Newly Developed Multiparameter Bulk Cloud Schemes. Part I: A New Triple-Moment Condensation Scheme and Tests

Jun Zhang aLaboratory of Cloud-Precipitation Physics and Severe Storms, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
cUniversity of Chinese Academy of Sciences, Beijing, China

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Jiming Sun aLaboratory of Cloud-Precipitation Physics and Severe Storms, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
bNanjing University of Information and Technology, Nanjing, China
cUniversity of Chinese Academy of Sciences, Beijing, China

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Wei Deng aLaboratory of Cloud-Precipitation Physics and Severe Storms, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
cUniversity of Chinese Academy of Sciences, Beijing, China

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Yuxia Ma dCollege of Atmospheric Sciences, Lanzhou University, Lanzhou, Gansu, China

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Abstract

Double-moment schemes cannot accurately describe the evolution of the cloud droplet spectrum during condensation. Hence, a new triple-moment condensation scheme is developed to describe the evolution of cloud droplet spectra. In this scheme, a three-parameter gamma distribution function of the cloud droplet mass is adopted, and the prognostic equations of the spectral shape parameter and slope parameter are derived by means of the number concentration, cloud water content, and reflectivity factor of cloud droplets. The new parameterization scheme is compared with high-resolution Lagrangian and Eulerian bin schemes, double-moment schemes, and existing triple-moment schemes by performing simulations under different supersaturation values. The new scheme can reduce the cloud spectral error in the cloud water content and reflectivity factor caused by the fixed shape parameter in some bulk schemes. The spectra simulated with the new scheme match the Lagrangian analytical solutions well, with errors within approximately 1% in the cloud water content and reflectivity factor. The effects of curvature and solution on condensation growth are also tested using the new scheme, and a method of using multiple gamma distribution functions to characterize the multimodal spectrum of cloud droplets is proposed in the new condensation scheme. Ultimately, the formation of rain embryos from giant aerosols can be simulated via the new scheme.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jiming Sun, jimings@mail.iap.ac.cn

Abstract

Double-moment schemes cannot accurately describe the evolution of the cloud droplet spectrum during condensation. Hence, a new triple-moment condensation scheme is developed to describe the evolution of cloud droplet spectra. In this scheme, a three-parameter gamma distribution function of the cloud droplet mass is adopted, and the prognostic equations of the spectral shape parameter and slope parameter are derived by means of the number concentration, cloud water content, and reflectivity factor of cloud droplets. The new parameterization scheme is compared with high-resolution Lagrangian and Eulerian bin schemes, double-moment schemes, and existing triple-moment schemes by performing simulations under different supersaturation values. The new scheme can reduce the cloud spectral error in the cloud water content and reflectivity factor caused by the fixed shape parameter in some bulk schemes. The spectra simulated with the new scheme match the Lagrangian analytical solutions well, with errors within approximately 1% in the cloud water content and reflectivity factor. The effects of curvature and solution on condensation growth are also tested using the new scheme, and a method of using multiple gamma distribution functions to characterize the multimodal spectrum of cloud droplets is proposed in the new condensation scheme. Ultimately, the formation of rain embryos from giant aerosols can be simulated via the new scheme.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jiming Sun, jimings@mail.iap.ac.cn

1. Introduction

The condensation of water vapor is an important process in the formation and development of clouds (Ryan 1974; Rao and Feng 1977). In particular, the evolution of the droplet size distribution (DSD) strongly impacts the occurrence of warm rain, the nucleation of ice, the discharge of lightning, and even the emission of radiation. Nevertheless, numerically modeling the condensation growth for a population of cloud droplets remains a major challenge for weather simulations when utilizing cloud-resolving models (CRMs) and general circulation models (GCMs).

Numerical simulations of DSDs have long been a focus in the cloud physics community (Mordy 1959; Fitzgerald 1974; Clark 1974, hereafter C74; Milbrandt and Yau 2005a,b, hereafter MY05; Harrington et al. 2013; Dolan et al. 2018; Paukert et al. 2018; Milbrandt et al. 2021; Lee et al. 2021). Two types of cloud microphysics schemes are used to describe the evolutions of hydrometeor size spectra in CRMs. The first type includes bin microphysics schemes, which focus on the microphysical processes in each of various hydrometeor categories (Tzivion et al. 1994; Khain et al. 2000, 2011; Lee et al. 2019, 2021). The DSD can be described by tens to several hundreds of mass bins, and the properties of each bin are calculated by solving microphysical equations (Khain et al. 2015). Therefore, bin schemes in CRMs have many prognostic variables and are highly capable of simulating the evolutions of irregular cloud DSDs (Tzivion et al. 1999; Yin et al. 2000; Khain et al. 2004, 2008; Fan et al. 2009; Khain et al. 2015; Lee et al. 2019, 2021). However, bin schemes suffer from some shortcomings, such as numerical diffusion (in other words, artificial broadening of the DSD), and large-scale simulations using these schemes are expensive (Khain et al. 2000; Morrison et al. 2018; Lee et al. 2021).

The second type of mathematical method involves bulk parameterization microphysics schemes, in which the DSDs of hydrometeor species are approximated by mathematical functions. The distribution functions in most bulk schemes are described by gamma distribution functions (C74; Cohard and Pinty 2000; Milbrandt and Yau 2005a; Lim and Hong 2010) containing three parameters, namely, the intercept, shape, and slope, all of which must be predicted or prescribed to simulate the evolution of the DSD. According to the number of prognostic distribution moments, these bulk parameterization schemes can be classified as either single-moment schemes that predict the hydrometeor mass mixing ratio (Kessler 1969; Wisner et al. 1972; Lin et al. 1983; Rutledge and Hobbs 1983; Cotton et al. 1986; Dudhia 1989; Tao and Simpson 1993; Walko et al. 1995; Kong and Yau 1997; Bae et al. 2018) or double-moment schemes that additionally predict the particle number concentration (Ziegler 1985; Murakami 1990; Ikawa and Saito 1991; Levkov et al. 1992; Wang and Chang 1993; Ferrier 1994; Harrington et al. 1995; Meyers et al. 1997; Reisner et al. 1998; Cohard and Pinty 2000; Girard and Curry 2001; Seifert and Beheng 2001; Morrison et al. 2005; Milbrandt and Yau 2005a; Thompson et al. 2008; Morrison et al. 2009; Lim and Hong 2010; Mansell and Ziegler 2010). In double-moment schemes, the shape parameter is either fixed or diagnosed by an empirical relationship (Morrison and Grabowski 2007, hereafter MG07; Thompson et al. 2008). The shape parameter for precipitation particles has been found to increase from the stratiform phase to the convective phase (Uijlenhoet et al. 2003). Thus, surface precipitation simulations with constant shape parameters are significantly different from those with bin schemes (Milbrandt and Yau 2005a).

When the shape parameter is fixed, the width of the droplet spectrum remains nearly unchanged. However, absent the mixing of parcels during condensation, the width of the spectrum should narrow over time because small droplets grow faster than large droplets (Rogers and Yau 1989). Hence, it is more advantageous to diagnose the shape parameter by an empirical relationship than to adopt a fixed shape parameter (Milbrandt and Yau 2006; MG07). For instance, Milbrandt and Yau (2006) found that the hydrometeor fields simulated with a diagnosed shape parameter in a double-moment scheme matched both the observations and those produced by a triple-moment scheme. However, Loftus et al. (2014) pointed out that the empirical relationship used for such diagnoses is not universal.

As a result, double-moment schemes may not accurately describe the evolution of the DSD during condensation. Since observed DSDs are averaged over different temporal and spatial scales, a changeable shape parameter is necessary to account for different temporal and spatial resolutions. Moreover, fitting the measured size distributions of cloud droplets by gamma distribution functions has revealed substantial differences in the shape parameters among maritime, coastal, continental, and urban cumulus clouds (Costa et al. 2000). Therefore, a gamma distribution function with a fixed shape parameter cannot be used to microphysically describe the evolutions of hydrometeor size distributions.

In particular, the shape parameter is of key importance because it determines the relative dispersion of the simulated cloud spectra and thus the cloud optical properties and the rate at which precipitation forms (Meyers et al. 1997; Liu and Daum 2000; Uijlenhoet et al. 2003; Milbrandt and Yau 2005a; Igel and van den Heever 2017a,b). Therefore, accurate simulations of DSDs are essential for understanding several phenomena in clouds. For example, the heat and mass transfer coefficients of graupel during the riming process can be substantially modified by using different cloud DSDs (Avila et al. 2001). Additionally, studies have demonstrated that the most important impact of the shape parameter is on drizzle formation (Stevens et al. 1998; vanZanten et al. 2005; Petters et al. 2006). Nevertheless, the values of the shape parameter have typically been chosen between 1 and 10.

Consequently, recent studies have paid considerable attention to triple-moment schemes in which the shape parameter is taken as a prognostic variable (MY05; Milbrandt and Yau 2006; Loftus et al. 2014; Naumann and Seifert 2016; Chen et al. 2016; Milbrandt et al. 2021). C74 developed a triple-moment condensation scheme and considered the temporal evolution of the shape parameter; a recent study demonstrated that simulating cloud droplets by using this triple-moment scheme can greatly mitigate cloud spectrum broadening (Deng et al. 2018). Milbrandt and Yau (2005b) developed a triple-moment scheme with a closure formulation for calculating the source and sink terms of the reflectivity factor. In this scheme, the tendency of the reflectivity factor is derived via the tendencies of the mass content and number concentration with a fixed shape parameter at each time step during condensation. However, this treatment actually introduces a certain error; namely, when employing this scheme, the spectrum width cannot be correctly simulated due to the under- and overestimation of the cloud water content and reflectivity factor, respectively, as will be discussed in detail later. Therefore, it is necessary to develop a triple-moment condensation scheme to more accurately describe the evolution of the droplet spectrum.

Furthermore, the observations reported by Prabha et al. (2011) and the numerical simulations performed by Pinsky and Khain (2002) and Khain et al. (2012) indicate that the existence of the smallest droplets within clouds at any height above the cloud base forms a new second DSD mode. Giant aerosols also play an important role in the formation of rain embryos (Johnson 1982; Laird et al. 2000; Jensen 2008; Sun et al. 2022). However, the existing bulk parameterization schemes are unable to describe both the second mode and the mode induced by giant nuclei. This is mainly due to the use of a unimodal gamma distribution function to describe the evolution of the cloud DSD, in which freshly nucleated droplets must obey the same gamma distribution as other cloud droplets and are assumed to be immediately distributed over a wide range of sizes. Unfortunately, this treatment may cause the DSD to be erroneous (Khain et al. 2015).

On the basis of cloud microphysical processes, new multimoment cloud parameterization schemes have been developed by the Institute of Atmospheric Physics (IAP), Key Laboratory of Cloud-Precipitation Physics and Severe Storms (LACS), Chinese Academy of Sciences. In these schemes, the size distribution spectra of hydrometeors are expressed by gamma distribution functions on the mass size scale [f(m) = N0mα−1eβm]. In our research, we employ the mass distribution because the analytical solutions for the autoconversion rate and the accretion rate cannot be derived from the radius distribution function in gravitational stochastic collision and coalescence processes (Sun et al. 2019). Accordingly, the shape parameters of all kinds of hydrometeors can be predicted based on the combination of the number concentration, the water content, and the reflectivity. A series of our recent papers introduce our new microphysical parameterization schemes, including a new condensation scheme, a new analytical warm rain formation scheme (Sun et al. 2019; Deng and Sun 2019), and new four-parameter ice deposition and riming schemes (Zhang and Sun 2019). In this paper, we introduce a new triple-moment condensation scheme (hereafter referred to as the TM scheme). Moreover, to simulate multimodal cloud droplet distributions resulting from the primary and secondary activation of cloud condensation nuclei (CCN), three gamma distribution functions are coupled with our new nucleation scheme (Sun et al. 2022) to simulate the evolutions of cloud droplet spectra. Then, we further couple the new TM scheme into a 1.5D Eulerian model (Sun et al. 2012a) and perform a simulation of an in situ observation case from the Cooperative Convective Precipitation Experiment (CCOPE).

The remainder of this paper is structured as follows. The mathematical formulations for the TM parameterization are derived in section 2. Theoretical experiments of the TM parameterization under constant supersaturation are given in section 3, and comparisons of bulk schemes with bin schemes are performed in section 4. The in situ observation simulation is described in section 5. Finally, the results of our work are discussed and concluded in section 6.

2. Mathematical formulation of the triple-moment condensation scheme

a. Parameterization of condensation growth for large pure water droplets

One of the general analytical distribution functions used to describe the size distributions of hydrometeors is given in the following form (Seifert and Beheng 2001, 2006):
f(m)=N0mμα1e(βm)μ,
where m is the mass of a single cloud droplet, α is the shape parameter, β is the slope parameter, N0 is the intercept parameter, and μ is the tail parameter (here, μ is set to 1). If the cloud droplet mass (m) is replaced by the cloud droplet radius (r), f(r) represents the distribution function on the radius size scale (C74; Pruppacher and Klett 1997; Cohard and Pinty 2000; Milbrandt and Yau 2005a; Lim and Hong 2010).
The kinetic equation of condensation for the continuity of droplets with mass m can be written in the following form (Buikov 1961; Levin and Sedunov 1966; Cotton and Anthes 1989; Khvorostyanov and Curry 1999):
f(m,x,y,z)t+xi{[uiυ(m)δi3]f(m,x,y,z)}+m[dmdtf(m,x,y,z)]=J,
where ui denotes the velocity components in the x, y, and z directions, υ(m) is the terminal velocity for droplets with mass m, δi3 is the Kronecker symbol, J describes droplet sources and sinks, the usual summation convention over double indices is assumed, and i = 1, 2, 3. The last term /m[(dm/dt)f(m,x,y,z)] represents the divergence of f(m, x, y, z) due to condensation growth.
Under the assumption that the velocity divergence is zero and advection does not occur, the continuity equation for the droplet size spectra during condensation can be written in a 1D mass space:
f(m)t=m[dmdtf(m)]+J.
The change rate of the number of droplets during condensation and nucleation can be described by the above equation (Rogers and Yau 1989). Since additional physical variables are necessary to determine the evolution of the DSD, the multiple moments in this study are defined by
MP=Hp0mpf(m)dm.
This study applies three moments (p = 0, 1, 2) representing the number concentration (cm−3), cloud water content (g m−3), and reflectivity factor (mm6 m−3), which are equivalent to the zeroth, third, and sixth moments, respectively, in the radius–diameter space. H0=1, H1=106, H2=1012/(πρw/6)2, and ρw is the density of water. By substituting Eq. (1) into Eq. (4), the analytical solutions of the three moments can be written as follows:
M0=H00N0mα1eβmdm=N0Γ(α)βα,
M1=H10N0mαeβmdm=H1N0Γ(α+1)βα+1=H1M0αβ,
M2=H20N0mα+1eβmdm=H2N0Γ(α+2)βα+2=H2M0(α+1)αβ2.
Next, we exactly differentiate Eqs. (6) and (7) with respect to time:
dM1dt=H1M0βdαdtH1M0αβ2dβdt+H1αβdM0dt,
dM2dt=H2M0(2α+1)β2dαdt2H2M0(α+1)αβ3dβdt+H2(α+1)αβ2dM0dt.
The prognostic equations of the shape parameter and the slope parameter can be derived by combining Eqs. (8) and (9) as follows:
dαdt=2β(α+1)H1M0dM1dtβ2H2M0dM2dtα(α+1)M0dM0dt,
dβdt=β2(2α+1)H1M0αdM1dtβ3H2M0αdM2dtαβM0dM0dt.
For the growth of a single cloud droplet with a radius larger than 10 μm, the solution and curvature effects on the drop equilibrium vapor pressure can be neglected (Rogers and Yau 1989), and its mass condensation growth rate can be expressed as
dmdt=k(16πρw)1/3sm1/3,
where s is the ambient supersaturation and k is associated with heat conduction and vapor diffusion; k is given in the following form:
k=2π(LRυT1)LKaT+RυTDυEw,
where T is the ambient temperature, L is the latent heat, Rυ is the individual gas constant for water vapor, Dυ is the molecular diffusion coefficient, Ka is the coefficient of thermal conductivity of air, and Ew is the saturation vapor pressure.
For a population of cloud droplets, if nucleation is considered during condensation, the rates of change in the number concentration (M0), cloud water content (M1), and reflectivity factor (M2) can be obtained by combining Eqs. (1), (4), and (12) (the detailed derivations are shown in appendix A):
dM0dt=dM0dt|nucleation,
dM1dt=H10dmdtf(m)dm+dM1dt|nucleation=H1kM0(16πρw)1/3sΓ(α+13)Γ(α)β1/3+dM1dt|nucleation,
dM2dt=H202mdmdtf(m)dm+dM2dt|nucleation=2H2kM0(16πρw)1/3sΓ(α+43)Γ(α)β4/3+dM2dt|nucleation.

During condensation, the number concentration of cloud droplets remains the same, so any change in the number concentration must be caused by nucleation. dM0/dt|nucleation represents the cumulative number of cloud droplets nucleated by CCN per unit time. In contrast, the changes in the cloud water content and reflectivity factor are caused by both nucleation and condensation. The first terms on the right-hand sides of Eqs. (15) and (16) are the changes in the cloud water content and reflectivity factor during condensation, while dM1/dt|nucleation and dM2/dt|nucleation are the cumulative water content and reflectivity factor, respectively, of cloud droplets nucleated by CCN per unit time (the detailed derivations are shown in appendix D).

By substituting Eqs. (14)(16) into Eqs. (10) and (11), we can obtain the final prognostic equations for both the shape parameter and the slope parameter:
dαdt=4k3(16πρw)1/3sΓ(α+13)Γ(α)β2/3+2β(α+1)H1M0dM1dt|nucleationβ2H2M0dM2dt|nucleationα(α+1)dlnM0dt|nucleation,
dβdt=k3(16πρw)1/3sΓ(α+13)Γ(α+1)β5/3+β2(2α+1)H1M0αdM1dt|nucleationβ3H2M0αdM2dt|nucleationαβdlnM0dt|nucleation.
Alternatively, the shape parameter and slope parameter can be diagnosed by combining Eqs. (6) and (7):
α=H2M12H12M0M2H2M12,
β=H1H2M0M1H12M0M2H2M12.
The intercept parameter (N0) can be obtained by Eq. (5):
N0=M0βαΓ(α).

The above equation illustrates that the intercept parameter (N0) is related to the number concentration (M0), the shape parameter (α), and the slope parameter (β) and that both the shape parameter (α) and the slope parameter (β) are related to the number concentration (M0), the cloud water content (M1), and the reflectivity factor (M2). Thus, the intercept parameter (N0) depends on the number concentration (M0), the cloud water content (M1), and the reflectivity factor (M2).

b. Effects of curvature and solution on condensation growth

The curvature and solution effects are prominent factors influencing the condensation growth of droplets with radii of less than 10 μm. The growth rate of a single cloud droplet during condensation considering the effects of curvature and solution can be described as
dmdt=k[(16πρw)1/3sm1/32a+8(16πρw)2/3bm2/3],
where the second and third terms of the formula are called the “curvature term” and “solution term,” respectively, and a=2σs/(RυTρw), with σs representing the surface tension. The parameterization scheme represents the bulk or average effect, so b (4.3iMs¯/ms) of the “solution term” in the new TM scheme is related to the mean mass ( Ms¯) of the solutes. Furthermore, i is the degree of ionic dissociation, and ms is the molecular weight of a particular solute.
The change rate of the number concentration (M0) considering the effects of curvature and solution is the same as in Eq. (14), which depends on the ambient supersaturation, the hygroscopic behavior of aerosols, and the aerosol size distribution. The treatment of cloud droplet nucleation by a bin scheme is described in appendix D. In addition, the change rates of the cloud water content (M1) and the reflectivity factor (M2) considering the effects of curvature and solution can be written as follows (the detailed derivations are shown in appendix A):
dM1dt=H10dmdtf(m)dm+dM1dt|nucleation=H1kM0[(16πρw)1/3sΓ(α+13)Γ(α)β1/32a+(16πρw)2/38bΓ(α23)Γ(α)β2/3]+dM1dt|nucleation,
dM2dt=H202mdmdtf(m)dm+dM2dt|nucleation=2H2kM0[(16πρw)1/3sΓ(α+43)Γ(α)β4/32aΓ(α+1)Γ(α)β+(16πρw)2/38bΓ(α+13)Γ(α)β1/3]+dM2dt|nucleation.
By substituting Eqs. (14), (23), and (24) into Eqs. (10) and (11), we can obtain the final prognostic equations of the shape and the slope parameters considering the effects of curvature and solution:
dαdt=4k[13(16πρw)1/3sΓ(α+13)Γ(α)β2/3aβ+203(16πρw)2/3bΓ(α23)Γ(α)β5/3]+2β(α+1)H1M0dM1dt|nucleationβ2H2M0dM2dt|nucleationα(α+1)dlnM0dt|nucleation,
dβdt=k[13(16πρw)1/3sΓ(α+13)Γ(α+1)β5/32aβ2α+563(16πρw)2/3bΓ(α23)Γ(α+1)β8/3]+β2(2α+1)H1M0αdM1dt|nucleationβ3H2M0αdM2dt|nucleationαβdlnM0dt|nucleation.

In Eqs. (25) and (26), the second and third terms represent the “curvature term” and the “solution term,” respectively. The intercept parameter (N0) considering the effects of curvature and solution can also be determined by Eq. (21).

The ventilation effect is also considered in the TM scheme (the detailed derivation is shown in appendix A), but because simulations show that the impacts of ventilation on cloud droplet condensation growth are negligible, the relevant figures and discussion are omitted.

c. Comparison of the new triple-moment condensation scheme with existing schemes

To compare the simulated solutions of the proposed TM scheme with the analytical solutions, the solution and curvature effects on the droplet equilibrium vapor pressure are not considered; instead, these effects are analyzed separately. Here, the simulated solutions of other schemes, including two high-resolution bin schemes [the Eulerian bin scheme (hereafter EBS) and Lagrangian bin scheme (hereafter LBS)], a double-moment scheme (hereafter DM) and the triple-moment scheme of MY05, are compared with those of the proposed TM scheme.

As in the aforementioned gamma distribution function, the cloud droplet mass is adopted as a size scale in the TM scheme, but the cloud droplet radius (diameter) is often adopted in some bulk parameterization schemes. The adoption of different size scales results in a slight difference between spectra under the same moments (M0, M1, and M2). Thus, to demonstrate the performance of the new TM scheme under a wide range of conditions, we also modify our scheme to be applicable at the radius/diameter scale. In other words, by adjusting the tail parameter (μ) from 1 to 1/3, the distribution of the mass scale can be mapped to the radius/diameter scale (Khain et al. 2015). Therefore, the TM scheme with μ = 1/3 is also simulated, and the results are compared with the simulation results of the radius/diameter-scale schemes, including the double-moment scheme of MG07, the triple-moment scheme of C74 and the modified Clark scheme (hereafter C74–modified). A detailed description of the TM scheme with μ = 1/3 is introduced in appendix A.

With regard to bin schemes, the mass bin i + 1 initially equals 21/50 times the mass bin i. The mass increase between bins is fixed at 21/50 for EBS, which adopts the multidimensional positive definite advection transport algorithm (MPDATA) for numerical transport with the nonoscillatory option (Schar and Smolarkiewicz 1996). To mitigate Eulerian numerical diffusion (Morrison et al. 2018), we apply 3000 mass intervals to describe the cloud spectra between 1 and 1000 μm. LBS is based on an approach that solves analytical solutions for the evolution of the DSD (appendix D of Sun et al. 2022). If the solution and curvature effects on the droplet equilibrium vapor pressure are neglected, the differences between the squares of the cloud droplet radii are independent of time. Furthermore, the cloud droplet diameter of bin i at time step j can be determined analytically at time step j − 1 under constant ambient supersaturation. Moreover, the total number of cloud droplets is conserved for every size bin under the assumption that only condensation is considered without nucleation and coagulation at any time step, and thus, the size distribution at time step j can be resolved [Eq. (D5) of Sun et al. 2022].

For the MY05 scheme, although the diameter scale of the gamma distribution function is employed, the tail parameter μ is set to 3 (Milbrandt and Yau 2005b); due to this treatment, MY05 is consistent with TM. In contrast, DM assumes that the shape parameter remains constant at 1 and 5, while MG07 diagnoses the shape parameter by a formula related to the cloud droplet number concentration (MG07). The C74 and C74–modified schemes obtain the shape parameter by solving an exact differential equation derived by resorting to the average radius rather than the reflectivity factor. All differential equations are resolved through a sixth-order Runge–Kutta algorithm.

3. Condensation experiments under constant supersaturation

The temporal evolutions of the DSDs are simulated with initial cloud droplet spectra under a simplified scenario of constant supersaturation at T = 293.28 K and P = 94 479 Pa, which are the same conditions as in Sun et al. (2022).

a. Characteristics of DSDs with different schemes ignoring the effects of curvature and solution

Analytical solutions for the evolution of the DSD can be obtained if the curvature and solution effects on the growth rate of large droplets are ignored (Rogers and Yau 1989). Such a scenario assumes that the saturation vapor pressures over the surfaces of droplets are the same as the saturation vapor pressure over bulk water at the same temperature. By comparing the analytical solutions of different condensation schemes, we can determine the advantages and disadvantages of these algorithms based on their numerical simulation accuracies. To test the new TM scheme under a wide range of conditions, we establish 9 cases, as shown in Tables 1 and 2. These conditions include initial spectra excluding rain embryos with supersaturation values of 0.1%, 0.2%, and 0.3% (case 1, case 4, and case 7, respectively), initially narrow spectra including rain embryos with a maximum diameter of approximately 100.0 μm with supersaturation values of 0.1%, 0.2%, and 0.3% (case 2, case 5, and case 8, respectively) and initially wide spectra that include both small cloud droplets and rain embryos with supersaturation values of 0.1%, 0.2%, and 0.3% (case 3, case 6, and case 9, respectively). These cases are created both to simulate the formation of rain embryos by different schemes under different supersaturation values and to test the evolutions of cloud droplet spectra after rain embryo formation under different supersaturation values.

Table 1

The initial number concentrations, cloud water contents, and reflectivity factors for cases 1–9.

Table 1
Table 2

The corresponding initial shape parameters, slope parameters, and intercept parameter for cases 1–9 with tail parameters of μ = 1 and μ = 1/3.

Table 2

Figures 13 show the evolutions of both the cloud droplet spectra simulated with TM (μ = 1), MY05, DM, EBS, and LBS (ignoring the solution and curvature effects) and the DSD standard deviations in terms of the radius (σ) of each of these schemes over a 15-min interval for cases 1–9. The evolutions of the DSDs simulated with LBS (Figs. 13) indicate that the DSDs become narrow since the growth rates of cloud droplets are inversely proportional to their radii and because small droplets grow more quickly than large droplets. Correspondingly, the σ values of LBS decrease over time due to the shortened widths of the cloud droplet spectra. The DSDs simulated with EBS are wider than those obtained by LBS due to the numerical diffusion produced by the Eulerian advection algorithm (Morrison et al. 2018), which occurs even though 3000 mass intervals are adopted, especially in the cases where the initial cloud droplet distributions exclude rain embryos (Fig. 1d).

Fig. 1.
Fig. 1.

Evolutions of the cloud droplet spectra and the DSD standard deviations in terms of the radius (σ) simulated with TM (μ = 1), MY05, DM, EBS, and LBS while ignoring the solution and curvature effects with a supersaturation value of 0.1% for cases 1–3.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 2.
Fig. 2.

As in Fig. 1, but under a supersaturation value of 0.2%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 3.
Fig. 3.

As in Fig. 1, but under a supersaturation value of 0.3%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Regarding the simulations of the bulk schemes, the spectra simulated with MY05 are much wider than the Lagrangian analytical solutions, and the σ values are larger than those of LBS. Moreover, the σ values increase over time and with increasing supersaturation, especially in the cases including small droplets (Figs. 1d,l). Even though the differences in the cloud droplet spectra between MY05 and LBS are relatively small under the initially narrow spectra of large cloud droplets, including rain embryos, the values of σ in MY05 are still larger than those in LBS; the maximum value of σ simulated with MY05 can reach 10 μm. Furthermore, MY05 assumes that the shape parameter remains unchanged for condensation occurring within a small time step. As a result, the reflectivity factor is not an independent variable during condensation because the rate of change in the reflectivity factor can be obtained by the rate of change in the cloud water content and number concentration of cloud droplets [Eq. (B8) in appendix B]. Since the number concentration of cloud droplets remains unchanged if only condensation is taken into account, the rate of change in the reflectivity factor can be obtained only by the rate of change in the cloud water content. Using such a reflectivity factor and the cloud water content to predict the spectral shape parameter causes its rate of change to be zero [Eq. (B16) in appendix B]. Consequently, the spectra simulated with MY05 are the same as those simulated with DM. In contrast, the evolutions of the DSDs simulated with TM match the Lagrangian analytical solutions much better than those simulated with MY05, and the values of σ are close except in the cases with the initially wide spectra that include both small cloud droplets and rain embryos. In the cases of the initial spectra excluding rain embryos, TM can simulate small droplet growth even more accurately than EBS; this further illustrates the importance of the shape parameter for describing cloud droplet condensation. However, for the wide spectra including rain embryos, the small droplets on the left tail of the distribution in the TM simulations grow slower than those in the LBS and EBS simulations, and the values of σ obtained by TM are slightly larger than those obtained by LBS, but the difference in σ decreases over time and with increasing supersaturation. Moreover, contrary to the simulation results of MY05 and DM, the values of σ simulated with LBS and TM decrease over time and with increasing supersaturation, with the values of σ varying between 0 and 3 μm for the initial spectra both including and excluding rain embryos. In the cases of the initial spectra excluding rain embryos, the values of σ simulated with TM and LBS are close to 0 μm, which indicates that the spectra consisting of small cloud droplets must become very narrow through condensation alone. Figures 3c, 3g, and 3k show that the sizes of initially small droplets can exceed 40 μm after 15 min under a supersaturation value of 0.3. By neglecting the curvature and solution effects, the analytical growth rates of small droplets are overestimated because equilibrium supersaturation is underestimated. However, for droplets larger than 20 μm, the curvature and solution effects are negligible. Therefore, the TM simulations of rain embryo formation are reliable.

Figures 46 show the evolutions of the cloud droplet spectra and DSD standard deviations (σ) simulated with TM (μ = 1/3), C74, C74–modified, MG07, EBS, and LBS (ignoring the solution and curvature effects) for cases 1–9. Since the shape parameter of MG07 is diagnosed by the formula associated with the cloud droplet number concentration in each time step, the shape parameter of MG07 is adjusted to be different from the initial value in the second time step. Furthermore, the condensation simulation with MG07 without taking into account either nucleation or coagulation causes the shape parameter to remain unchanged. In other words, the σ values of MG07 are larger than those of LBS in the condensation process. The values of σ increase over time and with increasing supersaturation, and the maximum value of σ simulated with MG07 can even reach 12 μm. In contrast, the cloud droplets simulated with C74 grow slower than those simulated with LBS, which is especially prominent in the simulations with the initial cloud droplet spectra consisting of small cloud droplets (Figs. 4b,c). Since the values of σ characterize the spectrum width, the agreement of σ between C74 and LBS indicates that C74 can describe the main characteristics of the cloud droplet spectra during condensation even if the growth rates of droplets are underestimated. The C74–modified scheme improves this deficiency by modifying the growth rate of cloud droplets [Eq. (B31)]. The evolutions of the DSDs simulated with both C74–modified and TM show robust agreement with the Lagrangian analytical solutions except in the cases with the initially wide spectra that include both small and large cloud droplets.

Fig. 4.
Fig. 4.

As in Fig. 1, but for the TM (μ = 1/3), C74, C74–modified, MG07, EBS, and LBS simulations.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 5.
Fig. 5.

As in Fig. 4, but under a supersaturation value of 0.2%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 6.
Fig. 6.

As in Fig. 4, but under a supersaturation value of 0.3%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

The simulations with the aforementioned schemes also produce different cloud water contents and reflectivity factors; such discrepancies will be discussed in section 3c. In addition, note that although nine cases are simulated in this section, the above simulations cannot comprehensively describe the real condensation process because several other important supersaturation-related impact factors are ignored, such as the vertical velocity of air parcels and their acceleration (Segal et al. 2003; Sun et al. 2022), which results in secondary nucleation. This issue will be discussed in section 4.

Figure 7 shows the evolutions of the shape parameters simulated with TM (μ = 1), MY05 and DM in comparison with those simulated with LBS for cases 1–9. Although none of the simulations with the bin schemes have a shape parameter, the shape parameter can be uniquely determined by the combination of the number concentration, cloud water content and reflectivity factor by Eq. (19). Therefore, the shape parameters of LBS are also calculated by Eq. (19) to facilitate a comparison with those of the other schemes, whereas the shape parameters of EBS are not given because we focus on comparing the simulation accuracies of the bulk parameterization schemes with that of LBS. The shape parameters of DM and MY05 are constant. The evolutions of the shape parameters in the simulations with TM and LBS show that the former underestimates the shape parameter for the cases with the initially narrow cloud droplet spectra excluding rain embryos (Figs. 7a,d,g) and the initially wide spectra that include both small droplets and rain embryos (Figs. 7c,f,i). For the initially narrow spectra including rain embryos, the evolution of the shape parameter of TM matches that of LBS (Figs. 7b,e,h). In general, the shape parameter of TM is close to that of LBS. These findings demonstrate that the evolution of the shape parameter is quite flexible in our new TM scheme: the value of the shape parameter in our scheme can even reach approximately 33 000 after 900 s in case 7, but in case 3, it reaches only approximately 6 after 900 s. This flexibility of the shape parameter ensures that the DSD is accurately described.

Fig. 7.
Fig. 7.

Evolutions of the shape parameters simulated with TM (μ = 1), MY05, DM, and LBS for cases 1–9.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Figure 8 shows the evolutions of the shape parameters simulated with TM (μ = 1/3), C74, C74–modified, and MG07 in comparison with the Lagrangian analytical solutions for cases 1–9. The shape parameter of MG07 is constant, whereas the C74 scheme heavily underestimates the shape parameter. Contrary to the simulation results of C74, the C74–modified scheme overestimates the shape parameter. Similar to the simulations of TM with μ = 1, those of TM with μ = 1/3 also underestimate the shape parameter for the initially small cloud droplet spectra. Nevertheless, the differences relative to the Lagrangian analytical solutions in the simulation of the shape parameter with TM are smaller than those with the other schemes.

Fig. 8.
Fig. 8.

Evolutions of the shape parameters with TM (μ = 1/3), C74, C74–modified, MG07, and LBS for cases 1–9.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Since the relative dispersion (the ratio of the standard deviation of the radius to the average radius) reflects the spectrum width (C74; Liu and Daum 2002; Liu et al. 2006; Liu and Li 2015; Xie et al. 2017; Wang et al. 2020), we also analyze the relative dispersion of the cloud droplet spectrum; the derivation of the relative dispersion is shown in appendix C. Figure 9 shows the evolutions of the relative dispersions (ε) simulated with TM (μ = 1), MY05, DM, and EBS in comparison with the Lagrangian analytical solutions for cases 1–9. As mentioned above, the numerical diffusion of EBS is especially obvious given an initial spectrum of small cloud droplets, and the numerical diffusion of EBS causes large differences in the relative dispersions between EBS and LBS in case 1, case 4, and case 7, as shown in Figs. 9a, 9d, and 9g. The relative dispersions of DM and MY05 are constant, as their shape parameters and larger than those simulated with LBS over time. The evolutions of the relative dispersions of TM are generally consistent with those of LBS. However, in case 3, case 6, and case 9, there are also large differences between TM and LBS (Figs. 9c,f,i). The evolutions of the relative dispersions of TM in the simulations of the initially narrow cloud spectrum that includes rain embryos are consistent with their shape parameters, such as in case 2, case 5, and case 8. However, the evolutions of the relative dispersions are not always identical to those of the shape parameters even though the relative dispersion is a function of the shape parameter [Eqs. (C2), (C7), and (C9)]. The relatively large discrepancy in the high values of the shape parameter may result in small differences in the relative dispersion (such as the simulations of the initial cloud spectrum excluding rain embryos in case 1, case 4, and case 7) at late simulation times. On the other hand, the relatively small discrepancy in the low values of the shape parameter may lead to large differences in the relative dispersion (such as the simulations of the initially wide cloud spectrum that includes both small droplets and rain embryos in case 3, case 6, and case 9) at early simulation times. Such characteristics of the relative dispersions are consistent with the cloud spectra in Figs. 13. For example, the small droplets on the left tail of the distribution simulated with TM grow slower than those simulated with LBS under an initially wide spectrum that includes both small cloud droplets and rain embryos.

Fig. 9.
Fig. 9.

Evolutions of the relative dispersions (ε) simulated with TM (μ = 1), MY05, DM, EBS, and LBS for cases 1–9.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Figure 10 shows the evolutions of the relative dispersions (ε) simulated with TM (μ = 1/3), MG07, C74, C74–modified, and EBS in comparison with the Lagrangian analytical solutions for cases 1–9. The relative dispersions of EBS are larger than those of LBS for an initial spectrum of small cloud droplets in case 1, case 4, and case 7, as shown in Figs. 10a, 10d, and 10g. The relative dispersions of MG07 are constant as their shape parameters and larger than those simulated with LBS over time. The evolutions of the relative dispersions of TM are still larger than those of LBS (Figs. 10c,f,i) in the simulations of the initial cloud spectrum that includes both small cloud droplets and rain embryos. The relative dispersions of C74 are larger than those of LBS in all cases, but the relative dispersions of C74–modified match those of LBS very well in all cases.

Fig. 10.
Fig. 10.

Evolutions of the relative dispersions (ε) simulated with TM (μ = 1/3), MG07, C74, C74–modified, EBS, and LBS for cases 1–9.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

b. Sensitivity analysis of the double-moment scheme under different shape parameters

The above simulations imply that the fixed shape parameter of DM results in a large DSD error. To accurately address this issue, here, simulations are performed with DM given different shape parameters (α = 1, 10, 102, 103, 104, and 105). Figure 11 shows that the spectrum widths simulated with LBS become narrow and that the maximum distribution density correspondingly increases under different initial shape parameters. However, the spectrum widths simulated with DM become wider, and the maximum distribution density correspondingly decreases. Such results indicate that double-moment schemes with assumed values of shape parameters may produce cloud droplet size distributions broader than those adiabatically predicted in condensation.

Fig. 11.
Fig. 11.

Sensitivities of the cloud droplet spectrum evolutions for the DM scheme in comparison with the LBS evolutions under different initial shape parameters (1, 10, 102, 103, 104, and 105).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

c. Error analyses for the cloud water content (M1) and reflectivity factor (M2)

Figures 1214 show both the evolutions of M1 and M2 simulated with TM (μ = 1), MY05, DM, EBS, and LBS and the M1 and M2 errors with respect to the Lagrangian analytical solutions over a 15-min interval for cases 1–9. MY05 underestimates M1 for the cases including small cloud droplets. The M1 errors for case 1 and case 3 nearly reach −15% and −6%, respectively, with a supersaturation value of 0.1% at 900 s. In the cases of the initially narrow spectra that include rain embryos, the simulations of M1 with MY05 generally match those of LBS well. The M1 errors increase with increasing supersaturation for the cases including rain embryos. In addition, the values of M1 simulated with DM are similar to those simulated with MY05, and the values of M1 simulated with TM and EBS match the Lagrangian analytical solutions very well, with the errors all less than 1%.

Fig. 12.
Fig. 12.

Evolutions of (a),(e),(i) M1 and (c),(g),(k) M2 simulated with TM (μ = 1), MY05, DM, EBS, and LBS and the evolutions of the (b),(f),(j) M1 errors and (d),(h),(l) M2 errors with respect to the Lagrangian analytical solutions over a 15-min interval for cases 1–3 under a supersaturation value of 0.1%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 13.
Fig. 13.

As in Fig. 12, but under a supersaturation value of 0.2%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 14.
Fig. 14.

As in Fig. 12, but under a supersaturation value of 0.3%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

The reflectivity factor (M2) is highly sensitive to the large droplet component of DSDs and thus can be referred to as an indicator of large droplet production. In this study, M2 is simulated with the new TM scheme. Notably, double-moment schemes are not used to calculate the reflectivity factor (M2), so we do not simulate M2 with DM and MG07 in this study. The simulations show that MY05 overestimates M2, especially for the initial cloud droplet spectra that include small droplets. Under the initially wide spectra that include both small cloud droplets and rain embryos, the simulation of M2 with MY05 results in a large error of nearly 50% with a supersaturation value of 0.1% at 900 s, and the error increases with increasing supersaturation. In the cases with the initial spectra excluding rain embryos, the errors of the M2 simulated with MY05 remain at 40% for all three different supersaturation values. In the cases with the initially narrow spectra that include rain embryos, MY05 yields errors of nearly 9%–13% in the simulated M2 among the three different supersaturation values at 900 s. These overestimates of M2 originate from MY05 simulating too many large droplets, as shown in the DSD simulations of Figs. 13. Hence, cloud droplet simulations with MY05 may spuriously accelerate rain embryo formation speeds. In contrast, under all supersaturation conditions, the evolutions of M2 simulated with TM match those simulated with LBS and EBS very well, and the errors are all less than 1%.

Figures 1517 show the evolutions of M1 and M2 simulated with TM (μ = 1/3), C74, C74–modified, MG07, EBS, and LBS and their errors with respect to the Lagrangian analytical solutions over a 15-min interval for cases 1–9. Figure 15 shows that MG07 largely overestimates M1: the maximum M1 errors simulated by MG07 reach nearly 70.0%, 20.0%, and 45.0% after 900 s under a supersaturation value of 0.1% and the different initial spectra of cases 1, 2, and 3, respectively. Comparing Figs. 1517 reveals that the M1 error increases with increasing supersaturation, but in the cases with the initial spectra excluding rain embryos, the error does not change significantly. In contrast to MG07, C74 underestimates M1: the maximum M1 errors simulated with C74 reach nearly −60.0%, −20.0%, and −40.0% after 900 s under the three different initial spectra with a supersaturation value of 0.1%. The error also increases with increasing supersaturation, but in the cases with the initial spectra of small cloud droplets, the error changes slowly. These simulations show that C74 also underestimates M2: the M2 errors in C74 are nearly −80%, −35%, and −60% under the three different initial spectra with a supersaturation value of 0.1% after 900 s. Larger supersaturation values similarly lead to greater errors in M2, and such large errors originate from C74 underestimating the droplet growth rates, as shown in Figs. 46. The simulations of M2 with C74–modified lead to a maximum error of only 7% for the initially wide spectra that include both small cloud droplets and rain embryos. Moreover, the M1 and M2 simulated with TM match the Lagrangian analytical solutions very well under all initial spectrum conditions, and the errors are less than 1%.

Fig. 15.
Fig. 15.

Evolutions of (a),(e),(i) M1 and (c),(g),(k) M2 simulated with TM (μ = 1/3), C74, C74–modified, MG07, EBS, and LBS and evolutions of the (b),(f),(j) M1 errors and (d),(h),(l) M2 errors with respect to the Lagrangian analytical solutions over a 15-min interval for cases 1–3 under a supersaturation value of 0.1%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 16.
Fig. 16.

As in Fig. 15, but under a supersaturation value of 0.2%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Fig. 17.
Fig. 17.

As in Fig. 15, but under a supersaturation value of 0.3%.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

d. Effects of droplet curvature and solution on condensation growth

The above simulations confirm that the C74–modified scheme and our new TM scheme perform well, but the curvature and solution effects are both ignored in all of the above simulations. Both C74–modified and our new TM scheme can consider the effects of both curvature and solution simultaneously. To illustrate the effects of curvature and solution on condensation growth for the new TM scheme, Eqs. (25) and (26) are adopted to simulate the evolution of the cloud droplet spectrum. Since the effects of curvature and solution act mainly on small cloud droplets, the initial spectrum of small cloud droplets with a supersaturation value of 0.3% (case 6) is chosen as the initial experimental condition, and the solute is set to be sodium chloride (NaCl).

Figure 18 shows the experimental results of the curvature and solution effects on the evolution of cloud droplet spectra with sodium chloride particles given different mean diameters { DNaCl=[6Ms¯/(πρNaCl)]1/3=50 nm, 100 nm, 200 nm, and 300 nm}. The combined effect of curvature and solution on small droplet growth depends on the mean mass diameter of the solute. For a small solute mass, the curvature effect dominates droplet growth and broadens the DSD compared with the simulation without considering the effects of curvature and solution (Figs. 18a–c). By enhancing the solute mass, the solution effect becomes stronger and even narrows the DSD (Fig. 18d). After 60 s, as the sizes of the droplets are larger than 10 μm, the two effects are negligible. Nevertheless, the growth rates of small cloud droplets are highly influenced by the combination of these two effects and even their surface temperature (Sun et al. 2022), and small cloud droplet growth affects the tail of the cloud droplet distribution, which further impacts rain formation.

Fig. 18.
Fig. 18.

Experiments involving the curvature and solution effects on the evolutions of cloud droplet spectra with sodium chloride particles given different mean diameters: (a) DNaCl = 50 nm, (b) DNaCl = 100 nm, (c) DNaCl = 200 nm, and (d) DNaCl = 300 nm. “TM+solute+curve” denotes the simulation considering the effects of both solution and curvature.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

4. Simulation of rain embryo formation produced by giant sodium chloride particles and secondary nucleation with the newly developed scheme

In situ cloud spectrum measurements show that cloud droplet distributions with multiple modes have been observed (Khain et al. 2015). These distributions may be the result of the primary activation or even secondary activation of multimodal CCN. Therefore, bimodal and trimodal distributions of cloud droplets cannot be described by the existing double-moment schemes based on a unimodal gamma distribution function. To solve this problem, we use three gamma distribution functions for the newly developed TM scheme to describe the trimodal distributions of cloud droplets generated by different CCN modes and secondary activation. We assume that the cloud droplets nucleated by CCN whose radii are larger than 0.55 μm belong to the first mode of the gamma distribution function, while those nucleated by CCN whose radii are less than 0.55 μm are attributed to the second mode (except those nucleated by secondary nucleation, as these particles are taken as the third mode). The criterion for secondary activation is that the ambient supersaturation surpasses its extremum. The threshold of 0.55 μm is the dry aerosol radius corresponding to the intersection point of the first-mode aerosol spectrum and the second-mode aerosol spectrum, as described by Sun et al. (2022).

To accurately address this issue, we perform simulations with TM under different nucleation-mode thresholds of the dry aerosol radius (Rtd = 0.35, 0.45, 0.55, 0.65, and 0.75 μm). Then, the cloud droplets of each mode are grown separately with the new TM condensation scheme. The three gamma modes are independent of one another during droplet growth. However, each mode affects the other two modes by some factors, such as the latent heat released by condensation and water vapor consumption. Thus, the evolution of the cloud droplet spectrum in each mode is simulated separately, and then the simulated spectra are accumulated. To simulate both the formation of rain embryos by giant sodium chloride particles and the occurrence of secondary nucleation with the newly developed scheme, the nucleation process must be taken into account. Therefore, we couple our new nucleation parameterization scheme (Sun et al. 2022) with the new TM condensation scheme (the details of this coupling are provided in appendix D). We also test the new nucleation parameterization scheme by using the nucleation scheme to obtain the initial M0, M1, and M2 of the cloud droplets for each mode. The initial shape, slope, and intercept parameters of the gamma distribution functions with the three modes can be diagnosed by Eqs. (19)(21) with three different sets of equations concerning M0, M1, and M2 [Eqs. (D1)(D3)]. Aerosol activation is treated from large particles to small particles, and three gamma distributions appear at different times in the simulation with a Lagrangian parcel bulk model. The dynamic frame is the same as the Lagrangian bin model based on Maxwell theory (LBMBM) of Sun et al. (2022), whereas the microphysical treatment is replaced by the newly developed bulk scheme.

a. Initial conditions

The ascending parcel is triggered by a temperature disturbance of 0.25 K under initial parcel conditions of 293.28 K and 94 479 Pa with a relative humidity of 95%. The ambient temperature lapse rate is 6.4 K km−1, and the aerosol size distribution is characterized by three modes, as described by Sun et al. (2022).

b. Comparison between the bin and bulk schemes

Figure 19 shows the temporal evolutions of the ambient supersaturation and vertical velocity of the ascending parcel simulated by TM with different Rtd in comparison with those simulated by LBMBM over 900 s. The simulations show that the evolutions of the ambient supersaturation and vertical velocity are not sensitive to the threshold. The results simulated with TM are close to those simulated with LBMBM, but the ambient supersaturation simulated with TM suddenly decreases at approximately 160 s. This is due to the discontinuous production of cloud water from the initially sudden activation of giant nuclei in a few time steps, which consumes an equivalent mass of water vapor. However, the consumption of water vapor simulated with LBMBM is a continuous condensation process from haze droplets to cloud droplets.

Fig. 19.
Fig. 19.

Temporal evolutions of the ambient supersaturation (s) and vertical velocity (w) of the ascending parcel simulated by LBMBM and TM with different dry aerosol radius thresholds (Rtd).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Figure 20 shows the evolutions of the cloud droplet spectra simulated by TM with different Rtd in comparison with those simulated by LBMBM over 900 s. Sodium chloride particles with diameters equal to or larger than 0.55 μm are all activated at 190 s. Consequently, the first gamma distribution of droplets appears to match the distribution of large droplets in the first mode simulated with LBMBM (Fig. 20a), and the second mode of the gamma distribution of droplets appears once the threshold Rtd exceeds 0.55 μm. Since the ambient supersaturation has already passed its peak value at 220 s for both TM and LBMBM, the concentration of cloud droplets reaches its maximum, and no new cloud droplets nucleate until secondary nucleation occurs.

Fig. 20.
Fig. 20.

Evolutions of the cloud droplet spectra simulated by LBMBM and TM with different Rtd over 900 s.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

A comparison of the cloud spectra after 220 s shows that the distributions become increasingly narrow for both schemes. However, the spectra simulated with LBMBM become wider than those simulated with TM over time. This phenomenon further indicates that the vapor diffusion formula [Eq. (22) or Eq. (7.18) in Rogers and Yau 1989] makes the cloud droplet spectrum narrower than Maxwell theory does. Figures 20h and 20i show the third mode of production owing to secondary nucleation, which broadens these narrow spectra. These results illustrate that the multimodal spectrum of cloud droplets produced by giant sodium chloride particles and secondary nucleation can be simulated by the newly developed TM scheme. Moreover, different nucleation-mode thresholds of the dry aerosol radius result in different production times of the second mode spectrum in the TM scheme, and a lower (higher) threshold results in more (fewer) cloud droplets in the first mode. If the threshold exceeds 0.8 μm or is less than 0.3 μm, the concentration and size errors in the first mode are relatively large compared to the Lagrangian analytical solutions.

In the current bulk parameterization schemes, freshly nucleated droplets must obey the same gamma distribution as other cloud droplets. In other words, new droplets are assumed to be immediately distributed over a wide range of sizes. Khain et al. (2015) pointed out that such treatment may erroneously simulate large cloud droplets in the DSD. The above simulation results illustrate that our new nucleation scheme and the newly developed TM scheme can resolve this cloud simulation issue when employed for more than one gamma distribution.

5. Simulations of cloud droplet nucleation and condensation with a 1.5D Eulerian model

To further verify the vertical kinematic characteristics of DSDs simulated with the new TM scheme in cumulus clouds, we couple the scheme with a 1.5D Eulerian model (Sun et al. 2012a) and ignore other microphysical processes. Similar models consisting of two cylinders have been used to explore the impacts of aerosol particles on cloud properties and dynamic structures (Leroy et al. 2006; Sun et al. 2012b; Hiron and Flossmann 2015; Simmel et al. 2015; Yang et al. 2020, 2022). The model of Sun et al. (2012a) considers both the gradient force of perturbation pressures and the buoyancy force as triggers of convection to mitigate unrealistically sharp gradients of variables at the cloud top. In this model, the region of updraft is inside the inner cylinder, and the region of downdraft is the space between the inner and outer cylinders. The lateral exchanges of entrainment and detrainment between the external environment and the cloud are performed by horizontal velocities diagnosed by the continuity equation. The dynamics of the model are described in detail in Sun et al. (2010, 2012a,b). In the following simulation, the vertical resolution is set to 100 m, the dynamic time step is set to 2 s, and the microphysical time step is 0.1 s. The dynamic tendencies are linearly input to the microphysical time steps, and the terminal velocities of both aerosols and cloud droplets are ignored. The concentration of aerosol particles remains unchanged after nucleation, and the nucleation sizes of aerosols are determined by both the cloud droplet concentration and the ambient supersaturation. If the cloud droplet concentration is less than the concentration determined by the critical supersaturation equal to the ambient supersaturation, cloud droplets are considered to nucleate, and vice versa (Deng and Sun 2019). The initial shape parameters for the first mode and second mode are 0.65 and 1.0, respectively, in this simulation.

a. Initial conditions

The profiles of the initial temperature and humidity fields were measured during CCOPE on 19 July 1981 in Miles City, Montana (Dye et al. 1986). This in situ case has also been simulated with 1.5D bin models (Leroy et al. 2006; Hiron and Flossmann 2015; Yang et al. 2020, 2022). The aforementioned trimodal aerosol distribution excluding giant aerosols is also adopted, and the aerosol concentration is multiplied by a factor of 6 but decreases exponentially with increasing height above 700 m.

b. Simulation results

Figure 21 shows the temporal evolutions of the vertical velocity, cloud water content and supersaturation. The maximum values of both the vertical velocity and the cloud water content are lower than those simulated by the bin scheme (Yang et al. 2022) because the numerical diffusion of the Eulerian bin scheme leads to the enhanced production of cloud water and the increased release of latent heat. The cloud water contents that condense on large aerosols greater than 1.1 μm in diameter are far lower than those that condense on smaller aerosol particles. Hence, the maximum cloud water contents of the first-mode spectra are less than 0.02 g m−3 (Fig. 21b). The evolution of supersaturation indicates that nucleation occurs at both boundaries, namely, the cloud base and the cloud top. However, secondary nucleation does not occur due to a lack of maximum supersaturation in the cloud. The high vertical velocities at the cloud base indicate low vertical velocities and accelerations above the cloud base; in other words, the gradient force of perturbation pressures inhibits the vertical velocity from accelerating due to buoyancy in the cloud. On the other hand, supersaturation is not directly diagnosed after each dynamic time step and calculated after microphysical adjustment. Instead, the supersaturation employed in the condensation calculation at each microphysical time step is the value diagnosed by the variables at the last time step after both dynamic and physical processes. This supersaturation treatment is highly precise if the microphysical time step is no more than 0.1 s (Deng and Sun 2019). Most simulations may overestimate secondary nucleation because the supersaturation diagnosed after a dynamic step without a microphysical adjustment can result in maximum supersaturation in the cloud. Therefore, it is not necessary to include the third mode of cloud spectra in the simulation.

Fig. 21.
Fig. 21.

Temporal evolutions of the vertical velocity (m s−1), cloud water content (g m−3), and supersaturation (%): (a) vertical velocity (m s−1) and cloud water content (g m−3), where the solid lines denote the vertical velocity and the shaded area represents the cloud water content; (b) cloud water contents (g m−3) for the first and second modes of the cloud droplet spectra, where the solid lines denote the cloud water content of the first-mode cloud droplets and the shaded area represents the cloud water content of the second-mode cloud droplets; and (c) supersaturation (%).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

The King airplane crossed the cloud at approximately 6 km. Comparing the observed cloud water content and vertical velocity with the simulated values (Fig. 22a and Fig. 22b, respectively), the observed values are generally comparable to the simulated values. Nevertheless, considering the ice phase in future simulations may improve the result.

Fig. 22.
Fig. 22.

Evolutions of the measured and simulated (a) cloud water contents (g m−3) and (b) vertical velocities (m s−1) at an altitude of approximately 6 km.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

Figure 23 shows the cloud spectra for different heights at 50 min. Figure 23a shows the spectra at the cloud base. The newly nucleated droplet spectra have a low shape parameter and a wide droplet diameter for both the first mode and the second mode. The width of the spectra should decrease with height and become increasingly narrow, as in the above Lagrangian parcel model simulations (Figs. 23b,c). However, a vertical increase in updrafts leads to entrainment due to atmospheric continuity, and vice versa. The entrainment of dry air from the outer cylinder below the maximum vertical velocity results in new droplet activation, and these newly formed droplets, in addition to the growth of old droplets, slow the rate at which the cloud spectra narrow toward both sides (Figs. 23d,e,f). The upper part of the cloud spectra is influenced by both droplets advecting vertically from the lower levels (Figs. 23g,h) and particle nucleation at the upper boundary (Fig. 23i). From 6.9 to 7.9 km, the cloud depth is attributed mainly to cloud boundary nucleation. Indeed, upper boundary nucleation is important for the formation of cloud droplets and the broadening of the cloud droplet spectrum (Sun et al. 2012b). Since high vertical velocity gradients lead to large horizontal detrainment velocities, many droplets nucleating at the upper boundary are horizontally transported into the outer cylinder and form a liquid upper-level cloud anvil.

Fig. 23.
Fig. 23.

Cloud droplet spectra at different heights at 50 min. The solid black lines denote the combined distribution of cloud spectra, the red dashed lines represent the cloud spectra of the first mode, and the blue dotted lines denote the cloud spectra of the second mode.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0039.1

The cumulus cloud spectra simulated with the 1.5D model depict the average distribution of cloud spectra at different heights. However, the entrainment and detrainment triggered by the cloud top vortex complicate the DSD. Thus, our upcoming paper will feature a comparison of the 3D simulation by the Weather Research and Forecasting (WRF) Model with the observations from the Rain in Shallow Cumulus over the Ocean (RICO) campaign.

6. Discussion and conclusions

In this study, we develop a new TM scheme based on a three-parameter gamma distribution function with the cloud droplet mass, and prognostic differential equations for the slope and shape parameters are derived by means of the number concentration, water content, and reflectivity factor of cloud droplets.

The new TM scheme with a tail parameter of μ = 1 was compared with various existing schemes, including LBS, EBS, DM, and MY05. The cloud droplet spectra simulated with TM ignoring the curvature and solution effects become increasingly narrow since the diffusion growth rates of cloud droplets are inversely proportional to their radii and because small droplets grow more quickly than large droplets, as in the simulations with LBS. In the cases with the initial spectrum of small cloud droplets, the standard deviation of σ simulated with LBS is close to 0 μm, which indicates that the spectra are very narrow. However, the spectra simulated with EBS based on the Eulerian advection algorithm are not as narrow as those simulated with LBS due to numerical diffusion even though 3000 mass intervals are adopted. This numerical diffusion is especially significant in the simulation of spectra consisting of small cloud droplets.

In addition, the MY05 scheme yields large errors in the spectrum simulations compared with the Lagrangian analytical solutions; the triple-moment MY05 scheme assumes that the shape parameter is constant within a small time step during condensation, which is equivalent to the scenario described by double-moment schemes if condensation is the only process involved. This assumption of MY05 causes the spectrum width to remain unchanged during condensation, and thus, the spectrum is much wider than in the Lagrangian analytical solutions. Moreover, in contrast to the Lagrangian analytical solutions, the standard deviation of the DSD in MY05 increases over time. In addition, MY05 underestimates the cloud water content and overestimates the reflectivity factor. These errors increase over time and with increasing supersaturation. Consequently, the maximum errors in the cloud water content and reflectivity factor reach −16% and 50%, respectively. The simulation of an excessive number of large droplets by MY05 results in an overestimated reflectivity factor and may cause considerable errors in the calculation of autoconversion rates in the formation of warm rain.

The simulations with DM are similar to those with MY05. There are also large errors in the simulations compared with the Lagrangian analytical solutions except in the cases with the initially narrow spectra that include rain embryos. The fixed shape parameter in the DM scheme results in a constant spectral width over time and causes the cloud water content to be underestimated with a maximum error of −16% among all the simulated cases. The spectra simulated with TM match those simulated with LBS very well except for the cases of the initially wide spectra that include both small droplets and rain embryos. In addition, the small droplets on the left tail of the distribution simulated with TM grow slower than those simulated with LBS. Under all conditions, the evolutions of M1 and M2 simulated with TM are consistent with those simulated with LBS, and their errors are all less than 1%.

For a comparison with the schemes described by the gamma distribution function on the radius scale, the tail parameter μ = 1/3 is also adopted for the new TM scheme. TM still achieves small and steady errors (less than 1%) for M1 and M2. The simulated spectra of TM with μ = 1/3 are also consistent with those of LBS. For MG07, although the shape parameter is diagnosed by the formula related to the cloud droplet number concentration, the shape parameter remains unchanged due to the constant supersaturation without nucleation, which further results in a constant spectral width over time, but the standard deviation of σ increases over time. Moreover, the cloud water content can be overestimated by as much as 70% after 900 s. The simulations with C74 underestimate both the cloud water content and the reflectivity factor, the maximum errors in which reach −50% and −75%, respectively. In contrast, the C74-modified scheme improves the simulation results. The evolutions of the DSDs simulated with C74-modified match those simulated with LBS very well except in the cases with the initially wide spectra that include both small cloud droplets and rain embryos.

The above simulations further illustrate the important role that the shape parameter plays in the evolution of the DSD. For double-moment schemes, the sensitivity analysis under different shape parameters confirms that a constant shape parameter during condensation broadens the cloud droplet spectrum in comparison with both the initial spectra and the analytical solutions. In contrast, the shape parameters of triple-moment schemes are not fixed during condensation, and the lack of a constraint on the shape parameter reflects the flexibility of these schemes to physically respond to the evolution of the DSD and ensures an accurate description of the DSD despite some remaining errors in the shape parameter simulations. However, the simulations with the C74 scheme underestimate the shape parameter, and the simulations with C74-modified overestimate the shape parameter. Nevertheless, the C74-modified scheme greatly improves the C74 simulation results, especially for the cloud spectrum widths. The TM simulations also underestimate the shape parameter, but the simulated errors are the smallest among all the triple-moment schemes. However, the TM simulations underestimate the growth of small cloud droplets in the cloud spectrum that includes both small cloud droplets and rain embryos, which leads to larger cloud spectrum widths than those of both C74-modified and LBS.

To correctly describe the evolution of the DSD, the curvature and solution effects are considered in the C74-modified scheme and the new TM scheme. Simulations incorporating both of these effects are performed with TM. For a small solute mass, the curvature effect dominates droplet growth and causes spectrum broadening toward small sizes. Upon enhancing the solute mass, the solution effect is strengthened, and small droplets grow even faster than the droplets simulated while ignoring the solution and curvature effects. Since the growth of small droplets can influence the tail of the cloud droplet distribution, which could impact rain formation, it is necessary to account for the curvature and solution effects in the TM scheme. In contrast, the ventilation effect is negligible for droplets below 20 μm in size and is also incredibly small on rain embryo formation, so simulation results considering this effect are not given. However, the ventilation effect is significant for large droplets and is important for raindrop evaporation, so this effect will be considered in future raindrop simulations.

Multimodal cloud droplet distributions have been observed by in situ cloud spectrum measurements. However, based on the assumption that freshly nucleated droplets must obey a unimodal gamma distribution function similar to other cloud droplets, neither double-moment nor triple-moment parameterization schemes can properly describe the bimodal and trimodal distributions of cloud droplets. Khain et al. (2015) also pointed out that a unimodal gamma distribution function leads to the erroneous appearance of large particles in DSDs. The newly developed TM scheme with more than one gamma distribution function can capture the features of multimodal cloud droplet distributions. In the newly developed TM scheme, freshly nucleated droplets no longer obey the same gamma distribution as other cloud droplets. We assume that the cloud droplets nucleated by CCN whose radii are larger than 0.55 μm belong to the first mode of the cloud droplet distribution, while those nucleated by CCN whose radii are less than 0.55 μm are attributed to the second mode (excepting particles nucleated by secondary nucleation, as these particles are taken as the third mode). In comparison with the simulation with LBMBM (Sun et al. 2022), the multimodal spectra of cloud droplets produced by giant sodium chloride particles and secondary nucleation can be effectively captured by our newly developed TM scheme. Notably, the cloud spectrum widths of each mode simulated with TM based on the current simplified vapor diffusion formula are narrower than those simulated with LBMBM governed by Maxwell theory. Nevertheless, while the temporal evolutions of the three gamma distributions are independent of each other during condensation, the proceeding gamma distribution can affect the subsequent distributions through the release of latent heat and the consumption of water vapor. Taking this line of thinking one step further, during the rain formation process, cloud droplets from multimodal distributions collide with each other. This complex process will be discussed in our future research.

The Eulerian kinematic structure of DSDs simulated with the 1.5D model also shows that the first DSD mode is necessary for simulating rain embryo formation. However, the third DSD mode can be ignored for secondary nucleation if it is negligible. Secondary nucleation is highly dependent on the supersaturation treatment. If supersaturation is diagnosed after a dynamic time step without a microphysical adjustment, secondary nucleation is prominent; however, microphysical adjustments are essential for cloud simulation. Otherwise, supersaturation should be predicted (Deng and Sun 2019).

Accurately describing the formation of rain embryos from giant sodium chloride particles with the newly developed scheme indicates that the new TM scheme may improve the calculation accuracy for the autoconversion rate of rain droplets in both CRMs and GCMs. However, the double integral of the quasi-stochastic equation for cloud droplets with gamma distribution functions in the calculation of the autoconversion rate is difficult to solve analytically (Cohard and Pinty 2000), so some simplified approaches must be adopted to parameterize the autoconversion rate. Nevertheless, this double integral can be solved analytically based on the gamma distribution function at the droplet mass scale (Zhang and Sun 2019). This is another reason why we employ the gamma distribution function at the mass scale to represent the cloud droplet spectrum in our series of papers. The analytical solution of the autoconversion rate based on more than one gamma distribution function will be shown in future work.

Acknowledgments.

This study is supported by the National Natural Science Foundation of China (Grants 41275147, 41905127, 41830966, 41875173) and by the National Program on Key Basic Research Project (973 Program) (Grant 2014CB441403). The authors thank the two anonymous reviewers for their helpful comments that greatly improved the clarity of this manuscript.

Data availability statement.

The simulation dataset is massive and not publicly available, but readers can contact author Jun Zhang (zhangjun@mail.iap.ac.cn) to obtain the data.

APPENDIX A

Equation Derivations of the Newly Developed Triple-Moment Condensation Scheme

a. Rates of change in the number concentration (M0), cloud water content (M1), and reflectivity factor (M2) in the new triple-moment scheme

The change rate of Mp can be given by the definition of Mp in Eq. (4):
dMpdt=ddtHp0mpf(m)dm=Hp0[mptf(m)+mpf(m)t]dm=Hp0[pmp1mtf(m)+mpf(m)t]dm.
Since ∂m/∂t = 0, the change rate of Mp can be written as
dMpdt=Hp0mpf(m)tdm.

The change rate of Mp in Eq. (A2) includes both the condensation (evaporation) process and the nucleation process.

Accordingly, by substituting Eq. (3) into Eq. (A2), the change rate of Mp in Eq. (A2) during condensation (evaporation) can be written as follows:
dMpdt|condensation/evaporation=Hp0mp{m[dmdtf(m)]}dm.
The growth rate of a single cloud droplet can be expressed in the following general form:
dmdt=xi=1yicximdxi,
where if i = 1 and y1 = 1, Eq. (A4) represents the mass growth rate of a single cloud droplet during condensation taking into account only the droplet equilibrium vapor pressure; if i = 2 and y2 = 3, Eq. (A4) represents the mass growth rate of a single cloud droplet considering the solution and curvature effects on the droplet equilibrium vapor pressure; and if i = 3 and y3 = 6, Eq. (A4) represents the mass growth rate of a single cloud droplet considering the ventilation effect. cxi and dxi are coefficients independent of the cloud droplet mass.
Substituting Eqs. (1) and (A4) into Eq. (A3) yields
dMpdt|condensation/evaporation=Hp0mp[m(N0xi=1yicximα1+dxieβm)]dm=Hpxi=1yicxi0N0[(α1+dxi)mα+p2+dxieβmβmα+p1+dxieβm]dm=Hpxi=1yicxi[(α1+dxi)M0Γ(α+p1+dxi)Γ(α)βp1+dxiM0Γ(α+p+dxi)Γ(α)βp1+dxi]=Hpxi=1yicxipM0Γ(α+p1+dxi)Γ(α)βp1+dxi=Hpxi=1yicxi0pmp1+dxif(m)dm=Hp0pmp1xi=1yicximdxif(m)dm.
Substituting Eq. (A4) into Eq. (A5) gives the following relation:
dMpdt|condensation/evaporation=Hp0pmp1dmdtf(m)dm.
Thus,
dMpdt=dMpdt|condensation/evaporation+dMpdt|nucleation=Hp0pmp1dmdtf(m)dm+dMpdt|nucleation.
Therefore, the change rate of the number concentration (M0) can be written as Eq. (14). Similarly, by substituting Eqs. (1) and (12) into Eq. (A7), the change rates of the cloud water content (M1) and the reflectivity factor (M2) can be written as
dM1dt=H10dmdtf(m)dm+dM1dt|nucleation=H10k(16πρw)1/3sN0mα2/3eβmdm+dM1dt|nucleation=H1kM0(16πρw)1/3sΓ(α+13)Γ(α)β1/3+dM1dt|nucleation,
dM2dt=H202mdmdtf(m)dm+dM2dt|nucleation=H202k(16πρw)1/3sN0mα+1/3eβmdm+dM2dt|nucleation=2H2kM0(16πρw)1/3sΓ(α+43)Γ(α)β4/3+dM2dt|nucleation.

b. Effects of curvature and solution on condensation growth

The change rate of the number concentration is the same as in Eq. (14) when the effects of curvature and solution on condensation growth are taken into account. By substituting Eqs. (1) and (22) into Eq. (A7), the change rates of the cloud water content (M1) and reflectivity factor (M2) can be written as follows:
dM1dt=H10dmdtf(m)dm+dM1dt|nucleation=H10k[(16πρw)1/3sm1/32a+8(16πρw)2/3bm2/3]N0mα1eβmdm+dM1dt|nucleation=H1[0k(16πρw)1/3sN0mα2/3eβmdm02kaN0mα1eβmdm+08k(16πρw)2/3bN0mα5/3eβmdm]+dM1dt|nucleation,
dM2dt=H202mdmdtf(m)dm+dM2dt|nucleation=H202mk[(16πρw)1/3sm1/32a+8(16πρw)2/3bm2/3]N0mα1eβmdm+dM2dt|nucleation=H2[02k(16πρw)1/3sN0mα+1/3eβmdm04kaN0mαeβmdm+016k(16πρw)2/3bN0mα2/3eβmdm]+dM2dt|nucleation.
The aerosol mass in a droplet can be tracked by bin microphysical schemes with two dimensions: the water mass and the aerosol mass. However, the aerosol masses in droplets cannot be described by a function of the water mass and aerosol mass. Since bulk parameterization schemes describe the change rates for an entire size distribution of droplets, we can consider the solution effect by obtaining the aerosol mean mass. The “solution term” b in the new TM scheme is determined by the mean mass of solutes ( Ms¯) and can be expressed in the following form:
b4.3iMs¯ms.
From Eq. (A12), b is independent of the cloud droplet mass, and Eqs. (A10) and (A11) can be given as
dM1dt=H1[k(16πρw)1/3sN00mα2/3eβmdm2kaN00mα1eβmdm+8k(16πρw)2/3bN00mα5/3eβmdm]+dM1dt|nucleation=H1kM0[(16πρw)1/3sΓ(α+13)Γ(α)β1/32a+(16πρw)2/38bΓ(α23)Γ(α)β2/3]+dM1dt|nucleation,
dM2dt=H2[2k(16πρw)1/3sN00mα+1/3eβmdm4kaN00mαeβmdm+16k(16πρw)23bN00mα2/3eβmdm]+dM2dt|nucleation=2H2kM0[(16πρw)1/3sΓ(α+43)Γ(α)β4/32aΓ(α+1)Γ(α)β+(16πρw)2/38bΓ(α+13)Γ(α)β1/3]+dM2dt|nucleation.

c. Ventilation effect on condensation growth

The vapor field surrounding each drop is spherically symmetrical for droplets at rest, but when droplets acquire significant descent velocities, the rates of heat and mass transfer increase and reach the maximum on the upstream side of each drop (Rogers and Yau 1989). These effects should be taken into account by the appropriate ventilation coefficients once the droplet radius exceeds 10 μm. The ventilation coefficient is given as follows Pruppacher and Klett (1997):
fυ=i=0jbi(Nsc1/3NRE1/2)i=i=0jbiNsci/3a1i/2(8gρaπη2)a2i/2(m)a2i/2,
where bi is the coefficient of the ith term, j is the maximum number of fitting terms, NRE is the Reynolds number of the flow around the drop, Nsc is the Schmidt number of the flow around the drop, and ρa and η denote the density and dynamic viscosity of the air, respectively. The coefficients a1 and a2 are given by Mitchell and Heymsfield (2005).
Considering the ventilation effect, we can calculate the growth rate of a single cloud droplet by
dmdt=ki=0jbiNsci/3a1i/2(8gρaπη2)a2i/2[(16πρw)1/3sma2i/2+1/32ama2i/2+8(16πρw)2/3bma2i/22/3].
Therefore, the change rate of the number concentration (M0) is also the same as Eq. (14), and the change rates of the cloud water content (M1) and the reflectivity factor (M2) considering the ventilation effect can be written as
dM1dt=H10dmdtf(m)dm+dM1dt|nucleation=H1kM0i=0jbiNsci/3a1i/2(8gρaπη2)a2i/2[(16πρw)1/3sΓ(α+a2i2+13)Γ(α)βa2i/2+1/32aΓ(α+a2i2)Γ(α)βa2i/2+8(16πρw)2/3bΓ(α+a2i223)Γ(α)βa2i/22/3]+dM1dt|nucleation,
dM2dt=H202mdmdtf(m)dm+dM2dt|nucleation=2H2kM0i=0jbiNsci/3a1i/2(8gρaπη2)a2i/2[(16πρw)1/3sΓ(α+a2i2+43)Γ(α)βa2i/2+4/32aΓ(α+a2i2+1)Γ(α)βa2i/2+1+8(16πρw)2/3bΓ(α+a2i2+13)Γ(α)βa2i/2+1/3]+dM2dt|nucleation.
By substituting Eqs. (14), (A17), and (A18) into Eqs. (10) and (11), we can obtain the final prognostic equations for the shape parameter and the slope parameter considering the ventilation effect:
dαdt=2ki=0jbiNsci/3a1i/2(8gρaπη2)a2i/2[(16πρw)1/3s(23a2i2)Γ(α+a2i2+13)Γ(α)βa2i/22/32a(1a2i2)Γ(α+a2i2)Γ(α)βa2i/21+8(16πρw)2/3b(53a2i2)Γ(α+a2i223)Γ(α)βa2i/25/3]+2β(α+1)H1M0dM1dt|nucleationβ2H2M0dM2dt|nucleationα(α+1)dlnM0dt|nucleation,
dβdt=ki=0jbiNsci/3a1i/2(8gρaπη2)a2i/2[(16πρw)1/3s(13a2i)Γ(α+a2i2+13)Γ(α+1)βa2i/25/32a(1a2i)Γ(α+a2i2)Γ(α+1)βa2i/22+8(16πρw)2/3b(73a2i)Γ(α+a2i223)Γ(α+1)βa2i/28/3]+β2(2α+1)H1M0αdM1dt|nucleationβ3H2M0αdM2dt|nucleationαβdlnM0dt|nucleation.

d. TM scheme with μ = 1/3

Note that although the same initial number concentration, cloud water content, and reflectivity factor are used, there are slightly different initial spectra between the bulk schemes based on the gamma mass distribution and those based on the gamma radius (diameter) distribution. Fortunately, this difference can be eliminated by adjusting the tail parameter (μ). If μ is set to 1/3 in the TM scheme, the spectra simulated with TM can be compared with those of the bulk schemes based on the gamma radius (diameter) distribution.

With μ = 1/3, Eq. (1) can be written as
f(m)=N0m(1/3)α1e(βm)1/3.
By substituting Eq. (A21) into Eq. (4), the analytical solutions of the three moments can be written as follows:
M0=H00N0m(1/3)α1e(βm)1/3dm=3N0Γ(α)βα/3,
M1=H10N0m(1/3)αe(βm)1/3dm=3H1N0Γ(α+3)β(α+3)/3=H1M0Γ(α+3)Γ(α)β,
M2=H20N0m(1/3)α+1e(βm)1/3dm=3H2N0Γ(α+6)β(α+6)/3=H2M0Γ(α+6)Γ(α)β2.
We exactly differentiate Eqs. (A23) and (A24) with respect to time:
dM1dt=H1M0k1βdαdtH1M0k2β2dβdt+H1k2βdM0dt,
dM2dt=H2M0k3β2dαdt2H2M0k4β3dβdt+H2k4β2dM0dt,
where
k1=3α2+6α+2,
k2=α3+3α2+2α,
k3=6α5+75α4+340α3+675α2+548α+120,
k4=α6+15α5+85α4+225α3+274α2+120α.
The prognostic equations of the shape parameter and the slope parameter can be derived by combining Eqs. (A25) and (A26) as follows:
dαdt=2k4βH1M0(2k1k4k2k3)dM1dtk2β2H2M0(2k1k4k2k3)dM2dtk2k4M0(2k1k4k2k3)dM0dt,
dβdt=k3β2H1M0(2k1k4k2k3)dM1dtk1β3H2M0(2k1k4k2k3)dM2dt(k2k3k1k4)βM0(2k1k4k2k3)dM0dt.
The change rate of the number concentration is the same as in Eq. (14). By substituting Eqs. (A21) and (12) into Eq. (A7), the change rates of the cloud water content (M1) and reflectivity factor (M2) can be written as follows:
dM1dt=H10dmdtf(m)dm+dM1dt|nucleation=H10k(16πρw)1/3sN0m(1/3)α2/3e(βm)1/3dm+dM1dt|nucleation=H1kM0(16πρw)1/3sΓ(α+1)Γ(α)β1/3+dM1dt|nucleation,
dM2dt=H202mdmdtf(m)dm+dM2dt|nucleation=H202k(16πρw)1/3sN0m(1/3)α+1/3e(βm)1/3dm+dM2dt|nucleation=2H2kM0(16πρw)1/3sΓ(α+4)Γ(α)β4/3+dM2dt|nucleation.

By substituting Eqs. (14), (A33), and (A34) into Eqs. (A31) and (A32), we can obtain the final prognostic equations for both the shape and the slope parameters:

dαdt=2k2k1k4k2k3(16πρw)1/3sβ2/3Γ(α)[k4Γ(α+1)k2Γ(α+4)]+2k4βH1M0(2k1k4k2k3)dM1dt|nucleationk2β2H2M0(2k1k4k2k3)dM2dt|nucleationk2k42k1k4k2k3dlnM0dt|nucleation,
dβdt=k2k1k4k2k3(16πρw)1/3sβ5/3Γ(α)[k3Γ(α+1)2k1Γ(α+4)]+k3β2H1M0(2k1k4k2k3)dM1dt|nucleationk1β3H2M0(2k1k4k2k3)dM2dt|nucleation(k2k3k1k4)β2k1k4k2k3dlnM0dt|nucleation.

The initial shape parameter, slope parameter, and intercept parameter can be calculated by combining Eqs. (A22)(A24).

APPENDIX B

Other Triple-Moment Condensation Schemes

a. MY05 scheme

The size distribution of cloud droplets is expressed by the gamma distribution function in terms of the particle diameter (MY05):
f(D)=N0Dμα1e(βD)μ,
where D is the diameter of the cloud droplet. The tail parameter (μ) for the cloud droplets is equal to 3 in MY05.
In MY05, the sixth moment (the radar reflectivity factor) is applied based on the number concentration and the cloud water content:
M0=0f(D)dD=N0Γ(α)3β3α,
M1=H116πρw0D3f(D)dD=16πρwH1N0Γ(α+1)3β3(α+1)=16πρwH1M0αβ3,
M2=H2(16πρw)20D6f(D)dD=(16πρw)2H2N0Γ(α+2)3β3(α+2)=(16πρw)2H2M0(α+1)αβ6.
The change rate of the cloud water content during the condensation process can be given as follows:
dM1dt=ddt16πρwH10D3f(D)dD=12πρwH10D2dDd<