Different entrainment–mixing processes can occur in clouds; however, a homogeneous mixing mechanism is often implicitly assumed in most commonly used microphysics schemes. Here, we first present a new entrainment–mixing parameterization that uses the grid mean relative humidity without requiring
the relative humidity of the entrained air. Then, the parameterization is
implemented in a microphysics scheme in a large eddy simulation model, and
sensitivity experiments are conducted to compare the new parameterization
with the default homogeneous entrainment–mixing parameterization. The
results indicate that the new entrainment–mixing parameterization has a
larger impact on the number concentration, volume mean radius, and cloud
optical depth in the stratocumulus case than in the cumulus case. This is
because inhomogeneous and homogeneous mixing mechanisms dominate in the
stratocumulus and cumulus cases, respectively, which is mainly due to the
larger turbulence dissipation rate in the cumulus case. Because
stratocumulus clouds break up during the dissipation stage to form cumulus
clouds, the effects of this new entrainment–mixing parameterization during
the stratocumulus dissipation stage are between those during the
stratocumulus mature stage and the cumulus case. A large aerosol
concentration can enhance the effects of this new entrainment–mixing
parameterization by decreasing the cloud droplet size and evaporation timescale. The results of this new entrainment–mixing parameterization with
grid mean relative humidity are validated by the use of a different
entrainment–mixing parameterization that uses parameterized entrained air
properties. This study sheds new light on the improvement of
entrainment–mixing parameterizations in models.
Introduction
The process of entrainment and subsequent mixing between clouds and their
environment is one of the most uncertain processes in cloud physics, which
is thought to be crucial to many outstanding issues, including warm rain
initiation and subsequent precipitation characteristics, cloud–climate
feedback, and evaluating the indirect effects of aerosol
(Paluch and Baumgardner, 1989; Yum, 1998;
Ackerman et al., 2004; Kim et al., 2008; Huang
et al., 2008; Del Genio and Wu, 2010; Lu et al., 2011, 2014; Kumar et al., 2013; Zheng and Rosenfeld, 2015; Fan et al., 2016; Gao et al., 2020, 2021;
Zhu et al., 2021; Xu et al., 2021; Yang et al., 2016, 2021). The most well-studied
concepts are homogeneous/inhomogeneous entrainment–mixing mechanisms. During
homogeneous mixing, all droplets experience evaporation, and no droplet is
evaporated completely. During extremely inhomogeneous mixing, some droplets
near the entrained air evaporate completely, while the remaining droplets
maintain their original sizes. If the situation is somewhere between these
two extreme scenarios, an inhomogeneous mixing process occurs. Some studies
suggest that homogeneous mixing is likely to be typical
(Jensen et al., 1985; Burnet and Brenguier,
2007; Lehmann et al., 2009), whereas others have claimed that
an extremely inhomogeneous scenario is dominant (Pawlowska et al., 2000;
Burnet and Brenguier, 2007; Haman et al., 2007; Freud et al., 2008, 2011). Different mechanisms can be undistinguishable when the relative
humidity in the entrained air is high (Gerber et al., 2008).
Some sensitivity studies assuming homogeneous or extremely inhomogeneous
mixing have found that different mixing mechanisms can significantly
influence the microphysics and radiative properties of clouds
(Lasher-Trapp et al., 2005; Grabowski, 2006;
Chosson et al., 2007; Slawinska et al., 2008). For
example, Grabowski (2006) used a cloud-resolving model and found
that the amount of solar energy reaching the surface in the pristine case,
assuming the homogeneous mixing scenario, is the same as in the polluted
case with extremely inhomogeneous mixing. This result was verified by
Slawinska et al. (2008) using a large-eddy simulation (LES)
model. Although the influence of different mixing mechanisms in simulations
is lower when two-moment microphysics schemes are used (Hill et
al., 2009; Grabowski and Morrison, 2011; Slawinska et al.,
2012; Xu et al., 2020), Hill et al. (2009) also claimed that
there are still many uncertainties in the entrainment–mixing process, and
the effect of different mixing mechanisms can be more important over the
entire cloud life-cycle.
In recent years, methods have been developed to describe general
entrainment–mixing processes, with homogeneous and extremely inhomogeneous
scenarios as special cases (Andrejczuk et al., 2006, 2009; Lehmann et al.,
2009; Lu et al., 2011). Hoffmann et al. (2019) and
Hoffmann and Feingold (2019) conducted LESs at the
subgrid scale with turbulent mixing using a linear eddy model.
Andrejczuk et al. (2009) used the results of direct
numerical simulation (DNS) to establish a relationship between instantaneous
microphysical properties and Damköhler number (Da;
Burnet and Brenguier, 2007) and developed a parameterization of
the entrainment–mixing process. Lu et al. (2013) developed a
parameterization of the entrainment–mixing process based on the relationship
between the homogeneous mixing degree (ψ) and transition scale number
(NL) in the explicit mixing parcel model (EMPM), as well as aircraft observation data. Gao et al. (2018) investigated how ψ is
related to Da and NL in a DNS to improve the parameterization of the entrainment–mixing process. Luo et al. (2020) simulated more than 12 000 cases with EMPM by changing a variety of parameters affecting entrainment–mixing processes and developed a parameterization that improved the one proposed by Lu et al. (2013).
Although several entrainment–mixing parametrizations have been proposed, to
the best of our knowledge, only one study (Jarecka et al.,
2013) has coupled an entrainment–mixing parameterization with cloud
microphysics to consider the change in cloud droplet concentration during
the entrainment–mixing process. Jarecka et al. (2013) applied
an entrainment–mixing parameterization, in terms of the Damköhler
number, to a two-moment microphysics scheme and found small impacts of
entrainment–mixing parameterization in shallow cumulus clouds. To further
explore the influences of entrainment–mixing processes, this study first
modifies the entrainment–mixing parameterization in terms of the transition
scale number proposed by Luo et al. (2020) to couple it more easily with
microphysics schemes. The parameterization is then implemented in the
two-moment Thompson aerosol-aware scheme (Thompson and Eidhammer, 2014).
Finally, the effects of parameterization on the physical properties of
clouds are examined in both cumulus and stratocumulus clouds.
The rest of this paper is organized as follows: Sect. 2 describes the new
entrainment–mixing parameterization, simulated cases, and modeling setup.
The major results are presented and discussed in Sect. 3. The influences
of the new entrainment–mixing parameterization on cloud physics and the
underlying mechanisms are examined, and the effects of turbulence
dissipation rate (ε) and aerosol concentration are also discussed. Some concluding remarks are presented in Sect. 4.
Parameterization, simulated cases, and modeling setupThe new entrainment–mixing parameterization
According to Morrison and Grabowski (2008), the effect of
the entrainment–mixing process on cloud microphysical properties can be
expressed as follows:
Nc=Nc0qcqc0α,
where Nc and Nc0 are the cloud droplet number concentrations after and before the evaporation process, respectively, and qc and qc0
represent the corresponding cloud water mixing ratios. It is noteworthy that
when a new saturation is achieved after evaporation, qc is determined by
qc0, relative humidity (RH), air pressure, and temperature. The
parameter α can be pre-set to any value between 0 (homogeneous
mixing) and 1 (extremely inhomogeneous mixing) to represent a different
degree of subgrid-scale mixing homogeneity. In this study, instead of
specifying α as a predetermined constant, here it is determined through the following expressions (Lu et al., 2013; Luo et al., 2020):
2aα=1-ψ,2bψ=cexpaNLb,
where a, b, and c are the three fitting parameters (Luo et al., 2020). The dimensionless number NL is a dynamical measure of the degree of subgrid-scale mixing homogeneity (Lu et al., 2011) defined by
3aNL=L∗η,3bη=ν3/ε1/4,3cL∗=ε1/2τevap3/2,
where L∗ is the transition length (Lehmann et al.,
2009), η is the Kolmogorov microscale, ν is the kinematic
viscosity, and ε is calculated from the subgrid turbulent kinetic energy (Deardorff, 1980):
ε=CE3/2/L,
where C=0.70 is an empirical constant, E is the subgrid turbulent kinetic
energy, and L is the model grid size. The evaporation timescale (τevap) is defined as the time taken for droplets to evaporate
completely in an unsaturated environment and is calculated as
5aτevap=-r22ASe,5bA=1LhRvT-1LhρLKT+ρLRvTDes(T),
where r is the volume mean radius of cloud droplets, A is a function of pressure and temperature, Se is the supersaturation (RH - 1) of entrained air, Lh is the latent heat, Rv is the specific gas constant for water vapor, T is air temperature, ρL is the density of liquid water,
K is the coefficient of thermal conductivity of air, D is the diffusion
coefficient of water vapor in the air, and es(T) is the saturation vapor pressure over a plane water surface at temperature T.
Unfortunately, Se in Eq. (5a) is generally unavailable in
atmospheric models, including LES models. Thus, the entrainment–mixing
parameterization developed by Luo et al. (2020) based on the properties
of entrained air cannot be used directly. To solve this problem, we modify
the entrainment–mixing parameterization of Luo et al. (2020) by
replacing Se with the domain mean RH in the EMPM, after entrainment but before evaporation, based on 12 218 cases:
ψ=107.19exp-1.99NL-0.29.
Figure 1 shows the fitting results of the modified new entrainment–mixing
parameterization. Compared to the parametrization proposed by Luo et al. (2020), the modified parameterization has similar ψ-NL
distributions but with a larger NL for the same ψ because the EMPM domain mean RH is larger than the entrained air RH. With this
modification, NL, ψ, and thus the effect of the entrainment–mixing
processes on droplet concentration can be directly calculated using the LES
grid mean RH. It is important to note that the parameterization does not
mean that the entrained air RH is equal to that of the LES grid mean RH. It
is also worth noting that a wide range of ε, Se, and fraction of entrained air (f) are taken into account when establishing the
parameterization with the EMPM. The details of the EMPM simulations and
related calculations are provided by Luo et al. (2020).
Parameterization of cloud entrainment–mixing mechanisms by
relating homogeneous mixing degree (ψ) to transition scale number
(NL) from EMPM. The contours represent the joint probability
distribution function (PDF) of ψ vs. NL. The magenta dots and error bars are mean values and standard deviations of ψ in each NL bin, respectively. The mean values are fitted using a weighted least squares method with the number of data points in each NL bin as with the weight. The fitting equation, coefficient of determination (R2), and p values are also given. NL is calculated with the domain-averaged relative humidity after entrainment but before evaporation in the EMPM.
LES model, simulation cases, and modeling setup
The LES model is built by applying the large-scale forcing module presented
in Endo et al. (2015) to the Weather Research and Forecasting (WRF)
model tailored for solar irradiance forecasting (WRF-Solar; Hacker
et al., 2016; Haupt et al., 2016). The large-scale forcing
data (VARANAL) used in this process are derived from the constrained
variational analysis (CVA) approach developed by
Zhang et al. (2001) and provided by the U.S.
Department of Energy's Atmospheric Radiation Measurement Program (http://www.arm.gov, last access: 20 March 2022). The modified entrainment–mixing parameterization is
implemented in the two-moment Thompson aerosol-aware scheme (Thompson and
Eidhammer, 2014).
To investigate the behaviors of the new entrainment–mixing parameterization
in different cloud types, cumulus and stratocumulus cases are simulated. For
both the cumulus and stratocumulus cases, the horizontal resolution of the
model is 100 m × 100 m with a domain area of 14.4 km × 14.4 km. The vertical direction is divided into 225 layers with a resolution of 30 m.
Summary of names and corresponding descriptions of the four
experiments for each case of cumulus and stratocumulus. The meaning of each
symbol for each experiment can be found in the text.
For each cloud case, ψ is first set to 1 for the default experiment because
most LES models assume a homogeneous entrainment–mixing mechanism. The
simulation with the new entrainment–mixing parameterization (Eqs. 1–6) is hereafter referred to as new. First, NL is diagnosed for each grid, and ψ is then calculated using Eq. (6). Finally, the
variation in Nc during entrainment–mixing is obtained using Eqs. (1) and (2a). To examine the influence of the aerosol number concentration on the entrainment–mixing process, we conduct the numerical experiments default_10 and new_10 by multiplying the initial aerosol number concentrations for the
default and new models, respectively, by a factor of 10. Thus, four sets of numerical
experiments are conducted for both the cumulus and stratocumulus cases; the
names of the experiments and corresponding descriptions are summarized in
Table 1.
ResultsCumulus case
For the cumulus case, the simulation starts at 12:00 UTC on 11 June 2016 and
ends at 03:00 UTC on 12 June 2016 with an output interval of 10 min and
spin times of 3 h. To demonstrate the utility of the model, Fig. 2
compares the temporal evolution of the observed and simulated cloud fraction
(Fig. 2a) and solar irradiance (Fig. 2b) from the default experiment. Grid points with qc larger than 0.01 g kg-1 are defined as “cloudy areas”. Also shown for
comparison are observational data with a 1 h temporal resolution, which
are provided by the LES Atmospheric Radiation Measurement Symbiotic
Simulation and Observation (LASSO) campaign (Gustafson et al.,
2020). The observations show that the cloud forms at 12:00 UTC on 11 June
and dissipates completely by 01:00 UTC on 12 June with a maximum cloud
fraction of 0.47 at 16:00 UTC on 11 June. Considering the difference between
the solar irradiances obtained from point measurements and the value
representing the simulation domain, the observed solar irradiance at the
Southern Great Plains (SGP) Central Facility are compared with the results
of the central grid point in simulation (Fig. 2b). Evidently, although the
results of the simulation do not fluctuate as much as the observations, the
model captures the general behaviors of both cloud fraction and solar
irradiance. The general agreement between the simulations and observations
lends credence to using the model in further study.
Time series of (a) domain-averaged cloud fraction and (b) total downward irradiance at the central point from the observation and the default experiment in the cumulus case.
The temporal evolutions of main cloud microphysical and optical
properties in all simulation experiments for the cumulus case, including (a) cloud water mixing ratio (qc) (g kg-1), (b) cloud droplet number concentration (Nc) (cm-3), (c) cloud droplet volume mean radius (rv) (µm), (d) cloud water path (CWP) (g m-2), and (e) cloud
optical depth (τ). In (b), Nc values in the experiments default and new are normalized by the maximum cloud droplet concentration (Ncmax) from default; Nc values in the experiments default_10 and new_10 are normalized by Ncmax from default_10. The four experiments are detailed in Table 1.
Figure 3 shows the evolution of the microphysical and optical properties of
clouds in the cloudy areas of all simulation experiments, including
qc, Nc, droplet volume mean radius (rv), cloud water path (CWP),
and cloud optical depth (τ). To visually and simultaneously compare
the change in cloud droplet concentration under different aerosol
concentrations, the maximum cloud droplet concentration (Ncmax) from default is used to normalize Nc in default and new, while Ncmax from
new_10 is used to normalize Nc in default_10 and new_10. The CWP is calculated as follows:
CWP=∫0Hρaqc(z)dz,
where ρa is the air density, qc(z) is the cloud water mixing ratio at each height (z), and H is the cloud thickness. The optical depth τ is estimated with
τ=321ρw∫0Hρaqc(z)re(z)dz,
where ρw is the water density, and re(z) is the effective radius of the cloud droplets at each height (z). The time-averaged values of these physical properties of the clouds are listed in Table 2 for convenience.
Summary of the case mean values of the key quantities in all the
simulations of the cumulus case, containing cloud water mixing ratio
(qc), cloud droplet number concentration (Nc), cloud droplet volume mean radius (rv), cloud water path (CWP), and cloud optical depth (τ). The experiments are detailed in Table 1.
For the low aerosol number concentration, the simulations with the new
entrainment–mixing parameterization have smaller Nc (35.53 cm-3) and larger rv (13.29 µm) than the default homogeneous simulation (35.78 cm-3 for Nc and 13.27 µm for rv in default). However,
comparing new to default, the relative changes in Nc, rv, and τ are very
small. When the aerosol concentration increases 10-fold
(default_10 and new_10), qc, CWP, and τ increase according to the aerosol indirect
effect (Peng et al., 2002; Wang et al., 2019; Li et al., 2011; Wang et al., 2011). Meanwhile, rv decreases significantly owing to the
larger cloud number concentration. The effects of the new entrainment–mixing
parameterization also increase; for example, the change in Nc increases from -0.70 % (new compared to default) to -2.74 % (new_10 compared to default_10), rv increases from +0.15 % to +0.57 %, and τ increases from -0.38 % to -0.58 %; the reasons for these changes are discussed later.
These small changes are similar to those identified in previous cumulus
studies (Jarecka et al., 2013; Hoffmann et al., 2019).
Stratocumulus case
The stratocumulus case is simulated from 09:00 UTC on 19 April 2009 to 03:00 UTC on 20 April 2009; the first 3 h are set to be spin-up times. We examine the stratocumulus region of the cloud base at ∼ 2.1 km and the cloud top at ∼ 2.3 km (cloud thickness of
∼ 200 m). Figure 4 shows the time series of the
domain-averaged cloud fraction and total downward irradiance at the central
point in the observation and the default experiment from 12:00 to 24:00 UTC.
Similar to the cumulus case, the simulations compare favorably with the
observations, which further reinforces the utility of the LES model. The
observed data show that the cloud fraction increases with time and peaks at
16:00 UTC. The simulated cloud fraction has a value of 1 before 16:00 UTC,
fluctuates from 16:00 to 21:00 UTC, and decreases sharply after 21:00 UTC. This period can be divided into three stages, namely the mature stage,
pre-dissipation stage, and dissipation stage.
Time series of (a) domain-averaged cloud fraction and (b) total downward irradiance at the central point from the observation and the default experiment in the stratocumulus case.
The temporal evolutions of main cloud microphysical and optical
properties in all simulation experiments for the stratocumulus case,
including (a) cloud water mixing ratio (qc) (g kg-1), (b) cloud droplet number concentration (Nc) (cm-3), (c) cloud droplet volume mean radius (rv) (µm), (d) cloud water path (CWP) (g m-2), and (e) cloud optical depth (τ). In (b), Nc values in the experiments default and new are normalized by the maximum cloud droplet number concentration (Ncmax) from default; Nc values in the experiments default_10 and new_10 are normalized by Ncmax from default_10. The four experiments are detailed in Table 1.
As with the cumulus case, the temporal evolutions of the physical properties
(qc, Nc, rv, CWP, and τ) of the clouds are shown in Fig. 5. In contrast to the oscillating changes exhibited by the physical
quantities in the cumulus case (Fig. 3), the physical properties in the
stratocumulus case exhibit a mostly smooth temporal evolution. Furthermore,
default and new exhibit clear distinctions during the early periods, but these
differences decrease during the dissipation stage. This is also the case
with default_10 and new_10.
Summary of the case mean values of the key quantities in all the
simulations of the stratocumulus case, including cloud water mixing ratio
(qc), cloud droplet number concentration (Nc), cloud droplet volume mean radius (rv), cloud water path (CWP), and cloud optical depth (τ). The numbers in and out of the parentheses are the results at the mature and dissipation stages, respectively. The experiments are detailed in Table 1.
To compare the different behaviors of the simulation experiments at
different stages, the results at the mature and dissipation stages are
analyzed in detail. The mean values of the main microphysical and optical
properties of the clouds are summarized in Table 3. As expected, the cloud
microphysical and optical properties at the mature stage are all larger than
those at the dissipation stage. The effects of the new entrainment–mixing
parametrization are also more significant at the mature stage. Compared to
default, the new model results in a 7.36 % smaller Nc, 3.20 % larger rv, and 5.98 % smaller τ during the mature stage. During the dissipation stage, the changes in Nc, rv, and τ are
-4.76 %, +2.12 %, and -2.56 %, respectively. The largest
influence of the new entrainment–mixing parametrization occurs during the
mature stage when the aerosol concentration is 10 times greater. The
differences in Nc, rv, and τ between new_10 and default_10 are -9.69 %, +3.88 %, and -5.85 %, respectively, averaged over
the mature stage. These differences are much larger than those reported by
Hill et al. (2009), who found that assuming extremely
inhomogeneous mixing has a negligible effect on stratocumulus simulations.
Our results also prove the speculation of Hill et al. (2009)
that the mixing process might play an important role when the stratocumulus
is thin (∼ 200 m in this study). Furthermore, implementing the
new entrainment–mixing parameterization has similar effects on cloud
properties to those described by Hoffmann and Feingold (2019),
who used the linear eddy model to represent subgrid-scale turbulent mixing.
Note that stratocumulus clouds occur in most regions around the world and
are important contributors to the surface radiation budget (Wood, 2012; Zheng et al., 2016; Wang et al., 2021; Wang and Feingold, 2009).
Stratocumulus clouds dominate in some regions and occur over 60 % of the
time as vast long-lived sheets, such as the semipermanent subtropical marine stratocumulus sheets (Wood, 2012). In these
regions, a nearly 6 % decrease in τ caused by the new
entrainment–mixing parameterization is expected to have significant effects
on the simulation of regional radiative properties and climate change.
The averaged influences of the new entrainment–mixing parametrization over
all the simulation periods are also examined (Table 4). Quantitatively, the
effect of the new entrainment–mixing parameterization is much greater on
stratocumulus clouds than on cumulus clouds. Compared to default, new has an average change of -6.20 % in Nc, +2.01 % in rv, and -3.23 % in
τ. When the aerosol concentration increases 10-fold, the differences
in Nc, rv, and τ between default_10 and new_10 are -9.00 %, +3.16 %,
and -4.14 %, respectively. These differences are larger than the largest changes in the cumulus case.
Cloud water mixing ratio (qc), cloud droplet number
concentration (Nc), cloud droplet volume mean radius (rv), cloud water path (CWP), and cloud optical depth (τ) in all simulations for the entire lifetime of the stratocumulus case. The experiments are detailed in Table 1.
defaultnewdefault_10new_10qc(g kg-1)0.110.110.130.13Nc (cm-3)29.9828.12223.65203.50rv (µm)9.389.575.065.22CWP (g m-2)30.7829.9242.2243.13τ4.023.8910.399.96Mechanisms of the effects of the new entrainment–mixing parameterization
The different effects of the new entrainment–mixing parameterization on
different types of clouds and even on different stages of stratocumulus
clouds are likely be related to variations in the dominant mixing mechanism.
To confirm this, we calculate the average ψ at all grid points
experiencing evaporation, the proportion of inhomogeneous mixing grid points
to all grid points experiencing evaporation, and the average ψ at the
inhomogeneous mixing grid points in new and new_10 (Table 5) for the cumulus case, as well as the
mature and dissipation stages in the stratocumulus case.
Homogeneous mixing degree (ψ) at all grid points experiencing
evaporation, the proportion of inhomogeneous mixing grid points to all grid
points experiencing evaporation, and ψ at the inhomogeneous mixing
grid points in the experiments new and new_10 (Table 1) for the cumulus (Cu) and
stratocumulus (St) cases. The numbers in and out of the parentheses are the
results at the mature and dissipation stages in the stratocumulus (St) case,
respectively. The experiments are detailed in Table 1.
ψ at all gridsProportion ofψ at the(%)inhomogeneousinhomogeneousmixing gridsmixing grids(%)(%)new (Cu)99.934.5298.62new_10 (Cu)95.3325.1092.96new (St)78.56 (94.68)63.07 (40.61)71.56 (89.33)new_10 (St)68.20 (88.11)97.31 (73.54)65.01 (84.99)
For the cumulus case, simulations exhibit large ψ and a small
proportion of inhomogeneous mixing, indicating that homogeneous mixing is
the dominant entrainment–mixing mechanism (Luo et al., 2020; Lu et
al., 2013). Correspondingly, the influences of the new entrainment–mixing
parameterization on the cloud physical properties are not significant, as
shown in Fig. 3 and Table 2. The new_10 model exhibits a smaller average ψ and a larger proportion of inhomogeneous mixing than new, which results in
larger changes in cloud physics, as mentioned in Sect. 3.1.
Probability distribution functions (PDFs) of (a) turbulence dissipation rate (ε), (b) cloud droplet volume mean radius (rv), (c) evaporation timescale (τevap), and (d) transition scale number (NL) of cloud grids experiencing the entrainment–mixing process in the simulations with the new entrainment–mixing parameterization for the cumulus case (Cu, the solid lines) and the stratocumulus case (St, the dashed lines), respectively. The experiments are detailed in Table 1.
For the stratocumulus case, Table 5 shows the average ψ at all grid
points experiencing evaporation, the proportion of inhomogeneous mixing grid
points to all grid points experiencing evaporation, and the average ψ at
the inhomogeneous mixing grid points during the two stages. The mature stage
always has a smaller ψ but a larger proportion of inhomogeneous mixing
than the dissipation stage. The inhomogeneous mixing process dominates the
mature stage in new because more than 60 % of the grid points experience inhomogeneous mixing. The inhomogeneous mixing process is more dominant in new_10 because less than 3 % of the cloudy grid points experience a homogeneous mixing process during the mature stage, which explains why new_10 has the largest influence when implementing the new entrainment–mixing
parametrization. Meanwhile, the average ψ in both stages is smaller
than that in the cumulus case for the same simulation configuration. Thus,
the effects of the new entrainment–mixing parameterization are more
significant for stratocumulus than for cumulus clouds, especially at the
mature stage. It is noted that the average ψ and the proportion of
inhomogeneous mixing at the dissipation stage of new in the stratocumulus case are very close to the results of new_10 in the cumulus case. This is because the
cloud fraction decreases sharply during the dissipation stage; the
stratocumulus clouds break up and produce cumulus clouds with small cloud
droplet radii.
The effects of dissipation rate and aerosol concentration on the
entrainment–mixing process
Previous studies have shown the notable effects of the dissipation rate and
aerosol concentration on the entrainment–mixing process. For example,
Luo et al. (2020) changed ε from 10-5 to
10-2 m2 s-3 and noted huge differences in the corresponding
ψ. Small et al. (2013) compared aircraft observations with
different background concentrations and found that higher-pollution flights
tended to slightly more inhomogeneous mixing; Jarecka et al. (2013) also showed various homogeneities of subgrid mixing when aerosol
concentration increases 10-fold. To explain the different behaviors of
different simulations with the new entrainment–mixing parameterization, the
influences of ε and aerosol concentration are examined. Figure 6 shows the probability distribution functions (PDFs) of ε, rv, τevap, and NL for cloud grids experiencing entrainment–mixing processes in new and new_10 for
the cumulus and stratocumulus cases, respectively. The PDFs from the mature
and dissipation stages of the stratocumulus case are shown in Fig. 7.
Probability distribution functions (PDFs) of (a) turbulence
dissipation rate (ε), (b) cloud droplet volume mean radius (rv), (c) evaporation timescale (τevap), and (d) transition scale number (NL) of cloud grids experiencing the entrainment–mixing process at the mature stage from 12:00 to 16:00 UTC (the red lines) and the dissipation stage from 21:00 to 24:00 UTC (the green lines) in new for the stratocumulus case. The experiment is detailed in Table 1.
Dissipation rate
According to Eq. (3), NL is a function of ε3/4; hence, the PDF of ε directly affects NL and further results in different ψ. For the cumulus case, the mean ε of 0.0043 m2 s-3 in new is similar to
those obtained for cumulus clouds in previous studies (e.g.
Lu et al., 2016; Hoffmann et al., 2019). As shown in Fig. 1, cloud grids experience a homogeneous mixing process if NL is larger
than ∼ 105, the limited distribution of NL values
less than 105 in new results in a very small number of cloud grid points
undergoing an inhomogeneous mixing process. Even at the cloud grid points that
undergo inhomogeneous mixing, the average ψ is large (98.62 %)
because most of the NL values are larger than 103. Therefore, the cloud properties in new are close to those in default.
For the stratocumulus case, the mean value of ε (2.9 × 10-4 m2 s-3 in new) is an order of magnitude less than those in the cumulus case. Therefore, compared with the cumulus case, NL is reduced in the stratocumulus case, while the peak value of new almost reaches the criterion of inhomogeneous mixing (∼ 105). For the two stages of stratocumulus clouds, ε is an order of magnitude smaller, but
rv was larger (Fig. 7) during the mature stage than during the
dissipation stage. According to Eq. (5a), droplets with smaller
rv are more prone to complete evaporation and have a smaller τevap. The combination of smaller ε and larger rv results in a smaller NL (Eq. 3). This is the reason for the new entrainment–mixing parametrization having more significant effects during the mature stage than during the dissipation stage. In addition, the similarity of the ε and rv values during the dissipation stage of the stratocumulus case in new, compared to the cumulus case in new_10 (Fig. 6a and b),
explains the similar average ψ values of these scenarios and the
proportion of inhomogeneous mixing (Table 5).
Therefore, the distribution of ε has a vital impact on the influence of the new entrainment–mixing parameterization. Smaller values of ε result in the new entrainment–mixing parameterization having a more significant influence. Moreover, the rv in the stratocumulus case is smaller than that in the
cumulus case, which is also conducive to a more inhomogeneous mixing
process. These are the reasons why the implementation of the new
entrainment–mixing parameterization has a larger influence in the
stratocumulus case than in the cumulus case when compared to a homogeneous
mixing mechanism.
Aerosol concentration
The aerosol concentration affects the entrainment–mixing process by
decreasing the cloud droplet radius. As rv decreases, the distributions of τevap in new_10 move to a smaller overall value, while the mean value is an order of magnitude smaller than that in new, which causes a much smaller NL because NL is proportional to τevap3/2
(Eqs. 3a and 3b). The larger percentage of smaller NL values
indicates that in new_10, more grid points undergo an inhomogeneous mixing process, and the proportion of such grid points is much larger than in the new model (Table 5). Therefore, compared to new, new_10 exhibits a smaller ψ, and the
effects of the new entrainment–mixing parameterization on cloud properties
are more obvious for both the cumulus and stratocumulus cases.
Verification by the simulations with a different parameterization using entrained air relative humidity
In the above simulations, the new entrainment–mixing parameterization is
based on the grid mean RH. This section serves to verify these simulations
using the entrainment–mixing parameterization proposed by Luo et al. (2020),
ψ=107.96exp-0.95NL-0.35,
which was developed using the entrained air RH in the EMPM. This
parameterization needs the entrained air RH within each grid in WRF, which
is estimated following Grabowski (2007) and Jarecka et al. (2009).
Briefly, assuming that RH mixes linearly when the dry air entrains into the
cloud, then entrained air RH can be simply calculated by
RHentrained=RHgrid-(1-f)RHcloudf,
where the subscripts entrained, cloud, and grid indicate the RH of the entrained, cloudy, and
grid point air, respectively. In Eq. (10), although the cloudy air RH
is approximately 100 % and grid mean RH is predicted in the model, the
entrained air fraction f needs to be further parameterized. To obtain f at 100 m, a parameterization of f is developed based on the simulations for both the cumulus and stratocumulus cases with a higher resolution of 10 m; the other configurations are the same as those in the experiment default. The 10 m resolution
simulation results are then averaged to the resolution of 100 m. Following
Xu and Randall (1996), “1-f” can be fitted by the function
1-f=RHgridγ1-exp-βqc,
where γ and β are empirical parameters. Figure 8 shows that
the parameterization with γ=8.72 and β=1.47× 104 can reproduce well the simulated values of “1-f”, with the correlation coefficient (R) of 0.89 and significant level p value < 0.01. Considering that local shear (dw/dz) and buoyancy (B) may drive
turbulence generation and entrainment for a microscale process, the two
quantities are also used to fit “1-f” except for RHgrid and
qc. However, the addition of dw/dz and B to Eq. (11) does not increase R. Therefore, using RHgrid and qc to parametrize “1-f” is good
and reasonable for a microscale process.
The fitted 1-f as a function of the calculated 1-f. The fitted 1-f is obtained by the fitting functions with grid mean relative humidity
(RHgrid) and cloud water mixing ratio (qc). The black line denotes the 1 : 1 line.
The temporal evolutions of main cloud microphysical and optical
properties in all simulation experiments for the cumulus case, including (a) cloud water mixing ratio (qc) (g kg-1), (b) cloud droplet number concentration (Nc) (cm-3), (c) cloud droplet volume mean radius (rv) (µm), (d) cloud water path (CWP) (g m-2), and (e) cloud optical depth (τ). In (b), Nc values in the experiments default and new_f are normalized by the maximum cloud droplet concentration (Ncmax) from default; Nc values in the experiments default_10 and new_f_10 are normalized by Ncmax from default_10; new_f and new_f_10 are the experiments using entrained air relative humidity.
The temporal evolutions of main cloud microphysical and optical
properties in all simulation experiments for the stratocumulus case,
including (a) cloud water mixing ratio (qc) (g kg-1), (b) cloud droplet number concentration (Nc) (cm-3), (c) cloud droplet volume mean radius (rv) (µm), (d) cloud water path (CWP) (g m-2), and (e) cloud optical depth (τ). In (b), Nc values in the experiments default and new are normalized by the maximum cloud droplet number concentration (Ncmax) from default; Nc values in the experiments default_10 and new_10 are normalized by Ncmax from default_10; new_f and new_f_10 are the experiments using entrained air relative humidity.
Cloud water mixing ratio (qc), cloud droplet number
concentration (Nc), cloud droplet volume mean radius (rv), cloud water path (CWP), and cloud optical depth (τ) in new_f and new_f_10 for the cumulus
(Cu) and stratocumulus (Sc) cases. The results of new and new_10 in Tables 2 and 4 are shown in the parentheses.
Equations (9)–(11) are applied in the simulations for both the cumulus and
stratocumulus cases with different aerosol background (hereafter
new_f and new_f_10). The same as for Figs. 3 and 5, the temporal evolutions of the cloud physical properties (qc, Nc, rv, CWP, and τ) in default, default_10, new_f, and new_f_10 are shown in Figs. 9 and 10. The results are similar to Figs. 3 and 5. The mean values of these properties of new_f and new_f_10 for the cumulus and stratocumulus cases are also shown in Table 6, and the results of new and
new_10 are also shown in the parentheses for the convenience of comparison. The results of new_f and new are very similar, with the maximum difference being no more
than 1 %, and so are the results of 10-fold aerosol background. Such a close
agreement suggests that the results of the new entrainment–mixing
parametrization with grid mean RH are reliable.
It is worth noting that instead of parameterizing f, Jarecka et al. (2009, 2013) added an equation to predict
f for each grid. In principle, this is a good choice if this method is
available in models. Our method shown here is an alternative way to
represent the entrainment–mixing process when the prognostic f is not available.
Concluding remarks
The entrainment–mixing process near cloud edges has important effects on
cloud microphysics, but the most commonly used microphysics schemes simply
assume one extreme mechanism, that is, homogeneous entrainment–mixing. This
study first improves the entrainment–mixing parameterization proposed by
Luo et al. (2020), which connects the homogeneous mixing degree and
transition scale number to estimate the homogeneity of the subgrid mixing
process and its impact on the droplet number concentration. The improved
parameterization uses grid mean relative humidity and can be implemented
directly into microphysics schemes; there is no need to know the relative
humidity of the entrained air. Second, the modified entrainment–mixing
parameterization is implemented in the two-moment Thompson aerosol-aware
scheme of the LES version of WRF-Solar to examine its effects on the
microphysical and optical properties of cumulus and stratocumulus clouds.
Third, several sensitivity experiments are conducted to investigate the
effects of the new entrainment–mixing parameterization under different
conditions of turbulence dissipation rate and aerosol number concentration.
The results of implementing the new entrainment–mixing parameterization are
finally verified by the results using entrained air properties.
Unlike the commonly assumed homogeneous mixing scenario, the new
entrainment–mixing parameterization produces a smaller cloud droplet number
concentration and larger cloud droplet radius, with the degree of difference
depending on cloud types and stages. Sensitivity tests show that in the
cumulus case, the largest average influence of the new entrainment–mixing
parameterization occurs under a high aerosol background but results in only
a 2.74 % decrease in cloud droplet number concentration and a 0.57 %
increase in cloud droplet volume mean radius. The changes become even
smaller with a low aerosol background because of the larger cloud droplet
radius. In contrast, the new entrainment–mixing parameterization has a
larger influence on the microphysical and optical properties of
stratocumulus clouds, especially under a high aerosol background and during
the mature stage, with a cloud fraction equal to 1. The largest changes
resulting from the new entrainment–mixing parameterization are -9.69 %,
+3.88 %, and -5.85 % for cloud number concentration, cloud droplet volume mean radius, and cloud optical depth, respectively. The new
entrainment–mixing parameterization has less of an influence on the
dissipation stage than on the mature stage of the stratocumulus case but
affects this case more than the cumulus case.
The varying effects of the new entrainment–mixing parameterization are
caused by variations in the dominant entrainment–mixing mechanism between
different cloud types and stages. Compared to the cumulus case, the
stratocumulus case has a much smaller homogeneous mixing degree and a larger
proportion of inhomogeneous mixing grid points, especially during the mature
stage, which indicates that the inhomogeneous mixing mechanism dominates in
the stratocumulus case, while the homogeneous mixing mechanism dominates in
the cumulus case. As mentioned above, the changes in physical properties of
stratocumulus clouds in the dissipation stage are between those in the
mature stage and those of the cumulus case; this is because stratocumulus
clouds dissipate sharply to form small cumulus clouds, and the degree of
homogeneous mixing during the dissipation stage is therefore between that
which occurs during the mature stage and the cumulus case.
Sensitivity studies show that turbulence dissipation rate and aerosol
concentration can have notable effects on the subgrid homogeneity of the
mixing process. A larger dissipation rate can accelerate the mixing process,
which results in a larger transition scale number and homogeneous mixing
degree and, therefore, a mostly homogenous mixing mechanism. This is why the
cumulus case exhibits smaller changes than the stratocumulus case after the
new entrainment–mixing parameterization is implemented. Larger aerosol
number concentrations cause a smaller cloud droplet radius. Smaller droplets
evaporate more easily, which leads to a smaller transition scale number and
a smaller homogeneous mixing degree. Thus, the entrainment–mixing mechanism
tends to be inhomogeneous. Therefore, a larger aerosol number concentration
increases the influence of the new entrainment–mixing parameterization in
both the cumulus and stratocumulus cases.
The influences of implementing the new entrainment–mixing parameterization
with grid mean relative humidity have been verified by simulations with
entrained air properties. The entrained air properties are obtained and
calculated from simulations with a finer resolution (10 m). Sensitivity
tests show similar cloud microphysical and optical properties in the two
different methods, which suggests that the new entrainment–mixing
parameterization with grid mean relative humidity is convincing.
Note that the new entrainment–mixing parameterization could be more
important in the models if the relative humidity near the cloud is more
accurately simulated because numerical diffusion may spuriously humidify
the entrained air (Hoffmann and Feingold, 2019). The
artificially increased relative humidity limits the influences of the new
entrainment–mixing parameterization because homogeneous and inhomogeneous
entrainment–mixing processes are close to each other under conditions of
high relative humidity.
Data availability
The large-scale forcing data used in this study can be downloaded from Atmospheric Radiation Measurement (ARM) user facility (10.5439/1647300, Tao and Xie, 2004; 10.5439/1647174, Tao and Xie, 2012). The LASSO data can be downloaded from 10.5439/1342961 (Gustafson et al., 2017).
Author contributions
XX, CL, and YL designed the experiments. XX carried out
the experiments and conducted the data analysis with contributions from all
coauthors. XX, CL, XZ, and SE developed the model code. XX prepared the
paper with help from CL, YL, YW, SL, and LZ.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
This research is supported by the National Natural Science
Foundation of China (41822504, 42175099, 42027804, 41975181, 42075073). Yangang Liu, Xin Zhou, and Satoshi Endo are supported by the U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under Solar Energy Technologies Office (SETO) award number 33504, as well as the Office of Science Biological and Environmental Research program as part of the Atmospheric Systems
Research (ASR) program. Brookhaven National Laboratory is operated by
Battelle for the U.S. Department of Energy under contract DE-SC00112704.
Financial support
This research has been supported by the National Natural Science Foundation of China (grant nos. 41822504, 42175099, 42027804, 41975181, and 42075073) and the U.S. Department of Energy (Solar Energy Technologies Office (SETO), award number 33504; Atmospheric System Research (ASR) program, grant no. DE-SC00112704).
Review statement
This paper was edited by Hailong Wang and reviewed by two anonymous referees.
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