1. Introduction: Microphysical parameters determining the radiative properties of cloud ensembles
Small, warm cumulus clouds (Cu) in the boundary layer (BL) play an important role in the atmospheric radiation and moisture budgets (Trenberth 2011; Stephens et al. 2012). These clouds are frequent over both the oceans and continents (Norris 1998) and are responsible for the largest uncertainty in tropical cloud feedbacks in climate models (Bony and Dufresne 2005). Hence, an accurate calculation of radiative properties of such clouds is of crucial importance for weather prediction and simulation of global circulation and climate, including climatic changes.
The dynamical and microphysical properties of small maritime Cu were measured in situ in several field experiments: the Small Cumulus Microphysics Study (SCMS) (Gerber 2000), the Barbados Oceanographic and Meteorological Experiment (BOMEX) (Holland and Rasmusson 1973; Siebesma et al. 2003); Rain in Cumulus over the Ocean (RICO) (Gerber et al. 2008; Arabas et al. 2009); and Cloud, Aerosol, Radiation and Turbulence in the Trade Wind Regime over Barbados (CARRIBA) (Katzwinkel et al. 2014; Schmeissner et al. 2015).
Small Cu over the land were investigated in the Gulf of Mexico Atmospheric Composition and Climate Study (GoMACCS) (Jiang et al. 2008; Lu et al. 2008) and the Routine Atmospheric Radiation Measurement Aerial Facility Clouds with Low Optical Water Depths Optical Radiative Observations (RACORO) field campaign (Lu et al. 2014).
Comparison of cloud properties in clean air over ocean and in polluted air over land allows for investigation of the effects of aerosols on dynamics, geometrical structure of cloud fields, and cloud microphysics. Aerosols act as cloud condensation nuclei (CCN), on which droplets can form. Polluted clouds have initially smaller and more numerous droplets (Squires 1958; Squires and Twomey 1960; Warner and Twomey 1967; Twomey 1977). Cloud microphysical and dynamical processes are coupled, making the cloud system complex. Changes in the initial droplet size distribution (DSD) (driven by changes in the aerosol number concentration) affect processes like condensation efficiency (Pinsky et al. 2013; Seiki and Nakajima 2014; Dagan et al. 2015), latent heat fluxes and vertical velocities (Pinsky and Khain 2002; Pinsky et al. 2013; 2014; Koren et al. 2014; Dagan et al. 2015), the ability of the droplets to move with the ambient air (Koren et al. 2015), collision–coalescence (Shaw 2003; Benmoshe et al. 2012), sedimentation, and rain production (see reviews in Rosenfeld et al. 2008; Khain 2009; Levin and Cotton 2009; Tao et al. 2012; Khain et al. 2015). In the early stages of the cloud development in polluted air, more and smaller droplets provide more surface area for condensation than in clouds developing in clean air. At these stages, the enhanced condensation yields more latent heat release and enhancement of the updrafts. Moreover, the smaller droplets have better mobility and, therefore, will be pushed higher in the atmosphere by the updrafts, all of which form the theoretical basis for warm cloud invigoration (Koren et al. 2014). Convective invigoration was observed in warm clouds (Kaufman et al. 2005; Yuan et al. 2011; Koren et al. 2005, 2014). At the same time, faster evaporation increases turbulence on the cloud edge and, therefore, the entrainment of dry ambient air into the cloud, which reduces the cloud fraction (CF) (Xue and Feingold 2006; Small et al. 2009; Dagan et al. 2017).
The radiative properties of warm clouds depend on their extent and optical properties. Therefore, changes in the aerosol properties imply changes in the clouds’ radiative effects. The reflectance of polluted clouds is likely to increase (Twomey 1974, 1977). Moreover, delay in the onset of drizzle implies changes in the shallow clouds’ liquid water path, optical depth, lifetime, and coverage (Albrecht 1989; Jiang et al. 2006; Small et al. 2009). When considering aerosol-induced feedbacks in the cloud field scale, the picture becomes even more complicated as the aerosol properties affect the way by which clouds change the environmental thermodynamics (Khain 2009; Heiblum et al. 2016 a,b; Dagan et al. 2016).
Large-scale models in which small, warm clouds are subgrid phenomena require a parameterization of small Cu properties, including the microphysical and dynamical response of the Cu fields to changes in aerosol properties in the BL.
This study shows that despite the high complexity of microphysical processes, vertical profiles of many basic quantities may be parameterized using results of LES.
There are three basic parameters that are used in large-scale models for characterizing radiation and microphysical properties of small clouds: CF, liquid water content (LWC), and effective radius 
In polluted clouds, the effective radius grows with height slower than in clean clouds. This aerosol dependence allowed Rosenfeld et al. (2014) to use 
If the concept of “a robust vertical profile” of 
Formation of raindrops makes the problem of determination of robust 
In section 2 we describe the theoretical background concerning calculation of basic microphysical parameters in cloud-resolving numerical weather prediction (NWP) models and the role of large-eddy simulation (LES) utilization. Model setup is presented in section 3. The LES results are given in section 4. In section 5 we suggest a new parameterization of microphysical parameters of cloud ensembles. Section 6 is dedicated to comparison of results of calculations with in situ observations, as well as to analysis of the sensitivity of results (and parameterization) to thermodynamic conditions and model grid spacing. The paper ends with discussion and conclusions.
2. Theoretical background
a. Calculation of shallow cumulus LWC and re in NWP models
Large-scale models with grid spacing of tens of kilometers do not resolve clouds, and hence, they use convective parameterization schemes. The values of effective radius and LWC are prescribed in such models, which inevitably leads to errors in calculation of related quantities. The treatment of clouds in mesoscale cloud-resolving models (CRMs) with grid spacing of a few kilometers is quite complicated because only the largest (grid scale) clouds are resolved in such models. Small clouds and especially shallow warm Cu, with a typical size smaller or similar to the grid spacing of CRM, remain subgrid phenomena.
In CRM, calculations of effective drop radii at resolvable scales are performed when the relative humidity (RH) in the grid point exceeds 100%. When the grid spacing is larger than the size of shallow Cu, the conditions for small-cloud formation can be suitable even when the calculated mean grid RH is lower than 100%. These clouds may, however, affect the radiative budget substantially, and the knowledge of 
In the limited-area NWP, 
b. Utilization of LES for simulation of small Cu and parameterization goals
An efficient method to simulate fields of small Cu and investigate Cu dynamics and microphysics, or to develop new parameterizations, is LES. In LES clouds are simulated explicitly using a high model resolution that varies from 250 m (Abel and Shipway 2007) to, in rare cases, 10–25 m (Matheou et al. 2011; Dawe and Austin 2012; Seifert et al. 2015). Typical grid spacing used for simulation of small Cu is 100 m (Siebesma et al. 2003; Jiang et al. 2008; Heiblum et al. 2016 a,b; Dagan et al. 2016). Taking into account sharp changes of microphysical values in the vertical direction, the vertical grid spacing is usually smaller (typically 40 m) than in the horizontal.
To simulate cloud microphysical processes in such models, two main microphysical methods are used: bulk parameterization and bin microphysics [see a review by Khain et al. (2015)]. The bulk schemes solve microphysical equations for a few moments of the DSD, assuming a specific DSD shape.
LESs using bulk parameterization microphysical schemes typically focus on investigating effects of thermodynamical and microphysical parameters on the Cu fields, cloud fraction, surface fluxes and mass fluxes, and precipitation (e.g., Siebesma et al. 2003; Abel and Shipway 2007). Seifert and Heus (2013) and Seifert et al. (2015) used the University of California, Los Angeles, large-eddy simulation (UCLA-LES) model (Stevens et al. 1999) with a two-moment bulk parameterization scheme to investigate the spatial organization of precipitating trade wind cumulus clouds and the evolution of cloud size distributions under different aerosol concentration conditions. They stressed substantial effect of environment air humidity on the structure of simulated cloud fields.
In bulk schemes the characteristic radius of drops in particular grid points is calculated using the predicted values of LWC and predicted or diagnosed droplet concentration Nd. The serious difficulties arising in such calculations using one-, two-, and three-moment bulk parameterization schemes are discussed by Milbrandt and Yau (2005) and Milbrandt and McTaggart-Cowan (2010).
The more detailed microphysical approach is the bin microphysics, which solves the microphysical equations for determining the DSDs. This method is expensive computationally, which hinders its applicability in large-scale CRM.
In LES simulations the bin microphysical approach is often used for investigation of thermodynamic factors and aerosols on the parameters of droplet size distributions, drizzle formation, and cloud dynamics (e.g., Jiang et al. 2008). Such parameters as LWC, droplet concentration, and effective radius are determined directly from the calculated DSD.
Small cumulus cloud fields were simulated by Zhang et al. (2011) and Heiblum et al. (2016a,b) using the System for Atmospheric Modeling (SAM) (Khairoutdinov and Randall 2003). Zhang et al. (2011) analyzed the simulated vertical profiles of effective radius (using the SAM with microphysical method of moments; Tzivion et al. 1987; Reisin et al. 1996) and confirmed the validity of the hypothesis made by Rosenfeld and Lensky (1998) concerning the possibility to retrieve re(z) using the values of effective radius measured remotely (from satellites) in tops of developing Cu. Using the SAM with spectral bin microphysics (SBM) described by Khain et al. (2004, 2013), Heiblum et al. (2016a,b) found substantial dependence of center of gravity of cloud mass on aerosol concentrations.
In addition to bin microphysics LES models, a new type of model, sometimes referred to as Lagrangian cloud model (LCM), has been described in studies by Andrejczuk et al. (2008), Shima et al. (2009), and Riechelmann et al. (2012). In this approach, the motion of a great number of individual droplets within a flow field generated in LES is calculated. Turbulent mixing in such models is treated explicitly, without any parameterization. This potentially powerful approach is currently in the development stage, and further efforts are required to properly take into account processes of collisions, droplet nucleation, and the formation of raindrops.
In addition to simulation of cloud fields, LESs are used to develop parameterization schemes for the large-scale models. Using statistical analysis of LES results (with bin microphysics), Khairoutdinov and Kogan (1999, 2000) and Kogan (2013) developed a parameterization of autoconversion and accretion rates in drizzling stratocumulus clouds as functions of droplet concentration and LWC. Kogan and Kogan (2001) derived parameterization formulas for drop effective radius in stratocumulus clouds. Wong and Ovchinnikov (2017) used LES for parameterization of subgrid-scale fluxes caused by deep convection.
In the present study, we use LES results (the model details are presented below) to investigate the behavior of microphysical properties of nonprecipitating and slightly precipitating shallow Cu fields under different aerosol conditions. The goal of the study is twofold. First, we will check the variability of effective radius in clouds of different sizes and the robustness of utilization of one mean vertical profile of 
3. Model setup
LESs were performed using the SAM (Khairoutdinov and Randall 2003) with SBM (Khain et al. 2004, 2013; for details, see http://rossby.msrc.sunysb.edu/~marat/SAM.html). SAM is a nonhydrostatic, inelastic model with cyclic boundary conditions in the horizontal direction. The SBM is based on solving kinetic equations for size distribution functions of water drops and aerosol particles (APs). Aerosols and droplet size distribution functions are defined on the doubling mass grids containing 33 bins. The drops radii range between 2 μm and 3.2 mm. The size of APs serving as CCN ranges between 0.005 and 2 μm. Using the values of supersaturation with respect to water, the critical CCN radius is calculated (using the Köhler theory) and the APs larger than the critical size are nucleated to droplets as described by Khain et al. (2000). The rest of the APs are advected with air motion. Diffusional growth and evaporation of droplets are calculated based on the changes in the supersaturation during a model time step. The time steps are chosen to be smaller than drop relaxation time. This is the physical condition of accurate calculation of diffusion growth/evaporation. Collision–coalescence is solved by the stochastic collision equation using the accurate method of Bott (1998). The collision kernels were calculated using an exact method described by Pinsky et al. (2001). Drop sedimentation is calculated using fall velocities determined by Beard (1976).
Fields of small trade cumulus clouds were simulated within a domain of 12.8 km × 12.8 km × 5.1 km using a horizontal resolution of 100 m and a vertical resolution of 40 m with a dynamical time step of 1 s. As a case study, small Cu observed during BOMEX were chosen for simulation (Siebesma et al. 2003). The vertical profiles of temperature and dewpoint used in simulations are shown in Fig. 1. The original BOMEX profile is that with the inversion at 1500 m; the other two profiles with the inversion bases at 1000 and 2000 m were used for analyzing the effect of the inversion height on the microphysical properties of simulated cloud fields.
Initial vertical profiles of temperature (solid lines) and dewpoint (dot–dashed) used in simulations: low inversion at 1000 m (black), high inversion at 2000 m (blue), and middle inversion at 1500 m (red) (the original BOMEX profile). Green and cyan profiles denote sensitivity tests with decreased relative humidity (by 5% and 10%, respectively) with respect to high inversion profile.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Note that the vertical profiles of temperature and dewpoint shown in Fig. 1 are not specific only to the BOMEX. Similar profiles can be observed both over ocean and over land at any time during warm seasons (Garratt 1992; Jiang and Feingold 2006).
Following Jaenicke (1988), Altaratz et al. (2008), and Ghan et al. (2011), size distribution of aerosols is designed as sum of three modes of lognormal distributions describing fine, accumulated, and coarse aerosols. According to Ghan et al. (2011), typical total concentration of condensational nuclei (CN) is about 400 cm−3, with the main fraction of this amount in the fine mode. In the simulations we used concentrations of CN within the range from 500 to 5000 cm−3, which lead to mean droplet concentrations at cloud base from ~50 to ~500 cm−3. According to the accepted definition, this range of droplet concentrations includes three cloud types: clean (maritime), intermediate (clean continental), and polluted (continental) (Ghan et al. 2011). Note that comparatively small mean cloud droplet concentration at cloud base at such CN concentration is determined by low vertical velocities at cloud base and by prescribed initial CN size distributions containing large fraction of small CN, which cannot be activated at such vertical velocities. We do not consider clouds developing in the extremely clean atmosphere, and producing drizzle immediately after their formation.
We will refer to the simulations according to the CN concentration and inversion height: H for high, M for medium, and L for low, for example, E5000H, where E denotes “experiment” and 5000 shows CCN concentration. The list of simulations is presented in Table 1.
List of simulations. Total CF is time-averaged total cloud fraction, STD of total CF is standard deviation of total CF, 〈Nd〉 is time- and space-averaged Nd, 〈CWC〉 max is maximum (over height) value of the mean profile of CWC, and 〈re〉 max is maximum (over height) value of the mean profile of re.
In addition to simulations listed in Table 1, three supplemental simulations were performed. The simulation E2000H-50 is similar to E2000H but has horizontal grid spacing of 50 m. It was performed to analyze effects of model resolution on cloud thermodynamics and microphysics. Two other simulations, E2000H-RH1 and E2000H-RH2, have air relative humidity within the layer from the surface to the inversion base by 5% and 10% lower than in E2000H, respectively. The profiles of Td(z) in these simulations are shown in Fig. 1. These simulations aim to investigate effects of environment humidity on cloud properties and to check universality of the parameterization proposed (see below) under different thermodynamic characteristics of the BL.
Convection is triggered by random temperature perturbations at the beginning of the simulations. For isolating the aerosol effect on the thermodynamic conditions, the radiative effects (as included in the large-scale forcing) as well as the surface fluxes were prescribed in all simulations (see Dagan et al. 2016 for details). The surface fluxes and the large-scale forcing of BOMEX have been used for this study following Siebesma et al. (2003). All simulations were performed for 8.3 h.
4. Results
a. Time dependencies and mean profiles
We will first examine the time evolution of the simulations in order to select the period for analysis. We have chosen the time periods when the cloud fields were quasi stationary; that is, the statistics and mean microphysical properties of the clouds do not change significantly with time. Figure 2 shows time dependence of total CF (determined by the area where liquid water path is positive, that is, calculated as a ratio of the area of the projection of the cloud on the surface to the computational area) for simulations with the inversion-base height of 1500 m.
Time evolution of the cloud fraction in simulations with 1500-m inversion-base height for different CN concentrations.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Figure 2 shows that during the time period 2–6 h the changes in total CF are not significant. The moderate decrease in CF at t > 6 h can be attributed to thermodynamical effects of clouds on the temperature and humidity profiles in the boundary layer (Dagan et al. 2017). Hence, our analysis will be performed for the period between 2 and 6 h of each simulation. It allows us to relate microphysical and dynamical characteristics of the cloud fields to the initial aerosol and thermodynamic conditions. Note that the CF increases with the decrease in CCN concentration. This behavior can be explained by more efficient evaporation of cloud droplets in polluted clouds, because of smaller droplet sizes than in cleaner clouds (Xue and Feingold 2006).
To give an idea of how the cloud field simulated by SAM looks, which can simplify further interpretations of the results, we present Fig. 3 showing 3D snapshots of the LWC, Nd, and re in E5000H. In the particular case rainwater content is negligible, so LWC is equal to CWC, determined by cloud droplets with radii below 25 μm. This is the approximate maximum drop radius that can be reached by diffusion growth and nonintense collisions between small cloud droplets (Pinsky and Khain 2002).
A snapshot of (a) CWC, (b) Nd, and (c) 
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
The cloud field consists of clouds of different sizes and shapes. The smallest clouds rapidly evaporate by mixing with surrounding air, while the few largest clouds penetrate the inversion layer. The largest clouds contain cores with high CWC. As expected, the CWC increases with height, reaching its maximum near the cloud top, while Nd is nearly constant with height. Hence, drop size, and in particular, the effective radius 
Vertical profiles of cloud-averaged cloud water content 
The vertical profiles of (a) cloud water content 
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
One can see that cloud base is at about 500 m. In Fig. 4 the values of CWC, droplet concentration, and effective radii were plotted at cloudy grid points with CWC > 0.01 g m−3, and 
The cloud-top heights are determined mainly by the location of inversion layer. Above the inversion base there is intense “cloud sorting” when smaller clouds rapidly lose their buoyancy and dissipate, so only the air in the largest clouds continues ascending. The largest clouds penetrate inversion layers by up to 500 m. With the exception of E500H and E1000H, 
Cloud-averaged droplet concentration 
Figure 5 presents the vertical profiles of the time-averaged CF in different simulations. CF(z) is calculated at each model level as the ratio of number of grid points with LWC > 10−5 g m−3 to total number of grid points at the model level. Figure 5a shows CF calculated using total LWC, including cloud and raindrops. Figure 5b shows CF determined by cloud droplets only.
Vertical profiles of mean CF in the different simulations determined by (a) total LWC and (b) cloud droplets only.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Analysis of Fig. 5 shows the following features of the CF:
CF reaches its maximum at about 200–300 m above cloud base in all simulations. Above this level, CF decreases because of evaporation of the smallest clouds, as clearly seen in Fig. 3. Droplet evaporation in these smallest clouds makes them negatively buoyant, which prevents their further growth. The strength of entrainment, which determines the maximum height of clouds, is maximum in smallest clouds because smaller clouds entrain more strongly due to their larger surface to volume ratio (e.g., De Rooy et al. 2013). Similar behavior of CF with height was reported in LESs by Siebesma et al. (2003) and observations (Nuijens et al. 2015).
In the presence of inversion layer, the cloud-top heights are determined largely by the height of the inversion base and do not depend significantly on the CN concentration.
Raindrops increase cloud cover significantly only in E500H, where raindrops increase CF above z = 1000 m by more than 2 times. In simulations with CN = 500 cm−3 CF is maximum. This result can be attributed to the fact that at low droplet concentrations, the droplets are larger and do not evaporate in the course of mixing with dry surrounding air as fast as small cloud droplets in polluted clouds. This effect was discussed by Dagan et al. (2017), among others. The decrease in CF with the increase in the CN concentration is a specific feature of small Cu. In case of deep convective clouds, the CF increases with the increase in the CN concentration (Khain et al. 2010; Fan et al. 2018).
Figure 5 shows that the CF changes significantly with height. Note that in the current operational COSMO version, the CF estimated for small (subgrid) Cu cloud field is assumed to be independent of height and proportional to the depth of the cloudy layer. Moreover, in current NWP models, including the current COSMO scheme, CF is not sensitive to aerosol loadings. This clearly contradicts the accurate LES results. The recently implemented COSMO parameterization of shallow convection assumes the CF to be proportional to the mass fluxes in the cloud cores (Böing et al. 2012).
The goal of this study is to parameterize cloud-averaged profiles shown in Fig. 4 for calculation of the radiative fluxes in cloud-resolving models. Not all cloud layers contribute to the radiative fluxes equally. Two simplifications will be made for the following parameterizations. First, within the inversion layer, the total mass of liquid water is small because of a dramatic decrease of CF with height (Fig. 5). Accordingly, we will simplify the parameterization by neglecting the contribution of liquid water within the inversion layer. Second, the formation of raindrops involves consideration of complicated microphysical processes of collisions, which hinders development of the parameterization. Accordingly, for developing the parameterization, it is of importance to evaluate the contribution of the upper parts of clouds as well as raindrops to radiative cloud properties.
Expression (1) allows calculating the vertical profile of optical thickness of a narrow cloud column within which 
In real clouds, microphysical quantities of clouds are spatially inhomogeneous. Using Eq. (1), one can calculate “effective” optical thickness 
Vertical profiles of “effective” optical depth in different simulations calculated using Eq. (1) for (a) cloud droplets and (b) raindrops. Thin dotted lines at optical depths lower than ~0.01 in (a) show the upper parts of cloud layers where contribution to the total optical depth is less than 5%.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Thin lines in Fig. 6a show that the upper parts of cloud layers with contributions to the total optical depth are below 5%. One can see that the parts of clouds penetrating the inversion layers contribute to the total optical thickness only slightly. This result will be taken into account when deriving parameterization expressions for profiles of cloud-averaged quantities. Note that the optical depth for cloud droplets is only slightly sensitive to aerosol concentration (Fig. 6a) due to low CF. The effect of aerosols is seen better in Fig. 6b because aerosols affect rain formation. Figure 6 shows also that contribution of raindrops to optical thickness is much lower than that of cloud droplets. This small effect is explained by large effective radius of raindrops [see Eq. (1)] and low CF covered by them.
b. Variability of LWC and droplet concentration
Figure 7 shows vertical changes of the CWC (Figs. 7a,c,e) and RWC (Figs. 7b,d,f) occurrence in the E500H, E2000H, and E5000H simulations. The cases with high inversion are chosen because clouds in these cases are more developed and RWC is larger than in cases with lower inversion. The color scale shows the number of cloudy grid points with specific CWC or RWC plotted in logarithmic scale. Note first that significant RWC appears only in E500H above 2000 m. At each level on Figs. 7a, 7c, and 7e, we find the grid points (in space and time) where the CWC is maximal (black curves). Then in Figs. 7b, 7d, and 7f, RWC is calculated in these specific grid points (denoted also by black curves). Thus, the grid points of maximal CWC generally coincide with the maximal RWC.
Frequency by altitude diagrams of the (a),(c),(e) CWC and (b),(d),(f) RWC occurrence in the E500H, E2000H, and E5000H simulations, respectively. The color scale shows the number of cloudy grid points with specific CWC or RWC plotted in logarithmic scale. Black lines show maximum values of CWC that were found in the cloud interior. Brown lines denote the modeled approximation to adiabatic LWC. Blue lines show the vertical profiles of cloud-averaged values.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Brown lines denote adiabatic LWC, which represents parcels rising adiabatically from the lifting condensation level (LCL), assuming conversion of all water vapor in excess of saturation to liquid water and neglecting collisions and drop settling. The LCL was determined by empirical formula relating the LCL to the surface temperature and humidity (Bolton 1980). Simulations show that this formula determines the LCL level close to that of cloud base calculated in LES. We note several points in analysis of Fig. 7. First, Fig. 7 shows high variability of LWC per horizontal level. At each level LWC varies from zero at cloud edges to the value relatively close to adiabatic value, LWCad. This variability is caused by the competition between condensation and evaporation in different locations in the clouds that are exposed to different intensities of entrainment and mixing with the drier outside environment. Second, they show that there exist slightly diluted cloud volumes in the lowest 1000 m above cloud base. The fraction of such slightly diluted cloud volumes (grid points) with high adiabatic fraction (AF = LWC/LWCad) is very small and decreases with height, as can be seen by the logarithmic scale of the plots. Note that the existence of slightly diluted cloud volumes with sizes supposedly smaller than those resolved in the simulations at such distances from cloud base was reported in high-frequency in situ measurements (see Fig. 2 in Gerber 2000). The possible reasons of appearance of such volumes in 100-m resolution simulations are discussed in section 6 below. Deviations of the maximum LWC from the adiabatic value LWCad increase with height. Substantial deviations of LWC from LWCad take place within the inversion layer. Note that the existence of weakly diluted cloud volumes was reported in 1-Hz frequency in situ measurements in developing convective clouds with maximum AF values of 0.93 at distances from 1 km to 3.4 km above cloud base (Prabha et al. 2011; Khain et al. 2013). In E500H the maximum values of sum of CWC and RWC are comparatively close to adiabatic values. Slightly diluted cloud volumes were also simulated in 100-m-resolution LES by Xue and Feingold (2006), Zhang et al. (2011), and Dawe and Austin (2012).
Figure 8 presents frequency by altitude diagrams of droplet concentration for the same simulations discussed in Fig. 7. The maximum droplet concentrations in these simulations are of about 100, 420, and 900 cm−3, respectively. The color reflects the number of cloudy grid points with specific Nd in logarithmic scale. Black lines show droplet concentration in cloud interior in points where CWC was maximum. One can see that zones of CWC maximum are also zones of maximum droplet concentration. In E2000H and E5000H the maximum droplet concentration within the layer of 1000–1500 m above cloud base is nearly constant with height, as could be expected in ascending adiabatic volumes. Some growth of maximum droplet concentration within the few hundred meters above cloud base is caused by in-cloud nucleation. The decrease in the droplet concentration maximum (i.e., deviation from “adiabatic” values) is related to two mechanisms: accretion by raindrops (the most pronounced in E500H) and the effects of mixing–dilution. The first mechanism is dominating in cloud interior.
Height–droplet concentration occurrence (frequency) diagrams for the same simulations discussed in Fig. 7. The color reflects the number of cloudy grid points with specific Nd in logarithmic scale. Black lines show droplet concentration in cloud cores in points where CWC is maximal. Blue lines show the vertical profiles of cloud-averaged values; brown lines show the modeled approximation to adiabatic values. The green dots show the re,ad = 12-μm level.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Similarly to LWC, Nd changes strongly in the horizontal direction (per height level). The dispersion of the concentration near cloud base can be attributed to fluctuations of the vertical velocities at cloud base as well as by fluctuations of LCL. The decrease in the droplet concentration values is related to the mixing process, when concentration changes from zero at cloud edge to its maximum in cloud cores.
c. Volume mean and effective droplet radii in cloud ensemble
Volume mean 
Scattering diagram rυ vs re obtained in simulations E500H, E2000H, and E5000H. The color reflects the number of cloudy grid points with specific rυ and re in logarithmic scale. Black line denotes the linear fit. Black dashed lines show 1:1 relation.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Freud and Rosenfeld (2012) found that in tops of nonprecipitating developing deep convective clouds k ≈ 1.08. A similar relationship was numerically found by Benmoshe et al. (2012) for deep convective clouds using the Hebrew University Cloud Model (HUCM). Measurements in stratocumulus clouds show that k varies between 1.14 and 1.21 (Reid et al. 1999; Martin et al. 1994). Using numerical results of LES of nondrizzling stratocumulus clouds, Kogan and Kogan (2001) and Magaritz-Ronen et al. (2016b) found that the effective radius is about 10% larger than the mean volume radius that agrees well with the observations.
According to our simulations k ≈ 1.15–1.17. Thus, our LES simulations support these observational and numerical results, showing a general character of such relationship. The lowest values of coefficient k take place for polluted clouds in which DSD are the narrowest. Accordingly, the largest values of k take place for clean clouds, where DSD width is largest, and the existence of large cloud droplets increases 
Main properties of effective radius in cloud field of small Cu can be derived from Fig. 10 showing height versus re scattering diagrams for selected simulations with different inversion layer heights and different aerosol concentrations. These simulations reflect the basic properties of effective radius.
The height–re scattering diagrams for simulations with different CN concentrations. Cases with (left) low (base at 1000 m), (center) middle (base at 1500 m), and (right) high (base at 2000 m) inversion. The color scale reflects the number of cloudy grid points with specific 
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
Because of some uncertainties in evaluation of 
An analysis of Fig. 10 shows several important features of effective radius in clouds forming in cloud fields, which are described next.
1) Strong effect of aerosols
A well-known effect is the faster effective radius growth with height in the case of low CCN concentrations. In the case of clean environment, Nd is small, and therefore the fewer droplets that form near the cloud base have little competition on the available supersaturation 
2) Low horizontal variability of effective radius
A fundamental property of effective radius is low horizontal variability. This variability is much lower than that of LWC as seen in Fig. 7. The variability of effective radius is relatively large near cloud base. This can be attributed to the fact that 
Low variability of 
3) The maximum effective radius is in cloud interior
The important feature is that the effective radius is maximal in cloud interior (black lines), where LWC are maximum. In cloud cores the profiles of 
4) Limiting value in case of raindrop formation
As seen in Fig. 10, 
The physical reasons for low variability of effective radius were analyzed by Pinsky et al. (2016b) and Pinsky and Khain (2018a) using a semi-analytical diffusion-mixing model and by Magaritz-Ronen et al. (2016b), who used a Lagrangian–Eulerian model of Sc with a precise description of microphysical processes of diffusion growth and turbulent mixing. Relative humidity in cloudy volumes, even well-diluted ones, remains high, which does not allow effective radius to decrease significantly anyhow. A small decrease in 
5. Parameterization of microphysical parameters of cloud ensembles
Vertical profiles of the cloud-averaged effective radius calculated directly from LES and parameterized using Eqs. (5)–(7) are shown in Fig. 11.
Vertical profiles of the cloud-averaged effective radius calculated directly from LES (solid lines) and parameterized using Eqs. (5)–(7) (dashed lines). Each panel is plotted for a different inversion base height: (left) low, (center) medium, and (right) high.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
One can see a good agreement of parameterized 
The parameters in Eqs. (7)–(9) are obtained minimizing the root-mean-square over all the simulations. Equations (8) and (9) allow one to calculate the averaged concentration profiles using the droplet concentration at cloud base and the height of the level where 
Figure 12 shows profiles of cloud-averaged droplet concentration obtained in LES and using Eqs. (8) and (9). One can see a good agreement between the approximations and profiles directly calculated using LES.
Profiles of cloud-averaged droplet concentration obtained in LES (solid lines) and using Eqs. (8) and (9) (dashed lines). Each panel is plotted for different inversion base height: (left) low, (center) medium, and (right) high.
Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1
