1. Introduction
Recent advances in supercomputing have introduced a “gray zone” in the representation of cumulus convection in general circulation models (GCMs), in which this physical process is getting partially resolved (Wyngaard 2004). Recent studies have defined the gray zone as the range of resolutions within which the subgrid and resolved contribution to turbulence and transport are of the same order of magnitude (e.g., Dorrestijn et al. 2013; Honnert et al. 2011). Existing parameterization schemes, built on the assumption that cumulus populations are fully sampled in the GCM grid box, often lack the scale awareness and scale adaptivity to conceptually and practically deal with this situation. How to best achieve this is still an open research question.
Research into the spatial structure of cumulus cloud populations goes back decades (e.g., Cahalan and Joseph 1989; Sengupta et al. 1990; Nair et al. 1998) but has recently intensified because of the arrival of the gray zone problem. A natural way of characterizing the scale dependence within a cumulus population is the size distribution of a cloud field, which has been scientifically established for many cloud regimes using a large variety of instrumentation (e.g., Plank 1969; Raga et al. 1990; Benner and Curry 1998; Zhao and Di Girolamo 2007; Yuan 2011) and finescale cloud-resolving simulation (e.g., Neggers et al. 2003; Rieck et al. 2014; Senf et al. 2018). A characteristic feature of the size density of cumulus cloud number [a cloud size density (CSD)] is its shape, which has been described by lognormal, exponential, and power-law functions. These shapes are argued to reflect the underlying physical–dynamical processes of cloud formation, interaction, and subcritical percolation (Newman 2005; Cohen and Craig 2006; Ding et al. 2014; Garrett et al. 2018). The size range covered by the distribution has been found to differ between marine and continental cloud populations (Sakradzija and Hohenegger 2017). Over land, the maximum size of the distribution exhibits a distinct diurnal dependence. With many studies relying on finescale simulations of cumulus cloud populations, the confrontation of simulated and observed cumulus populations has recently become an important focus point. Both area-covering measurements (Kassianov et al. 2005) and vertically pointing measurements (Lareau et al. 2018) have started to provide a wealth of new information about cumulus cloud populations at high resolutions and frequencies.
Building on these new results and insights, a new class of convection schemes has recently emerged in which the transport is explicitly formulated in terms of discretized CSDs (Wagner and Graf 2010; Park 2014; Neggers 2015). In essence, these schemes adopt the spectral approach in convective modeling as first proposed by Arakawa and Schubert (1974). Compared to bulk convection schemes, the CSD is a new variable that requires closure, concerning both its functional form and its range. Differences exist among schemes in how the CSD is reconstructed. But a general benefit is that the CSD is in general well observable, allowing such models to be constrained by both existing and new datasets. Furthermore, scale awareness is by definition present at the foundation of the scheme (Neggers 2015). The reconstructed CSD can be size filtered, which introduces scale adaptivity in the parameterized convective transport and clouds (Brast et al. 2018).
A complicating factor is that the CSD is not a global constant and exhibits both external and internal variability. External variability arises from various sources such as large-scale synoptic conditions or surface properties (Rieck et al. 2014). In contrast, internal variability arises because of spatial sampling issues. Given a well-defined CSD reflecting a cloud population covering an infinitely large area, a limited spatial sampling leads to stochastic variability (e.g., Nair et al. 1998; Cohen and Craig 2006). This situation applies to convective modeling in the gray zone, in which the grid box has become too small to include the full population. In CSDs diagnosed in LES realizations of a relatively small domain, this stochastic signal is sometimes present at the largest cloud sizes, which are often poorly sampled (Neggers et al. 2003). The spatial organization in a cloud field can add to this stochastic variability by introducing perturbations in the cloud populations on scales much larger than the boundary layer depth (e.g., Seifert and Heus 2013). Physical–dynamical processes that drive spatial organization include cold pool formation (e.g., Schlemmer and Hohenegger 2014) and oscillations (Sakradzija et al. 2015; Feingold et al. 2017).
In recent years, super-large-domain LES has become computationally feasible (Khairoutdinov and Randall 2006; Satoh et al. 2008; Heinze et al. 2017) and has offered new ways to study the CSD and its variability. The availability of high spatial resolution in combination with a large domain size yields cloud size distributions that can freely form and evolve across a range of scales that is less and less artificially constrained at both ends. The large domain allows more complete sampling of the cumulus population, as well as better coverage of the rarely occurring largest clouds. With the cumulus clouds still resolved, the ever-larger domain of the simulation also allows mesoscale fluctuations to form naturally and start to affect the CSD. The opportunities created by superlarge LES to investigate the behavior of cumulus CSDs and their interaction with mesoscale organization have only recently begun to be explored (e.g., Senf et al. 2018).
The goal of this study is to use large-domain simulations to investigate and quantify the internal variability in a cumulus CSD. We ask how this variability depends on (i) the domain size of the analysis and (ii) the degree of spatial organization within the cloud population. The aim is to work with domain sizes large enough to approach full sample size when diagnosing the CSDs. While this study is exclusively limited to shallow cumulus convection, two distinctive convective regimes are investigated. The first is a slowly organizing marine cumulus case based on the Rain in Cumulus over the Ocean (RICO) field campaign (Rauber et al. 2007). The second regime reflects continental summertime conditions at the Southern Great Plains (SGP) site of the Atmospheric Radiation Measurement (ARM; Stokes and Schwartz 1994) program. A subdomain analysis is performed for all fields to quantify the impact of subsampling on the CSD. The difference in organization speed between the two regimes then provides insight on how this process affects the internal variability of shallow cumulus CSDs. The implications of the obtained results for our understanding of cumulus populations in general and for stochastic cumulus parameterization in the gray zone in particular will be discussed.
2. Method
a. Large-eddy simulations
This study makes use of LES results for two well-defined shallow cumulus regimes, in which the pace of the spatial organization differs considerably. The first, a slowly organizing cloud regime, reflects marine subtropical subsidence conditions as observed during the RICO field campaign (Rauber et al. 2007). The spatial organization in this case is related to warm precipitation processes, producing cold pools with low cloud mass that are separated by convergence lines where clouds tend to congregate. The spatial organization in this case is a slow process, taking multiple days. The second, faster organizing cloud regime reflects diurnal cycles of continental shallow cumulus at the ARM SGP site. For this regime, five day-long cases are chosen, which are all part of the archive of simulations for the ARM SGP site in 2016 generated by the LES ARM Symbiotic Simulation and Observation Workflow (LASSO; Gustafson et al. 2017b). The cases include 18 May, 19 and 25 June, 16 July, and 18 August 2016 (hereinafter referred to as the 20160518, 20160619, 20160625, 20160716, and 20160818 cases). These days are selected because they best reflect the prototype view of diurnal cycles of shallow convection, with most daytime cloudiness associated with surface-driven convective clouds and with more or less clear-sky conditions before and afterward.
The LES experiments for these six cases have already been described in detail in the literature, and only the details relevant for this study will be summarized here. The RICO composite shallow cumulus case (vanZanten et al. 2011) is simulated using the University of California, Los Angeles, LES (UCLA-LES) code, as described in detail by Seifert and Heus (2013); their simulations are also used in this study. The simulated domain size is 
Although two different LES codes are used, the experiment setups share some important aspects. A spatial resolution of 25 m × 25 m × 25 m is always applied, as is adaptive time stepping. Both the UCLA-LES and the MicroHH codes apply the Smagorinsky scheme for subgrid transport of momentum, energy, and heat, and both make use of the warm-cloud double-moment microphysics scheme of Seifert and Beheng (2001). Horizontally periodic boundary conditions are applied, as well as a sponge layer in the top third of the domain to dampen any gravity waves. A prescribed surface temperature is used in both the RICO and LASSO cases, the only difference being the presence of a diurnal cycle in the latter.
The evolution of the simulated cloud population for the RICO case is illustrated in Fig. 1, showing the vertically projected cloud mask at each storage time point. The cloud field at the first two time points still looks more or less homogeneously distributed, but increasingly, the cloud field becomes more organized, with large cloud structures appearing after about one day of simulation associated with cold pool development (Seifert and Heus 2013). Figure 2 shows the same for the LASSO 20160518 case, revealing the distinct diurnal evolution of the cloud population. The spacing between the largest clouds increases during the day, likely reflecting the continuous ongoing deepening of the boundary layer that is typical of such diurnal cycles (Brown et al. 2002; Zhang et al. 2017). Organization in the cloud field becomes apparent during the final period, with small clouds increasingly clustering around the larger ones, being separated by large areas without any cloud mass. This may be related to the decay of the largest clouds at the end of the day, when the surface forcing of the turbulence weakens.
Vertically projected cloud masks for the six time points during the LES of the RICO case that are considered in this study. The domain size shown is 51.2 km by 51.2 km. A grid point is masked (black) when any condensate exists in the vertical column.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
As in Fig. 1, but for the cloud mask for six time points during the LASSO 20160518 case. The domain size is 25 km by 25 km.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
To provide further confidence in the realism of the simulations, the LES cloud cover for the five LASSO cases is compared to two ARM observational products. The Active Remote Sensing of Clouds (ARSCL) value-added product (Clothiaux et al. 2000, 2001) combines lidar and radar measurements, while the total sky imager (TSI; Kassianov et al. 2005) retrieves cloud cover from wide-angle imagery. The comparison for the five cases is shown in Fig. 3, only showing daytime hours. While some differences exist between the two measurements, they agree reasonably well on the general evolution and amplitude of the cloud cover. The model does reproduce these trends; however, the amplitude seems to be a bit underestimated. This slight bias has been revealed in other recent LES evaluation studies at supersites (Schalkwijk et al. 2015; Zhang et al. 2017). In addition, on 18 May and 18 August, the LES seems to underestimate cloud cover in the morning, which we speculate is due to either (i) high-altitude cloudiness or (ii) spinup effects. Despite these minor shortcomings, the results do suggest that the experiment setup captures the prototype diurnal variation in cloudiness present in the observations to a reasonable degree, as well as its modulation due to large-scale forcing. This agreement suggests the simulations are representative of nature and justifies their use for subsequent cloud studies.
Simulated and observed total cloud cover for the five LASSO cases investigated in this study. TSI is an estimate of the total sky cover and is only available during daytime, while ARSCL represents cloud cover as diagnosed over only the lowest 5 km. The gray bars denote the period before sunrise and after sunset.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
b. Deriving cloud size distributions
Cloud size distributions can be derived in many ways, involving choices concerning what defines a cloud and its size. In this study, a simple definition is adopted, to be applicable to both observed and simulated cloud populations, and that allows the use of past data records as much as possible. The first simplification is to disregard cloud life cycles by only analyzing standalone instantaneous three-dimensional fields. Often, the time frequency of observations of cloud populations is simply too low to allow detection of cloud life cycle properties. Also, we hypothesize that the CSD internal variability should already be detectable in instantaneous snapshots. The second choice is to use the presence of any cloud condensate mass to label a grid box as cloudy or noncloudy. This allows the analysis to be used with any LES, no matter how complex its representation of cloud microphysics. The third choice is to define cloud size as the square root of the projected area fraction of the object defined on the 3D grid. This choice is made to allow comparison to previous CSD studies, which often relied on this definition.
Cloud number densities 
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
c. Subdomain analysis
The subdomain analysis method used by Dorrestijn et al. (2013) to investigate the scale dependence of turbulent transport is applied here to cloud number N. To this purpose, the full horizontal LES domain 
3. Results
a. Number densities in subdomains
The normalized cloud number density 
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
Figure 5a shows the analysis for n = 32 so that the subdomain equals the full domain. As a result, the plotted data are equivalent to that shown in Fig. 4a. A few aspects catch the eye. First, the power-law exponential fit reproduces the shape of the CSD in this case to a reasonable degree. Second, the horizontal spacing in the left tail of the PDF reflects the horizontal discretization of the LES, allowing only a few specific cloud sizes. Third, the vertical spread in the data increases toward the largest sizes, which is an expression of an increase in the CSD internal variability. Also, the data tend to organize along horizontal lines, with the lowest line of points (with lowest 
The following panels show the analysis for decreasing subdomain size L, for 
The probability P that no clouds are present in a subdomain, as a function of cloud size l, for a range of subdomain sizes L for the first snapshot of the RICO case.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
b. Stochastic variability
The next step is to quantify the internal variability of the CSD due to subsampling as analyzed in the previous section. This is achieved by calculating the variance in cloud number 
Log–log plot of the normalized standard deviation in cloud number 
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
The fact that the power-law scaling is similar for all cloud sizes, and that the lines of constant cloud size are more or less equidistant in the vertical, motivates reformulation in terms of two dimensionless groups. Figure 7b shows the same log–log plot but now with subdomain size L divided by cloud size l. This transformation yields a data collapse. A first-order estimation of the slope suggests a power-law exponent 
c. The impact of subsampling
Schematic illustration of the subsampling of an equally spaced single-size cloud population. The red box indicates the subdomain of size L, while the blue grid boxes represent the spatial unit occupied by a single cloud and its spacing 
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
What is also clear is that the effect of subsampling is first experienced by larger clouds. We speculate that this is explained by the observation that larger clouds typically have larger spacing 
d. The impact of spatial organization
The analysis is now repeated for all six RICO fields, which become increasingly organized as time progresses, as shown in Fig. 1. The time evolution of the relation between 
(a) As in Fig. 7a, but for only a single cloud size l = 25 m as a function of time in the RICO case. (b) Power-law exponent b as a function of cloud size and time in the RICO case, resulting from linear fits in 
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
More insight is provided by Fig. 9b, showing the power-law exponent for all cloud sizes l as a function of time. Two features stand out. First, the maximum cloud size in the domain increases with time, approaching 10 km at the t = {24, 32}-h time points. This reflects the emergence of large cloud structures as visible in Fig. 1, although their number is still very small. Second, the impact of the spatial organization in the population on the power-law exponent appears limited to the smaller end of the distribution and gradually disappears above about l = 250 m. These two findings suggest that spatial organization apparently affects both ends of the CSD. On the one end, the size of the largest cloud increases, while on the other, the variability in the number of small clouds is enhanced. This reflects that small clouds then “live” on top of larger-scale thermodynamic structures in the boundary layer, such as cold pools or convergence lines, which favor or inhibit their formation.
To investigate the robustness of the impact of spatial organization on the CSD variability, the analysis is now repeated for all LASSO cases, as shown in Figs. 10a–e. The time evolution of the CSD is similar in all LASSO cases, with a gradually increasing maximum cloud size in the period after cloud onset. A reduction in the power-law exponent at the smaller cloud sizes is also present; however, it is only visible toward the end of the day. A clear difference with the RICO case is the much shorter time span during which this happens. This suggests that the physical process of spatial organization causing this behavior is of a slightly different nature. After inspecting the cloud mask fields in Fig. 2, one notices that, toward the end of the day, small clouds start to surround the bigger clouds in the population. One speculates that this is caused by big clouds falling apart toward sunset, when turbulence dies. In that sense, one expects that this behavior is a typical and prototype feature of diurnal cycles of shallow cumulus over land. This motivates compositing the five cases to enhance the statistical significance, as shown in Fig. 10f. The compositing indeed makes the contoured field smoother. In addition, the compositing emphasizes that the reduction of the power-law exponent in the last hours of convective cloud existence is a robust feature.
As in Fig. 9b, but for the LASSO cases (a) 20160518, (b) 20160619, (c) 20160625, (d) 20160716, and (e) 20160818 and (f) a composite average of (a)–(e). Nonsimulated time periods are shaded light gray.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
4. Discussion
The results obtained so far for the RICO and LASSO cases suggest that the impact of spatial organization on the variability in cloud number is superimposed on the impact of subsampling alone. Two different mechanisms of spatial organization have been encountered in this study, including cold pool dynamics in the RICO case and cloud decay in the LASSO cases. But apparently, the impact of subsampling always takes place in the same way. This is further illustrated by Fig. 11, showing the scaled variances of all time points from all cases in one frame. All cloud sizes l > 250 m are included, thus excluding the size range that is most affected by organization in the cases considered. Combining all fields from all cases still yields a data collapse for this size range. A power-law fit is included in Fig. 11 (dashed line), yielding 
As in Fig. 7b, but for all snapshots from all six cases. All combinations of 
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
The scaling relation (10) has various potential uses. For example, it can inform the development of parameterizations in the gray zone of moist convection by acting as a benchmark relation that stochastic parameterizations of cumulus cloud populations have to reproduce. This is in particularly applicable to schemes that are formulated in terms of reconstructed CSDs. Complex population dynamics models might let the number density grow from interaction between sizes, while simpler ones using assumed functional forms for the CSD could have the impact of subsampling and spatial organization as established in this study superimposed. Regardless of the design of the stochastic parameterization, they can be called successful when they reproduce the 
The results obtained in this study highlight the important role of cloud spacing in controlling the stochastic variability. Normalizing subdomain size by cloud size yields a data collapse in the scaled variance, argued here to reflect that cloud size is proportional to the cloud spacing. The data collapse suggests that the relation between cloud size and cloud spacing could be pretty robust. More research is needed to confirm and quantify this proportionality. Previous observational studies have indeed hinted at its existence in nature (e.g., Joseph and Cahalan 1990; Sengupta et al. 1990), but typically, these relations still carried considerable scatter. In addition, the definition of cloud spacing used in most previous studies is not equivalent to what is used in this study, which is the spacing between clouds of equal size. Exploring both observed and simulated datasets to investigate various definitions of cloud spacing is a future research topic.
This study also sheds more light on the nature of spatial organization in shallow cumulus cloud fields. Two types were encountered in the cases covered, including cold pool formation in the marine case and large cloud decay in the diurnal cycle cases. While this study does not explain how the organization takes place, what the results do reveal is that both types of spatial organization are associated with a similar shift in the power-law exponent in the relation between 
Some aspects of our analysis could affect the scaling. The first is numerical and concerns the LES discretization, which could artificially affect the variability of the smallest clouds. This is investigated in more detail in appendix A. We find that while the power-law scaling in the cloud number variability (slope) is unaffected, the amplitude increases at cloud sizes smaller than 4Δx. A close proximity to discretization thus slightly enhances the variability in cloud number. Note that these numerically affected clouds were not included in the fitting exercise shown in Fig. 11. Nevertheless, because the spatial organization affects the smallest clouds most, additional simulations at even higher resolution are still advisable to investigate at which point convergence takes place. A second important aspect concerns the definition of cloud size, for which more complex alternatives can be used. For example, cloud objects could be tracked through time, which might change how their variability behaves. Perhaps this study, based on a simple definition of size, can serve as a starting point for such analyses.
5. Summary, conclusions, and outlook
The main results of this study can be summarized as follows:
Multiple three-dimensional snapshots from various large-domain LESs of marine and continental shallow cumulus cases are used to investigate the behavior of variability in cumulus cloud number due to subsampling and spatial organization.
Strong power-law scaling is found in the relation between cloud number variability and subdomain size, reflecting an inverse linear relation.
Cloud spacing as a function of cloud size is a crucial parameter, explaining the data collapse across cases and time points.
The impact of spatial organization on the variability in the CSD is found to act on top of this baseline model by enhancing its amplitude, in particular at the smaller cloud sizes. The power-law scaling is still preserved but with a reduced exponent. This impact reflects that the small clouds start to live on top of larger-scale thermodynamic structures, favoring or inhibiting their formation.
A simple expression for the CSD internal variability is proposed that is formulated in terms of two dimensionless groups and that captures the impact of both subsampling and spatial organization.
This scaling relation provides a new way to objectively estimate the degree of organization Borg in simulated or observed cumulus cloud populations as the deviation from a theoretical power-law scaling.
Compositing all continental cumulus cases suggests the existence of a prototype diurnal time dependence in the spatial organization Borg, partially reflecting the decay of large convective cumulus clouds toward sunset.
The subdomain analysis adopted in this study needs to be repeated for many more cumulus cloud scenes in order to broaden the parameter space of environmental large-scale conditions and thus enhance the statistical significance of the obtained results. This is required to better calibrate constants of proportionality and also to better understand the nature of spatial organization in cumulus cloud populations. Of particular relevance is the behavior of nearest-neighbor spacing between clouds of a certain size, as this dependence is at the foundation of the scaling relation found in this study. In an ongoing effort by the authors, data from super-large-domain LES experiments with the new Icosahedral Nonhydrostatic (ICON; Zängl et al. 2015) model in the Caribbean dry season are used for this purpose. In addition, existing observational datasets of shallow cumulus cloud populations are revisited to seek observational support for the scaling found in the simulations. These data consist of satellite imagery as well as data from cloud-detecting ground-based imaging and remote sensing instrumentation at the ARM SGP site.
Acknowledgments
This research was supported in part by the U.S. Department of Energy’s Atmospheric System Research, an Office of Science, Office of Biological and Environmental Research program, under Grant DE-SC0017999. Data from the DOE as part of the Atmospheric Radiation Measurement (ARM) Climate Research Facility Southern Great Plains site were used to support the simulations. We thank Andy Vogelmann and the LASSO team for providing the forcing data defining the five LES cases at the ARM SGP site and for kindly providing the ARSCL data. The resimulations of the LASSO cases with the MicroHH code were performed using the supercomputing facilities of the Ohio Supercomputer Center (2016). We finally thank three anonymous reviewers for their constructive comments, which greatly helped to improve this manuscript.
APPENDIX A
Impacts of Discretization
In Fig. A1, the σ/N scaling is investigated for cloud sizes smaller than 10 times the LES discretization. For the clouds smaller than about 4Δx, the scaling relation starts to be situated above the baseline model (dotted line) but remains more or less parallel. This implies that, at sizes near the discretization scale, the CSD variability is somewhat larger than can be expected from subsampling alone. We speculate that this numerical artifact reflects on–off cloud behavior in single grid boxes that artificially boosts the variability in their number. However, most importantly, the power-law exponent is more or less unaffected by the proximity to discretization, meaning that the scaling is maintained even for barely resolved clouds.
As in Fig. 7b, but for clouds with l < 250 m.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
APPENDIX B
Sensitivity to CSD Bin Width
The bin width that was chosen to discretize the CSD might affect the scaling relation in (10). To examine this possible impact in more detail, the scaling is now recalculated for a specific size range of the CSD in the first snapshot of the RICO case for various bin widths Δl. The size range 200 < l < 400 m is chosen to this purpose, as it is far enough removed from discretization (see appendix A) but covers sizes still small enough to be well sampled. For Δl ∈ {25, 50, 100, 200} m, this yields a number of {8, 4, 2, 1} estimates of relation (10), respectively. The averages over these estimates are shown in Fig. B1. Increasing the bin width shifts the relation downward. This means that while cloud number N increases with larger bin width, the variability in cloud number σ also increases but to a lesser extent. This probably reflects the smaller chance of zero cloud occurrence at a larger cloud number per bin. However, most importantly, the power-law exponent is always preserved and is independent of the bin width.
As in Fig. 7b, but for the sensitivity to CSD bin width Δl. Only the size range 200 < l < 400 m is considered.
Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-18-0194.1
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