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  • View in gallery

    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.

  • View in gallery

    Time evolution of the cloud fraction in simulations with 1500-m inversion-base height for different CN concentrations.

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    A snapshot of (a) CWC, (b) Nd, and (c) fields for E5000H experiment: inversion is at 2 km, the CN concentration is 5000 cm−3, t = 208 min. The impression of “noodle”-shaped clouds and not the usual “cotton ball” shape are just due to the difference between the horizontal and vertical scales.

  • View in gallery

    The vertical profiles of (a) cloud water content averaged over all clouds, (b) rainwater content , (c) droplet concentration , and (d) effective radius in different simulations. , , and are plotted for CWC > 0.01 g m−3. is plotted for RWC > 0.01 g m−3.

  • View in gallery

    Vertical profiles of mean CF in the different simulations determined by (a) total LWC and (b) cloud droplets only.

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    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%.

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    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.

  • View in gallery

    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.

  • View in gallery

    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.

  • View in gallery

    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 in logarithmic scale. Black curves denote effective radii in clouds cores, purple curves denote the modeled approximation to adiabatic profiles, and blue curves denote profiles of horizontally averaged . Black dashed lines denote the inversion-base heights.

  • View in gallery

    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.

  • View in gallery

    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.

  • View in gallery

    Vertical profiles of the cloud-averaged CWC calculated directly from LES (solid lines) and Eq. (11) (dashed lines) for different aerosol loadings and for different inversion levels: (left) low, (center) medium, and (right) high. Cyan solid and dashed lines indicate the profiles obtained using different options within the current parameterization in COSMO.

  • View in gallery

    Vertical profiles of optical depth calculated by Eq. (1) using Eqs. (7) and (11) (dashed lines) and the average values calculated directly in LES (solid lines). Each panel corresponds to a certain inversion level: (left) low, (center) medium, and (right) high.

  • View in gallery

    Comparison of mean effective radius profiles simulated using SAM with observations. Simulated mean re profiles: E5000H (black line), E2000H (blue solid line), E2000H-50 (blue dashed line), and E500H (magenta). Blue empty squares show results of in situ observations during GoMACCS in polluted atmosphere (Jiang et al. 2008, Lu et al. 2008), and black empty circles (dissolving Cu) and black filled circles (developing Cu) show results of in situ observations during CARRIBA (Schmeissner et al. 2015). Green filled squares show results reported by Gerber et al. (2008) obtained in flight RF12 during RICO. Brown–red–yellow color map shows re occurrence frequency from flight RF07 during RICO (Arabas et al. 2009). The extremely low values of re at heights around 1500 m in the color map are likely related to dissipating cloud fragments.

  • View in gallery

    Comparison of mean LWC profile simulated using SAM with observations. Red solid and dashed lines show results of E2000H and E2000H-50, respectively, using 0.01 g m−3 as threshold of minimum LWC. Green solid and dashed lines show results of E2000H and E2000H-50, respectively, but using 0.05 g m−3 as a threshold of minimum LWC. Blue solid line shows the results of E2000L (threshold of 0.01 g m−3). Black dotted line shows results of Warner (1955) taken from Gerber (2000). Black dashed line is calculated by averaging the LWC measured during SCMS (Gerber 2000). Black filled squares show results reported by Gerber et al. (2008). Blue empty squares show results of in situ measurements during GoMACCS (Jiang et al. 2008). Cyan symbols denote values of cloud-averaged LWC obtained during RICO at different aircrafts: crosses (Bac146), diamonds (C-130), squares (King Air). Combined values on 19 Jan 2005 are shown by cyan filled circles (Abel and Shipway 2007).

  • View in gallery

    PDF of 90th percentile of the adiabatic fraction within clouds in (a) CARRIBA measurements (Katzwinkel et al. 2014, their Fig. 3a) and in (b) E2000H simulation around a height of 1500 m. Similarly, a PDF of 90th percentile of vertical velocity is presented for (c) measurements (Katzwinkel et al. 2014, their Fig. 3a) and (d) simulation. The measurements were performed around 100 m below cloud tops, with effective resolution of 20 cm. E2000H simulations results are presented for heights of 1380–1540 m.

  • View in gallery

    Vertical profiles of averaged simulated (solid lines) and parameterized (dashed lines) effective radii. Blue: E2000H; green: E2000H-50; red: E2000H-RH1; and black: E2000H-RH2.

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Parameterization of Vertical Profiles of Governing Microphysical Parameters of Shallow Cumulus Cloud Ensembles Using LES with Bin Microphysics

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  • 1 Israel Meteorological Service, Bet-Dagan, Israel
  • | 2 Weizmann Institute of Science, Rehovot, Israel
  • | 3 Deutscher Wetterdienst, Offenbach am Main, Germany
  • | 4 Hebrew University of Jerusalem, Jerusalem, Israel
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Abstract

Shallow convection is a subgrid process in cloud-resolving models for which their grid box is larger than the size of small cumulus clouds (Cu). At the same time such Cu substantially affect radiation properties and thermodynamic parameters of the low atmosphere. The main microphysical parameters used for calculation of radiative properties of Cu in cloud-resolving models are liquid water content (LWC), effective droplet radius, and cloud fraction (CF). In this study, these parameters of fields of small, warm Cu are calculated using large-eddy simulations (LESs) performed using the System for Atmospheric Modeling (SAM) with spectral bin microphysics. Despite the complexity of microphysical processes, several fundamental properties of Cu were found. First, despite the high variability of LWC and droplet concentration within clouds and between different clouds, the volume mean and effective radii per specific level vary only slightly. Second, the values of effective radius are close to those forming during adiabatic ascent of air parcels from cloud base. These findings allow for characterization of a cloud field by specific vertical profiles of effective radius and of mean liquid water content, which can be calculated using the theoretical profile of adiabatic liquid water content and the droplet concentration at cloud base. Using the results of these LESs, a simple parameterization of cloud-field-averaged vertical profiles of effective radius and of liquid water content is proposed for different aerosol and thermodynamic conditions. These profiles can be used for calculation of radiation properties of Cu fields in large-scale models. The role of adiabatic processes in the formation of microstructure of Cu is discussed.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pavel Khain, pavelkh_il@yahoo.com

Abstract

Shallow convection is a subgrid process in cloud-resolving models for which their grid box is larger than the size of small cumulus clouds (Cu). At the same time such Cu substantially affect radiation properties and thermodynamic parameters of the low atmosphere. The main microphysical parameters used for calculation of radiative properties of Cu in cloud-resolving models are liquid water content (LWC), effective droplet radius, and cloud fraction (CF). In this study, these parameters of fields of small, warm Cu are calculated using large-eddy simulations (LESs) performed using the System for Atmospheric Modeling (SAM) with spectral bin microphysics. Despite the complexity of microphysical processes, several fundamental properties of Cu were found. First, despite the high variability of LWC and droplet concentration within clouds and between different clouds, the volume mean and effective radii per specific level vary only slightly. Second, the values of effective radius are close to those forming during adiabatic ascent of air parcels from cloud base. These findings allow for characterization of a cloud field by specific vertical profiles of effective radius and of mean liquid water content, which can be calculated using the theoretical profile of adiabatic liquid water content and the droplet concentration at cloud base. Using the results of these LESs, a simple parameterization of cloud-field-averaged vertical profiles of effective radius and of liquid water content is proposed for different aerosol and thermodynamic conditions. These profiles can be used for calculation of radiation properties of Cu fields in large-scale models. The role of adiabatic processes in the formation of microstructure of Cu is discussed.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Pavel Khain, pavelkh_il@yahoo.com

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 , which is the ratio between the third and second moments of the DSD (e.g., Twomey 1977; Nakajima and King 1990; Rosenfeld and Lensky 1998; Blahak and Ritter 2013). Effective radius together with LWC allows for evaluation of the integral cross section of droplets in clouds, which is needed for radiation calculations. Observational studies of cloud tops of growing nonprecipitating clouds (Gerber 2000; Gerber et al. 2008; Freud et al. 2008, 2011; Rosenfeld et al. 2008; Prabha et al. 2011; Katzwinkel et al. 2014; Schmeissner et al. 2015) show a low variability of along horizontal levels. The existence of “a robust vertical profile” of re(z) for certain environmental (thermodynamic and aerosol) conditions follows also from observations that the first radar echo appears for different clouds within a cloud field nearly at the same height (Andreae et al. 2004; Freud and Rosenfeld 2012). The same conclusion was reached from numerical simulations with bin microphysics cloud models (Benmoshe et al. 2012; Khain et al. 2013). This may indicate the existence of a mean vertical profile of in a cloud field that can well represent nonprecipitating clouds of different sizes and lifetime stages.

In polluted clouds, the effective radius grows with height slower than in clean clouds. This aerosol dependence allowed Rosenfeld et al. (2014) to use measured from satellites near cloud tops for evaluating the aerosol concentration below cloud base.

If the concept of “a robust vertical profile” of for certain environmental (thermodynamic and aerosol) conditions is valid, it can substantially simplify the problem of parameterization of radiative fluxes from fields of cumulus clouds of different sizes and at different stages of their evolution.

Formation of raindrops makes the problem of determination of robust profile more complicated because precipitation affects vertical profile of effective radius and decreases droplet concentration and cloud water content (CWC). Note, however, that optical properties of Cu cloud field are determined largely by small cloud droplets, and not by raindrops (Wiscombe et al. 1984; Savijarvi et al. 1997; Savijarvi and Räisänen 1998).

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 , LWC, and CF values is necessary for such calculations.

In the limited-area NWP, of unresolved clouds is a tuning parameter. For instance, in the model COSMO (Doms and Schättler 2002; Steppeler et al. 2003; for details see http://cosmo-model.org) re = 5 μm by default. Since on average over the grid box there is subsaturation, the LWC in these shallow cumuli is crudely parameterized as a function of temperature (reduces with a decrease in temperature), mimicking the reduction of available water vapor for condensation with height. Recently, a new cloud radiation scheme was developed for COSMO (Blahak and Ritter 2013) and is currently under evaluation. Among other developments, of unresolved clouds is calculated in this scheme from the ratio of LWC and droplet concentration. While LWC is evaluated using the parameterization described above, the droplet concentration is determined using the assumed aerosol concentration and the effective (unresolved) vertical velocity at cloud base (see section 4c). The large uncertainty in LWC causes to be highly uncertain as well. As a result, such parameterizations may lead to significant errors in evaluation of effects of shallow convection on radiation and on other thermodynamic atmospheric properties.

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 to characterize all the clouds within the field. We explore the reasons for the comparatively low horizontal variability of effective radius despite the high variability of LWC both within each cloud and between different clouds. Second, using the results of the LES, we propose a simple parameterization of the mean vertical profiles of effective radius and LWC that uses adiabatic LWC profile and droplet concentration at cloud base.

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.

Fig. 1.
Fig. 1.

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.

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.

Table 1.

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.

Fig. 2.
Fig. 2.

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).

Fig. 3.
Fig. 3.

A snapshot of (a) CWC, (b) Nd, and (c) fields for E5000H experiment: inversion is at 2 km, the CN concentration is 5000 cm−3, t = 208 min. The impression of “noodle”-shaped clouds and not the usual “cotton ball” shape are just due to the difference between the horizontal and vertical scales.

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 , is increasing with height, as can be seen in Fig. 3c. It is possible to see that changes in the horizontal direction within the range 9–11 μm in the upper parts of clouds; that is, the relative changes of the effective radius are much lower than those of CWC and Nd.

Vertical profiles of cloud-averaged cloud water content , rainwater content , droplet concentration , and effective radius of the cloud droplet mode of drop size distribution (2–25 μm) in representative numerical experiments are shown in Figs. 4a–d, respectively. Cloud-averaged values are calculated as the averaged values of corresponding quantities over all clouds at any given height level.

Fig. 4.
Fig. 4.

The vertical profiles of (a) cloud water content averaged over all clouds, (b) rainwater content , (c) droplet concentration , and (d) effective radius in different simulations. , , and are plotted for CWC > 0.01 g m−3. is plotted for RWC > 0.01 g m−3.

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 was calculated for cloud points with RWC > 0.01 g m−3.

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, increases with height. A strong increase in above the inversion base is a reflection of the cloud sorting. is significant only in E500H (Fig. 4b).

Cloud-averaged droplet concentration (Fig. 4c) remains nearly constant with height in E2000, E3000, and E5000, but decreases in E500 and E1000. The reason of the decrease in and in experiments with the concentration of CN of 500 and 1000 cm−3 can be derived from the analysis of the profiles of and of the cloud-averaged effective radius (Fig. 4d). One can see that in these experiments exceeds 12 μm already 500 m above cloud base. This value can be considered as the threshold value for first drizzle or raindrop formation (vanZanten et al. 2005; Benmoshe et al. 2012; Khain et al. 2013; Magaritz-Ronen et al. 2016a,b). Above this level collisions become intense, which in addition to increased mixing with the environment, leads to decrease in and . The values of do not exceed 20 μm even in precipitating clouds, which agrees with the estimations by Rosenfeld and Lensky (1998), who referred the regime with ~ 20 μm to as washout regime.

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.

Fig. 5.
Fig. 5.

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:

  1. 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).
  2. 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.
  3. 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.

An important characteristic of clouds being used for estimation of their radiation properties in a short wavelength range is optical thickness, which is related to liquid water content and effective radius as (Szczodrak et al. 2001)
e1
where is the water density and Q is the scattering efficiency. The mean scattering efficiency for cloud drops was taken equal to 2 (independent of wavelength).

Expression (1) allows calculating the vertical profile of optical thickness of a narrow cloud column within which and effective radius can be considered as horizontally homogeneous.

In real clouds, microphysical quantities of clouds are spatially inhomogeneous. Using Eq. (1), one can calculate “effective” optical thickness of the entire atmospheric layer with embedded cloud field.

The effective optical thickness that provides the same attenuation of direct radiative flux within the atmospheric layer as the attenuation calculated within each atmospheric column can be calculated as
e2
where ztop is the height of the upper boundary of the layer, I0(x, y, ztop) is the flux at this boundary, and x and y are horizontal coordinates. Assuming that at the upper boundary the radiative short-wavelength flux does not depend on x and y, one can get
e3
where is calculated using Eq. (1). Figure 6 shows a vertical profile of the effective optical depth averaged over a 4-h period (from 2 to 6 h) in different simulations calculated using Eq. (3). Calculations are performed over the entire computational area, so contributions of cloud-free regions were taken into account. Effective optical depth was calculated for cloud droplets (Fig. 6a) and drizzle and raindrops with radii exceeding 25 μm (Fig. 6b). The effective optical thickness under the simulation conditions is small because of low cloud cover. For the goals of the study, it is important that the optical thickness due to raindrops is substantially lower than that due to cloud droplets, even in E500, where RWC is substantial. For higher CN concentrations, contribution of raindrops to the optical depth is negligible. This result agrees well with that reported by Savijarvi et al. (1997). Savijarvi and Räisänen (1998) showed also that contribution of raindrops to the longwave radiation fluxes is also much lower than that of cloud droplets, even under strong rain rates. Moreover, consideration of RWC contribution to the optical depth would complicate the analysis without getting useful information. Therefore, we will focus below on the aerosol effects on CWC and of cloud droplets.
Fig. 6.
Fig. 6.

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.

Fig. 7.
Fig. 7.

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.

Fig. 8.
Fig. 8.

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 and effective droplet radii are among the most important microphysical characteristics of clouds. Two-moment bulk schemes are able to calculate mean volume radius. At the same time, radiation fluxes are calculated using effective radius. Figure 9 shows scattering diagram versus as obtained in simulations E500H, E2000H, and E5000H. The diagrams plotted using results of other simulations are similar to those shown in Fig. 9. The diagrams show that the values of and are well correlated. Note that according to the definition (see appendix). The 1:1 line reflects grid points with very narrow DSD where , since for monodisperse DSDs . One can see that the lower the CN concentration, the larger the maximum values of and . The difference between and increases with the increase in the DSD width. Formation of large cloud droplets in DSD by drop–drop collisions leads to a larger increase in than of . Accordingly, the dispersion of scattering diagram versus increases with the decrease in the droplet (or CN) concentration.

Fig. 9.
Fig. 9.

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

Accordingly, in several studies the mean volume and effective radii are related as
e4

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.

Fig. 10.
Fig. 10.

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 in logarithmic scale. Black curves denote effective radii in clouds cores, purple curves denote the modeled approximation to adiabatic profiles, and blue curves denote profiles of horizontally averaged . Black dashed lines denote the inversion-base heights.

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

Purple lines in Fig. 10 denote the profiles of “adiabatic” effective radius of cloud droplets , which could be observed in adiabatically ascending cloud volume with droplet concentration calculated at cloud base. In Fig. 10 is calculated as
e5
where is adiabatic LWC that can be calculated as described, for example, by Pontikis (1996) or Pinsky et al. (2012). The adiabatic cloud number concentration is the concentration in an adiabatic cloud core. As can be seen from Fig. 7, the cloud cores are adiabatic only at the first few hundreds of meters above the cloud base. Hence, in these levels in the cloud cores should be equal to . We have chosen so that will nearly coincide with in the cloud cores. The values of are shown by solid brown line segments in Fig. 8. The value of can be calculated in large-scale models using, for instance, lookup tables as a function of aerosol concentration and cloud base vertical velocities (Segal and Khain 2006), theoretical formulas (Pinsky et al. 2012), or other approaches (Ghan et al. 2011). For instance, the cloud base vertical velocities in the COSMO model are calculated as a sum of gridscale updraft, vertical component of turbulent fluctuations using turbulent kinetic energy (with isotropy assumption), radiative cooling effect (Khvorostyanov and Curry 1999), and the convective velocity scale (Deardorff 1970). Other NWP models calculate these velocities using the intensity of turbulence in the BL and the BL height (e.g., Zheng and Rosenfeld 2015). In NWP models these parameters are calculated in boundary layer schemes.

Because of some uncertainties in evaluation of , we will refer to as “modeled” adiabatic effective radius. Although the Nd and LWC in cloud cores become smaller with height than and , respectively, still represents well the effective radius in cloud cores. The advantage of is that it can be calculated in any model knowing the LCL (together with the cloud-base temperature and pressure) and the droplet concentration at cloud base.

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 (Pinsky et al. 2012, 2014; Dagan et al. 2016). As a result, remains relatively high and the droplets (and their effective radius) grow rapidly with height. At CN concentration of 5000 cm−3, the maximum of the effective radius remains below 13–15 μm, so these clouds do not produce either raindrops or drizzle. At a CN concentration of 2000 cm−3, the maximum effective radius reaches 14–15 μm at z = 2000 m. So, light drizzle arises above this level (Fig. 7). In E500 the maximum of exceeds 15 μm and raindrops arise (Fig. 7). The reason for the existence of critical or the threshold value of effective radius is discussed in several studies (Freud and Rosenfeld 2012; Pinsky and Khain 2002; Benmoshe et al. 2012; Khain et al. 2013; Magaritz-Ronen et al. 2016b). Long (1974) showed that collision kernel for small droplets is proportional to the sixth power of droplet radius. Freud and Rosenfeld (2012) evaluated that the collision kernel is proportional to . These evaluations show a dramatic increase in the rate of collisions when the largest cloud droplets reach the radii of 20–21 μm, which correspond to the critical value of effective radius mentioned above.

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 rapidly grows above the LCL, so even small local fluctuations of LCL lead to significant fluctuations of effective radius at low levels. Generally, the averaged (over all heights) relative variability varies between 10% and 15%. This result is in agreement with observations in small cumulus clouds (Arabas et al. 2009; Gerber et al. 2008; Schmeissner et al. 2015) and in deep convective clouds (Prabha et al. 2011; Khain et al. 2013). In addition, low variability in deep nonprecipitating clouds under different geographical locations observed in situ was reported previously (Tas et al. 2012, 2015; Liu and Daum 2000; Freud and Rosenfeld 2012; Rosenfeld et al. 2014). Rosenfeld et al. (2016) used the low variability of found in observations to propose a method to determine CCN concentration in the boundary layer. Note that the low variability of in small cumuli should be considered as somehow surprising, since the influence of mixing with environment is more significant than in large deep convective clouds and in a horizontally homogeneous stratocumulus cloud (Sc). Schmeissner et al. (2015) stress that despite the fact that in dissolving small Cu at their dissipating stage LWC and droplet concentration are decreased by about 50% compared to growing Cu, droplet size remains almost constant. Our LESs reproduce these observation findings. Schmeissner et al. (2015) found also that some fraction of DSD are bimodal because of in-cloud nucleation. SAM microphysics obviously takes into account all the mechanisms leading to in-cloud nucleation and bimodal DSD formation. However, as mentioned above, our results show low variability of . We partially attribute this insensitivity to the following. The mean volume radius is proportional to . As the in-cloud nucleation leads to formation of smallest droplets, the LWC is defined by the first mode of droplets activated at cloud base. Because of the 1/3 power, the in-cloud nucleation cannot change Nd strongly enough to affect the mean volume radius (as well as ) significantly.

Low variability of indicates that cloud field can be characterized by robust vertical profile of effective radius that is close to the adiabatic one and depends on the aerosol loading.

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 almost coincide with the adiabatic values (purple lines) calculated according Eq. (5). This fact shows that cloud droplets reach maximum size in cloud cores, together with the maximum LWC and the maximum droplet concentration.

4) Limiting value in case of raindrop formation

As seen in Fig. 10, in cloud core in simulations with CN = 500 cm−3 is maximum in cloud cores and close to the adiabatic value. Formation of raindrops is seen by termination of the growth with height. In case of raindrop formation, effective radius determined within the range of cloud droplet radii (<25 μm) does not exceed about 22 μm and remains height independent (Fig. 10, E500H). Such a regime is known as rainout (Rosenfeld and Lensky 1998). The reason for low dependence of with height in the case of raindrop formation is that raindrops collect cloud droplets of all sizes, which leads to a decrease in droplet concentration, but does not change the effective radius of the cloud droplet mode.

According to the results, in case of rain formation, the maximum of cloud droplets can be calculated as
e6
Because of low variability of , the values of are concentrated around , which is slightly smaller than .

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 reflects the contribution of air volumes newly penetrating clouds and located near cloud edges, as was observed by, for example, Kumar et al. (2017). In such volumes partial evaporation of drops penetrating from cloudy volumes leads to formation of DSD with lower effective radii. Rapid increase of relative humidity in such volumes with time leads to termination of evaporation of the largest droplets penetrating from the cloudy volumes. As a result, the effective radius in the volumes penetrating clouds rapidly reaches values typical of cloudy volumes, while LWC and droplet concentration remain much lower than in the cloud core. This process of rapid increase of until the value is close to that of cloud interior is investigated in detail by Pinsky et al. (2016b), Magaritz-Ronen et al. (2016b), and Pinsky and Khain (2018a). Note that turbulent mixing leads to formation of a humid shell of moist air around the cloud (Heus and Jonker 2008; Lehmann et al. 2009; Bar-Or et al. 2012; Schmeissner et al. 2015; Pinsky and Khain 2018a). The mixing with humid air, which entrains clouds, does not cause strong changes in drop sizes near cloud edge.

5. Parameterization of microphysical parameters of cloud ensembles

For practical goals, it is desirable to characterize cloud field by some mean vertical profile . Analysis of results presented in Fig. 10 shows that is smaller than the maximum value by only ~10%–15%. The difference between and reaches its maximum at the inversion base level. This increase is related to the fact that cloud ensemble contains clouds of different sizes, and clouds of lower width experience stronger effects of mixing because the interface zone affected by mixing in such clouds occupies a significant fraction of the cloud volume. Above the base of inversion the difference decreases again, since only the largest clouds remain. This behavior retards simple approximation of via . We will approximate dependence between these two quantities (as well as other approximation of averaged values), ignoring the upper part of clouds within inversion layer, which contribution to the optical depth of cloud layer is less than 5% (see Fig. 6a). Within the layer below inversion base, the cloud-averaged effective radius can be approximated as
e7
where α(z) =0.95–1.2 × 10−4(zzcb), where z is in meters and where zcb is the cloud base level, determined by the minimal distance from the surface where supersaturation for the first time becomes positive and droplet nucleation takes place. To take into account the effect of raindrops, the values of in Eq. (7) were replaced by calculated using Eq. (6). According to Eq. (7), is closer to than as reported by Gerber et al. (2008). We attribute this to different approaches to define and hence [see comments about Eq. (5)]. At the same time, the values of correspond well to Gerber et al. (2008, their Tables 2 and 3).

Vertical profiles of the cloud-averaged effective radius calculated directly from LES and parameterized using Eqs. (5)(7) are shown in Fig. 11.

Fig. 11.
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 with those calculated directly in the LES within an entire range of aerosol loadings and different levels of the inversion. Standard deviation is of 0.39 μm.

The next step is parameterization of mean droplet concentration as function of droplet concentration at cloud base that, as was mentioned above, can be determined using different approaches in different kinds of atmospheric models, including NWP models. The concentration at cloud base can be referred to as adiabatic concentration , which is being formed at cloud base and does not change with height within ascending adiabatic parcels (in small Cu the changes of air density with height are small). Profiles of depend on whether clouds produce raindrops or not. In the case of nonprecipitating clouds, and are equal to the maximum value , which can be considered as constant with height. Analysis of Fig. 8 shows that the maximum droplet concentration in clouds producing raindrops can be approximated as a constant until the level at which and linearly decreasing above this level. Statistical analysis shows that the profile of maximal droplet concentration can be written in the form
e8
where γ = 0.45 m−1 and z is in meters, and
e9
where on average (over all heights and simulations) with a standard deviation of 0.03.

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 reaches the precipitating threshold.

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.

Fig. 12.
Fig. 12.

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

In addition to , the parameterizations of radiation transfer through unresolved cloudiness in many NWP and climatic models require a prognostic calculation of . As mentioned in the introduction, of unresolved clouds in the COSMO model is currently crudely parameterized as a function of the temperature. Now, using the parameterization of and as mentioned above, we are able to parameterize as well. Strictly speaking, the practically required values of should be determined as
e10
However, since variability of effective radius is low, the last equality can be rewritten as
e11
which is substantially simpler than Eq. (10).

Figure 13 presents the vertical profiles of calculated directly from the results of the LES simulations and the profiles of calculated using Eq. (11).

Fig. 13.
Fig. 13.

Vertical profiles of the cloud-averaged CWC calculated directly from LES (solid lines) and Eq. (11) (dashed lines) for different aerosol loadings and for different inversion levels: (left) low, (center) medium, and (right) high. Cyan solid and dashed lines indicate the profiles obtained using different options within the current parameterization in COSMO.

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

One can see that Eq. (11) allows reproducing the LES profiles of quite accurately, especially in comparison with COSMO approximations (cyan solid and dashed lines). Deviations of CWC calculated using Eq. (11) from that calculated directly in LES seen at higher levels are not of high importance for calculation of radiation fluxes because CF of these clouds at such altitudes are low (see Fig. 5).

To evaluate the errors in calculation of optical depth using the approximation formulas given in Eqs. (7) and (11) instead of the averaged values calculated in LES, we present Fig. 14. The figure shows vertical profiles of the optical depth calculated by Eq. (1) using the approximation expressions and the average values calculated directly in LES. One can see a very good agreement in the approximation and directly calculated values.

Fig. 14.
Fig. 14.

Vertical profiles of optical depth calculated by Eq. (1) using Eqs. (7) and (11) (dashed lines) and the average values calculated directly in LES (solid lines). Each panel corresponds to a certain inversion level: (left) low, (center) medium, and (right) high.

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

We would like to stress that calculation of optical depth of cloud layer requires the utilization of a more complicated procedure than just application of Eq. (1), but a good agreement of vertical profiles of optical depth seen in Fig. 14 indicates that approximated formulas are accurate enough to be applied in such calculations.

6. Comparison with observations and sensitivity analysis

Our parameterization aims to reproduce first of all the average vertical profiles of effective radius and of LWC. Accordingly, we first compare the vertical profiles of these values with in situ observations in small Cu.

The grid spacing of 100 m × 100 m × 40 m used in this study was applied also by Siebesma et al. (2003), Neggers et al. (2003), Xue and Feingold (2006), Zhang et al. (2011), and Heiblum et al. (2016a,b), who showed that this grid spacing is suitable for simulation of the main properties of small cumulus. To evaluate the role of grid spacing, we also present results of E2000H-50 with the horizontal grid spacing of 50 m (see section 3). Figure 15 shows vertical profiles of cloud-averaged re obtained in simulations E5000H, E2000H, E2000H-50, and E500H, which differ by aerosol concentrations. The mean values of re obtained during in situ measurements in polluted atmosphere in GoMACCS (Jiang et al. 2008; Lu et al. 2008), in CARRIBA (Schmeissner et al. 2015), and in RICO (Gerber et al. 2008; Arabas et al. 2009) are presented for comparison. Note that Nd in E5000H (~380 cm−3) was slightly lower than that observed in GoMACCS (about 500 cm−3) which explains slightly larger re in E5000H. The values of Nd in E2000H and E2000H-50 (110–130 cm−3) were similar to those observed during CARRIBA (70–135 cm−3) and slightly larger than in flight RF12 during RICO (54–116 cm−3) (Gerber et al. 2008). Accordingly, re in these simulations is in an excellent agreement with that in CARRIBA and is slightly smaller than that in RICO. The value of Nd in E500H and in flight RF07 during RICO was below 50 cm−3 (Arabas et al. 2009). Again, there is an excellent agreement between results of simulations and observations. In flights RF06, RF09, and RF12, Nd was 2–3 times larger than in RF07. Consequently, re values in these flights were smaller and close to those simulated in E2000H (blue line in Fig. 15). Note that the profiles of mean re(z) in E2000H and E2000H-50 actually coincide, showing low sensitivity to the grid spacing changes.

Fig. 15.
Fig. 15.

Comparison of mean effective radius profiles simulated using SAM with observations. Simulated mean re profiles: E5000H (black line), E2000H (blue solid line), E2000H-50 (blue dashed line), and E500H (magenta). Blue empty squares show results of in situ observations during GoMACCS in polluted atmosphere (Jiang et al. 2008, Lu et al. 2008), and black empty circles (dissolving Cu) and black filled circles (developing Cu) show results of in situ observations during CARRIBA (Schmeissner et al. 2015). Green filled squares show results reported by Gerber et al. (2008) obtained in flight RF12 during RICO. Brown–red–yellow color map shows re occurrence frequency from flight RF07 during RICO (Arabas et al. 2009). The extremely low values of re at heights around 1500 m in the color map are likely related to dissipating cloud fragments.

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

Note that re is the most robust value having low horizontal variability. Other variables experience larger variability. Figure 16 presents comparison of averaged LWC simulated using SAM with observations. It also compares the profiles of averaged LWC obtained in simulations with horizontal resolutions of 100 versus 50 m (E2000H and E2000H-50, respectively). Note first that profiles obtained using resolutions of 50 and 100 m are very close. These results may indicate that mean profiles of LWC are determined by motions with scales of 100 m or larger. Consequently, resolution (at least within the range of 50–100 m) does not play an important role.

Fig. 16.
Fig. 16.

Comparison of mean LWC profile simulated using SAM with observations. Red solid and dashed lines show results of E2000H and E2000H-50, respectively, using 0.01 g m−3 as threshold of minimum LWC. Green solid and dashed lines show results of E2000H and E2000H-50, respectively, but using 0.05 g m−3 as a threshold of minimum LWC. Blue solid line shows the results of E2000L (threshold of 0.01 g m−3). Black dotted line shows results of Warner (1955) taken from Gerber (2000). Black dashed line is calculated by averaging the LWC measured during SCMS (Gerber 2000). Black filled squares show results reported by Gerber et al. (2008). Blue empty squares show results of in situ measurements during GoMACCS (Jiang et al. 2008). Cyan symbols denote values of cloud-averaged LWC obtained during RICO at different aircrafts: crosses (Bac146), diamonds (C-130), squares (King Air). Combined values on 19 Jan 2005 are shown by cyan filled circles (Abel and Shipway 2007).

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

As mentioned by Jiang et al. (2008), “owing to the human element, aircraft sampling biases are inherently difficult to evaluate.” Besides human element, cloud-averaged LWC depends on many factors, including the minimum LWC value that is chosen to define cloud edge and the criteria of the choice of clouds for statistical analysis of observations (e.g., Raga et al. 1990; Gerber et al. 2008). Modern probes have different detectable levels of LWC from 0.002 (PVM) to about 0.05 g m−3 (the King probe). Katzwinkel et al. (2014) define cloud as the region with LWC > 0.2 g m−3 in order to separate cloud interior from the interface zone. Raga et al. (1990) and Gerber et al. (2008) have chosen clouds with significant updrafts, in which vertical velocities are positive in at least 80% of the Cu traverse. Moreover, Cu are typically traversed near cloud tops.

Following Bar-Or et al. (2012), Tas et al. (2015), Dagan et al. (2015, 2016), and many other studies, we used the threshold of minimum LWC of 0.01 g m−3 (which corresponds to about 10 small droplets per cubic centimeter) to determine the cloud-averaged LWC profile. This is in agreement with the criteria of Deng et al. (2009) for the determination of a cloudy region. Figure 16 also shows profiles of averaged LWC obtained using the criteria of cloud as LWC > 0.05 g m−3, which was used, for instance, by Abel and Shipway (2007) and Jiang et al. (2008). One can see that the choice of the larger threshold leads to small increase in LWC. Note that utilization of 50-m-resolution instead of 100 m makes the mean LWC profile even smaller.

In general, the comparison with the observations shows that the model reproduces the averaged LWC quite reasonably.

Note that the distributions of vertical velocity are in agreement with observations, and AF values increase with vertical velocity (not shown). These results agree with those observed by Schmeissner et al. (2015), who showed that the values of AF are larger in developing Cu, characterized by larger vertical velocity. Figure 17 shows PDF of 90th percentile of the AF and vertical velocity W within clouds in CARRIBA measurements (Katzwinkel et al. 2014) and in E2000H simulation around a height of 1500 m. The measurements were performed ~100 m below cloud tops, with effective resolution of 20 cm. E2000H simulation results are presented for heights of 1380–1540 m.

Fig. 17.
Fig. 17.

PDF of 90th percentile of the adiabatic fraction within clouds in (a) CARRIBA measurements (Katzwinkel et al. 2014, their Fig. 3a) and in (b) E2000H simulation around a height of 1500 m. Similarly, a PDF of 90th percentile of vertical velocity is presented for (c) measurements (Katzwinkel et al. 2014, their Fig. 3a) and (d) simulation. The measurements were performed around 100 m below cloud tops, with effective resolution of 20 cm. E2000H simulations results are presented for heights of 1380–1540 m.

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

One can see that the 90th percentiles are distributed similarly in measurements and the simulation. More cases with larger AF and W in E2000H can be attributed to slightly different stratifications and filtering out the largest Cu in measurements.

Comparison of Figs. 7 and 17 with corresponding figures obtained from high-frequency in situ measurements, that is, Fig. 2 from Gerber (2000), Fig. 1 from Gerber et al. (2008), and Fig. 3 from Katzwinkel et al. (2014), show surprising similarity between simulations and high-frequency observations. It is clear that 100-m-resolution numerical results cannot be directly compared with high-resolution measurements of 10–20 cm. Such high resolutions are not available in any cloud model, including the Lagrangian models mentioned above. However, both simulations and measurements indicate the existence of slightly diluted cloud volumes at comparatively high altitudes. We interpret this similarity as follows:

  1. Figure 1 from Gerber (2000) and Fig. 5.34 from (Brenguier et al. 2013) show that width of high AF zones can be on the order of 100 m, which can be detected by the LES. We attribute the existence of such zones to the fact that the most energetic LWC fluctuations are related to energetic W fluctuations with scales on the order of 100 m or larger (Magaritz-Ronen et al. 2014). We believe that PDF of W in Fig. 17c is determined by such large fluctuations.
  2. Statistics available in the model are much larger than in measurements. Since the fraction of cloud volumes with high AF is very small at high levels, in situ measurements can miss such volumes, while the model detects all such volumes.
  3. In situ measurements often performed near cloud tops, where AF is small. Moreover, typically they avoid measurements in vigorous clouds. AF is calculated either by averaging over clouds or over wide zones with W > 0 (Lu et al. 2008). Such averaging masks the possible existence of large values of AF. As a result, the measurements may miss by one or another reason the volumes with high AF. In contrast, the model detects all such volumes.
So, we believe that 1-Hz in situ measurements underestimate the number of volumes with high AF. This underestimation is partially compensated by using high-frequency measurements (Gerber 2000; Gerber et al. 2008; Katzwinkel et al. 2014; Schmeissner et al. 2015). As a result, we obtain these surprising similarities of AF and LWC profiles with high-frequency measurements.

Note that in any case the amount of high AF volumes at high levels is so small that they do not affect averaged profiles of corresponding values.

In several studies (e.g., Stevens and Seifert 2008; Seifert and Heus 2013; Seifert et al. 2015) the importance of environmental air humidity on clouds as well as on the structure of cloud fields was reported. To show that the parameterization proposed remains suitable for another environmental RH, two sensitivity runs E2000H-RH1 and E2000H-RH2 (see section 3) were performed with RH decreased by 5% and 10%, respectively, in the layer from the surface to the inversion base. Figure 18 shows average re profiles of E2000H, E2000H-RH1, E2000H-RH2, and E2000H-50. Similarly to Fig. 11, the parameterized profiles are also presented using Eqs. (5)(7).

Fig. 18.
Fig. 18.

Vertical profiles of averaged simulated (solid lines) and parameterized (dashed lines) effective radii. Blue: E2000H; green: E2000H-50; red: E2000H-RH1; and black: E2000H-RH2.

Citation: Journal of the Atmospheric Sciences 76, 2; 10.1175/JAS-D-18-0046.1

One can see that clouds are lower and cloud bases are higher with the decrease in RH. Despite the significant changes in the cloud structure, the parameterization using Eqs. (5)(7) works well. The 50-m-resolution averaged re profile almost coincides with that of 100 m.

We see therefore that the proposed parameterizations for mean profiles of re (Fig. 11) and LWC (Fig. 13) are robust with regard to model resolution, thermodynamical conditions, and aerosol concentrations.

7. Discussion and conclusions

Small Cu are still subgrid phenomena for large-scale NWP models and even for CRM. Taking into account the important role of small Cu in radiation budget and in BL structure, there is a strong necessity to parameterize their effects in such models. Consequently, we need to characterize the cloud field by a comparatively small number of governing parameters that can be used in large-scale models. Among such parameters are effective radius and LWC.

The goal of the study was twofold: 1) showing the robustness of characterizing all the clouds within the field by one mean vertical profile of effective radius and 2) proposing a parameterization of the mean vertical profiles of re and LWC in small, warm Cu.

To accomplish the goal, the fields of small Cu in an unstable boundary layer below an inversion were simulated using SAM with spectral bin microphysics. LESs were performed for different thermodynamic conditions and a wide range of aerosol concentrations.

The main results can be formulated as follows:

  1. Despite the high variability of LWC and droplet concentration within clouds and between different clouds, the mean volume and effective radius of droplets per specific level vary only slightly. This indicates similarity of DSDs along the horizontal direction within clouds and between different clouds. Such similarity with the unchanged modal radius of DSD was reported in in situ observations in Cu (Paluch and Knight 1984; Paluch 1986; Gerber 2000; Gerber et al. 2008; Paluch and Baumgardner 1989; Prabha et al. 2011). Low variability of re is a consequence of the facts that the mixing within clouds and at cloud edges is inhomogeneous and that the entrained air is humid (Pinsky and Khain 2018a). Theoretical and numerical studies with spectral bin models also show that the polydispersivity of droplet size distributions substantially decreases the variability of effective radius during mixing (Pinsky et al. 2016a; Pinsky and Khain 2018b). Thus, even intense mixing and strong dilution typical of small Cu do not increase the variability of re significantly. These findings allow us to characterize the cloud field by a specific vertical profile of . Note, that inhomogeneous and homogeneous mixing types are separated by scales less than 1 m (Pinsky et al. 2016a). This fact brings up the question of why spectral microphysics models with resolution of 100 m are able to reproduce observed DSD and their moments. Pinsky et al. (2016a) showed that the microphysical quantities in clouds are not sensitive to that separation scale if the model grid spacing is less than the spatial radius of correlation of microphysical quantities in clouds, which is on the order of 150–250 m.
  2. We have used LES to parameterize the vertical profiles of and for different aerosol concentrations and thermodynamical conditions. Assuming that Nd at cloud base can be deduced from the CCN concentration (e.g., Segal and Khain 2006), the adiabatic mean volume radius profile was calculated using the theoretical profile of LWCad. Using the found relation between rv and re, we deduced the adiabatic profile . We then used to parameterize , assuming increased effect of mixing with height. We have parameterized in terms of Nd at cloud base, taking into account decrease of Nd with height in cases of drizzle formation. Then we parameterized using and assuming low variability of re in horizontal.
  3. Universality of the parameterization formulas was justified in LES with different grid spacing (100 and 50 m) and different environmental humidity, as well as by detailed comparison with in situ measurements (Gerber 2000; Gerber et al. 2008; Abel and Shipway 2007; Lu et al. 2008; Jiang et al. 2008; Arabas et al. 2009; Katzwinkel et al. 2014; Schmeissner et al. 2015). The universality follows from the fact that the parameterized values depend only on the adiabatic values, which are universal.

Thus, we suppose that the parameterization formulas derived in this study are suitable for nonprecipitating warm Cu developing under a wide range of thermodynamic and aerosol conditions. For example, transition from maritime to continental conditions is partially described by an increase in CCN concentrations. BL diurnal cycle is partially described by changes in inversion height and accordingly of cloud depth. When implemented into NWP models, the parameterization specific values would adjust themselves continuously in space and time (depending on local cloud base and inversion heights, temperature, and CCN concentration in the model). We believe, therefore, that these parameterization profiles can be used for calculation of radiation properties of Cu fields in large-scale models. That will allow better estimation of optical properties and radiation transfer through such clouds. High sensitivity of microphysical parameters to aerosols indicate the necessity to determine better aerosol concentrations in large-scale models, including NWP ones. For sake of more generality, it is worthwhile to test the proposed parameterizations for different temperatures and Bowen ratios (Zhang et al. 2017) in future studies. In addition to LWC and re, the cloud cover of small Cu fields and vertical overlap of cloudy layers should be parameterized in a future study.

At the end of the discussion section, we would like to address the role of adiabatic processes in formation of the universal profiles of effective radius re(z) and in first raindrop formation:

  1. As was mentioned above, the values of re(z) are close to those that can be obtained in an adiabatically ascending parcel. We do not see any mechanism that could lead to such a universal profile except the dominating adiabatic ascent. As was shown in Figs. 7 and 17, LESs indicate the existence of adiabatic or only slightly diluted cloud volumes up to heights of 1–1.5 km above cloud base. Mixing of such volumes with surrounding air does not lead to change in re even in the dilution zone. The effective radius decreases only within a narrow zone of a few tens of meters at cloud edges (Pinsky and Khain 2018a). Because of the low width of these zones, the cloud-averaged re does not differ much from its maximum adiabatic value.
  2. According to our simulations, cloud volumes (grid points) with maximum LWC contain the maximum number concentration. Technically, we have first found the time–space coordinates with maximum LWC, and then marked them on the RWC, re, and Nd scatterplots (Figs. 7b, 8, and 10, respectively). This means that the collision rate in these volumes is maximum. It allows assuming that first raindrops form in cloud volumes, where dilution with surrounding clouds is minimal.

Earlier such findings were reported in LESs by Seifert et al. (2010) for trade wind Cu, and by Benmoshe et al. (2012) and Khain et al. (2013) for developing deep convective clouds. Formation of raindrops in slightly diluted cores explains a success of dynamically simple 1D parcel models in prediction of the conditions and the height of the first raindrop formation (Pinsky and Khain 2002; Freud and Rosenfeld 2012). Therefore, the results indicate a dominating role of basic adiabatic processes in cloud cores in formation of microphysical cloud structure. We found that this conclusion is valid even in small Cu, although the mixing there is very significant (because of the big ratio of surface area to volume).

Note that, although not analyzed in this study, other mechanisms of rain formation are also discussed in the literature. Among them are the effects of giant and ultragiant CCN (Baker et al. 1980; Blyth et al. 2003); decrease in droplet concentration in the course of inhomogeneous mixing leading to more intense diffusional growth and formation of superadiabatic droplets (Devenish et al. 2012); and turbulence-induced collision rate enhancement (Pinsky et al. 2008; Wang and Grabowski 2009; Seifert et al. 2010; Benmoshe et al. 2012; Hoffmann et al. 2017). Grabowski and Abade (2017) investigated DSD broadening in turbulent flow in the course of diffusional growth. Hence, more studies are required to establish the mechanisms of rain formation in Cu.

Acknowledgments

The authors express their gratitude to M. Ovchnnikov for support with SAM. The study is performed within the Test and Tuning of Revised Cloud Radiation Coupling (T2RC2) Project of the COSMO consortium. The scientific group of the Israel Meteorological Service is supported by the Israeli Ministry of Transport and Road Safety. Dr. A. Khan from the Hebrew University of Jerusalem is supported by the U.S. Department of Energy Atmospheric System Research program (Grants DE-SC0014295 and DE-FOA-0001638) and the Israel Science Foundation (Grants 1393/14 and 2027/17). Scientific group of the Weizmann Institute of Science has been supported by the Minerva foundation with funding from the Federal German Ministry of Education and Research.

APPENDIX

Relationship between Mean Volume and Effective Radius

According to definition, the expressions for effective radius and mean volume radius are
ea1
ea2
where f(r) is DSD with normalization. This DSD can be interpreted as probability distribution function at . The ratio can be written as
ea3
According to inequality for absolute moments (Korn and Korn 2000),
ea4
In particular for k = 2, since , . This inequality and Eq. (A3) lead to the following inequality: , which means that .

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