The sub-adiabatic model as a concept for evaluating the representation and radiative effects of low-level clouds in a high-resolution atmospheric model

. The realistic representation of low-level clouds, including their radiative effects, in atmospheric models remains challenging. A sensitivity study is presented to establish a conceptual approach for the evaluation of low-level clouds and their radiative impact in a highly resolved atmo-5 spheric model. Considering simulations for six case days, the analysis supports that the properties of clouds more closely match the assumptions of the sub-adiabatic rather than the vertically homogeneous cloud model, suggesting its use as basis for evaluation. For the considered cases, 95.7 % of 10 the variance in cloud optical thickness is explained by the variance in the liquid water path, while the droplet number concentration and the sub-adiabatic fraction contribute only 3.5 % and 0.2 % to the total variance, respectively. A mean sub-adiabatic fraction of 0.45 is found, which exhibits 15 strong inter-day variability. Applying a principal component analysis and subsequent varimax rotation to the considered set of nine properties, four dominating modes of variability are identiﬁed, which explain 97.7 % of the total variance. The ﬁrst and second components correspond to the cloud 20 base and top height, and to liquid water path, optical thickness, and cloud geometrical extent, respectively, while the cloud droplet number concentration and the sub-adiabatic fraction are the strongest contributors to the third and fourth components. Using idealized ofﬂine radiative transfer cal-25 culations, it is conﬁrmed that the shortwave and longwave cloud radiative effect exhibits little sensitivity to the vertical structure of clouds. This reconﬁrms, based on an unprecedented large set of highly resolved vertical cloud proﬁles, that the cloud optical thickness and the cloud top

the variance in cloud optical thickness is explained by the variance in the liquid water path, while the droplet number concentration and the sub-adiabatic fraction contribute only 3.5 % and 0.2 % to the total variance, respectively. A mean sub-adiabatic fraction of 0.45 is found, which exhibits 15 strong inter-day variability. Applying a principal component analysis and subsequent varimax rotation to the considered set of nine properties, four dominating modes of variability are identified, which explain 97.7 % of the total variance. The first and second components correspond to the cloud 20 base and top height, and to liquid water path, optical thickness, and cloud geometrical extent, respectively, while the cloud droplet number concentration and the sub-adiabatic fraction are the strongest contributors to the third and fourth components. Using idealized offline radiative transfer cal-25 culations, it is confirmed that the shortwave and longwave cloud radiative effect exhibits little sensitivity to the vertical structure of clouds. This reconfirms, based on an unprecedented large set of highly resolved vertical cloud profiles, that the cloud optical thickness and the cloud top and bot-30 tom heights are the main factors dominating the shortwave and longwave radiative effect of clouds, and should be evaluated together with radiative fluxes using observations, to at-tribute model deficiencies in the radiative fluxes to deficiencies in the representation of clouds. Considering the different 35 representations of cloud microphysical processes in atmospheric models, the analysis has been further extended and the deviations between the radiative impact of the singleand double-moment schemes are assessed. Contrasting the shortwave cloud radiative effect obtained from the double-40 moment scheme to that of a single moment scheme, differences of about ∼ 40 W m −2 and significant scatter are observed. The differences are attributable to a higher cloud albedo resulting from the high values of droplet number concentration in particular in the boundary layer predicted by 45 the double-moment scheme, which reach median values of around ∼ 600 cm −3 .

Introduction
Clouds play a crucial role in the global energy budget and climate. One important aspect is their strong influence on the 50 shortwave (SW) and longwave (LW) radiation budget. Despite significant progress over the past decades, the relevant processes and resulting climate feedbacks of clouds have not been fully understood, and cannot be reliably represented in climate projections (IPCC, 2013). The representation of 55 boundary layer clouds (i.e., shallow cumulus, stratiform) is particularly problematic (Turner et al., 2007), due to their high spatio-temporal variability. In addition, the coarse resolution of general circulation models (∼ 100 km) is not sufficient to resolve processes taking place at sub-grid scale, nor 60 allows to explicitly take vertical and horizontal heterogeneity into consideration.
Clouds are characterized by complicated threedimensional (3D) shapes with highly variable macrophysical, microphysical, and radiative properties. Full 3D radiative transfer calculations in complex cloudy atmospheres are computationally expensive and, hence, a number 5 of simplifications are commonly adopted for calculating their radiative effect in atmospheric models. The planeparallel (PP) approximation is often utilized, which implies that radiative transfer simulations are conducted assuming horizontally homogeneous clouds covering a fraction of the 10 model grid (Di Giuseppe and Tompkins, 2003;Chosson et al., 2007). One particular shortcoming of this assumption is the so-called plane-parallel albedo bias, which refers to the fact that inhomogeneous clouds reflect less solar radiation than otherwise identical homogeneous clouds (Werner et al.,15 2014). To account for this bias, and to consider horizontal heterogeneities in GCMs, several correction schemes have been developed over the last years, e.g., scaling the liquid water path by a constant reduction factor, renormalization techniques, among others (e.g., Cahalan et al., 1994;Barker, 20 2000; Cairns et al., 2000;Barker and Räisänen, 2004;Pincus et al., 2003;Shonk and Hogan, 2008).
The optical properties of a cloudy layer are largely determined by two of their physical properties, the liquid water content (q L ) and the effective radius (r eff ) (Slingo, 1989; 25 Collins et al., 2006). The latter is mostly obtained by assuming a fixed droplet size distribution (Chosson et al., 2007). Double-moment cloud microphysical schemes, which also constrain the effective radius through prognostic equations, are only recently becoming more widespread in use in oper-30 ational forecasting.
To improve the scientific understanding of clouds and their representation in models, high-quality observations from active (i.e., lidar and cloud radar) and passive (i.e., radiometers) instruments from both ground and space are essential.  rently, such instrumentation is available, i.e., from the Cloudnet program (Illingworth et al., 2007), the A-train constellation (Stephens et al., 2002), the geostationary satellite Meteosat Second Generation (MSG) (Roebeling et al., 2006), while upcoming missions comprise the Earth Cloud Aerosol 40 Radiation Explorer (EarthCARE) satellite mission (Illingworth et al., 2015) and Meteosat Third Generation (MTG) (Stuhlmann et al., 2005). A variety of algorithms have been developed for inferring cloud properties from these observations (e.g., Nakajima and King, 1990;Bennartz, 2007;Roe-45 beling et al., 2013). However, the underlying observational techniques often rely heavily on assumptions about the cloud vertical structure.
High-resolution atmospheric models at cloud-resolving scales are another promising avenue to gain insights into 50 cloud processes and the effects of small-scale cloud variability, and to improve their representation in GCMs. They can resolve relevant processes up to a much smaller scale (∼ 100 m for Large Eddy Simulations), and can, thus, serve as basis for developing more accurate parameterizations. En- 55 abled by the exponential growth in computer power over the past decades, they are increasingly utilized for simulations covering larger domains and longer time periods. In contrast to observations, they also offer the opportunity to assess the interplay of all relevant state variables simultaneously, while 60 instrumental capabilities are generally limited to a small subset, sometimes affected by large measurement uncertainties (Miller et al., 2016).
It is, however, crucial to also critically evaluate the performance of high-resolution models with observations. Like 65 coarse-resolution models, they include various assumptions and parameterizations, and their shortcomings need to be identified and mitigated. Given the complexity of atmospheric models and the level of detail available from the output of such models, it is, though, often a daunting task to 70 identify the physical reasons for model shortcomings. Inconsistent or even conflicting assumptions made in observationbased products add further complications to the evaluation of models with observations. Examples for such assumptions include a vertically homogeneous or a sub-adiabatic cloud 75 that is often made in satellite retrievals (Brenguier et al., 2000;Chosson et al., 2007), or the assumption of a vertically constant cloud droplet number concentration commonly used in ground-based remote sensing of clouds, which is a significant simplification of the profiles available from in situ ob-80 servations or double-moment cloud microphysical schemes.
In this work, the highly resolved ICON-LEM atmospheric model (ICOsahedral Non-hydrostatic Large-Eddy Model) is employed that was recently developed within the HD(CP) 2 (High Definition Clouds and Precipitation for advancing Cli-85 mate Prediction) project (Dipankar et al., 2015;Heinze et al., 2017). We introduce a conceptual approach for evaluating the representation of low-level clouds in this and other highresolution atmospheric models, with particular focus on the correct representation of their radiative effect. A sensitiv-90 ity study is conducted in order to investigate the relevance of the vertical distribution of microphysical properties for their radiative effect, aiming for the identification of suitable column-effective cloud properties for the purpose of model evaluation. The suitability of the sub-adiabatic cloud 95 model is compared to that of the vertically homogeneous cloud model, both of which are commonly used in remote sensing. In addition, differences in cloud radiative properties arising from the availability of the cloud droplet number concentration provided by the double-moment cloud micro-100 physical scheme of Seifert and Beheng (2006) compared to a single-moment scheme are highlighted.

ICON-LEM
The ICON unified modeling framework was co-developed 105 by the German meteorological service (DWD) and the Max Planck institute for meteorology (MPI-M) in order to support climate research and weather forecasting. Within the HD(CP) 2 project, ICON was further extended towards large eddy simulations with realistic topography and open boundary conditions. This resulted in ICON-LEM deployed in restricted areas that are centered on Germany and the Tropical Atlantic (Heinze et al., 2017). The equations utilized by the model are based on the prognostic variables given by Gassmann and Herzog (2008). Concerning turbulence parameterization, the three-dimensional Smagorinsky scheme is employed (Dipankar et al., 2015). These variables comprise the horizontal and vertical velocity components, the density of moist air, the virtual potential temperature, and the mass and number densities of traces, e.g., specific humidity, liquid water, and different ice hydrometeors. A comprehen-15 sive description of the model and its governing equations is found in Dipankar et al. (2015) and Wan et al. (2013). The activation of cloud condensation nuclei (CCN) is based on the parameterization of Seifert and Beheng (2006) and modified in order to account for the consumption of CCNs due to 20 their activation into cloud droplets. The CCN concentration is then parameterized following the pressure profile and the vertical velocity (Hande et al., 2016).
ICON-LEM utilizes the double-moment mixed-phase bulk microphysical parameterization scheme introduced by 25 Seifert and Beheng (2006). Following their comprehensive description, a generalized gamma distribution is utilized to describe the mass (x m ) of hydrometeors, The coefficients ν, ξ are constants taken from Table 1 in 30 Seifert and Beheng (2006) while the coefficients A m and B m are prognostic quantities expressed by the number and mass densities (see Appendix A).
Simulations are carried out for three different domains with 624 m, 312 m, and 156 m horizontal resolution. The 35 model domains consist of 150 vertical levels, with resolutions ranging from ∼25 m to 70 m within the boundary layer, and from 70 m to 355 m further up until the top of the domain at 21 km. For each of the aforementioned grids, data is stored as one-dimensional (1D) profiles every 10 sec, two-(2D), and 40 3D snapshots (Heinze et al., 2017). In case of the 3D output, the simulation data is interpolated from the original grids (e.g., 156 m) to a 1 km grid, the 3D coarse data, and 300 m grid, the so-called HOPE data. The latter output has been created for the purpose of model evaluation with ground-based 45 observations from the HD(CP) 2 Observational Prototype Experiment (HOPE) that took place near Jülich (Macke et al., 2017) and is limited to a domain size of about ∼ 45 km 2 . Note here that for the 2D and 3D output, data is stored at day-and night-time frequency. Day-time frequency begins 50 at 06:00 UTC and lasts until 00:00 UTC, while night-time starts at midnight and lasts until 06:00 UTC. The 2D data is stored with a day-time and night-time frequency of 10 sec and 5 min, respectively. The 3D coarse data has day-time frequency of 10 min (1 hour at night-time). In this study, the 3D 55 HOPE data has been used that is stored only at a day-time frequency of 15 min.

RRTMG
For radiative transfer simulations, ICON-LEM employs the rapid radiative transfer model (RRTM) for GCM applications 60 (RRTMG) (Mlawer et al., 1997;Iacono et al., 2008). For the purpose of this investigation, an interface of the RRTMG for use with the Python programming language has been developed, which allows the offline calculation of the radiative fluxes using ICON-LEM outputs as basis.
RRTMG is a fast and accurate broadband radiative transfer model developed by the Atmospheric Environmental Inc. The model employs the correlated-k approach for efficient fluxes and heating rates computations (Mlawer et al., 1997). Molecular absorption information for the k-70 distributions is taken from the line-by-line radiative transfer model (LBLRTM) (Clough et al., 2005). Fluxes and heating rates are derived for 14 bands in the SW and 16 bands in the LW. RRTMG considers major absorbing gases, i.e., water vapor, ozone, and carbon dioxide, but also minor absorbing 75 species, i.e., methane, oxygen, nitrogen, and aerosols. Optical properties (optical thickness, single-scattering albedo, and asymmetry parameter) of liquid water clouds are parameterized according to Hu and Stamnes (1993). Note that the RRTMG is a 1D plane-parallel radiative transfer model. 80 For the representation of the sub-grid cloud variability, a Monte Carlo independent column approximation (McICA) method is used (Pincus et al., 2003). Multiple-scattering is considered employing a two-stream algorithm (Oreopoulos and Barker, 2006).

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RRTMG provides the SW and LW radiative fluxes for both the upward (F ↑ ) and downward (F ↓ ) radiation. These two components can be combined to define the net flux (F net ), Accordingly, the cloud radiative effect (CRE) is defined as 90 the difference between the cloudy and clear sky net radiative fluxes, The CRE can be computed for the LW, SW, or the net CRE, defined by the sum of the SW and LW radiation.

Case days
In this study, the 3D HOPE data has been used and a set of 6 days of simulations has been considered, including: 24-25 April 2013, 5 May 2013, 29 July 2014, 14 August 2014, and 3 June 2016. Only a limited subset of variables is stored 100 including the specific humidity, cloud water, ice, rain and snow mixing ratio, wind, vertical velocity, temperature, pressure, cloud cover, and turbulent diffusion coefficient for heat. These days have been selected from the total set of available case days by the presence of suitable liquid water cloud fields and no known bugs in the used model version, which affect 5 the representation of low-level clouds.

Column selection
In order to investigate the characteristics of liquid water clouds in ICON-LEM, only idealized cloud profiles (i.e., stratiform and cumulus) are considered, corresponding to 10 single-layer non-drizzling clouds. The selection of such cloudy columns has been conducted according to requiring the following threshold criteria: -For each cloudy layer, a liquid water content of q L > 0.01 g m −3 and a liquid water path (Q L ) larger than 15 20 g m −2 .

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-A cloud geometrical extent (H) larger than 100 m (at least two subsequent model layers).
-Clouds located between 300 m and 4000 m.
-No vertical gaps are allowed.
-Mixed-phase clouds are excluded. The ice water content 25 for the first 4000 m must be zero.
The cloud bottom height (CBH) and cloud top height (CTH) are determined by the bottom and top of the lowermost and uppermost layers for the aforementioned ideal low-level 30 clouds, respectively.

Cloud property diagnostics
The model outputs the droplet number concentration and liquid water content for each model layer representing the zeroth and the first moments of the mass size distribution 35 (MSD, see Eq. 1). Following Petty and Huang (2011), the mass size distribution is transformed into a droplet size distribution (DSD). For details on the derivation of the moments of DSD and the cloud microphysical properties, the reader is referred to Appendix A. 40 Following Hansen and Travis (1974), the effective radius, r eff , is defined as the ratio of the third to the second moments of the DSD, The division by 2 is carried out for diameter-to-radius 45 conversion. The effective radius is linked to the volumeequivalent radius (r V ) by the k 2 factor, which depends only on the effective variance (υ) of the droplet size distribution, For ICON-LEM, the effective variance of the reconstructed 50 Gamma DSD is υ = 0.052, corresponding to k 2 = 0.849. Typical values of k 2 reported in the literature vary between 0.5 and 1 (e.g., Brenguier et al., 2000;Zeng et al., 2014;Merk et al., 2016). Furthermore, the radar reflectivity is defined as the sixth moment of the size distribution, Note, that in ICON-LEM, the droplet number concentration varies with height, but the width of the DSD is assumed invariant.
2.6 Cloud models 60 2.6.1 Vertically homogeneous cloud model A widely used assumption for passive satellite and groundbased retrievals is the vertically homogenous cloud model. Accordingly, a vertically homogeneous DSD is assumed, meaning vertically constant microphysical properties. It fol-65 lows that the cloud liquid water path is given by, describing a positive linear relationship between Q L and both the cloud optical thickness (τ ) and effective radius (r eff ).
Here, ρ w stands for the water density, while the factor 2/3 is 70 a scale factor resulting from the constant liquid water content and effective radius with height (Lebsock and Su, 2014). Assuming a vertically constant cloud droplet number concentration additionally implies that the cloud geometric extent depends linearly on the cloud water path for a fixed effective 75 radius.

Sub-adiabatic cloud model
The sub-adiabatic cloud model describes the evolution of a convective closed parcel of moist air. According to Albrecht et al. (1990), the liquid water content (q L ) of such an air par-80 cel increases linearly with height, where Γ ad is the adiabatic increase of the liquid water content (Bennartz, 2007), z is the height over the cloud base, f ad denotes the sub-adiabatic fraction, T is the temperature, and 85 P is the pressure. f ad describes the deviation from the linear increase with height of q L caused by entrainment of dry air resulting in evaporation and f ad < 1 (sub-adiabaticity). In case of a pure adiabatic cloud, f ad = 1 and Eq. (8) yields to the adiabatic liquid water content (q L,ad ). For low-level liq-5 uid water clouds, typical values of f ad found in the literature are in the range of 0.3 to 0.9 (Boers et al., 2006). An alternative definition for the liquid water content accounting for the depletion of the liquid water content due to entrainment, precipitation, and freezing drops, is described by, following a modified sub-adiabatic profile (Karstens et al., 1994;Foth and Pospichal, 2017). Γ ad depends on temperature (weak function of pressure) following the first law of thermodynamics and the Clausius-15 Clapeyron relationship. For low-level clouds, Γ ad varies slightly (∼ 20 %). Consequently, in most studies, Γ ad is assumed constant (e.g., Albrecht et al., 1990;Boers et al., 2006) or it is calculated from cloud bottom temperature and pressure (e.g., Merk et al., 2016) or cloud top information 20 (e.g., Zeng et al., 2014). For this study, an average value of Γ ad between cloud bottom and cloud top has been used.
Integrating the liquid water content between cloud base height and cloud top height, the cloud liquid water path is obtained, Hereby, H denotes the cloud geometrical extent. Compared to Eq. (7), Eq. (10) leads to a factor of 5/9, meaning that the sub-adiabatic liquid water path is 5/6 times the one of to the vertically homogeneous model (Wood and Hartmann,30 2006). Dividing Q L by its adiabatic value (inserting f ad = 1 into Eq. 10), the sub-adiabatic fraction can be computed, For low-level liquid water clouds, the droplet number concentration (N d ) depends on the availability of cloud conden-35 sation nuclei (CCN) that could get activated at cloud base (Bennartz, 2007). Considering the adiabatic increase of the liquid water content, it follows that at any given height, q L is distributed over the activated CCN (per unit volume). Consequently, there is no dependency of the mean volume radius 40 r V on the shape of the droplet size distribution, but only on N d and q L , Combining Eqs. (5) and (12), the effective radius for the uppermost cloud layer can be written in terms of the liquid wa-45 ter path, the droplet number concentration, and the adiabatic fraction, In the geometric optics regime, the extinction coefficient, b ext , can be written as a function of the liquid water con-50 tent and the effective radius. Consequently, the cloud optical thickness can be computed by integrating b ext over the cloud geometrical extent, i.e., from cloud base height to cloud top height, For vertically constant q L and r eff , this can be interpreted as the cloud optical thickness coming from the vertical homogeneous model (see Eq. 7). According to the sub-adiabatic cloud model, the cloud optical thickness is linked to the liquid water path and the effective radius (Wood, 2006), Alternatively, substituting r eff from Eq. (13) in Eq. (15), the cloud optical thickness is given by, 3 Cloud characteristics 65 3.1 General features Table 1 lists the statistics of the cloud properties for all the case days individually and on average as simulated by ICON-LEM, while Fig. 1 illustrates the corresponding histograms for the latter case only. Note that for the droplet number con-70 centration and the effective radius, results are presented as follows: droplet number concentration weighted over the cloud geometrical extent, given by, 75 effective radius weighted over the extinction coefficient at each layer, It can be shown that the latter equation reduces to Eq. (7), which implies that the calculated effective radius corresponds to that of a vertically homogeneous cloud with identical liquid water path and optical thickness. The different cloud properties are characterized by a large variability from 5 day to day, but even within the same day driven by entrainment processes. In addition, the differences are also subject to the sample size (n) for each day depending on the column selection filter that applied to ICON-LEM output. Recall here that a cloudy column is taken under consideration 10 when q L > 0.01 g m −3 for each cloud model level while the liquid water path for the entire column should be larger than 20 g m −2 . Subsequently, the fraction of clouds (FC) selected in this study is quite low (FC < 3 %). Alternatively, if only a liquid water path filter is applied to the data, defining as 15 cloudy the columns with Q L larger than 1 g m −2 , the actual cloud fraction (CF) is obtained. The rather large value of the CF found for 3 June 2016 is associated with very low (with 100 < CBH < 200 m) overcast cloudy conditions in the early hours.

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Looking at the mean histograms of CTH and CBH, one can identify multimodal distributions. Note here that, in this study, all low-level clouds are considered (i.e., cumuli-like, stratiform) increasing the variability of the different properties.

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The double-moment microphysical scheme adopted in ICON-LEM is reflected on the histograms of the droplet number concentration. The mean histogram of N int for all the case days on average suggests a bimodal distribution with peaks centered around 200 cm −3 and 450 cm −3 . These 30 two modes are clearly found for 29 July 2014, 14 August 2014, and 3 June 2016. Especially, for 3 June 2016, the peak around 200 cm −3 is even more notable (not shown here). Note here that this value is close to the fixed droplet number concentration profile suggested by single-moment micro-35 physical schemes adopted by atmospheric models, such as ECHAM  and ICON-NWP, which is the global Numerical Weather Prediction (NWP) version of the ICON model (Heinze et al., 2017). For 5 May 2013, the corresponding histrogram is characterized by a right-skewed 40 distribution, with a rather long tail towards large values of N int and a very small peak that appears around 800 cm −3 .
On the contrary, for 24-25 April 2013, the distributions of N int are described by skewed distributions (not shown here) with well-defined single peaks. For the 24 th , the peak veers 45 towards large N int values (left-skewed), while, for the 25 th the peak is located at small N int values (right-skewed), which are centered around 686 cm −3 and 380 cm −3 , respectively. A close relation between the effective radius and the droplet number concentration exists. On average, the larger the N int 50 the smaller the r int . Figure 2 shows a box-whisker plot of the droplet number concentration for all the case days on average, describing the histograms of N d simulated for different model levels 55 by the double moment scheme of ICON-LEM. For comparison, the red line shows the climatology-based droplet number concentration profile adopted by ECHAM . While above 2 km altitude, the modeled values match the climatology well, much larger me-60 dian values up to 600 cm −3 are found in the boundary layer. Compared to satellite estimates of N d , these values seem rather high (Quaas et al., 2006;Grosvenor et al., 2018). On the contrary, in situ observations suggest higher values of N d and, accordingly, closer to those simulated by ICON-65 LEM. Hence, efforts should be undertaken to further validate the cloud droplet number concentrations predicted by the double-moment scheme. Figure 3 depicts the mean profiles of q L and N d normalized over the cloud geometrical extent (from CBH to CTH) 70 for all the case days on average. The ICON-LEM simulated liquid water profile follows a linear increase from cloud bottom to around 50-60 % of the cloud height in agreement with the adiabatic cloud model. Thereafter, the liquid water content decreases towards the cloud top due to evaporation in-75 duced by entrainment of dry air mass from cloud top. Furthermore, the mean profile of the droplet number concentration is found roughly constant at verticals depths between 30 and 70 % of H (∼ 480 cm −3 ) and decreases towards the cloud top at values ∼ 100 cm −3 characterized by a large vari-80 ability.

Adiabaticity of liquid water clouds
Following the sub-adiabatic cloud model, higher values of the liquid water path are linked with geometrically thicker clouds (see Eq. 10). For all the days, the distribution of the cloud geometrical extent follows a similar pattern, except

Cloud optical thickness
One of the fundamental cloud properties describing the SW radiative effect is the cloud optical thickness. In this section, we focus on its derivation and its dependencies. With this intention, an effort has been conducted to predict 35 the cloud optical thickness derived from Eq. (14) by employing the sub-adiabatic model and Eq. (16). On a logarithmic scale, Eq. (16) suggests that τ is a linear function of Q L , f ad , and N d and it can be seen as a linear regression model. Here, the droplet number concentration weighted over the cloud 40 geometrical extent (N int ) is used. An advantage of the logarithmic scale is that the variance of the cloud optical thickness can be decomposed into the contributions from each of the regressors (Q L , f ad , and N int ). This enables us to attribute the relative importance of the regressors in explaining 45 the variance in τ . In our framework, we employed the ordinary least squares (OLS) regression method. This method Table 2. Prediction of cloud optical thickness by an ordinary least squares regression method: Regressor coefficients (a), Y -intercept (a0), squared correlations (R 2 ), and root mean square error (RMSE). Theoretical (Th.) values according to the sub-adiabatic model are also included.  Comparing the models Y 2 (Q L , N int ) and Y 3 (Q L , f ad ), Y 2 has a higher R 2 value (0.992 compared to 0.959), a lower RMSE (0.075 compared to 0.172), while the regression co-15 efficients are much closer to the sub-adiabatic theory. All in all, the liquid water path is able to explain 95.7 % of the variance in cloud optical thickness, while the droplet number concentration and the sub-adiabatic fraction additionally contribute 3.5 % and 0.2 % to the variance, respec-20 tively.
Variability caused by Γ ad is insignificant and, thus, is not shown here. This is confirmed by model Y 4 (Q L , f ad , N int ), which, even though it excludes Γ ad , explains 99.9 % of the variance in cloud optical thickness. In fact, model 25 Y 4 (Q L , f ad , N int ) supports the applicability of the subadiabatic model since it is able to approximate the cloud optical thickness with high accuracy (RMSE = 0.027).

Principal component analysis
To identify the minimum set of parameters for the represen-30 tation of low-level clouds towards the computation of the CREs, the dominating modes of variability among the different cloud properties have been investigated. Cloud properties from all the case days have been considered. Γ ad is not a cloud property, but since it is considered by the sub-adiabatic 35 model, we decided to include it in the analysis. Towards this direction, one should first map the correlation of the different properties. Figure 4 identifies groups of variables that tend to covary together. The first group comprises τ , Q L , and H that are strongly positively correlated with one another 40 (Pearson > 0.837), while in the second group, CTH, CBH are positively correlated (Pearson > 0.934) albeit inversely correlated with Γ ad (Pearson < −0.85). Alternatively, these two groups could be partly noted as the SW and LW (excluding Γ ad ) properties, respectively. Last but not least, only a 45 weak to mediocre correlation was found between r int , N int , f ad and the other properties. serving the maximum amount of information towards redundancy. This analysis has been conducted by employing the  logarithm of the properties. Since our aim is to retain as few degrees of freedom as possible, the first step is to estimate the optimal number of components needed. As a primary so-5 lution, we used the same number of components as the original variables (nine in number) and we estimated the fraction of variance explained by each component. Table 3 illustrates the resulting cumulative explained variance as a function of each rotational component (RC). The cumulative explained 10 variance suggests the use of four RCs (97.7 %), going from a nine-dimensional space to a four-dimensional space; the variance contributed by the fifth component is 2.1 %. The interpretation of the principal components (not shown here) is based on finding which properties are correlated with each 15 principal component (PC). However, PCs are hard to interpret. Although each new dimension is clearly dominated by some of the cloud properties, the PCs are found moderately or strongly correlated with other properties. However, the rotational component analysis associates each cloud property 20 to at most one rotational component (RC) by maximizing the sum of the variances of the squared correlations between the cloud properties and the PCs (Stegmann et al., 2006). Table 4 summarizes the quality of reduction in Pearson correlations by comparing the residual correlations (RCs) to 25 the logarithm of the original cloud properties. These correlations are either close to unity or zero, allowing only a few moderate correlations and pointing to how each cloud prop-    Table 4), pointing to two clear degrees of freedom. Effective radius is the only property that shows a moderate or strong importance in more than 15 one RCs, namely RC-2, and RC-4, but it could be substituted as a degree of freedom from a well defined DSD, with N int as a primary component and k 2 . Note here that the first two components account for more than 69.3 % of the variance of the cloud properties with the first component related to those 20 that dominate in the SW CRE while, the second component, with those that are of great importance in the LW CRE. The aforementioned analysis points to the reduced set of parameters for the representation of low-level clouds towards the computation of the CREs: N int , Q L , f ad , H, and one of 25 the CTH or CBH.
5 Cloud radiative effects of low-level clouds

Radiative transfer simulations
The input for the radiative transfer simulations was constructed on the basis of ICON-LEM. In other words, temper-30 ature, pressure, and water vapour profiles, surface temperature and pressure, and cloud liquid water content and droplet number concentration are taken from the high-resolution model. For ozone, the profile of the US standard atmosphere is adopted (Anderson et al., 1986). Note here that ICON-35 LEM profiles reach approximately 21 km altitude. Hence, we further extended the atmosphere up to 120 km height according to the US standard atmosphere. Considering the focus of this study the effects of aerosols are neglected and a maximum cloud overlap of cloudy layers is assumed, since only 40 idealized single-layer water clouds are considered. Table 5 compiles the rest input parameters for the radiative transfer simulations that are not adopted by ICON-LEM.

Simulated scenarios
In order to estimate the effects of the bulk microphysical 45 parameterizations and the vertical stratification of the cloud properties on the CREs, the double-moment scheme (ICON-LEM; hereafter reference simulation, Ref.) is confronted against the following scenarios: S1, single-moment scheme, whereby the droplet number concentration follows a fixed 50 profile that varies according to pressure profile (P ), sharing the same liquid water content profiles as in Ref., with, 55 Here, N d,2 is the droplet number concentration in the boundary layer, N d,1 = 50 cm −3 denotes the corresponding value in the free troposphere, and P b is the boundary layer height (800 hPa) . Two different scenarios are considered, where the liquid water path is preserved within the vertical column, but the water content profile is redistributed: In S2, a constant liquid water content profile is used with a fixed droplet number concentration representing the vertically homogeneous cloud model and scenario S3 denotes the equivalent sub-adiabatic profile. Finally, scenario S4 employs the mean vertical profile of N d over all case days (see Fig. 2). For scenarios S1-S3, three individual simula-10 tions have been conducted according to the following droplet number concentration: -N d following the climatology of ECHAM, 220 cm −3 .
-N d weighted over H, N int .
-N d = 480 cm −3 , employing the mean N int for all case 15 days.
Note here that all scenarios share the same Q L and k 2 parameter. The different scenarios are summarized in Table 6.

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For the reference run, the mean and the standard deviation of the modeled CREs for the SW, LW, and NET (SW +LW) radiation are summarized in Table 7. The atmospheric cloud radiative effect (ATM) defined as the difference between CREs at the TOA and BOA is also included. Results are presented 25 for all the case days. Low-level clouds induce a strong negative SW CRE, driven by vigorous scattering, and a positive LW CRE, due to absorption of upward radiation, resulting in a net cooling effect. The warming of the atmosphere due to absorption of SW radiation (∼ 32.9 W m −2 ) is recompensed 30 by the atmospheric LW cooling (∼ −39.2 W m −2 ), leading to a net cooling of the atmosphere (∼ −6.22 W m −2 ). The net CRE is characterized by high variability depending on the distribution of the microphysical and optical cloud properties (see Sect. 5.1.3). Table 8 lists the difference of the mean CREs between the reference and the rest of the simulated scenarios for the SW radiation for both TOA and BOA. In the LW, all the scenarios are able to reproduce the reference mean CREs (see Table   Table 6. Simulated scenarios. For scenarios S1-S3, three individual simulations (sub-cases) have been conducted according to different values for the droplet number concentration.
Double-moment scheme S1 Single-moment scheme S2 Vertical homogeneous model S3 Sub-adiabatic model S4 Mean vertical Nd profile Sub-cases a. 220 cm −3 b. Nint c. 480 cm −3 Overall, the single-moment radiative transfer simulations 45 underestimate the SW CREs for both the TOA and BOA. Starting from S1a (220 cm −3 ), the CREs in the singlemoment run is −40.1 W m −2 less than the double-moment one, with a root mean square error (RMSE) up to 47 W m −2 . The latter differences are attributed to the very low droplet 50 number climatology adopted by coarse climate models (such as ECHAM, ICON-NWP) as compared to ICON-LEM. For a given liquid water path, the smaller the droplet number concentration the larger the resulting effective radius and, accordingly, the smaller the cloud reflectance. In other words, 55 this can be seen as the magnitude of the cloud albedo effect, the so-called first indirect effect (e.g., Twomey, 1977;Ackerman et al., 2000;Werner et al., 2014). For all the case days, a mean value of 480±232 cm −3 is found for the droplet number concentration and a fixed N d profile of 220 cm −3 60 (in the boundary layer) can only represent a small fraction of the bimodal distribution of the droplet number concentration yielded from ICON-LEM (see also Fig. 1). A single-moment run with a more representative value for the droplet number concentration approximates the SW CRE with more accu-65 racy. By employing the mean N int (S1c), the differences in the CRE between the single-and the double-moment runs are considerably smaller, but with quite large scatter; for the BOA (TOA), a RMSE of 23.4 W m −2 (24.3W m −2 ) and a Pearson correlation of 0.964 (0.951) is yielded. The best sce-70 nario is found to be S1b, which is supplied by the droplet number concentration weighted over the cloud geometrical extent, i.e., N int . The differences of the mean CRES between S1b and the reference simulations leads to a RMSE of 11.7 W m −2 and a Pearson correlation of at least 0.994 75 for both the BOA and TOA. The latter small differences are no surprise considering the quite realistic representation of the droplet number concentration in each profile.
Having preserved the liquid water path profile (but redistributed, scenarios 2-3), one can regard the changes in 80 the CREs to the vertical stratification of low-level clouds within ICON-LEM. Comparing the SW CREs yielded by ations on the different droplet number concentration values follows the same pattern as that for the single-vs doublemoment schemes. For instance, in case of the sub-adiabatic scenario (S3) and, going from the least to the most accurate ones, errors (in terms of the RMSE) up to 42.9 W m −2 20 for S3a (220 cm −3 ), 29 W m −2 for S3c (mean N int ), and 10.6 W m −2 for S3b (N int ) are found for both BOA and TOA. Last but not least, by replacing the vertical profile of N d by the mean profile of N d over all case days (see Fig. 2), em-25 ulates the cloud radiative effects of the reference simulation quite well. Accordingly, scenario S4 slightly underestimates the mean SW CREs, with a mean error up to −3.16 W m −2 and a RMSE up to 17.2 W m −2 for both BOA and TOA. In fact, this scenario outperforms the rest scenarios (S1-S3), ex-30 cept from the sub-case b (N int ) in all scenarios. For an illustration of the excellent linear correlation between the reference simulation and S4 by means of a bivariate kernel density (BKD) plot, the reader is referred to Fig. B1 in Appendix B. One can see that the CREs computed by these scenarios 35 are in a very good agreement almost everywhere except towards larger values of the CREs in case of the SW radiation, with Pearson correlations larger than 0.977 for both BOA and TOA.
Note here that the RRTMG model is able to derive the ra-40 diative fluxes only for effective radius between 2.5 µm and 60 µm. For all scenarios, all columns with effective radius outside this range have been excluded.

Impact of the cloud properties on the CREs
For a better assessment of the impact of the different cloud 45 properties on both the SW and LW CREs their correlations have been investigated (in case of Ref.). Table 9 summarizes the corresponding correlations. Due to the monotonic relation between the SW CREs and the cloud properties and the linear relation between the LW CREs and the cloud prop-50 erties, results are presented only in terms of the Spearman (monotonic) and Pearson (linear) correlations, respectively. To demonstrate, Fig. 5 and Fig. 6 illustrate the resulting bivariate kernel density between the cloud radiative effects and the cloud properties that are essential to describe the SW and 55 LW radiation, respectively. Considering the small differences between BOA and TOA, results are only presented for the latter one.
In the SW radiation, there is an excellent monotonic relation between the CREs and τ , Q L , and H for both BOA 60 and TOA, with Spearman correlations higher than −0.987, −0.955, and −0.795, respectively (see Table 9 and Fig. 5), following the second rotational component (RC-2, see Table 4). In particular, the SW CREs increase monotonically with the liquid water path. The latter monotonic relation that 65 is found stronger for lower values of the liquid water path saturates at Q L > 300 g m −2 . In the same direction are the findings for τ (not shown here) and H with the saturation occurring at ∼ 60 and ∼ 0.75 km, respectively. This is no surprise considering their relation to Q L (see Eqs. 10 and 70 16). From Eq. (14), one could expect a similar correlation between the SW cloud radiative effect and the effective ra-dius, but a Spearman correlation below 0.46 (in absolute values) is found for both the BOA and TOA. The latter can be explained by the way the droplet number concentration is derived (see Eq. 4) and the two modes that can be seen in panel (c) of Fig. 5. The Spearman correlations of the SW CRE with 5 the cloud borders and f ad are very weak.
In the LW radiation, changes in Q L (and, thus, in τ and H) possess only a minor influence on CREs (see Table 9) with Pearson correlations below 0.226 (in absolute values). In addition, effective radius and droplet number concentra-10 tion have a moderate effect on the CRE; Pearson correlations are below 0.428 (in absolute values). The cloud radiative effect in the LW is mostly dependent on the macrophysical cloud properties, namely the cloud position and vertical extension that impacts the cloud temperature, following the first 15 rotational component (RC-1, see Table 4). Thus, we would expect a strong linear correlation with CBH and CTH. This holds true, but only in case of the TOA, whereby a Pearson correlation above 0.752 was yielded (see Table 9 and Fig. 6). For the BOA, the correlations are below 0.428 (in absolute 20 values) for both CBH and CTH. This can be explained by the large variability in CBH and CTH among the different case days (see Table 1). It follows that CRE at the BOA is much more sensitive to the macrophysical cloud properties as compared to the CRE at the TOA.

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Finally, we further examined the relation between the first two rotational components and the cloud radiative effects. Confirming our assumption, in Fig. 7, an excellent monotonic relation is found between SW CRE and RC-2 that is comprised by τ , Q L , and H, while a strong linear relation 30 is obtained between LW CRE and RC-1 in case of the TOA, which is described by CBH and CTH. The resulting Spearman and Pearson correlations are larger than 0.948 for the SW and 0.86 for the LW (for TOA only), respectively. Once again, low linear correlation is found between the LW CRE 35 and RC-1 for the BOA. In Fig. 7, panel (d), one can clearly identify several clusters that correspond to different days. With this in mind, we further investigated the latter correlation for each day individually (not shown here). For example, the two lower clusters, with CRE LW,B < 50 W m −2 , To this end, based on the robust evidence over all case days, such a statistical approach, i.e., rotational component analysis, can be employed as an alternative concept for describing the low-level clouds and, consequently, their radiative impact.

Discussion and conclusions
By analyzing simulations of the high-resolution model ICON-LEM, a sensitivity study has been carried out to in- 50 vestigate the suitability of the vertically homogeneous and the sub-adiabatic cloud models to, firstly, serve as conceptual models for the evaluation of the representation of low-level clouds in ICON-LEM and similar high-resolution models, and to, secondly, capture the relevant properties which de-55 termine the cloud radiative effect. Considering the representation of the cloud microphysical processes in ICON-LEM, we additionally have highlighted the differences in cloud radiative effect resulting from the use of a double-instead of a single-moment cloud microphysics scheme.

60
ICON-LEM, with its high vertical resolution, ranging from 25 m to 70 m within the boundary layer, and from 70 m to 100 m further up to the altitude limit for the occurrence of low-level clouds selected for this study (4000 m), enables a significantly improved investigation of the vertical distri-65 bution of microphysical properties of these clouds. Based on six case days, we find that the behavior of modeled liquid water clouds over Germany more closely resembles the subadiabatic than the vertically homogeneous one, in agreement with ground-based observational studies over the same are of 70 interest (Merk et al., 2016). A rather large number of vertical profiles of modeled low-level clouds has been considered in this study and supports the use of the sub-adiabatic model as a conceptual tool for the evaluation of these profiles in high-resolution models, in agreement with previous studies 75 that supported its use in parameterizations in GCMs (Brenguier et al., 2000). According to the sub-adiabatic model, the key cloud properties which determine the cloud optical thickness and, thus, the SW CRE are the liquid water path, the vertically integrated droplet number concentration (over 80 the cloud geometrical extend, in agreement with Han et al., 1998), the sub-adiabatic fraction, and the cloud geometrical extent, which provide a simplified approximation of the vertical structure of clouds. Consistent with this model, we have demonstrated that the cloud optical thickness varies pro-85 portionally to Q 5/6 L and not linearly with Q L , as predicted by the vertically homogeneous model that further supports both observational and theoretical studies (e.g., Brenguier et al., 2000;Merk et al., 2016). In addition, an effort has been conducted to predict the cloud optical thickness result-90 ing from ICON-LEM by the formulation suggested by the sub-adiabatic model. We employed the ordinary least squares (OLS) regression method and we show that, for all case days, the sub-adiabatic model approximates the cloud optical thickness with high accuracy (RMSE = 0.027). In brief, in 95 this prediction, 95.7 % of the variance in cloud optical thickness is explained by the variance in the liquid water path, while the droplet number concentration and the sub-adiabatic fraction contribute 3.5 % and 0.2 % to the total variance, respectively. The sub-adiabatic fraction of clouds is character-100 ized by a large variability (f ad = 0.45 ± 0.21) that strongly varies from day-to-day, but also within the same day, likely driven by entrainment processes. The latter is in agreement with previous studies based on ground-based observations (e.g., Boers et al., 2006;Kim et al., 2008;Merk et al., 2016)  values of f ad , which are often adopted in satellite retrievals of cloud droplet number concentration or cloud geometric thickness (e.g., Zeng et al., 2014) are not supported, and might lead to discrepancies in model validation. Therefore, a much lower value of f ad ranging from 0.4 to 0.6 should be 10 utilized in the sub-adiabatic model to link the cloud optical thickness to the prognostic quantities utilized in GCM parameterizations and determine the indirect effect and cloud feedbacks. The latter value of the sub-adiabatic fraction is close to the one adopted by Grosvenor et al. (2018) for the 15 error assessment of the retrieved N d . The vertical variability of the droplet number concentration was examined. For all the case days, above an altitude of about 2 km, values of N d are about 200 cm −3 and are, thus, close to climatological values, while in the boundary 20 layer, the double moment scheme predicts N d values above 600 cm −3 . Such values are regarded as rather high compared to satellite remote sensing estimates (Quaas et al., 2006;Grosvenor et al., 2018), but such comparison is rather vague considering, firstly, the large uncertainties of the satellitederived estimates of cloud droplet number concentration (Grosvenor et al., 2018) and, secondly, they are not available in high resolution. However, in situ observations, which are considered to be the most accurate approach to determine N d , suggest higher values and, hence, lie closer to those simulated by ICON-LEM. Thus by means of in situ observations, evaluation activities should be conducted for a better charac-10 terization of the droplet number concentration from remote sensing techniques. The latter will scrutinize the doublemoment scheme implemented in ICON-LEM and could potentially lead to better simulations of cloud processes and radiation.

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A principal component analysis and a subsequent varimax rotation (rotational component analysis) of cloud properties have been conducted to explore the covariance of cloud properties and radiative effects, and to identify the dominating modes of variability. The goal was ultimately to uncover po-20 tential shortcomings in the representation of clouds towards the computation of the cloud radiative effects. This analysis reveals that, out of the set of nine parameters considered by us, only four components are sufficient to explain 97.7 % of the total variance. The first component comprises the cloud 25 bottom and top heights, and thus corresponds to the vertical location of the cloud layer in the atmosphere. The second component combines liquid water path, optical thickness, and geometric extent of the clouds, while the third and fourth component are functions of the sub-adiabatic fraction and the 30 cloud droplet number concentration, respectively. By means of such a statistical approach, we offer an alternative concept for describing the CREs, with the first and second component representing the main modes of variability determining the LW and SW CREs explaining 33.8 % and 35.5 %, respec-35 tively. The third and fourth component, while having smaller contributions to the total variance (14.8 % and 13.6 %, respectively), point to clear degrees of freedom. Moreover, they potentially capture signatures of the second (cloud geometric extent, Pincus and Baker, 1994) and first indirect aerosol 40 effects (e.g., Twomey, 1977;Ackerman et al., 2000;Werner et al., 2014). This analysis points to the reduced set of parameters for the representation of low-level clouds towards the computation of the CREs: the column effective properties, i.e., N int , Q L , f ad , H, and one of the CTH or CBH. A similar attempt to provide an alternative concept for the description of the CREs was reported by Schewski and Macke (2003); they tried to correlate domain averaged radiative fluxes from 3D fields with domain averaged properties of cloudy atmospheres.
and Beheng, 2006) has been compared to that of a singlemoment scheme. Special emphasis was given on the characterization of the droplet number concentration and, thus, an effective radius, that could approximate the microphysical and radiative properties of the modeled low-level clouds 60 as simulated by ICON-LEM (reference scenario). Utilizing a droplet number concentration profile that follows the climatology of a coarse atmospheric model (ECHAM), the single-moment scheme would yield values of the SW CRE which are up to ∼ 40.1 W m −2 less than those of the double-65 moment scheme, with a RMSE of ∼ 47 W m −2 . By employing a more representative profile for the N d , i.e., a mean vertical profile of N d for all case days leads to a rather good approximation; the RMSE is below 17.2 W m −2 . This points to the need to better account for prognostic N d calculations. 70 Finally, we investigated the reliability of the vertically homogeneous and the sub-adiabatic model to determine the cloud radiative effects. Overall, the sub-adiabatic cloud model outperforms the vertically homogeneous one for the representation of low-level clouds for calculating their radia-75 tive effects.
Based on our results, the following approach is recommended to evaluate the representation of clouds and their radiative effects as simulated by high-resolution atmospheric models: for the shortwave, the vertically integrated water 80 path should be targeted primarily, which is quite reliably retrieved from remote sensing; recent advances in correcting the PP bias enable the retrieval of the liquid water path with high accuracy Werner et al., 2018). In addition, the cloud droplet number concentration and the 85 sub-adiabatic fraction are of relevance and deserve attention, but their reliable derivation remains challenging both due to the limitations of current remote sensing methods and the lack of validation data on the basis of in situ observations (Grosvenor et al., 2018). In this respect, the rather large val-90 ues of cloud droplet number concentration reported here as predicted by the two-moment scheme of Seifert and Beheng (2006), N d should be scrutinized on the basis of in situ observations. For the computation of the cloud radiative effects, a more representative vertical profile for the droplet number 95 concentration could be used, as long as they can represent the different magnitudes in N d within and above the boundary layer as shown here. For the LW CRE, the cloud base and top heights are the determining factors that are rather well derived from ground-and satellite-based observations, respec-100 tively. It has be noted, however, that the reliable determination of cloud base height from satellites remains challenging. The sub-adiabatic fraction is also of interest, as it controls the geometric extent of clouds for a given value of liquid water path. Based on our findings, the sub-adiabatic model seems 105 to be better suited than the vertically homogeneous model for the evaluation of the representation of clouds in models.
In future work, the results presented here should be combined with efforts to also take into account the impact of horizontal cloud variability, and in particular of the cloud frac-110 tion, which are well-known factors of relevance for the cloud radiative effect. In order to link deficiencies in the CRE to the model representation of cloud properties, an effort should be made to simultaneously evaluate the ICON-LEM-based fluxes and cloud properties discussed here to observations, (A1) Γ stands for the gamma function. For cloud droplets ν = ξ = 1 (see Table 1 in Seifert and Beheng, 2006), the zeroth and first moments of the mass size distribution that denote the droplet number concentration and the liquid water content, 10 respectively, are derived, According to Seifert and Beheng (2006) and Petty and Huang (2011), a power law is applied for the mass-size re-20 lation, D denotes the geometrical diameter. In case of spherical particles, α = π·ρw 6 and b = 3, with ρ w being the water density. In Table 2 in Petty and Huang (2011), one can find the trans-25 formation factors between the mass of hydrometers and the diameter of the hydrometers, Given the aforementioned relations, the formula describing the modified gamma distribution of the DSD is, Accordingly, the ηth moments of the DSD are given by,
For the reconstructed DSD, n (D), the zeroth moment (M 0 ) stands for the droplet number concentration. The volumeequivalent radius, r V , is derived from the third moment, Appendix B: Correlation between reference simulation and scenario S4 In sect. 5.1.2, by conducting idealized radiative transfer simulations, we estimated the impact of the representation of cloud properties in ICON-LEM on the cloud radiative ef-45 fects (CREs). Special emphasis was given on identifying the droplet number concentration (N d ), which approximates the microphysical and radiative properties of low-level clouds as simulated by ICON-LEM (reference scenario). A radiative transfer simulation, which employs a mean vertical profile of 50 N d of all case days (scenario S4), approximates the CREs of the reference scenario quite well. Figure B1 depicts the excellent linear correlation between the reference simulation and S4 by means of a bivariate kernel density (BKD). Appendix C: Differences of the mean CREs between the reference simulation and the new scenarios for the LW radiation Table C1 lists the difference of the mean CREs between the reference and the rest of the simulated scenarios for the LW 5 radiation for both TOA and BOA. All scenarios are able to reproduce the reference mean CREs; the difference of the mean CREs is below ∼ 0.55 W m −2 (in absolute values). (DKRZ) as part of the HD(CP)2 project.
Code and data availability. The Python RRTMG interface (pyRRTMG) used in this study is available at https://github.com/hdeneke/pyRRTMG. Author contributions. VB conceived and refined the overall struc-15 ture of the investigation, based on discussions with and feedback from all co-authors. VB carried out and refined the data analysis. HD implemented the Python interface to RRTMG used in the analysis. VB wrote the draft manuscript, with all authors contributing to the interpretation of the results and to its improvement.