The relationship between cloud-top entrainment and cloud hole size at the top of stratocumulus clouds is explored with large-eddy simulations and a Lagrangian parcel tracking model. The cloud-hole size distribution follows a negative power law, in excellent agreement with satellite observation at 15 m resolution. As a result of the steep decrease in the number of holes with increasing size, the number of entrained Lagrangian parcels also decreases with increasing hole size (negative power law), even though the number of entrainment events per hole increases with increasing hole size (positive power law). Thus, entrainment preferentially occurs in small holes. Further analysis shows that the domain averaged entrainment velocity is a reasonable approximation to the domain averaged cloud-hole vertical velocity, and dominated by contributions from the small holes.
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 Stratocumulus clouds play a significant role in shortwave cloud forcing [e.g., Randall et al., 2007] and there is significant uncertainty in stratocumulus cloud feedback in a warmer climate [e.g., Bony and Dufresne, 2005; Caldwell and Bretherton, 2009; Xu et al., 2010; Caldwell et al., 2013]. Cloud-top entrainment in the stratocumulus topped boundary layer (STBL) has been recognized as one of the critical processes regulating the maintenance of stratocumulus [e.g., Lilly, 1968; Wood, 2012] and is the focus of this work. Entrainment is considered here to be a one-way mass transport process from the free atmosphere into the STBL through the entrainment interface layer (EIL, or inversion layer); hence, the STBL rises with entrainment. Within the EIL, “cloud holes” (i.e., regions of depleted liquid water) have been shown to be the preferred locations of entrainment [e.g., Gerber et al., 2005; Yamaguchi and Randall, 2012, hereafter YR12].
 Although the physical properties of cloud holes have received significant scrutiny [e.g., Nicholls, 1989; Wang and Albrecht, 1994], cloud-top entrainment in relation to cloud hole size distribution for STBL has rarely been studied. Analyses with aircraft observations have shown that the cloud hole size distribution approximately follows a negative power law [Duroure and Guillemet, 1990; Korolev and Mazin, 1993; Gerber et al., 2005]. Recently, Gerber et al. [, manuscript submitted to J. Geophys. Res.] show that the cumulative entrainment flux as a function of cloud hole size also follows a power law, with significant contribution from smaller holes.
 The goal of this study is to explore entrainment in the STBL in terms of cloud hole size using large-eddy simulation (LES). The fundamental question posed is whether entrainment occurs at some preferential scale. To identify entrainment, a Lagrangian parcel tracking model (LPTM) [YR12] is utilized, and a large number, O(107) of mass-less parcels are tracked during runtime. The LPTM simplifies the analysis of entrainment, and provides credible statistics, provided that the number of entrained parcels is sufficiently large.
 We prepared two LES data sets based on the nonprecipitating nocturnal stratocumulus case of Stevens et al. : a small domain LES (hereafter, SD) performed by YR12, and a large domain LES (hereafter LD). Simulations were integrated with the System for Atmospheric Modeling [Khairoutdinov and Randall, 2003]. SD covers a 3.2 × 3.2 km2 horizontal domain on a 5 m horizontal grid, (∆x), while LD covers over 23 km in each horizontal direction with 1944 grid points and ∆x = 12 m, which is much larger than the peak wavelength of the energy spectrum for horizontal velocity and scalars [de Roode et al., 2004; Yamaguchi et al., 2013, manuscript submitted to J. Atmos. Sci.]. The vertical resolution for SD is 2.5 m while LD uses a stretched grid; 12 m for the subcloud layer and 5 m for the cloud layer and EIL. Because SD is performed at higher spatial resolution, it represents small scale turbulence and small cloud holes better than LD, while LD is advantageous for characterization of the larger scale structure and larger cloud holes.
 The LPTM is used for the last hour: parcels are placed near the horizontal mean EIL top, (zET), and then tracked with time. With the data saved every minute, entrained parcels are extracted based on the method outlined by YR12. Over its history, a parcel is classified as entrained when it reaches a level below the mean EIL bottom, (zEB), for the first time. The heights for zET and zEB are diagnosed with the method outlined in YR12, which uses the characteristic shape of the second moment profile of thermodynamic scalars (e.g., liquid water static energy) in the EIL. An instantaneous picture of the liquid water path (LWP) superimposed on the entrained parcels is shown in Figure 1a. A fractal structure of holes on the cloud surface is clearly evident.
 Cloud holes are identified based on the EIL LWP (i.e., LWP between zEB and zET). Grids with EIL LWP smaller than 3 g m–2 (2 g m–2) for SD (LD) are connected by a four-neighbor searching method to form unique cloud holes and the information is recorded every minute over the last hour. The EIL LWP thresholds are based on the 1 h, horizontal mean minus 1 standard deviation. We chose this threshold because the probability density functions of EIL LWP for the two simulations are very similar. Changing the threshold has only marginal influence on the shape of the cloud hole size distribution. A cloud hole map with entrained parcels is presented in Figure 1b. The area of a cloud hole is calculated as ahole = n(∆x)2, where n is number of grid points in the hole. For the large cloud holes, it is of note that the entrained parcels tend to be preferentially located near the edge of the holes.
 Entrained parcels located outside the cloud holes when they are classified as entrained are discarded in further analysis. Due to the 1 min output interval, these parcels may have moved away from the cloud holes, or the cloud holes may have closed, as seen in the movie provided by YR12. For this reason, the number of entrained parcels still in cloud holes is sensitive to the threshold used to identify the cloud hole grid. As a result, the number of entrained parcels is approximately 2 × 106 for LD, while it is only 9 × 104 for SD.
3 Power Laws
 The distributions of the number of the cloud holes, (nhole), the number of entrained parcels, (nparcel), and the number of parcels per hole, (npph ≡ nparcel / nhole), are plotted as a function of ahole for the two simulations (Figure 2). The best fit lines for nhole and npph are obtained using a least-squares linear fit to the logarithm of the variables (y ~ xk) for cloud holes satisfying nhole ≥ 10 and nparcel ≥ 10 to avoid a bias arising from small samples for large holes. The curve-fit for nparcel is derived as nparcel = nholenpph. The emphasis in the following discussion is on the power (k).
 The power law relationship for nhole is clearly evident with k ≈ –1.8 for both simulations; Small holes are much more numerous than large holes. The npph distribution shows that large holes attract many more parcels per hole than small holes. The distribution follows a power law with k ≈ 1. The suitability of these power-law fits for larger holes is better for LD than SD. For the derived nparcel curve-fit, the exponents are different for two simulations, probably due to the relatively small number of entrained parcels analyzed for SD; each nparcel distribution, however, follows the derived power law well for smaller holes. On the other hand, at larger hole sizes the distribution forms a “V” shape. Because with increasing domain size the power law fit improves for larger holes, we argue that poor sampling statistics at larger hole sizes create the “V” shape and that points would tend to a straight line (i.e., power law) if the domain were much larger and progressively more points were to meet the criterion of nhole>10. The negative power law behavior of nparcel shows that on balance, entrainment occurs preferentially in smaller holes. We further argue that the power laws would hold for unresolved small holes because the deviation from the power law for the small holes becomes smaller as resolution increases. We note that the same conclusion (with different exponents) is obtained with cloud hole perimeter.
 The cloud hole size distribution is further evaluated with an image taken by the visible and near-infrared (VNIR) instrument of the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) [Kahle et al., 1991]. Images taken by VNIR cover a 60 km footprint with ∆x = 15 m. An image off the coast of San Diego, California is shown in Figure 3a using the VNIR band 1 (0.52–0.60 µm) of the Level-1B registered radiance data. As shown in Figure 3b, the cloud hole size distribution for the ASTER image follows a power law, in good agreement with our simulations. In the figure, the mean minus 1 standard deviation of the radiance data is used as a threshold to diagnose holes. We note that the power is sensitive to the threshold, and varies from –1.5 to –1.9 depending on the threshold. We also note that direct comparison of powers between LES and ASTER is problematic because in LES, holes are based on a column LWP over a shallow layer (EIL) whereas ASTER radiances respond to cloud depth and morphology, which might hide some holes.
4 Entrainment Velocity
 We define wEIL (bEIL) as the mass-weighted EIL-mean vertical velocity (buoyancy) at each horizontal grid point averaged over the total number of grids occupied by ahole:
where φ is either w or , where Tv is virtual temperature and the overbar denotes horizontal mean, and ρ is air density. We also define σhole as the areal fraction occupied by ahole, thus, the domain (and 1 h) mean wEIL is given as .
 The distributions for wEIL, bEIL, and σholewEIL are plotted in Figure 4. All parameters depend on ahole. The wEIL and bEIL plots show that, on average, the smallest holes have the fastest downdraft and are neutrally buoyant, whereas the larger holes have very slow downdrafts and are positively buoyant. The two simulations overlap quite well, suggesting a robust relationship. Plots of the EIL-mean w field (not presented) reveal that the smaller holes are occupied mainly by downdrafts while the larger holes are filled with a mixture of updrafts and downdrafts, such that on average, wEIL is negligible for large holes. Plots of the EIL-mean b field (not shown) clearly show that the negative buoyancy is located at cloud top, and holes are generally neutrally or positively buoyant. For large holes, we expect bEIL ≈ 0, if the average were to be computed only for the edge region where the majority of entrainment occurs (Figure 1). The plot for σholewEIL suggests that the contributions to from cloud holes larger than some size are negligible, and the finite contributions for larger holes come from the sampling bias due to the domain size. The distribution of σholewEIL on a logarithmic abscissa forms a shape close to a power law, and matches well with the computed fit line.
 Lastly, we demonstrate that if is subsidence, then is a reasonable approximation to the domain mean entrainment velocity (). In other words, . As described in Yamaguchi et al. , is estimated based on the relaxed form of the EIL budget equations of a mixed layer [Lilly, 1968]. For the 1 h analysis period, mm s–1 and mm s–1 for SD, and mm s–1 and mm s–1 for LD. The fact that is not surprising because cloud holes are the dominant locations for entrainment, and thus for . We note that the model-derived values of are reasonable, but tend to be a little larger than observations [e.g., Stevens et al., 2005].
 Both LES and satellite data show that the size distribution of cloud holes at the top of stratocumulus follows a negative power law. Because of the steepness of the slope of this power law, the number of entrainment sites, and their contribution to bulk entrainment velocity also follow negative power laws. The negative power law distribution of cloud hole sizes should come as no surprise given the ubiquity of power laws observed in nature, the earlier results from aircraft observations, and negative power law distributions of cloud sizes for shallow cumulus clouds [Benner and Curry, 1998].
 Analysis of two simulations has allowed us to explore the effects of resolution, domain size, and number of entrained parcels. Both simulations point to entrainment occurring preferentially in small holes. An approximation to the bulk entrainment velocity based on the domain averaged cloud-hole vertical velocity shows reasonable agreement and reinforces the dominance of entrainment in small holes.
 Although the power law distribution of cloud holes and the dominance of entrainment in small holes appear to be solid, further research is desired to enhance the robustness of the results, refine the exponents, and examine their applicability to wide range of STBLs. The potential for better realizations of entrainment in models parameterizing the STBL turbulence such as cloud system resolving models should also be explored.
 This study was supported by the NOAA Climate Program Office. The authors acknowledge G. Hulley and K. Suzuki for their help with the ASTER product.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.