## Abstract

Cloud microphysics parameterizations for shallow cumulus clouds are analyzed based on Lagrangian cloud model (LCM) data, focusing on autoconversion and accretion. The autoconversion and accretion rates, *A* and *C*, respectively, are calculated directly by capturing the moment of the conversion of individual Lagrangian droplets from cloud droplets to raindrops, and it results in the reproduction of the formulas of *A* and *C* for the first time. Comparison with various parameterizations reveals the closest agreement with Tripoli and Cotton, such as and , where and are the mixing ratio and the number concentration of cloud droplets, is the mixing ratio of raindrops, is the threshold volume radius, and *H* is the Heaviside function. Furthermore, it is found that increases linearly with the dissipation rate and the standard deviation of radius and that decreases rapidly with while disappearing at > 3.5 *μ*m. The LCM also reveals that and increase with time during the period of autoconversion, which helps to suppress the early precipitation by reducing *A* with smaller and larger in the initial stage. Finally, is found to be affected by the accumulated collisional growth, which determines the drop size distribution.

## 1. Introduction

Warm cloud microphysical parameterizations usually divide the droplet spectrum within a cloud into cloud droplets and raindrops by size and calculates their physical quantities separately, following Kessler (1969, hereafter K69). Cloud droplets with small terminal velocity are assumed to remain within a cloud, and larger raindrops with appreciable terminal velocities are assumed to settle gravitationally, causing precipitation. The value of a separation radius between cloud droplets and raindrops is in the range of 20–50 *μ*m.

The mass transfer from cloud water to rainwater plays a critical role in the cloud microphysics parameterization, and it is divided into autoconversion, which results from the coalescence of cloud droplets, and accretion, which results from the coalescence of cloud droplets and raindrops. Autoconversion and accretion rates, *A* and *C*, respectively, can be thus expressed (Beheng and Doms 1986) as

and

where *n*(*x*) is the number concentration of drops with mass between *x* and *x* + *dx*, , *K* is the collection kernel, and is the density of water. A collision event that does not change the category of the involved droplets is called self-collection.

Numerous parameterizations have been suggested for autoconversion. One of the most widely used parameterizations is the so-called Kessler-type parameterization, originally proposed by K69 as

where is the cloud water mixing ratio and *H* is the Heaviside step function. The proportional constant and the threshold value are used typically as = 10^{−3} s^{−1} and = 5 × 10^{−4}.

A more comprehensive expression was proposed by Manton and Cotton (1977) and Tripoli and Cotton (1980, hereafter TC80), which can be written as

with the empirical constant . Here, the mean volume radius *R* is used to determine the threshold condition instead of . The parameterization (4) can be obtained from (1) by assuming that , , and , based on the collection kernel *K* derived by Long (1974) and the terminal velocity of a droplet at small *R* by the Stokes law, where is the cloud droplet number concentration and *E* is the collection efficiency. TC80 suggested = 38.56 cm^{−1} s^{−1} by assuming *E* = 0.55. They also suggested = 10 *μ*m, but a smaller value is often used (Wood 2005). Liou and Ou (1989), Baker (1993), and Liu and Daum (2004) also suggested modified versions of the Kessler-type parameterization. Other functional forms of *A* that do not use the Kessler-type parameterization were also proposed (Berry and Reinhardt 1974; Beheng 1994, hereafter B94; Khairoutdinov and Kogan 2000, hereafter KK00; Seifert and Beheng 2001).

Meanwhile, various evidence suggests that autoconversion is also influenced by various other factors besides and , and attempts have been made to incorporate these factors into account. Seifert et al. (2010), Franklin (2008), and Seifert and Onishi (2016) attempted to include the effect of the turbulence-induced collection enhancement (TICE), that is, a larger *K* under the influence of turbulence compared to gravitational collisions. Berry and Reinhardt (1974), B94, Liu and Daum (2004), and Milbrandt and Yau (2005) considered the effect of the dispersion of the drop size distribution (DSD), which induces larger *K* by increasing the vertical velocity difference between two droplets. Meanwhile, Cotton and Anthes (1989) pointed out that the “aging period” is necessary to commence autoconversion in order to avoid the early production of rainwater too low in the cloud. Straka and Rasmussen (1997) attempted to include its effect in the parameterization. Similarly, Seifert and Beheng (2001) considered the internal time scale in their parameterization.

Accretion is usually parameterized by considering cloud droplets within a cylindrical volume swept out by a gravitationally settling raindrop while assuming a raindrop size distribution. The accretion rate *C* depends on raindrop mixing ratio as well as and is usually represented in the form as

Autoconversion rates vary much more between schemes than accretion rates, often causing a difference by several orders of magnitude for the same (Menon et al. 2003; Wood 2005; Hsieh et al. 2009). The contribution of accretion to total precipitation is much larger than that of autoconversion in general. Nonetheless, autoconversion still plays a critical role, because it generates initial raindrops required for accretion and subsequent precipitation. Accordingly, the proper parameterization of autoconversion still remains a key issue in cloud microphysics parameterization.

Considering the difficulty of obtaining reliable observation data, one valuable approach to evaluate cloud microphysics parameterizations is to analyze the results from a model that can simulate the variation of droplet spectrum directly, such as a spectral-bin model (SBM), which solves the stochastic collection equation (SCE). The results of the SBM initialized with observed DSD data (Wood 2005; Hsieh et al. 2009) or with the idealized DSD (Seifert and Beheng 2001; Franklin 2008; Lee and Baik 2017) were used to evaluate parameterizations of *A* and *C*. Meanwhile, KK00 and Kogan (2013) developed a formula for *A* and *C* from regression analysis of SBM data, when a stratocumulus or cumulus cloud is simulated by large-eddy simulation (LES). LES has an advantage of providing the dynamically balanced DSD within the fine structure of the cloud, which plays an important role in the calculation of *A* and *C* from (1) and (2) (Kogan 2013). Evaluations have been carried out usually by the comparison of *A* and *C* calculated from the SBM and the parameterization. However, the comparison can be affected by factors that are not represented in the parameterization, such as DSD, TICE, and aging time.

An Eulerian model, such as the SBM, calculates only the averaged values of *A* and *C* over the grid size and the time step. Moreover, the numerical diffusion of the droplet spectrum, in both physical and spectral space, can hinder the accurate calculation of *A* and *C*. Therefore, probably the ideal approach to calculate *A* and *C* is to capture the moment of each Lagrangian droplet growing to a raindrop together with the background condition, as suggested by Straka (2009). Nonetheless, it is possible only when cloud droplets are simulated by Lagrangian particles.

Recently, several groups developed Lagrangian cloud models (LCMs), in which the cloud microphysics of Lagrangian droplets and cloud dynamics are two-way coupled (e.g., Andrejczuk et al. 2010; Shima et al. 2009; Sölch and Kärcher 2010; Riechelmann et al. 2012; Hoffmann et al. 2017). In these models, the flow field is simulated by LES, and the droplets are treated as Lagrangian particles, which undergo cloud microphysics while interacting with the surrounding air.

Hoffmann et al. (2017) applied the LCM to clarify the mechanism of raindrop formation in a shallow cumulus cloud. They found that the rapid collisional growth, leading to raindrop formation, is triggered when droplets with a radius of 20 *μ*m appear in the region near the cloud top that is characterized by large liquid water content, strong turbulence, large mean droplet size, a broad DSD, and high supersaturations. They also found that the rapid collisional growth leading to precipitation can be delayed without the broadening of the DSD, when turbulence is weak. On the other hand, TICE does not accelerate the triggering of the rapid collisional growth, but it enhances the collisional growth rate greatly after the triggering and thus results in faster and stronger precipitation. These results imply that both TICE and the dispersion of DSD are important factors to determine autoconversion and accretion.

The present paper aims to investigate the characteristics of the parameterizations of autoconversion and accretion by analyzing LCM data. For this purpose, we first compare *A* and *C* from the existing parameterizations with LCM data. At the next step, we investigate the effects of various other factors, such as the dispersion of the DSD, TICE, and aging time and parameterize their effects with an aim to improve the parameterization.

## 2. Simulation and analysis

### a. Model description

The LCM in this study is coupled to the Parallelized Large-Eddy Simulation Model (PALM; Raasch and Schröter 2001; Maronga et al. 2015). To handle an extremely large number of droplets in a cloud, the concept of a superdroplet is introduced. Each superdroplet represents a large number of real droplets of identical features (e.g., their radius). The number of real droplets belonging to a superdroplet of radius is called the “weighting factor” , and the total mass of a superdroplet is then calculated by

In the present model, differs for each superdroplet and changes with time as a result of collision and coalescence. The liquid water mixing ratio for a given grid box of volume is then calculated by

where is the density of dry air and is the number of superdroplets in an LES grid box.

The velocity of each superdroplet is determined by

where is the LES resolved-scale velocity at the particle’s location and is a stochastic turbulent velocity component , computed in accordance with the LES subgrid-scale model (Sölch and Kärcher 2010). The terminal velocity follows Rogers et al. (1993).

The diffusional growth of each superdroplet is calculated from

where *S* is the supersaturation; and are the thermodynamic terms associated with heat conduction and vapor diffusion, respectively; and represents the ventilation effect. Their functional forms follow Rogers and Yau (1989).

The temporal change of due to condensation/evaporation is then calculated as

and it determines the sink/source for potential temperature and water vapor mixing ratio *q* in the LES model.

To calculate the droplet growth by collision–coalescence, a statistical approach is used in which the growth of a superdroplet is calculated from the droplet spectrum resulting from all superdroplets currently located in the same grid box. The collisional growth is described in terms of the modification of and , which can be summarized as

assuming that the particles are sorted that for . Here, the collection of a superdroplet pair with is realized by the collection of droplets of the superdroplet *m* by the superdroplet *n*. It results in the decrease of but no change of , thus leading to the decrease of [represented by the second terms in the rhs of (11) and (12)], and the increase of but no change of , thus leading to the increase of [represented by the first term in the rhs of (12)]. The first term on the rhs of (11) describes the decrease of due to internal collections of droplets within a superdroplet. If in the probabilistic binary function , where is a random number uniformly chosen from the interval [0, 1], the collection takes place . No collection takes place if ; is necessary to realize the stochastic collisional growth (Telford 1955). Small perturbation is given to the initial weighting factor of each superdroplet to help initiate the collision process. One can refer to Hoffmann et al. (2017) for the detailed explanation of the collision scheme. Unterstrasser et al. (2017) examined the performance of the present collision algorithm under various conditions, while comparing with analytical and SBM results, and confirmed that it can reproduce the realistic evolution of cloud droplet spectrum.

### b. Simulation setup

The simulation setup is the same as in Hoffmann et al. (2017). A shallow cumulus cloud is triggered by a two-dimensional rising bubble of warm air, which is homogeneous in the *x* direction. The bubble is prescribed by an initial potential temperature difference given by

where = 1920 m and = 150 m mark the center of the bubble, = 200 m and = 170 m the radius of the bubble, and = 0.4 K, the maximum temperature difference. The model domain is 1920 m × 5760 m × 3840 m along the *x*, *y*, and *z* directions with an isotropic grid spacing of 20 m. Periodic boundary conditions are applied laterally, and Dirichlet and Neumann boundary conditions are applied at the bottom and top, respectively. The initial profiles of and *q* are derived from the LES intercomparison of shallow cumulus convection by vanZanten et al. (2011; Fig. 1 in Hoffmann et al. 2017). They represent the average thermodynamic state of a cumulus-topped boundary layer, as observed during the Rain in Cumulus over the Ocean (RICO) field campaign (Rauber et al. 2007). No background winds, no large-scale forcings, and no surface fluxes are applied.

The average distance between superdroplets is initially 3.4 m, yielding a total number of 7.9 × 10^{8} superdroplets and about 200 superdroplets per grid box, which has been found to be sufficient to represent the collisional growth correctly (Riechelmann et al. 2012; Arabas and Shima 2013; Unterstrasser et al. 2017). Two different initial droplet number concentrations = 70 and 150 cm^{−3} are simulated by using = 2.8 × 10^{9} and 6.0 × 10^{9}. The radius of all superdroplets is initially given by *r* = 0.01 *μ*m, and the particles are not allowed to evaporate any smaller. A time step of = 0.2 s is used in both LCM and LES.

Two simulations are carried out for each with different collection kernel *K*, which either considers only gravitational collision and coalescence (Hall 1980) or includes also the effect of TICE (Ayala et al. 2008; Wang and Grabowski 2009). In the latter case, TICE is parameterized as a function of the dissipation rate *ε*, which is calculated from the subgrid-scale model of LES. These simulations are called GRAV and TURB, respectively.

### c. Calculation of autoconversion and accretion

First, we detect collision events during the time step ; that is, *P* = 1 in (11) and (12). The increased mass of a superdroplet *n* after a collision with other superdroplet *m *, , is calculated for these droplets by

Every collision event is assigned to autoconversion, accretion, and self-collection, depending on the radii and before collision, and the radius after collision (Table 1). The case of accretion with and is possible in principle but negligible, because mostly occurs with after the initial period. The consequent mass transfer from cloud droplets to raindrops after a collision event is then calculated for autoconversion and accretion; that is, autoconversion is calculated by , and accretion is calculated by for and for . The autoconversion and accretion rates at each grid box, and , respectively, can be obtained by adding up the contribution from every collision event belonging to the corresponding category of collision within a grid box per unit time. Only a very small fraction of superdroplets experience collision during (=0.2 s) in the simulation.

Here, the critical radius that separates a cloud droplet and a raindrop is given by = 25 *μ*m. It is the same used by KK00 for shallow clouds. Larger values about 40–50 *μ*m are often used for deep clouds (Berry and Reinhardt 1974; Seifert and Beheng 2001). Hoffmann et al. (2017) showed that the collisional growth, which generate autoconversion and accretion, starts as the droplet size reaches *r* = 20 *μ*m. It is therefore desirable to choose that is slightly larger than 20 *μ*m, considering that the collection of larger droplets should be characterized as accretion. Sensitivity of the results to is examined in the next section.

Since most autoconversion parameterizations are expressed as a function of , we calculate by the following formula:

where is the number of grid boxes with , using bins of a logarithmic width of within the cloud from the data obtained at every time step over the whole period of cloud evolution. The cloud is defined as the region where > 1.0 × 10^{−5} kg kg^{−1}.

Similarly, we calculate the accretion rate *C* as a function of , as adopted in most formulas (TC80; B94; KK00); that is,

where and is the number of grids with within a cloud. The bin width is .

It should be mentioned that the calculations of *A* and *C* from the LCM and the SBM are somewhat different in nature. First, *A* and *C* are calculated by the integral of SCE within a grid in the SBM, but they are calculated at every collision event of Lagrangian droplets in the LCM. It also implies that they are affected by the growth history of Lagrangian droplets in the LCM. Second, the occurrence of autoconversion and accretion is continuous and deterministic in the SBM, but it is intermittent and stochastic in the LCM. Accordingly, the values of and are zero in a large number of grids in the LCM, contrary to the SBM.

## 3. Results

### a. Distribution of autoconversion and accretion

Figure 1 shows the distributions of autoconversion, accretion, , and , averaged in the *x* direction, during the evolution of a cumulus cloud (*t* = 20 and 25 min).

Autoconversion is larger than accretion initially (*t* = 20 min), but accretion soon dominates the conversion to raindrops (*t* = 25 min). It also reveals that both autoconversion and accretion appear in the upper part of the cloud initially (*t* = 20 min), but they appear in the center in the later stage (*t* = 25 min). It reflects the fact that raindrop formation is triggered near the cloud top that is characterized by strong turbulence and a broad DSD (Hoffmann et al. 2017).

The dominance of autoconversion soon after the triggering of raindrop formation is clearly illustrated in the time series of the total amount of autoconversion and accretion per unit time within the cloud (Fig. 2a). As a result of autoconversion and accretion, decreases and increases (Fig. 2b). Ultimately, they disappear with time by precipitation and the dilution of the cloud. Both the time series of autoconversion and accretion and their distributions within a cloud are in agreement with previous results (Wood 2005; Franklin 2008).

Figure 2 also shows that both autoconversion and accretion are smaller in GRAV, although they start to appear at about the same time. It reflects the fact that TICE does not accelerate the timing of the raindrop formation, but it increases the amount of precipitation (Hoffmann et al. 2017). Seifert et al. (2010) also showed, using an SBM, that precipitation increases about 2 times, as increases from 0 to 100 cm^{2} s^{−3}, when = 100 cm^{−3}.

### b. Comparison of A and C with parameterizations

Figure 3 shows the variation of *A* with from LCM results with different (=70 and 150 cm^{−3}) and collection kernels (GRAV, TURB). The frequency distribution is also shown for reference; *A* is calculated only in the range where the number of grid boxes with , , is sufficiently large ( > 5 × 10^{2}), since the frequency of collision events during is very low. Autoconversion parameterizations by K69, TC80, B94, and KK00 are compared with LCM results, similar to Wood (2005) and Hsieh et al. (2009). Table 2 presents autoconversion and accretion formulations for the four parameterizations examined. In all schemes, we use for . The decreases by less than 20% during autoconversion (*t* < 25 min).

Remarkably, the results reproduce successfully the Kessler-type autoconversion parameterization, such as (3) and (4), in which the threshold exists, and *A* increases with . It reveals that autoconversion does not occur in a large volume of regions with small within a cloud (Fig. 3). We should mention that the relation has never been directly obtained so far. Previous works compared *A* from the parameterizations and SBMs (KK00; Seifert and Beheng 2001; Wood 2005; Franklin 2008; Hsieh et al. 2009; Kogan 2013; Lee and Baik 2017).

The closest agreement in the relation is found with TC80; that is, = 7/3, although the values of and in (4) are different. The value of is certainly larger than = 1 (K69) and smaller than = 3 (Liu and Daum 2004) or = 4.7 (B94). A better agreement with TC80 is found for in TURB and in smaller , although it is always overestimated. It is consistent with previous reports that TC80 overestimates *A* from one to two orders of magnitude in the case of shallow cumulus clouds (Baker 1993; Wood 2005; Hsieh et al. 2009). Figure 3 also reveals many features that are consistent with previous assessments (Wood 2005; Hsieh et al. 2009). For example, B94 overestimates the increasing rate of *A* with , and KK00 underestimates *A* except at low below the threshold value. The threshold value and are overestimated in K69. Considering that all previous comparisons are based on SBM data, the consistency with previous reports suggests the general agreement in the calculations of *A* and *C* from the LCM and the SBM.

Similarly, we examined the variation of *C* with from LCM results with different (=70 and 150 cm^{−3}) and collection kernels (GRAV and TURB; Fig. 4). Once again, the frequency distribution of is displayed for reference, and *C* is calculated only in the range where the number of grids with is sufficiently large ( > 50). Here, we consider only the schemes in which *C* varies with (KK00; TC80; B94). The differences between accretion schemes are much smaller than between autoconversion schemes, similar to previous comparisons (KK00; Wood 2005; Hsieh et al. 2009). All show relatively good agreements with LCM results. Even the proportional constant in matches very well in GRAV, although it is somewhat larger in TURB. Meanwhile, *C* tends to increase slightly faster than for = 150 cm^{−3}.

### c. Influence of other factors on A and C

As discussed in the introduction, various evidence indicates that autoconversion is influenced not only by and but also by various other factors, such as TICE, the dispersion of the DSD, and the aging time since the generation of a cloud.

To clarify the influences of these factors, we replot Fig. 3 based on the subgroup of data according to the values of the dissipation rate , the standard deviation of radius , and , where is the time at which a cloud is generated at the lifting condensation level (LCL; =10 min; Fig. 6). Here, and represent the values in each grid box. If , , and are not sufficiently large, the autoconversion tends to be suppressed, resulting in smaller and larger . It is found that is affected by all variables , , and . On the other hand, is affected only by and and insensitive to .

It is difficult, however, to identify the effects of , , and separately from the LCM results, because all variables vary simultaneously. For this purpose, we performed a large number of simulations of a simple box collision model, as in Hoffmann et al. (2017). Simulations were carried out under different (=0, 200, and 400 cm^{2} s^{−3}), starting with lognormally distributed droplet spectra with different (=40, 70, and 150 cm^{−3}), (=0.5, 1.0, …, 7.0 *μ*m), and (=1, 2, …, 18.0 *μ*m), where is the arithmetic mean radius. The ranges of and in the initial distributions are < 0.2 and 2.7 × 10^{−8} < < 1.47 × 10^{−3}. The collisional growth algorithm is the same as used in the LCM and represented by 200 superdroplets. The calculation of *A* is made only for the first time step ( = 5 s) so that we can assume that all initial variables remain unchanged.

There are at least five variables that can influence autoconversion, such as , , , , and *t*, and it makes it very difficult to identify their effects separately. Therefore, we assume the relation from (4) (TC80), based on Fig. 3. Analysis of data reveals that, when > 3.5 *μ*m, does not vary significantly with , and it never becomes smaller than 1/10 of its value at the largest (=18 *μ*m), as decreases down to 1 *μ*m (not shown). On the other hand, when < 3.5 *μ*m, decreases rapidly with decreasing . In this case, is determined by the radius at which becomes smaller than 1/10 of its value at the largest (=18 *μ*m) for given and . The case with > 3.5 *μ*m is regarded as = 0 *μ*m, that is, no threshold *R*. Finally, is calculated by averaging from the data with > 3.5 *μ*m for given and .

First, we examine how and are affected by . Figure 7 shows that both and are essentially independent of , or equivalently , although they vary widely with and . Note that a large number of data with > 3.5 *μ*m belong to = 0 *μ*m in Fig. 7b. Figure 7 also justifies the assumption of the relation .

The variations of and with and are shown in Figs. 8 and 9 . They show that increases with both and . On the other hand, decreases rapidly with , and the threshold *R* disappears when *σ* ≥ 3.5 *μ*m ( = 0 *μ*m). It also shows that is insensitive to , although it tends to increase slightly for smaller . The increase of with and and the decrease of with are consistent with the dependence on and in Fig. 6.

We can obtain the dependence of on and as

with *a* = 1.0 cm^{−1 }*μ*m^{−1} s^{−1}, *b* = 8.8 × 10^{−3} cm^{−2} s^{3}, and = 1.35 *μ*m. The dependence of on can be expressed as

where = 3.5 *μ*m, *m* = 0.25, and = 34.4 *μ*m. According to (18), = 10 *μ*m, employed by TC80, is expected at , which is the typical value during the initial stage of shallow cumulus clouds (see Fig. 11 below).

The existence of the threshold *R* is attributed to two factors. First, if both *R* and are very small, the collection of two small droplets can never produce a droplet larger than , regardless of or . Second, the rapid collisional growth is triggered when droplets larger than *r* = 20 *μ*m are present (Hoffmann et al. 2017). Therefore, if both *R* and are very small, very few droplets are larger than *r* = 20 *μ*m, and it makes the mean values of *K* very small.

Similar to the case of autoconversion, we replot Fig. 4 based on the data regrouped according to the values of , , and (Fig. 10). It shows that *C* tends to be larger for larger and , but it is rather insensitive to , as expected from the dominance of gravitational collision for large droplets. It suggests that the larger *C* in TURB than in GRAV, shown in Fig. 4, is mainly due to the DSD with larger *R* and rather than the direct effect of TICE. The larger *A* under the influence of TICE produces more raindrops and, consequently, the larger DSD for raindrops. Actually, the mass density distributions of droplets (Fig. 7 in Hoffmann et al. 2017) exhibits larger *R* and in TURB than in GRAV after the collisional growth dominates (*t* = 25 min).

### d. Variations of and

We showed in the previous section that autoconversion varies significantly with and . The information of and is therefore necessary in order to apply the new autoconversion parameterization to a large-scale atmospheric model, such as a numerical weather prediction (NWP) model. However, and are not the variables that are usually predicted in most NWP models. Nonetheless, observational evidence indicates that the magnitudes of and vary widely during the evolution of a cloud and differ depending on the cloud type (Uijlenhoet et al. 2003; Hsieh et al. 2009; Geoffroy et al. 2010; Seifert et al. 2010).

With an aim to provide the information on the evolution of and for shallow cumulus clouds, we investigate how the mean values of and in an entire cloud vary with time (Fig. 11). It shows that both and increase with time after the generation of the cloud at *t* = *t*_{0} (=10 min) at the LCL. After precipitation starts at *t* = 21 min (Fig. 2), decreases rapidly, but continues to increase for a while. The variation of is largely independent of and TICE until the initiation of precipitation, suggesting that they are mainly determined by cloud dynamics, insensitive to cloud microphysics. TICE makes larger after the initiation of precipitation because of the enhanced raindrop formation (Hoffmann et al. 2017). On the other hand, is smaller for larger . Larger suppresses not only the condensational growth of droplets but also the broadening of the DSD, as reported earlier (Thompson et al. 2008; Hudson et al. 2012; Chandrakar et al. 2016).

The aging process is naturally realized by the initial increase of and with *t*, combined with the dependence of and on and . Small values of and in the early stage make small and large and thus suppress autoconversion, as shown in Fig. 6a. It can help avoid the too-early production of rainwater too low in the cloud, which is common in existing parameterizations (Cotton and Anthes 1989).

Another approach to estimate and is to use the information on the known parameters, such as and , if correlation exists between them (e.g., Geoffroy et al. 2010). Figure 12 shows two-dimensional histograms of the frequencies of *ε*–*q*_{c} and *σ*–*q*_{c} for the periods < 10 min and > 10 min. It reveals the negative correlation between and and the positive correlation between and at the late stage ( > 10 min). The positive correlation between and reflects the fact that both and are the largest in the cloud core near the top (e.g., Seifert et al. 2010). On the other hand, entrainment and mixing decrease but increase near the cloud edge, leading to the negative correlation between and . One can refer to the corresponding distributions of , , and in Figs. 2 and 3 in Hoffmann et al. (2017). Figure 12 also reveals that the mean values of and in the late stage are larger than in the early stage, as expected from Fig. 11.

Contrary to the box collision model, in which , , and are independent variables, they can be correlated with each other in the LCM. The correlations can affect the exponent in the relation , because varies with in (4). However, the opposite tendency in the variations of and with (Fig. 12) may make the effects of and weak in the LCM results in the late stage ( > 10 min). As a result, the relation can be maintained in the late stage (Fig. 6a) and also over the whole period (Fig. 3), since the number of data with > 10^{−4} kg kg^{−1} is much larger in the late stage (Fig. 12). If the effects of and are not cancelled out, the relations will not be followed as shown in the cases with small and in Fig. 6.

The previous parameterizations only in terms of and , as shown in Table 2, can be thought to be based on the assumption that the effects of the realistic distributions of and are already included implicitly. It is therefore possible that the different in other parameterizations may reflect the different variations of and with depending on the cloud type. For example, Kogan (2013) found that the optimum is different depending on the cloud type (shallow cumulus clouds vs stratocumulus clouds). Nonetheless, the parameterizations neglecting the effects of and are unlikely to realize the aging effect.

## 4. Conclusions

In the present paper, we applied the LCM to investigate the cloud microphysics parameterization for shallow cumulus clouds, focusing on autoconversion and accretion. Autoconversion and accretion were calculated directly by capturing the moment of the conversion of individual Lagrangian droplets from cloud droplets to raindrops.

The autoconversion rate *A* and the accretion rate *C*, calculated from the LCM, were compared with various parameterizations (K69; TC80; B94; KK00). The calculation produced for the first time the formulas of autoconversion and accretion, such as and . The closest agreement is found with TC80, such as and , although coefficients , , and are different.

Furthermore, LCM results help to clarify how and are affected by the dissipation rate , the standard deviation of radius , and the age of the cloud . The value of is found to increase linearly with and . On the other hand, decreases rapidly with , and it disappears as becomes larger than 3.5 *μ*m. The effects of and on and are parameterized (Table 3). The LCM data also reveal that the values of and increase with time, during which autoconversion contributes significantly to the conversion to raindrops. It helps avoid the early precipitation, which is common in existing cloud microphysics parameterizations, because small and large , resulting from small and , suppress autoconversion. Accretion generally follows the expression well, but tends to be larger than suggested by TC80, especially when TICE is included. The increase of *C* under TICE is due to larger *R* and as a result of accumulated contribution of collisional growth rather than the direct effect of TICE, however.

It is important to mention that (1) and (2) to calculate *A* and *C* are universal, independent of cloud dynamics and nucleation. Cloud dynamics and nucleation affect the variation of turbulence and DSD, and their effects are realized only in terms of the variation of *K* and *n* in (1) and (2) through the variation of and . We obtained the formula for the parameterization of *A*, including the dependence on and , by analyzing a large number of box collision model results with wide ranges of independent variables , , , and . It implies that the formula for *A* with the dependence on and in Table 3 is independent of the cloud type. On the other hand, the temporal evolutions of and in *A* and in *C* may vary depending on the cloud type. If and are correlated with in the real cloud, *A* can modify in the relation because in (4) varies with . It is possible that the different in other parameterizations (Table 3) reflect the different variations of and with under different cloud conditions. In our LCM results of a shallow cumulus cloud, the positive correlation between and and the negative correlation between and tend to cancel out their effects, and the relation is still observed.

We hope that an improved cloud microphysics parameterization, which takes into account the effect of the dispersion of DSD, TICE, and aging time, can be developed in the future based on the information obtained from the present work. It will be necessary for the application of the parameterization, however, to develop a general method to predict the variation of and by using the variables that are calculated in the NWP model, such as , , and . Empirical constants, especially , may need optimization too, which depends not only on the cloud type but also on the resolution and scales of the NWP. The optimal parameterization can be obtained by examining a large number of NWP simulation results. The more realistic simulations also help us to obtain further information on , , and : for example, the inclusion of nucleation process, the inclusion of droplet breakup, cloud field simulations, and simulations under different thermodynamic sounding.

## Acknowledgments

This work was funded by the Korea Meteorological Administration Research and Development Program under Grants KMI 2015-10410 and KMI 2018-07210. This LES/LCM used in this study (revision 1891) is publicly available (https://palm.muk.uni-hannover.de/trac/browser/palm?rev=1891). For analysis, the model has been extended, and additional analysis tools have been developed. The code is available from the authors on request. Most of the simulations have been carried out on the Cray XC-30 systems of the North-German Supercomputing Alliance (HLRN) and the supercomputer system supported by the National Center for Meteorological Supercomputer of Korea Meteorological Administration (KMA).

## REFERENCES

*Storm and Cloud Dynamics.*Academic Press, 883 pp.