the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Impacts of the ice-particle size distribution shape parameter on climate simulations with the Community Atmosphere Model Version 6 (CAM6)

### Wentao Zhang

### Chunsong Lu

The impacts of the ice-crystal size distribution shape
parameter (*μ*_{i}) were considered in the two-moment bulk cloud
microphysics scheme of the Community Atmosphere Model Version 6 (CAM6). The
*μ*_{i}'s impact on the statistical mean radii of ice crystals can be
analyzed based on their calculating formulas. Under the same mass
(*q*_{i}) and number (*N*_{i}), the ratios of the mass-weighted radius
(${R}_{{q}_{\mathrm{i}}}$, not related to *μ*_{i}) to other statistical mean radii
(e.g., effective radiative radius) are completely determined by *μ*_{i}. Offline tests show that *μ*_{i} has a significant impact on
the cloud microphysical processes owing to the *μ*_{i}-induced changes
in ice-crystal size distribution and statistical mean radii (excluding
${R}_{{q}_{\mathrm{i}}}$). Climate simulations show that increasing *μ*_{i} would lead
to higher *q*_{i} and lower *N*_{i} in most regions, and these impacts can be
explained by the changes in cloud microphysical processes. After increasing
*μ*_{i} from 0 to 5, the longwave cloud radiative effect increases
(stronger warming effect) by 5.58 W m^{−2} (25.11 %), and the
convective precipitation rate decreases by −0.12 mm d^{−1} (7.64 %). In short, the impacts of *μ*_{i} on climate simulations are
significant, and the main influence mechanisms are also clear. This suggests
that the *μ*_{i}-related processes deserve to be parameterized in a
more realistic manner.

Clouds are an integral part of the Earth's radiation budget and global water cycle (Liou, 1986; Luo and Rossow, 2004; Bony et al., 2015; Zhou et al., 2016). Since cloud microphysical processes occur at scales that are much smaller than the resolution of commonly used atmospheric models, it remains a significant challenge for atmospheric models to represent cloud-related processes, especially ice-phase cloud microphysical processes (Mitchell et al., 2008; Spichtinger and Gierens, 2009; Wang and Penner, 2010; Erfani and Mitchell, 2016; Paukert et al., 2019; Morrison et al., 2020; Proske et al., 2022). Because it is impossible for commonly used atmospheric models (excluding the ideal model with the recently developed Lagrangian-particle-based scheme) to individually describe cloud particles (e.g., cloud droplets or ice crystals), only the macrostatistical features of cloud particles are represented in cloud microphysics schemes. From the outset, the development of cloud microphysics schemes has resulted in two distinct categories: bulk microphysics parameterization and spectral (bin) microphysics (Milbrandt and Yau, 2005; Khain et al., 2015). The spectral (bin) approach explicitly represents the cloud particle size distributions (PSDs) using tens to hundreds of bins. The computational cost of this approach is very high because of the massive interactions among different bins. The bulk microphysics scheme represents the PSDs by a semiempirical distribution function. Compared to the spectral (bin) scheme, the bulk microphysics scheme has high computational efficiency and has been widely used in climate models (Morrison et al., 2005; Lohmann et al., 2007; Salzmann et al., 2010; Gettelman and Morrison, 2015).

In the bulk cloud microphysics schemes used for climate models, the PSD is
usually described by the gamma distribution function with three parameters,
namely, the intercept parameter (*N*_{0}), the slope parameter (*λ*),
and the spectral shape parameter (*μ*) (Khain et al., 2015; Morrison et
al., 2020). Note that the commonly used two-moment bulk microphysics scheme
predicts only the mass and number of cloud particles, which cannot constrain
these three parameters (i.e., *N*_{0}, *λ*, and *μ*). Therefore,
one of these three parameters (typically *μ*) must be determined from an
empirical formula or set to a given value (e.g., Morrison and Gettelman,
2008; Barahona et al., 2014; Eidhammer et al., 2017). For instance, the
*μ*(*μ*_{i}) of ice crystals (ICs; only represent cloud ice in this
study) in the two-moment bulk stratiform cloud microphysics scheme developed
by Morrison and Gettelman (2008) (hereafter “MG scheme”) is set to zero (i.e.,
the *μ*_{i} is ignored). In recent years, offline tests and short-term
simulations (a few days or less) with high-resolution atmospheric models
(e.g., cloud-resolving models and mesoscale models) have shown that *μ*_{i} has a significant impact on cloud microphysical processes and
synoptic systems (Milbrandt and Yau, 2005; Milbrandt and McTaggart-Cowan,
2010; Loftus et al., 2014; Khain et al., 2015; Milbrandt et al., 2021).
Unlike short-term simulations, climate simulations pay more attention to the
equilibrium states or quasi-equilibrium states because the feedback
processes become important (Sherwood et al., 2015; King et al., 2020).
However, in terms of climate simulations, few studies have focused on the
influence of *μ*_{i}.

In this study, in order to investigate the impacts of *μ*_{i} on
climate simulations with the Community Atmosphere Model version 6 (CAM6)
model, the impacts of *μ*_{i} were considered in the MG scheme by a
tunable parameter. There were two major motivations behind this work. First,
are the impacts of *μ*_{i} notable? If yes, it is necessary for climate
models to represent the *μ*_{i} and *μ*_{i}-related processes in a
more realistic manner. And second, what are the main mechanisms for these
impacts? These would be helpful to understand the climate simulations with
the impacts of *μ*_{i}. This paper is organized as follows: the
modified MG scheme and experimental setup are described in Sect. 2; cloud
microphysical process offline tests and CAM6 model simulation results are
analyzed in Sect. 3; and finally, the summary and conclusions are provided
in Sect. 4.

## 2.1 The modified MG scheme

The CAM6 model, which is the atmospheric component of the Community Earth System Model Version 2.1.3 (CAM6; Bogenschutz et al., 2018; CESM2, Danabasoglu et al., 2020), was used in this study. It is noteworthy that the treatments of clouds in climate models are usually divided into two categories: convective cloud schemes with simplified cloud microphysics and larger-scale stratiform cloud schemes with relatively detailed cloud microphysics. In the CAM6 model, the convective cloud scheme does not consider the PSD of ICs (Zhang and McFarlane, 1995; Zhang et al., 1998; Bogenschutz et al., 2013; Larson, 2017). The stratiform cloud microphysics was represented by the updated MG scheme with prognostic precipitation (Gettelman and Morrison, 2015). In both versions of the MG scheme, the ICs are assumed to be spherical, and the PSD of ICs is described by the gamma distribution function:

where ${N}_{\mathrm{i}}^{\prime}\left(D\right)$ is the number density (i.e., $\mathit{\delta}{N}_{\mathrm{i}}/\mathit{\delta}D)$ of the
ICs with diameter *D*. *N*_{0i}, *λ*_{i}, and *μ*_{i} (nonnegative
values) are the intercept parameter, the slope parameter, and the spectral
shape parameter, respectively. Given that *μ*_{i} is known, *N*_{0i}
and *λ*_{i} can be determined by the local in-cloud IC mass and
number mixing ratio (*q*_{i} and *N*_{i}, prognostic variables in units
of kg kg^{−1} and kg^{−1}, respectively).

where the IC bulk density (*ρ*_{i}) is 500 kg m^{−3}, and *μ*_{i} is zero in the default MG scheme. $\mathrm{\Gamma}\left(x\right)={\int}_{\mathrm{0}}^{\mathrm{\infty}}{t}^{x-\mathrm{1}}{e}^{-t}\mathrm{d}t$ is the gamma function. It is noteworthy that the *k*th
moment of this size distribution (*M*_{k}) is found by integrating the
distribution in this form: ${M}_{k}={\int}_{\mathrm{0}}^{\mathrm{\infty}}{N}_{\mathrm{0}\mathrm{i}}{D}^{{\mathit{\mu}}_{\mathrm{i}}+k}{e}^{-{\mathit{\lambda}}_{\mathrm{i}}D}\mathrm{d}D={N}_{\mathrm{0}\mathrm{i}}\mathrm{\Gamma}(k+{\mathit{\mu}}_{\mathrm{i}}+\mathrm{1})/\phantom{\rule{0.125em}{0ex}}{\mathit{\lambda}}_{\mathrm{i}}^{(k+{\mathit{\mu}}_{\mathrm{i}}+\mathrm{1})}$ (Eidhammer et al., 2014). Furthermore, the recursive
property of the gamma function (i.e., $\mathrm{\Gamma}\left(x+\mathrm{1}\right)=x\mathrm{\Gamma}\left(x\right)$ ) is also used for the following formula derivation.

Equations (2) and (3) also indicate that, under the same *q*_{i} and
*N*_{i}, changes in *μ*_{i} could impact the other two parameters
regarding the PSD of ICs (i.e., *N*_{0i} and *λ*_{i}). Meanwhile,
the number-weighted radius (${R}_{{\mathrm{n}}_{\mathrm{i}}}$) related to the IC
deposition/sublimation process, the effective radiative radius (${R}_{{\mathrm{e}}_{\mathrm{i}}}$)
used for the radiative transfer scheme, and other statistical mean radii
might be influenced. To better understand the influence of *μ*_{i} on
the ice-phase cloud microphysical processes, the equations for calculating
the statistical mean radii are introduced first. The mass-weighted radius
(${R}_{{q}_{\mathrm{i}}}$) is calculated from Eq. (4). The number-weighted radius
(${R}_{{\mathrm{n}}_{\mathrm{i}}}$), which is the so-called mathematical mean value, is calculated
from Eq. (5). The area-weighted radius (${R}_{{\mathrm{a}}_{\mathrm{i}}}$) is calculated from Eq. (6).
${R}_{{\mathrm{e}}_{\mathrm{i}}}$, which is defined as the cross-section-weighted radius (Schumann et
al., 2011; Wyser, 1998), is calculated from
${R}_{{q}_{\mathrm{i}}}^{\mathrm{3}}/{R}_{{\mathrm{a}}_{\mathrm{i}}}^{\mathrm{2}}$ (Eq. 7). Note that ${R}_{{q}_{\mathrm{i}}}$ can be calculated by
*q*_{i} and *N*_{i} (the last term of Eq. 4, without *μ*_{i}),
and the other statistical mean radii (e.g., ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, ${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and
${R}_{{\mathrm{e}}_{\mathrm{i}}}$) can be calculated by ${R}_{{q}_{\mathrm{i}}}$ and *μ*_{i} (Eqs. 5–7). In other
words, the ratios of the other statistical mean radii (e.g., ${R}_{{\mathrm{n}}_{\mathrm{i}}}$,
${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$) to ${R}_{{q}_{\mathrm{i}}}$ are functions of *μ*_{i}. For
nonnegative *μ*_{i} values, Eqs. (5) and (6) indicate that ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ and
${R}_{{\mathrm{a}}_{\mathrm{i}}}$ are always less than ${R}_{{q}_{\mathrm{i}}}$. This can be explained by the physical
reason that larger ICs contribute more to ${R}_{{q}_{\mathrm{i}}}$ than to ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ and
${R}_{{\mathrm{a}}_{\mathrm{i}}}$. Similarly, ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ is always greater than ${R}_{{q}_{\mathrm{i}}}$ (Eq. 7).
Furthermore, Eqs. (5), (6), and (7) also indicate that with increasing *μ*_{i}, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, ${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ approach ${R}_{{q}_{\mathrm{i}}}$. In Sect. 3.1,
more analyses are provided by offline tests.

Because *μ*_{i} is zero in the default MG scheme, the equations for the
cloud microphysical processes are simplified by omitting *μ*_{i}
(Morrison and Gettelman, 2008; Gettelman and Morrison, 2015). In this study,
these equations are modified to consider the impact of *μ*_{i} (i.e.,
nonzero *μ*_{i}). In the default MG scheme, there are three cloud
microphysical processes, which are related to the PSD of ICs. They consist
of the deposition/sublimation of ICs, the autoconversion of IC to snow, and
the mass-weighted and number-weighted IC fall velocities (${V}_{{q}_{\mathrm{i}}}$ and
${V}_{{\mathrm{n}}_{\mathrm{i}}}$), respectively. Table 1 shows the original and modified equations
for these cloud microphysical processes. The d${q}_{\mathrm{i}}/\mathrm{d}t$ (i.e., the time
derivative of *q*_{i}) caused by the deposition/sublimation process (including
the Wegener-Bergeron process in mixed-phase clouds) is calculated from
d${q}_{\mathrm{i}}/\mathrm{d}t={S}_{\mathrm{i}}/({T}_{\mathrm{p}}{\mathit{\tau}}_{\mathrm{i}}$), where *S*_{i}, *T*_{p}, and *τ*_{i} are the ice supersaturation, a psychrometric correction to account
for the release of latent heat, and the supersaturation relaxation timescale, respectively (Morrison and Gettelman, 2008). Among them, *τ*_{i} is related to *μ*_{i}. In the original equation of *τ*_{i} (Table 1, left column), *N*_{0i}=*λ*_{i}*N*_{i} (Eq. 3) and
${\mathit{\lambda}}_{\mathrm{i}}^{-\mathrm{1}}=\mathrm{2}{R}_{{\mathrm{n}}_{\mathrm{i}}}$ at *μ*_{i}=0 (Eq. 5). Therefore,
the original equation for *τ*_{i} can be rewritten as the modified
equation (Table 1, right column). The modified equation indicates that *τ*_{i} is inversely proportional to ${N}_{\mathrm{i}}{R}_{{\mathrm{n}}_{\mathrm{i}}}$, which is consistent with
the equation obtained by Korolev and Mazin (2003). This modified equation also
indicates that, under the same *q*_{i} and *N*_{i} (${R}_{{q}_{\mathrm{i}}}$ is also fixed),
*μ*_{i} can affect *τ*_{i} (i.e., the IC deposition/sublimation
process) via the influence on ${R}_{{\mathrm{n}}_{\mathrm{i}}}$. In the MG scheme, ICs with radii
greater than the threshold (*R*_{cs}) are considered to be snow.
Correspondingly, the mass and number of ICs converted to snow (*q*_{iauto}
and *N*_{iauto}) are represented by the integration of those ICs with radii
greater than *R*_{cs}. Therefore, the incomplete gamma function, ($\mathrm{\Gamma}\left(s,x\right)={\int}_{x}^{\mathrm{\infty}}{t}^{s-\mathrm{1}}{e}^{-t}\mathrm{d}t)$, is used to calculate
*q*_{iauto} and *N*_{iauto} (right column). It is necessary to note that, at
*μ*_{i}=0, the modified equations for *q*_{iauto} and *N*_{iauto} can
be rewritten as the original equations (i.e., omitting *μ*_{i}, left
column) based on a property of the incomplete gamma function (i.e., $\mathrm{\Gamma}\left(s,x\right)=\left(s-\mathrm{1}\right)\mathrm{!}{e}^{-x}\sum _{k=\mathrm{0}}^{s-\mathrm{1}}\frac{{x}^{k}}{k\mathrm{!}}$,
where *s* is a positive integer). Based on the diameter–fall speed
relationship, *V*=*a**D*^{b} (*a* and *b* are empirical coefficients), and the
properties of the gamma function, *μ*_{i} is considered in the
equations for mass-weighted and number-weighted terminal fall speeds
(${V}_{{q}_{\mathrm{i}}}$ and ${V}_{{\mathrm{n}}_{\mathrm{i}}}$, Table 1).

^{*} where *D*_{v} is the diffusivity of water vapor in air (*D*_{v} is calculated
as a function of temperature and pressure, *D*_{v}=8.794 × 10${}^{-\mathrm{5}}\times {T}^{\mathrm{1.81}}/P$), *R*_{cs} is the threshold radius for the
autoconversion of IC to snow (*R*_{cs}=100 µm), *ρ*_{a} is the
air density, *ρ*_{a850} is the reference air density at 850 hPa, and *a* and *b* are empirical coefficients (*a*=700 m^{1−b} s^{−1},
*b*=1).

## 2.2 CAM6 experimental design

Observational studies have shown that *μ*_{i} is less than 5 under most
conditions (Heymsfield, 2003; McFarquhar et al., 2015). This study focuses
only on investigating the influence of *μ*_{i}. There are four *μ*_{i}-related processes (i.e., the radiative transfer process and three
cloud microphysical processes) in the modified CAM6 model. Note that *μ*_{i} can be set to different values for different processes with the
advantage of model simulations. Seven experiments were conducted in this
study (Table 2). The Mu0 experiment is considered to be the reference
experiment because *μ*_{i} is set to zero for all of the *μ*_{i}-related processes. The *μ*_{i} is set to 2 for all of the
*μ*_{i}-related processes in the Mu2 experiment, and the *μ*_{i}
is set to 5 for all of the *μ*_{i}-related processes in the Mu5
experiment. The comparison between the Mu2 (or Mu5) and Mu0 experiments
shows the influence of *μ*_{i} on climate simulations. Furthermore, to
investigate the influence of each *μ*_{i}-related process, an
additional four experiments, namely, Tao5, Auto5, Fall5, and Rei5, were
conducted. It is also necessary to investigate negative *μ*_{i} because
negative *μ*_{i} has also been reported by observational studies.
However, the bulk cloud microphysics schemes usually constrain *μ*_{i}
to be nonnegative (see Appendix A). Therefore, only nonnegative *μ*_{i}
was investigated. In this study, for ease of expression, “Δ” is
used to denote the difference from the Mu0 experiment (e.g., ΔTao5
= Tao5 − Mu0). Without specification, the comparisons between model
simulations are relative to the Mu0 experiment. When analyzing a cloud
property variable (e.g., *q*_{i}), it is necessary to know which experiment
the variable comes from. To show this information, the experiment name is
added as a superscript. For example, the *q*_{i} from the Mu5 experiment is
denoted as ${q}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{5}}$, the difference in *q*_{i} between the Mu5 and Mu0
experiments is denoted as ${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$, and the relative change of
*q*_{i} from the Mu5 experiment is denoted as ${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$.

In this study, all experiments were atmosphere-only simulations (i.e., sea
surface temperature and sea ice are prescribed) with a horizontal resolution
of 1.9^{∘} latitude × 2.5^{∘} longitude and 32 vertical layers. All experiments ran for 11 model years, and the last 10 years was used for the analyses. In addition, the standard deviation
calculated from the averages of each year (i.e., 10 averages) was used to
check the statistical significance of the multiyear average (i.e., 10-year
average).

## 3.1 Offline tests

To better understand the impact of *μ*_{i} on climate simulations, the
impacts of *μ*_{i} on the IC PSD and *μ*_{i}-related cloud
microphysical processes are first illustrated by offline tests. In these
offline tests, the impact of *μ*_{i} was analyzed at a given
${R}_{{q}_{\mathrm{i}}}$ (i.e., the ratio of *q*_{i} to *N*_{i} is fixed).

Equations (1), (2), and (3) indicate that the normalized IC size distribution (i.e.,
the relative contributions of each bin) can be calculated from ${R}_{{q}_{\mathrm{i}}}$ and
*μ*_{i}. Figure 1 shows the impact of *μ*_{i} on the normalized PSDs
of ICs. Under the same *μ*_{i}, the shapes of the PSDs (i.e., the
relative number or mass contributions of each bin) with ${R}_{{q}_{\mathrm{i}}}=\mathrm{20}$ µm (small IC scenario) are the same as those with ${R}_{{q}_{\mathrm{i}}}=\mathrm{60}$ µm
(large IC scenario). In other words, the shape of the PSD is completely
determined by *μ*_{i} (i.e., spectral shape parameter). As expected,
the PSDs move toward larger radii with increasing ${R}_{{q}_{\mathrm{i}}}$. As introduced in
the study of Milbrandt et al. (2021), the PSD becomes narrow with
increasing *μ*_{i}. Note that, in terms of number, the contributions of
the smaller size bins significantly decrease with increasing *μ*_{i}.
Unlike the number contributions, the mass contributions of the larger size
bins significantly decrease with increasing *μ*_{i} because the mass
contribution is more sensitive to the IC radius. Under the large IC scenario
(i.e., ${R}_{{q}_{\mathrm{i}}}=\mathrm{60}$ µm), the mass contribution of the ICs with radii
greater than *R*_{cs} is significantly decreased with increasing *μ*_{i}. The above analyses suggest that the cloud microphysical processes
that depend on the PSD of ICs (e.g., autoconversion of IC to snow) might be
significantly influenced by *μ*_{i}.

The offline tests were performed for the *μ*_{i}-related cloud
microphysical processes and statistical mean radii (Table 3). As introduced
in Sect. 2.1, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, ${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ can be calculated from
${R}_{{q}_{\mathrm{i}}}$ and *μ*_{i}. Both ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ and ${R}_{{\mathrm{a}}_{\mathrm{i}}}$ significantly increase
with increasing *μ*_{i} (Table 3). This is in agreement with their
calculation equations (Eqs. 5, 6). ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ is approximately half of
${R}_{{q}_{\mathrm{i}}}$ at *μ*_{i}=0 (i.e., $\mathrm{11.00}/\mathrm{20}$ and $\mathrm{33.02}/\mathrm{60}$), while ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ is
close to ${R}_{{q}_{\mathrm{i}}}$ at *μ*_{i}=5 (i.e., $\mathrm{17.26}/\mathrm{20}$ and $\mathrm{51.78}/\mathrm{60}$).
According to the calculation equation of ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ (Eq. 7), ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ decreases
with increasing *μ*_{i}. The ratios of ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ to ${R}_{{q}_{\mathrm{i}}}$ at *μ*_{i}=0, 2, and 5 are 1.65 (i.e., $\mathrm{33.02}/\mathrm{20}$ and $\mathrm{99.06}/\mathrm{60}$), 1.28 (i.e.,
$\mathrm{25.54}/\mathrm{20}$ and $\mathrm{76.63}/\mathrm{60}$), and 1.15 (i.e., $\mathrm{23.01}/\mathrm{20}$ and $\mathrm{69.04}/\mathrm{60}$),
respectively. It is necessary to point out that with increasing *μ*_{i}, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, ${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ approach ${R}_{{q}_{\mathrm{i}}}$ (Table 3)
because the PSD of ICs becomes narrow (Fig. 1). As expected, *τ*_{i}
decreases with increasing *μ*_{i} (Table 3) because *τ*_{i} is
inversely proportional to ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ (Table 2). The decrease in *τ*_{i}
suggests that the d${q}_{\mathrm{i}}/\mathrm{d}t$ caused by the deposition/sublimation process is
accelerated (Morrison and Gettelman, 2008). Compared to the $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ with
*μ*_{i}=0 (i.e., $\mathrm{3.35}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1} and 10.$\mathrm{04}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1}), the $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ with *μ*_{i}=2
($\mathrm{4.66}\times {\mathrm{10}}^{-\mathrm{4}}$ and $\mathrm{13.98}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1})
and *μ*_{i}=5 ($\mathrm{5.25}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1} and
$\mathrm{15.74}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1}) increase by 39.10 % and 56.72 %,
respectively. This is consistent with the previous finding (ICs vapor
deposition process is obviously accelerated by increasing *μ*_{i})
reported by Mitchell (1991). In Table 3, ${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$ and
${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$ indicate the portion of ICs that convert to snow in terms
of number and mass, respectively. Under the small IC scenario (i.e.,
${R}_{{q}_{\mathrm{i}}}=\mathrm{20}$ µm), regardless of the value of *μ*_{i}, both
${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$ and ${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$ are very small (< 2 %,
Table 3) because there are few ICs with radii greater than *R*_{cs} (Fig. 1).
Under the large IC scenario (i.e., ${R}_{{q}_{\mathrm{i}}}=\mathrm{60}$ µm), there is a
considerable portion of ICs with radii greater than *R*_{cs}, especially the
mass contribution (Fig. 1). The ${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$ at *μ*_{i}=0, 2,
and 5 is 64.08 %, 36.54 %, and 18.40 %, respectively (Table 3).
This suggests that the autoconversion of IC to snow becomes difficult with
increasing *μ*_{i}. Compared with the considerable values for ${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$,
the ${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$ is relatively small (i.e., 4.84 % at *μ*_{i}=0, 4.23 % at *μ*_{i}=2, and 2.63 % at *μ*_{i}=5). Therefore, the ${R}_{{q}_{\mathrm{i}}}$ of the residual ICs (*R*_{qi_afterauto}; 43.36 µm at *μ*_{i}=0, 52.31 µm at *μ*_{i}=2, and 56.57 µm at *μ*_{i}=5) is obviously lower
than the original ${R}_{{q}_{\mathrm{i}}}$ (60 µm). During the falling process, it is
inevitable that ${V}_{{q}_{\mathrm{i}}}$ is greater than ${V}_{{\mathrm{n}}_{\mathrm{i}}}$ because larger ICs
with faster falling contribute more in the ${V}_{{q}_{\mathrm{i}}}$. Thus, larger ICs appear
preferentially in the lower model layers. This is called the size-sorting
mechanism (Milbrandt and Yau, 2005). ${V}_{{q}_{\mathrm{i}}}$ decreases with increasing
*μ*_{i}, while ${V}_{{\mathrm{n}}_{\mathrm{i}}}$ increases with increasing *μ*_{i}
(Table 3). This could also be explained by their calculation equations (the
corresponding derivations are similar to those for ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, ${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and
${R}_{{\mathrm{e}}_{\mathrm{i}}}$, not shown). With increasing *μ*_{i}, the difference between
${V}_{{q}_{\mathrm{i}}}$ and ${V}_{{\mathrm{n}}_{\mathrm{i}}}$ decreases (Table 3) because the PSD of ICs becomes
narrow (Fig. 1). As a result, the size-sorting process becomes slow. For
instance, there are many ICs with ${R}_{{q}_{\mathrm{i}}}=\mathrm{60}$ µm in a model layer.
The height of each model layer is 200 m. After one model time step (10 min),
some ICs fall into the lower layer. For *μ*_{i}=0, the ${R}_{{q}_{\mathrm{i}}}$ of
the ICs that are still in the model layer (*R*_{qi_leftover})
is 42.11 µm, and the ${R}_{{q}_{\mathrm{i}}}$ of the ICs in the lower layer
(${R}_{{\mathrm{qi}}_{\mathrm{l}}\mathrm{owlayer}}$) is 95.24 µm. For *μ*_{i}=2,
*R*_{qi_leftover} is 52.45 µm, and
*R*_{qi_lowerlayer} is 75.60 µm. For *μ*_{i}= 5,
*R*_{qi_leftover} is 55.81 µm, and
*R*_{qi_lowerlayer} is 68.68 µm. It is clear that the
difference in ${R}_{{q}_{\mathrm{i}}}$ between these two adjacent layers that is caused by the
sedimentation process (i.e., the difference between *R*_{qi_leftover} and *R*_{qi_lowerlayer}) becomes small with
increasing *μ*_{i}. In short, the above analyses clearly suggest that
*μ*_{i} has a significant impact on the cloud microphysical processes
and statistical mean radii of ICs.

## 3.2 CAM6 simulations

During the evolution of stratiform clouds, the properties of ice clouds
(e.g., *q*_{i}, *N*_{i}, and ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, including mixed-phase clouds) largely
determine the ice-phase cloud microphysical processes. Meanwhile, these
cloud microphysical processes in turn change the cloud properties. They
interact as both cause and effect and finally reach equilibrium climate
states. To facilitate the subsequent analyses, the cloud properties and
*μ*_{i}-related cloud microphysical processes are shown together in one
figure. For ease of expression, “*δ*” is used to denote the changes
in cloud properties that are caused by the cloud microphysical process
during one model time step (tendency × one time step). For example,
the changes in *q*_{i} and *N*_{i} that are caused by the sedimentation
process during one model time step are denoted as *δ**q*_{ised} and
*δ**N*_{ised}, respectively.

Figure 2 shows the model results from the Mu0, Mu2, and Mu5 experiments. The
${q}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ is larger in the upper tropical troposphere (> 3 µg L^{−1}) and relatively larger in the lower troposphere over middle
latitudes in both hemispheres (> 1 µg L^{−1}). The spatial
pattern of ${q}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ is generally in agreement with the satellite
retrieval data (Li et al., 2012). Higher ${N}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ (> 200 L^{−1}) can be found in the tropopause region, where homogeneous freezing
produces a large number of ICs (not shown) due to sufficient soluble aerosol
particles, higher subgrid vertical velocity, and lower temperature (Shi et
al., 2015). All statistical mean radii (i.e., ${R}_{{q}_{\mathrm{i}}}$, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, and
${R}_{{\mathrm{e}}_{\mathrm{i}}}$) decrease with an altitude increase. One possible reason is that it
is hard for ICs to grow big in the upper troposphere because there the water vapor
density is very low (lower temperature). Furthermore, the
size-sorting effect (i.e., sedimentation process) could also be a
contributor to this phenomenon (Milbrandt and Yau, 2005; Khain et al.,
2015). As expected, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ is less than ${R}_{{q}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ is larger than
${R}_{{q}_{\mathrm{i}}}$. After considering the impact of *μ*_{i} (i.e., *μ*_{i}=2 or 5), the ΔMu2 and ΔMu5 experiments show that
*q*_{i} is significantly increased, while *N*_{i} is significantly decreased.
The ${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ is 30 %–100 % in nearly all regions, and the
${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ reaches even higher levels (> 100 %)
in most regions. Both ${N}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and ${N}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$
are $<-$20 % above the −37^{∘} isotherm and even reach
−50 % in the upper tropical troposphere. Consistent with the increase
in *q*_{i} and the decrease in *N*_{i}, the ${R}_{{q}_{\mathrm{i}}}$ significantly increases.
The ${R}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ is 30 %–100 % above the −37^{∘}
isotherm, and the ${R}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ is 30 %–100 % in most regions
and even reaches 100 % in a few regions of the upper tropical
troposphere. Because ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ increases with increasing *μ*_{i} at a
fixed ${R}_{{q}_{\mathrm{i}}}$ value (Sect. 3.1), the relative increases in ${R}_{{\mathrm{n}}_{\mathrm{i}}}$ from the
ΔMu2 and ΔMu5 experiments (i.e., ${R}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$
and ${R}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$) are obviously higher than the relative
increases in ${R}_{{q}_{\mathrm{i}}}$ (i.e., ${R}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and
${R}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$). The ${R}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ is > 100 % in some regions, and the ${R}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ is > 100 % in most regions. Compared with the relative increases in
${R}_{{q}_{\mathrm{i}}}$, the relative increases in ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ from the ΔMu2 and ΔMu5 experiments are obviously reduced or even negative because ${R}_{{\mathrm{e}}_{\mathrm{i}}}$
decreases with increasing *μ*_{i} at a fixed ${R}_{{q}_{\mathrm{i}}}$ value (Sect. 3.1). Overall, the impacts of *μ*_{i} on *q*_{i} and *N*_{i} are notable.
The changes in the statistical mean radii (i.e., ${R}_{{q}_{\mathrm{i}}}$, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, and
${R}_{{\mathrm{e}}_{\mathrm{i}}}$) can be explained by the changes in *q*_{i}, *N*_{i}, and *μ*_{i}.

This paragraph analyzes the interaction between the ice cloud properties
(*q*_{i}, *N*_{i}, and${R}_{{\mathrm{n}}_{\mathrm{i}}}$) and the IC deposition/sublimation process and
the influence of *μ*_{i} on this interaction. Since $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ is
proportional to ${N}_{\mathrm{i}}{R}_{{\mathrm{n}}_{\mathrm{i}}}$ (Table 1), the $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ is
larger in the upper tropical troposphere (> $\mathrm{20}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1}) due to the high ${N}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ (> 200 L^{−1}). Both the ΔMu2 and ΔMu5 experiments show that the
$\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ increases in most regions because the relative increase in
${R}_{{\mathrm{n}}_{\mathrm{i}}}$ (i.e., ${R}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and ${R}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$) is
stronger than the relative decrease in *N*_{i} (i.e., ${N}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$
and ${N}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$). However, the $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ is slightly
decreased in some regions of the upper tropical troposphere because the
relative decrease in *N*_{i} is remarkable ($<-$50 %) in these
regions. The *δ**q*_{idep} which indicates the change in
*q*_{i} caused by the deposition/sublimation process is mainly determined by
the $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ and in-cloud ice supersaturation (*S*_{i}) (Morrison and
Gettelman, 2008). Except for a very small region, the annual zonal mean
${S}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ is positive. This is consistent with the deposition events
being much more frequent than sublimation events (not shown). When
*S*_{i}>0, ice supersaturation (i.e., *S*_{i}>0)
towards ice saturation (i.e., *S*_{i}=0) occurs because the water vapor
is consumed by *δ**q*_{idep} (Korolev
and Mazin, 2003; Krämer et al.,
2009). The ${S}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ is lower (< 3 %) in the upper tropical
troposphere due to the high $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ (> $\mathrm{20}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1}). Both the ΔMu2 and ΔMu5
experiments show that *S*_{i} is increased in the upper tropical troposphere
due to the decreasing $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$, and *S*_{i} is decreased in the other
regions due to the increasing $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$. It is noteworthy that the
${S}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and ${S}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$ in the mixed-phase cloud
layers are obviously weaker than those in the pure ice cloud layers (i.e.,
above the −37^{∘} isotherm). This is consistent with the fact that the
*S*_{i} is relatively stable in mixed-phase clouds because liquid droplets
are often present. The $\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{Mu}\mathrm{0}}$ is generally decreased
with the altitude because the saturated vapor pressure significantly
decreases with decreasing air temperature. The comparison between *δ**q*_{idep} and *q*_{i} suggests that *δ**q*_{idep} is an
important source of *q*_{i}. The $\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and
$\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$ are greater than 0.1 µg L^{−1} in
most mixed-phase cloud layers due to the strongly increasing $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$
and relatively stable *S*_{i} values. This suggests that the increasing
*μ*_{i} could lead to a higher equilibrium state of *q*_{i} in the
mixed-phase cloud layers via the deposition process. The $\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and $\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$ are
negative between 200 and 300 hPa, mainly because the ${S}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and ${S}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$ are negative, and the $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ are
relatively small. The $\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and $\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$ are positive above 100 hPa, mainly because the
${S}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and ${S}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$ are positive. These
results indicate that the impact of *μ*_{i} on *δ**q*_{idep} becomes complex above the −37^{∘} isotherm, where *S*_{i} is more
susceptible to $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ and *δ**q*_{idep}. Meanwhile, the impact
of *μ*_{i} on *δ**q*_{idep} also becomes weak above the −37^{∘} isotherm because the feedback processes (i.e., the interaction
between *S*_{i} and *δ**q*_{idep}) become important. In short, the
*μ*_{i}-induced changes in the deposition/sublimation process (i.e.,
$\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ and *δ**q*_{idep}) can be largely explained by the
changes in *N*_{i} and ${R}_{{\mathrm{n}}_{\mathrm{i}}}$. One reason for the higher *q*_{i} in the
mixed-phase cloud layers from the Mu2 and Mu5 experiments is that *δ**q*_{idep} increases with increasing *μ*_{i}.

This paragraph analyzes the interaction between the ice cloud properties
(*q*_{i}, *N*_{i}, and ${R}_{{q}_{\mathrm{i}}}$) and the autoconversion process of IC to snow
(hereafter “the autoconversion process”) and the influence of *μ*_{i} on
this interaction. Both ${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ and
${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ are decreased with the altitude because the
${R}_{{q}_{\mathrm{i}}}^{\mathrm{Mu}\mathrm{0}}$ is decreased with the altitude. As expected, the
${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$ is considerable and much larger than the
${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}^{\mathrm{Mu}\mathrm{0}}$. It is clear that the autoconversion process is an important sink of *q*_{i}. However, the autoconversion process is not an important sink of *N*_{i} because the ${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$ is very
small. Both the ΔMu2 and ΔMu5 experiments show that the
${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$ is significantly decreased because the
autoconversion process obviously becomes difficult at higher *μ*_{i} values
(offline tests, Sect. 3.1). The difficult autoconversion process leads to
an equilibrium state with higher *q*_{i} and larger ${R}_{{q}_{\mathrm{i}}}$. Because of the
larger ${R}_{{q}_{\mathrm{i}}}$, the ${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$ from the Mu2 and Mu5
experiments is significantly increased. The increasing
${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$ from the Mu2 and Mu5 experiments might be a main reason
for the decrease in *N*_{i} in the mixed-phase cloud layers. However, the
remarkable decrease in *N*_{i} (mostly in the pure ice cloud layers) from the
Mu2 and Mu5 experiments is mainly due to the ice nucleation process. In the
MG scheme, the newly formed IC number density (excluding the ICs in
mixed-phase clouds) is calculated by a physically based ice nucleation
parameterization (Liu and Penner, 2005). Because the autoconversion process
becomes difficult in the Mu2 and Mu5 experiments, the in-cloud ICs should
have longer lifetimes and larger radii. As a result, *δ**N*_{inuc},
which denotes the newly formed IC number density from the nucleation
process, significantly decreases in the Mu2 and Mu5 experiments (Fig. 2).
The main reason is that the preexisting ICs would hinder the subsequent ice
nucleation process (especially for homogeneous freezing), owing to the
depletion of water vapor via deposition growth (Barahona et al., 2014; Shi
et al., 2015). *δ**N*_{inuc} is the main source of *N*_{i}. Therefore,
both the ΔMu2 and ΔMu5 experiments show that *N*_{i} is
significantly decreased. In short, the increase in *μ*_{i} causes the
autoconversion process to be difficult and then leads to a higher
equilibrium state of *q*_{i} and ${R}_{{q}_{\mathrm{i}}}$. Meanwhile, *N*_{i} is significantly
decreased due to the higher equilibrium state of *q*_{i} and ${R}_{{q}_{\mathrm{i}}}$ (i.e.,
the stronger suppression effect of the preexisting ICs on the ice
nucleation process).

This paragraph analyzes the interaction between the ice cloud properties and
the IC sedimentation process and the influence of *μ*_{i} on this
interaction. The sedimentation process is the last cloud microphysical
process in the MG scheme. The IC fall velocity is calculated based on the
updated cloud properties (i.e., the other cloud microphysical processes at
this model time step have been considered). Here, ${R}_{{q}_{\mathrm{i}}}^{\ast}$ denotes
the updated ${R}_{{q}_{\mathrm{i}}}$, which includes the changes caused by the
deposition/sublimation and autoconversion processes at this model time step.
In the mixed-phase cloud layers, the ${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{Mu}\mathrm{0}}$ is slightly less
than the ${R}_{{q}_{\mathrm{i}}}^{\mathrm{Mu}\mathrm{0}}$ because the sedimentation process has not occurred.
After considering the impacts of *μ*_{i} on the cloud microphysical
processes introduced above, the relative increases in ${R}_{{q}_{\mathrm{i}}}^{\ast}$ from
the ΔMu2 and ΔMu5 experiments (i.e., ${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and ${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$) are higher than the relative
increase in ${R}_{{q}_{\mathrm{i}}}$. As expected, both ${V}_{{q}_{\mathrm{i}}}^{\mathrm{Mu}\mathrm{0}}$ and
${V}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{Mu}\mathrm{0}}$ decrease with an altitude increase, and the
${V}_{{q}_{\mathrm{i}}}^{\mathrm{Mu}\mathrm{0}}$ is obviously larger than the ${V}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{Mu}\mathrm{0}}$. Although the
${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and ${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ are
positive, the ${V}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and ${V}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ are
negative in most regions because ${V}_{{q}_{\mathrm{i}}}$ decreases with increasing *μ*_{i} (offline tests, Sect. 3.1). However, both ${V}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and ${V}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ are positive in some layers over
the tropics and subtropics, where the ${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ and
${R}_{{q}_{\mathrm{i}}}^{\ast ,\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ are relatively higher. Because ${V}_{{\mathrm{n}}_{\mathrm{i}}}$
increases with increasing *μ*_{i} at a fixed ${R}_{{q}_{\mathrm{i}}}$ value (offline
tests, Table 3) and the ${R}_{{q}_{\mathrm{i}}}^{\ast}$ from the Mu2 and Mu5 experiments
are increased, the relative increases in ${V}_{{\mathrm{n}}_{\mathrm{i}}}$ from the ΔMu2 and
ΔMu5 experiments are remarkable. The ${V}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}/\mathrm{Mu}\mathrm{0}}$ is
> 100 % in some regions, and the ${V}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ is
> 100 % in most regions. *δ**q*_{ised} is mainly
determined by the gradient of ${V}_{{q}_{\mathrm{i}}}{q}_{\mathrm{i}}$ in the vertical direction.
Actually, the newly updated *q*_{i} between the substeps of the sedimentation
process is used for calculating *δ**q*_{ised}. Similarly, *δ**N*_{ised} is mainly determined by the gradient of ${V}_{{\mathrm{n}}_{\mathrm{i}}}{N}_{\mathrm{i}}$ in the
vertical direction. Furthermore, the ICs that fall into the clear portions
of the lower model layer sublimate instantly. Therefore, both $\mathit{\delta}{q}_{\mathrm{ised}}^{\mathrm{Mu}\mathrm{0}}$ and $\mathit{\delta}{N}_{\mathrm{ised}}^{\mathrm{Mu}\mathrm{0}}$ are negative in
most regions. This is consistent with sedimentation being a sink of clouds.
The *δ**q*_{ised} from the Mu2 and Mu5 experiments (i.e., $\mathit{\delta}{q}_{\mathrm{ised}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and $\mathit{\delta}{q}_{\mathrm{ised}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$) decreases
(negative, stronger sink) in most regions, mainly because of the increasing
${V}_{{q}_{\mathrm{i}}}$ and higher *q*_{i}. The *δ**N*_{ised} from the Mu2 and Mu5
experiments (i.e., $\mathit{\delta}{N}_{\mathrm{ised}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{2}}$ and $\mathit{\delta}{N}_{\mathrm{ised}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$) increases (positive, weaker sink) in a few layers
over the tropics. This is mainly due to the changes in the vertical gradient
of *N*_{i}. Both the ΔMu2 and ΔMu5 experiments show that the
changes in *δ**q*_{ised} and *δ**N*_{ised} are generally
weaker than the changes in *δ**q*_{idep}, *δ**q*_{iatuo} (i.e., ${q}_{\mathrm{i}}\times {q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$), and *δ**N*_{inuc}. In
short, the fall velocities (i.e., ${V}_{{q}_{\mathrm{i}}}$ and ${V}_{{\mathrm{n}}_{\mathrm{i}}}$) and their impacts on
ice clouds (i.e., *δ**q*_{ised} and *δ**N*_{ised}) are mainly
determined by the cloud properties (i.e., *q*_{i}, *N*_{i}, ${R}_{{q}_{\mathrm{i}}}$, and
${R}_{{q}_{\mathrm{i}}}^{\ast}$). Although the sedimentation process is also a main factor
that determines the cloud properties, the changes in the sedimentation
process that are caused by the increasing *μ*_{i} are not as strong as
those in the deposition/sublimation, autoconversion, and nucleation
processes.

Based on the analyses presented above, it can be concluded that increasing
*μ*_{i} would lead to a climate equilibrium state with higher
*q*_{i} and lower *N*_{i} in most regions. The changes in the statistical mean
radii (i.e., ${R}_{{q}_{\mathrm{i}}}$, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$) and ice-phase cloud
microphysical processes (i.e., *δ**q*_{idep}, ${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$,
${N}_{\mathrm{iauto}}/{N}_{\mathrm{i}}$, *δ**N*_{inuc}, *δ**q*_{ised}, and *δ**N*_{ised}) are mainly determined by the higher *q*_{i}, lower *N*_{i}, and
increasing *μ*_{i}. On the other hand, the higher *q*_{i} and lower *N*_{i}
can largely be explained by the changes in the ice-phase cloud microphysical
processes (i.e., *δ**q*_{idep}, ${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$, and *δ**N*_{inuc}) that are caused by the increasing *μ*_{i}. Furthermore, the
ΔMu2 and ΔMu5 experiments show very similar spatial patterns
for the *μ*_{i}-induced changes. This suggests that the impact of
*μ*_{i} on the simulated climate equilibrium state is stable.

Figure 3 shows the changes in the simulated climate equilibrium states that
are caused by each individual *μ*_{i}-related process. The $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ from the Tao5 experiment is significantly increased ($\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Tao}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$). Similar to the ΔMu2 and ΔMu5
experiments, this increasing $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ could lead to a higher
equilibrium state of *q*_{i} in the mixed-phase cloud layers
(${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Tao}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$) via the deposition/sublimation process ($\mathit{\delta}{q}_{\mathrm{idep}}^{\mathrm{\Delta}\mathrm{Tao}\mathrm{5}}$). However, this increasing $\mathrm{1}/{\mathit{\tau}}_{\mathrm{i}}$ leads
to lower *q*_{i} and lower *N*_{i} in most of the pure ice cloud layers. The
main reason might be that the ICs grow faster and their lifetimes become
shorter (Mitchell, 1991; DeMott et al., 2010; Storelvmo et al., 2013). The
${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}$ from the Auto5 experiment is significantly decreased
(${q}_{\mathrm{iauto}}/{q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Tao}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$). This could lead to a higher
*q*_{i} in nearly all regions (${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Auto}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$) and a lower *N*_{i}
in the pure ice cloud layers (${N}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Auto}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$). The mechanism is
the same as that introduced based on the ΔMu2 and ΔMu5
experiments. It is noteworthy that the *N*_{i} from the Auto5 experiment is
slightly increased in some mixed-phase cloud layers. The main reason might
be that the accretion of *N*_{i} by snow is significantly decreased in the
mixed-phase cloud layers (not shown) due to the difficult autoconversion
process. The${V}_{{q}_{\mathrm{i}}}$ from the Fall5 experiment is significantly decreased
(${V}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$), and the sink term of *q*_{i} due to
sedimentation becomes weaker (i.e., positive $\mathit{\delta}{q}_{\mathrm{ised}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}}$) in most regions. Unlike the ${V}_{{q}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$, the
${V}_{{\mathrm{n}}_{\mathrm{i}}}$ from the Fall5 experiment obviously increases (${V}_{{\mathrm{n}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$), and the sink term of *N*_{i} due to sedimentation becomes
stronger (i.e., negative $\mathit{\delta}{N}_{\mathrm{ised}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}}$) in the pure
ice cloud layers. These might be the main reasons for the increase in
*q*_{i} and the decrease in *N*_{i} in the pure ice cloud layers over the
tropics (${q}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$ and ${N}_{\mathrm{i}}^{\mathrm{\Delta}\mathrm{Fall}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$). It is
interesting to note that the ${R}_{{q}_{\mathrm{i}}}$, ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, and ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ from the Fall5
experiment all increase in pure ice cloud layers (i.e., upper layers) and
decrease in mixed-phase cloud layers (i.e., lower layers). This can be
explained by the *μ*_{i}-induced weaker size-sorting mechanism (Sect. 3.1). The ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ from the Rei5 experiment (${R}_{{\mathrm{e}}_{\mathrm{i}}}^{\mathrm{\Delta}\mathrm{Rei}\mathrm{5}/\mathrm{Mu}\mathrm{0}}$) is
significantly decreased. Because the change of ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ does not directly
affect the cloud microphysical processes, the changes in cloud properties
from the ΔRei5 experiment are not statistically significant in most
regions. Taken overall, the above analyses clarify the mechanism of *μ*_{i}'s impacts. Increasing *μ*_{i} in autoconversion impacts pure ice
clouds the most (i.e., significantly increased *q*_{i} and
significantly decreased *N*_{i} in the pure ice cloud layers). Furthermore,
increasing *μ*_{i} in autoconversion also leads to a much higher
*q*_{i} in the mixed-phase cloud layers. Increasing *μ*_{i} in
deposition/sublimation can also lead to a higher *q*_{i} in the mixed-phase
cloud layers. Increasing *μ*_{i} in sedimentation can lead to a higher
IC radius in the upper layers and lower IC radius in the lower layers. The
impacts from sedimentation and deposition/sublimation are obviously weaker
than those from autoconversion. The changes caused by increasing *μ*_{i} in the radiative process (i.e., ${R}_{{\mathrm{e}}_{\mathrm{i}}}$) are relatively chaotic.

The above analyses focus on cloud properties and cloud microphysical
processes (i.e., in-cloud variables). This paragraph discusses the impacts
of *μ*_{i} on radiation and precipitation. The annual zonal mean
distributions of the ice water path (IWP), column *N*_{i} (ColN_{i}),
longwave (CRE_{LW}) and shortwave (CRE_{SW}) cloud radiative effects,
and convective (RainC) and large-scale (RainL) precipitation rates are shown
in Fig. 4, and the corresponding global annual mean values are listed in
Table 4. The comparison of the Mu0, Mu2, and Mu5 experiments shows that the
zonal mean IWPs over all latitudes clearly increase with increasing *μ*_{i}. This is consistent with the changes in in-cloud *q*_{i} (Fig. 2).
The comparison of the ΔMu5, ΔTao5, ΔAuto5, ΔFall5, and ΔRei5 experiments shows that the *μ*_{i}-induced
increases in IWP are mainly provided by the autoconversion process. Compared
to the Mu0 experiment, the ColN_{i} from the Mu2 and Mu5 experiments
obviously decreases over tropical regions. It is clear that the
autoconversion process is also the main contributor to the decreases in
ColN_{i} (Fig. 4, right column). Compared to the Mu0 experiment, both
CRE_{LW} and CRE_{SW} from the Mu2 and Mu5 experiments are obviously
enhanced, mainly because of the increasing IWPs. It is clear that the
enhancements of CRE_{LW} and CRE_{SW} are also mainly contributed to
by the autoconversion process (Fig. 4). Both the CRE_{LW} and CRE_{SW}
from the Rei5 experiment are also obviously enhanced in terms of their zonal
mean values (Fig. 4) and global mean values (Table 4, CRE${}_{\mathrm{LW}}^{\mathrm{\Delta}\mathrm{Rei}\mathrm{5}}=\mathrm{1}$.29 W m^{−2} and CRE${}_{\mathrm{SW}}^{\mathrm{\Delta}\mathrm{Rei}\mathrm{5}}=-$1.79 W m^{−2}). This suggests that the impact of *μ*_{i} on ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ could lead to considerable changes in the Earth's radiation
budget. Compared to the impacts of *μ*_{i} on radiation, the impact on
large-scale precipitation (i.e., RainL) is not statistically significant (Fig. 4, left column). However, the convective precipitation from the ΔMu5
experiment (i.e., RainC^{ΔMu5}) is significantly reduced over the
tropics and subtropics (Fig. 4, right column). The reason is that the
increase in ice clouds (i.e., *q*_{i}) increases atmospheric stability via
the radiative budget and then leads to weaker convective precipitation
(Andrews et al., 2010; Wang et al., 2014). Overall, the impacts of *μ*_{i} on radiation and precipitation are considerable. The global mean CRE${}_{\mathrm{LW}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$, CRE${}_{\mathrm{SW}}^{\mathrm{\Delta}\mathrm{Mu}\mathrm{5}}$, and RainC^{ΔMu5} are 5.58 W m^{−2}, −5.34 W m^{−2}, and −0.12 mm d^{−1}, respectively. These
changes are mainly contributed to by the autoconversion process.
Furthermore, the comparisons between the ΔMu2 and ΔMu5
experiments (Fig. 4 and Table 4) show that, in most cases, the *μ*_{i}-induced changes are enhanced with increasing Δ*μ*_{i}.
This suggests that, in terms of the zonal mean and global mean values, the
impacts of *μ*_{i} are relatively stable.

This paper investigates the impacts of *μ*_{i} on climate simulations
with the CAM6 model. To achieve this, the two-moment bulk cloud microphysics
scheme used in CAM6 was modified to consider the *μ*_{i}'s impacts by a
tunable parameter. After that, the impacts of *μ*_{i} on the IC size
distribution and *μ*_{i}-related cloud microphysical processes are
illustrated first by calculation equations and offline tests, and the
impacts of *μ*_{i} on the climate simulations are then analyzed with
the CAM6 model.

The impacts of *μ*_{i} on the IC size distribution and various
statistical mean radii are clearly explained by the calculation equations.
${R}_{{q}_{\mathrm{i}}}$ can be calculated from *q*_{i} and *N*_{i}, and the normalized
IC size distribution (i.e., the relative contributions from each bin) can be
calculated from ${R}_{{q}_{\mathrm{i}}}$ and *μ*_{i}. The impact of *μ*_{i} on
mass-weighted size distribution is obviously different from that on the
commonly used number-weighted size distribution (Fig.1). Unlike the number
contributions, the mass contributions of the larger size bins significantly
decrease with increasing *μ*_{i} because the mass contribution is more
sensitive to the IC radius. In the bulk cloud microphysics scheme, the
physical processes are calculated based on various statistical mean radii.
The ratios of the other statistical mean radii (i.e., ${R}_{{\mathrm{n}}_{\mathrm{i}}}$, ${R}_{{\mathrm{a}}_{\mathrm{i}}}$, and
${R}_{{\mathrm{e}}_{\mathrm{i}}}$) to ${R}_{{q}_{\mathrm{i}}}$ are functions of *μ*_{i}. At *μ*_{i}≥0,
${R}_{{\mathrm{n}}_{\mathrm{i}}}$ and ${R}_{{\mathrm{a}}_{\mathrm{i}}}$ are always less than ${R}_{{q}_{\mathrm{i}}}$, whereas ${R}_{{\mathrm{e}}_{\mathrm{i}}}$ is always
greater than ${R}_{{q}_{\mathrm{i}}}$. The differences among these statistical mean radii
become small with increasing *μ*_{i}, which is consistent with the
narrower size distribution determined by higher *μ*_{i} values.

The impacts of *μ*_{i} on the *μ*_{i}-related cloud microphysical
processes are clearly illustrated by the offline tests. Under the same
*q*_{i} and *N*_{i} (${R}_{{q}_{\mathrm{i}}}$ is also fixed), the IC deposition/sublimation
process is considerably accelerated with increasing *μ*_{i} because the
${R}_{{\mathrm{n}}_{\mathrm{i}}}$ used for calculating deposition/sublimation is increasing with
increasing *μ*_{i}. Under the same ${R}_{{q}_{\mathrm{i}}}({R}_{{q}_{\mathrm{i}}}$ is much less than the
snow radius), the autoconversion of IC to snow obviously becomes difficult
with increasing *μ*_{i} because the portion of ICs with radii greater
than the threshold (> *R*_{cs}) decreases under the narrow size
distribution. A major effect of IC sedimentation is size-sorting because
${V}_{{q}_{\mathrm{i}}}$ is greater than ${V}_{{\mathrm{n}}_{\mathrm{i}}}$, and *μ*_{i} plays an important role in
determining the rate of size-sorting (Milbrandt and Yau, 2005). In this
study, the offline tests clearly show that the difference in IC radius
between two adjacent model layers caused by sedimentation becomes small with
increasing *μ*_{i} because the difference between ${V}_{{q}_{\mathrm{i}}}$ and
${V}_{{\mathrm{n}}_{\mathrm{i}}}$ becomes small (i.e., the size-sorting rate becomes slow).

The climate simulations show that the impacts of *μ*_{i} on the ice
cloud properties are notable, and the main corresponding mechanisms are
clear. After increasing *μ*_{i} from 0 to 2 and 5, *q*_{i} significantly
increases, while *N*_{i} significantly decreases. The accelerated
deposition process contributes to the higher *q*_{i} in the mixed-phase cloud
layers where the ice supersaturation (*S*_{i}>0) is relatively
stable. The difficult autoconversion process leads to longer IC lifetime and
higher *q*_{i}. Meanwhile, *N*_{i} significantly decreases because the newly
formed IC number density is significantly decreased, owing to the longer IC
lifetime and higher *q*_{i}. The experiments with only one modified *μ*_{i}-related process make the mechanisms of *μ*_{i}'s impacts more
clear. Autoconversion contributes the most. The *μ*_{i} also has
considerable impacts on radiation and precipitation. After increasing *μ*_{i} from 0 to 5, the global mean CRE_{LW} is increased (stronger
warming effect) by 5.58 W m^{−2} (25.11 %). Meanwhile, the CRE_{SW}
is decreased (less cooling effect) by −5.34 W m^{−2} (10.84 %). The
enhancement of the cloud radiative effects is largely provided by the higher
equilibrium state of *q*_{i}. The considerably stronger CRE_{LW} could
increase the atmospheric stability and then lead to weaker convective
precipitation (Andrews et al., 2010; Wang et al., 2014). As expected, after
increasing *μ*_{i} from 0 to 5, the global mean RainC is decreased by
−0.12 mm d^{−1} (7.64 %). In short, the impacts of *μ*_{i} on
climate simulations are significant. This suggests that the *μ*_{i}
(i.e., the PSD of ICs) and *μ*_{i}-related cloud microphysical
processes deserve a more realistic representation in climate models,
especially for cloud schemes with autoconversion. Fortunately, there have
been some studies that can help to address this issue. For example, *μ*_{i}
is described by an empirical formula (Eidhammer et al., 2017), *μ*_{i}
is predicted in a three-moment cloud scheme (Milbrandt et al., 2021), and
single-ice-category cloud schemes could obviate the need for autoconversion
process (e.g., Morrison and Milbrandt, 2015; Eidhammer et al., 2017; Zhao et
al., 2017).

This study only focuses on the impacts of *μ*_{i}; the default tunable
parameters (except for *μ*_{i}) are used in all the simulations. After
improving the representation of *μ*_{i}-related processes, further
model tuning and analyses are required based on the updated cloud scheme.
Therefore, this study does not estimate which value of *μ*_{i} could
lead to a better simulation. Finally, it is necessary to point out that the
main mechanism for *μ*_{i}'s impacts introduced in this study (i.e.,
autoconversion becomes difficult with increasing *μ*_{i}) is not
applicable to climate simulations with the single-ice-category cloud scheme.
However, similar to the *μ*_{i}'s impact on autoconversion, the
interaction between small ICs and large ICs (e.g., the accretion of small
ICs by large ICs) should become weaker with increasing *μ*_{i} (i.e.,
narrower size distribution). Therefore, we can speculate that the impacts of
*μ*_{i} on climate simulations with the single-ice-category cloud
scheme may still be worth noting.

Here, we show the PSDs represented by gamma functions of the form ${N}^{\prime}\left(D\right)=$
*N*_{0}*D*^{μ}*e*^{−λD} firstly and then discuss the reason why bulk
cloud microphysics schemes usually constrain *μ* to be nonnegative.

Figure A1 shows that the PSDs are relatively wide with negative *μ*. Under
negative *μ*, the particle number densities (*N*^{′}) are increased with
decreasing *D* and become very large at *D*<1 µm (Fig. A1 left).
In the real world, the cloud particles are usually not less than 1 µm. It is necessary to point out that the contribution from small particles
(e.g., *D*<50 µm) is usually neglected for getting the
gamma-fitted PSD from observations (Heymsfield, 2003). For instance, when
only considering the particles over sizes (*D*) from as small as 10 µm to
as large as 2000 µm (measured particle size), the uncertainty from the
extrapolation below 50 µm is negligible in the linear space of particle
size (Fig. A1 right). Therefore, some gamma-fitted PSDs from observations
might show negative *μ* values (e.g., Heymsfield, 2003; Heymsfield et
al., 2013; Schmitt and Heymsfield, 2009).

Unlike the gamma-fitted PSDs from observations (small particles might be
neglected), the gamma functions used in bulk cloud microphysics schemes
represent the particles with diameter from 0 to ∞ (hereafter
“mathematical size range”). For instance, the other two gamma distribution
parameters (*N*_{0} and *λ*) used in the bulk cloud scheme are
calculated by the particle's mass (*q*) and number (*N*) and some gamma
functions of *μ* ($\mathit{\lambda}=[\frac{\mathit{\pi}\mathit{\rho}}{\mathrm{6}}\frac{N}{q}\frac{\mathrm{\Gamma}\left(\mathrm{4}+\mathit{\mu}\right)}{\mathrm{\Gamma}\left(\mathrm{1}+\mathit{\mu}\right)}{]}^{\mathrm{1}/\mathrm{3}}$,
${N}_{\mathrm{0}}=\frac{N{\mathit{\lambda}}^{\left(\mathrm{1}+\mathit{\mu}\right)}}{\mathrm{\Gamma}\left(\mathrm{1}+\mathit{\mu}\right)}$). Because the gamma
function, $\mathrm{\Gamma}\left(x\right)={\int}_{\mathrm{0}}^{\mathrm{\infty}}{t}^{x-\mathrm{1}}{e}^{-t}\mathrm{d}t$, is used for
deriving these calculation formulas, the *q* and *N* in these calculation
formulas indicate the mass and number of particles from the mathematical
size range (i.e., 0 to ∞). Furthermore, the *μ* must be greater
than −1 in these two calculation formulas because the negative integer and
zero are the singularity of the gamma function. Under negative *μ* ($-\mathrm{1}<\mathit{\mu}<\mathrm{0}$; the *N*^{′} is very large at *D*<1 µm,
Fig. A1 left), more attention should be paid to using the gamma function
because it integrates from 0 to ∞. Figure A2 shows the relative number
contributions from each radius bin of ICs under different ${R}_{{q}_{\mathrm{i}}}$. Table A1
lists the contributions of ICs with a radius from 1 to 1000 µm (hereafter “realistic size range”) to the total number (i.e., the
*N*_{i} from the mathematical size range). At *μ*_{i}≥0 (i.e.,
*μ*_{i}=0, 2, and 5), the number contributions are mostly from the
realistic size range except for one case (*μ*_{i}=0 and ${R}_{{q}_{\mathrm{i}}}=\mathrm{5}$ µm). Under the small IC scenario (i.e., ${R}_{{q}_{\mathrm{i}}}=\mathrm{5}$ µm) and
${\mathit{\mu}}_{\mathrm{i}}=-$0.5, the number of ICs from the realistic size range
only contributes $\sim \mathrm{1}/\mathrm{2}$ to the total number. At ${\mathit{\mu}}_{\mathrm{i}}=-$0.9, the contributions of ICs from the realistic size range cannot
reach $\mathrm{1}/\mathrm{2}$, even for the large IC scenario (i.e., ${R}_{{q}_{\mathrm{i}}}=\mathrm{60}$ µm). In
other words, under negative *μ*_{i}, the gamma distribution functions
(mathematical size range) used in the bulk cloud schemes might be not suited
for representing realistic ICs (realistic size range). Therefore, our study
only evaluates the impacts of changing *μ*_{i} from 0 to 2 and 5.

The CAM6 model used in this study is the atmospheric component of the Community Earth System Model version 2.1.3 (CESM2.1.3). The CESM2.1.3 is a release version of CESM2. The model code, scripts, and input data are freely available through a public GitHub repository (https://escomp.github.io/CESM/versions/cesm2.1/html/downloading_cesm.html CESM Working Groups of National Center for Atmospheric Research (NCAR), 2022). More details about model workflow can be found in the corresponding quick-start guide. The modified model code, model run control scripts, and simulation results post-processing scripts covering every data processing action for all the model results reported in the paper are available online at https://doi.org/10.5281/zenodo.6409156 (Zhang et al., 2022). The FORTRAN code for offline tests is also archived at the same location. Furthermore, the NCL scripts and data used to make every figure are also available at the above DOI.

XS designed this study. WZ and XS designed the CAM6 model experiments and developed the modified model code. XS and WZ analyzed the results and wrote the original paper. All authors contributed to improving and reviewing the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank Yuxi Zeng and Jiaojiao Liu for checking the English expressions of an earlier draft of this paper. The model simulation was conducted at the High Performance Computing Center of Nanjing University of Information Science & Technology.

This research has been supported by the National Key Research and Development Program of China (grant no. 2017YFA0604001) and the National Natural Science Foundation of China (grant no. 41775095).

This paper was edited by Simon Unterstrasser and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Model and experiments
- Results and analysis
- Summary and conclusions
- Appendix A: The limitation in the use of gamma functions for representing ice-phase PSDs in bulk cloud microphysics schemes
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

*μ*

_{i}). After that, how the

*μ*

_{i}impacts cloud microphysical processes and then climate simulations is clearly illustrated by offline tests and CAM6 model experiments. Our results and findings are useful for the further development of

*μ*

_{i}-related parameterizations.

- Abstract
- Introduction
- Model and experiments
- Results and analysis
- Summary and conclusions
- Appendix A: The limitation in the use of gamma functions for representing ice-phase PSDs in bulk cloud microphysics schemes
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References