (SBM)scheme is used here to simulate convective processes.Unlike the bulk scheme, which often assumes that the distributions of hydrometeors follow gamma distributions, the distributions of hydrometeors in the SBM scheme are predicted, which is convenient for analyzing the relative dispersion of different hydrometeor spectra ( Khain and Lynn, 2009 ).Given the important influence of aerosol particles on the cloud microphysical processes and hydrometeor spectra ( Peng and Lohmann, 2003 ; Jia et al.,2019 ), we discuss the relationships between the relative dispersion of different spectra and their volume-mean diameters and the related microphysical processes, under both polluted and clean conditions.Then,based on these relationships, relative dispersion parameterizations for different hydrometeor spectra are developed.Finally, the impacts of the relative dispersion parameterizations in the terminal velocity parameterizations are preliminarily evaluated.To be consistent with previous studies in which the relationship between relative dispersion and volume-mean radius was analyzed ( Liu et al., 2008 ; Tas et al., 2012 ),relative dispersion is used here.

2.Model and methods

2.1. Numerical model experiments

The WRF model, version 3.2, coupled with a fast version of the SBM scheme (Fast-SBM) is used to simulate convective clouds over Shouxian (32.58°N, 116.78°E) of Anhui Province in China ( Chen et al., 2015 ).Each hydrometeor size distribution contains 33 bins, where the mass in bini+ 1 is twice that in bini.The mass of the initial hydrometeor bin is 3.35 × 10?14kg, and the corresponding diameter is 4μm (assuming that the hydrometeor is spherical).The simulation area is centered in Shouxian, complete with two-way nesting using two nested domains.The horizontal resolutions of the two domains are 12 km and 2.4 km,and there are 110 × 90 and 251 × 201 grid points, respectively.In the vertical direction, there are 41 levels.The simulations are run from 1800 UTC 16 July 2008 to 1200 UTC 17 July 2008, and the first 6 h of the simulation is taken as the spin-up time.The six-hourly National Centers for Environmental Prediction FNL (Final) global reanalysis data with a resolution of 1° × 1° are used to produce the initial and boundary conditions.The primary physics schemes include the Grell–Devenyi cumulus parameterization in the outer domain ( Grell and Dévényi, 2002 )(no cumulus parameterization in the inner domain), the Noah landsurface scheme ( Chen and Dudhia, 2001 ), the Yonsei University planetary boundary layer scheme ( Hong et al., 2006 ), and the Rapid Radiative Transfer Model for global climate models shortwave and longwave radiation schemes ( Iacono et al., 2008 ).The clean and polluted conditions have initial aerosol concentrations of 280 and 6 × 280 cm?3, respectively ( Chen et al., 2020 ).The model simulation outputs three types of particles –namely, liquid droplets, graupel particles, and snow particles.The liquid droplets are divided into cloud droplets and raindrops by the diameter of 50μm ( Lu et al., 2012 ), and the snow particles are divided into ice crystals and snow particles by the diameter of 250μm( Eidhammer et al., 2014 ).The simulation is sampled according to the mixing ratio (qx) and number concentration (Nx), where the subscript“x” represents cloud droplets (c), raindrops (r), ice crystals (i), graupel particles (g), and snow particles (s).The cloud grid points are selected with the criterionqc+qi>0.01 g kg?1.After that,qc>0.01 g kg?1andNc>10 cm?3are applied to select cloud droplet spectra; for raindrops, ice crystals, graupel particles, and snow particles, their mixing ratios and number concentrations need to be larger than 0.01 g kg?1and 10 m?3, respectively.

2.2. Calculation of microphysical quantities

Relative dispersion (εx) is defined as the ratio of the standard deviation (SDx) to the mean diameter

where

Here,Dxis the diameter of hydrometers in each bin,nx(Dx) is the number density corresponding to each hydrometeor bin,Nxis the total number concentration,Dxminis the minimum diameter, andDxmaxis the maximum diameter.

The volume-mean diameter (Dvx) is calculated by:

The expression of the autoconversion threshold function (T) is( Liu et al., 2005 )

whereDcis the critical diameter ofT.The range ofTis 0–1.The larger the value ofT, the greater the probability of the collision–coalescence process ( Niu et al., 2010 ; Lu et al., 2013 ).The analytical expression ofDcriis ( Liu et al., 2004 )

whereβcon= 1.15 × 1023is an empirical constant and LWCcis the cloud liquid water content.

2.3. Calculation of terminal velocity

The gamma distribution is the most commonly used particle size distribution in models ( Milbrandt and Yau, 2005a ):

wheren0xis the intercept parameter,μxis the shape parameter, andλxis the slope parameter.Combination of Eqs.(1) and (4) yields theλx:

The terminal velocity (Vx) is defined as ( Locatelli and Hobbs, 1974 )

whereaxandbxare the coefficients of the terminal velocity (Table S1)( Ferrier, 1994 ; Morrison and Gettelman, 2008 ).The particle mass (mx)is

wherecxanddxare constants.The gamma function is defined as

wheresandtare two parameters.

Combination of Eqs.(4) , (6) –(8) yields the equations for the massweighted terminal velocity (Vmx) and the number-weighted terminal velocity (Vnx) ( Ferrier, 1994 ; Morrison and Gettelman, 2008 ),

whereγis the ratio of the surface air density to the air density at a certain level andμxis related to theεx( Tas et al., 2012 ):

To optimize the terminal velocity parameterizations of hydrometeors,μxis firstly obtained from the parameterizedεxthrough Eq.(14) .Secondly,λxis calculated according to the relationship betweenλxandμx( Eq.(8) ).Finally,VmxandVnxare calculated, assumingγis equal to 1 ( Eqs.(12) and (13) ).

3.Results and discussion

3.1. Relationship between relative dispersion and volume-mean diameter

Fig.S1 shows the relationships betweenεxandDvxfor different hydrometeors under polluted and clean conditions.To ensure statistically significant conclusions and sufficient samples, only the data points in Fig.S1 with their average probability density greater than 0.05% are selected.

To identify the mechanisms responsible for the positive correlations for the liquid-phase hydrometeors (cloud and rain, Fig.S1(a–d)),Tis calculated according to Eqs.(5) and (6) .Previous studies indicate thatTcan represent the intensity of the collision–coalescence process and its impact on cloud microphysical quantities ( Liu et al., 2004 ; Lu et al.,2013 ).Here,Tis 0.82 for the polluted condition and 0.90 for the clean condition, which indicates the collision-coalescence process is more significant in clean air than in polluted air ( Albrecht, 1989 ).During the collision–coalescence process, the cloud droplet spectra expand toward the large droplets ( Lu et al., 2018 ), increasing theεc, which is consistent with the conclusion in Tas et al.(2012) .The collision–coalescence process also causes cloud droplets to grow into raindrops ( Dagan et al.,2015 ).Due to the collision–coalescence process of raindrops with cloud droplets, theεrof the raindrop spectra increases with the increasingDvr.In addition, the maximumDvcin the polluted condition is smaller than that in the clean condition ( Rosenfeld, 1999 ; Cheng et al., 2009 ),because more aerosol increases the cloud droplet number concentration( Albrecht, 1989 ; Shi et al., 2010 ).

Unlike the liquid-phase hydrometeors, different ice-phase hydrometeors have different relationships betweenεxandDvx(Fig.S1(e–j)).The negative correlation betweenεiandDvicould be caused by the deposition process of ice crystals.Small ice crystals grow faster than large ones during deposition, which narrows the spectral width.Therefore,εidecreases with the increase ofDvi.In contrast,εgis positively correlated withDvg.Riming is the main process for the growth of graupel ( Rutledge and Hobbs, 1984 ; Gayatri et al., 2017 ), which increasesεgandDvgat the same time.Gayatri et al.(2017) also found that the spectrum of graupel particles broadens during the riming process.The positive correlation betweenεsandDvscould be related to the aggregation growth.Similar to the effect of collision–coalescence and riming,the process of aggregation leads to a broadening of the snow particle spectrum ( Heymsfield et al., 2002 ) and an increase ofεs.

3.2. Development of relative dispersion parameterizations

After analyzing the relationships betweenεxandDvxof different hydrometeor spectra, we hope to quantify these relationships.An interesting point is that the relationship betweenεxandDvxis similar for the polluted and clean conditions, which facilitates the development of parameterizations ofεx.Fig.1 combines the results in both polluted and clean conditions, which are fitted using the equationy=axb+c, wherea,b,andcare three constant parameters; andxandyare independent and dependent variables, respectively.The relative dispersion parameterizations are:

for cloud droplets, where the unit ofDvcisμm;

for raindrops, where the unit ofDvris mm;

for ice crystals, where the unit ofDviis mm;

for graupel particles, where the unit ofDvgis mm; and

for snow particles, where the unit ofDvsis mm.The correlation coeffi-cients (R) of the above equations for raindrops, ice crystals, and snow particles show that the correlations between variables are all higher than 0.90, whereas theRvalues for cloud droplets and graupel particles are relatively lower, at 0.33 and 0.25, respectively.Thepvalues for indicating the statistical significance ( Hung et al., 1997 ) are smaller than 0.01 for all the fitting equations, which indicates high significance.

3.3. Application of the relative dispersion parameterizations in the terminal velocity parameterizations

Previous studies have shown thatμxplays an important role in the sedimentation process of hydrometeors ( Milbrandt and Yau, 2005a ).Fig.2 compares the terminal velocity calculated by the relative dispersion parameterizations with that calculated assumingμxis constant.As a first step,μx= 0, 1, 2, 3 is used for all hydrometeors; if the lines assuming theseμxvalues are still too far away from the line using the relative dispersion parameterization, then moreμxvalues are added.For example,μc= 4–5 is used for cloud.As expected,VmxandVnxincrease with the increase ofDvx, andVmxis larger thanVnx.For cloud droplets, ice crystals, graupel particles, and snow particles, theμxcalculated by the relative dispersion parameterizations is larger than 0; therefore, assumingμx= 0 in models underestimatesμx, and therefore overestimatesεx;while for raindrops, assumingμx= 0 is close to the relative dispersion parameterizations.

Fig.1.Joint probability density function of the relationships between the relative dispersion ( ε x ) and volume-mean diameter ( D v x ) of different hydrometeor spectra based on combined results under both clean and polluted conditions: (a) cloud droplets, (b) raindrops, (c) ice crystals, (d) graupel particles, and (e) snow particles.The black lines show the fitting functions, with R and the p -value representing the correlation coefficient and significance level, respectively.In the fitting functions,x and y represent the abscissa and ordinate, respectively.

The uncertainties of assuming constantμxandεxon terminal velocity are quantitatively evaluated with respect to using the relative dispersion parameterizations (Fig.S2).The relative deviation ofVmx,i.e.,100% , is calculated, whereis the terminal velocity calculated with the relative dispersion parameterization, andi s the terminal velocity calculated with the fixedμx.Similarly, the relative deviation ofVnxis also calculated.In Fig.S1, the relationships between relative deviation andDvxare positive for cloud, rain, graupel,and snow, and negative for ice –similar to the relationships betweenεxandDvxfor different hydrometeors.The comparisons confirm that the constantμxunderestimates theVmxandVnxin small volume-mean diameter bins, except for ice crystals.If we have to assume a constantμxin models, several suggestions based on the comparisons ( Figs.2 and S2) are given as follows:μc= 4 or 5,μr= 0,μi= 3 or 4,μg= 4 or 5, andμs= 2.

Since there is no observational terminal velocity for comparison, the results from the relative dispersion parameterization are taken to be the“true ” values in the above discussion.Whenin-situobservations of terminal velocity are available, it is necessary to evaluate the performance of the relative dispersion parameterization and fixed shape parameters.

4.Conclusion

Convective clouds in Shouxian of Anhui province in China are simulated with the fast version of WRF-SBM and the relationship between the relative dispersion and volume-mean diameter of each hydrometeor is analyzed for both polluted and clean conditions.Relative dispersion parameterizations are developed and their effects on terminal velocity evaluated.

It is found that the relationship between the two quantities is negative for ice crystals and positive for cloud droplets, raindrops, graupel particles, and snow particles.The reason for the different relationships is that different hydrometeors are affected by different microphysical processes.The ice crystals could be mainly affected by the deposition process, which tends to narrow the ice crystal spectra when the volumemean diameter increases.Cloud droplets and raindrops are affected by the process of collision–coalescence, and graupel and snow particles could be affected by riming and aggregation, respectively, increasing the volume-mean diameters and broadening the hydrometeor spectra.

The above relationships are similar under polluted and clean conditions; therefore, relative dispersion parameterizations are developed based on the data under both polluted and clean conditions.After applying the parameterizations, calculation of the mass-weighted terminal velocity and number-weighted terminal velocity indicates that for cloud droplets, ice crystals, graupel particles, and snow particles, assuming the constant shape parameter in the gamma distribution is equal to 0 underestimates the shape parameter and overestimates the relative dispersion;while for raindrops, assuming the shape parameter is equal to 0 is close to the relative dispersion parameterizations.If constant shape parameters have to be used, the most appropriate values are recommended for different hydrometeors.The relative dispersion parameterizations developed here can be used to optimize the terminal velocity parameterizations in models.

Although this simulation is dominated by convective cloud precipitation, it also includes stratiform cloud precipitation, according to the analysis of the same case in Fan et al.(2013) and Chen et al.(2020) .Furthermore, the results are based on a large dataset simulated with the SBM scheme; there are 44561, 261612, 100027, 10042, and 1171186 size distributions for cloud, rain, ice, graupel, and snow, respectively.Therefore, generally speaking, we think that the results can be applied in other precipitation events.It would be interesting to test the parameterizations, especially for simulations of extreme rain events.

Funding

This research was supported by the National Natural Science Foundation of China [Grant Nos.41822504 , 41775131 , 42027804 , 42075073 ,41975181, and 41775136 ].

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.aosl.2021.100141 .