Two types of Arctic mixed-phase clouds observed during the ISDAC and M-PACE field campaigns are simulated using a 3-dimensional cloud-resolving model (CRM) with size-resolved cloud microphysics. The modeled cloud properties agree reasonably well with aircraft measurements and surface-based retrievals. Cloud properties such as the probability density function (PDF) of vertical velocity (w), cloud liquid and ice, regimes of cloud particle growth, including the Wegener–Bergeron–Findeisen (WBF) process, and the relationships among properties/processes in mixed-phase clouds are examined to gain insights for improving their representation in General Circulation Models (GCMs). The PDF of the simulated w is well represented by a Gaussian function, validating, at least for arctic clouds, the subgrid treatment used in GCMs. The PDFs of liquid and ice water contents can be approximated by Gamma functions, and a Gaussian function can describe the total water distribution, but a fixed variance assumption should be avoided in both cases. The CRM results support the assumption frequently used in GCMs that mixed phase clouds maintain water vapor near liquid saturation. Thus, ice continues to grow throughout the stratiform cloud but the WBF process occurs in about 50% of cloud volume where liquid and ice co-exist, predominantly in downdrafts. In updrafts, liquid and ice particles grow simultaneously. The relationship between the ice depositional growth rate and cloud ice strongly depends on the capacitance of ice particles. The simplified size-independent capacitance of ice particles used in GCMs could lead to large deviations in ice depositional growth.
 The Arctic is a highly sensitive climate region [Comiso, 2006]. Average Arctic temperatures have increased at almost twice the global average rate in the past 100 years [Intergovernmental Panel on Climate Change (IPCC), 2007]. Current climate models greatly underestimate the observed decrease of sea ice cover. Much of the underestimation may come from uncertainties in the representation of clouds, aerosols and the cryosphere and their interactions in climate models, of which the cloud parameterizations are one of the largest sources of uncertainty [IPCC, 2007].
 Mixed-phase clouds (MPCs) commonly exist over the Arctic and play an important role there in precipitation formation, the energy budget and climate [Intrieri et al., 2002; Shupe and Intrieri, 2004; Turner, 2005]. In the Arctic, there are generally two types of MPCs, i.e., single layer mixed-phase clouds (SLMC) seen in the boundary layer and multilayer mixed-phase clouds (MLMC). Most of MPCs are single-layer, with very high frequencies in all seasons [Prenni et al., 2007]. MLMC also commonly occur in the Arctic during summer, comprising slightly less than half of the mixed-phase clouds then [Morrison et al., 2009; Shupe et al., 2006]. The high frequency of mixed-phase clouds in the Arctic suggests an important contribution to the cloud radiative forcing there, which may play a role in the rapid changes seen in Arctic climate [Garrett and Zhao, 2006; Kay and Gettelman, 2009].
 Unfortunately, mixed phase clouds are notoriously difficult to represent in numerical weather prediction and climate models, especially in the unique and complex Arctic environments [e.g., Illingworth et al., 2007; Klein et al., 2009]. Thus, more robust understanding of MPCs and better treatment of cloud processes are needed to improve the reliability of climate model simulations of Arctic climate change.
 Phase transformations of water between vapor, and liquid and ice particles can occur in the MPC at temperatures between 0°C and approximately −35°C, where ice can grow at the expense of liquid due to the lower saturation vapor pressure over ice compared with that over liquid. This process is known as the Wegener–Bergeron–Findeisen (WBF) mechanism [Wegener, 1911; Bergeron, 1935; Findeisen, 1938]. Recently, Korolev [Korolev, 2007; Korolev and Mazin, 2003] showed that the WBF process can only occur under a limited range of conditions, and that ice particles and liquid droplets in mixed-phase clouds do not always evolve via the WBF process. Storelvmo et al.  subsequently applied these constraints to a general circulation model and found better agreement with observations. Because the process can significantly impact cloud optical depth, cloud radiative forcing, precipitation rate, cloud coverage and lifetime [Fowler and Randall, 1996; Gregory and Morris, 1996; Tremblay et al., 1996; Jiang et al., 2000], it is necessary to accurately account for it in GCMs. However, it is challenging to physically represent the WBF process in GCMs due to subgrid variability of cloud properties and un-resolved supersaturation.
 Cloud simulations from Korolev's studies have provided some insights into mixed-phase regimes such as the limited conditions for the WBF process. However, those studies employed a simple parcel model framework with only diffusional growth in the microphysical processes and the conclusions were drawn using idealized clouds. To further understand the mixed-phase processes, it is helpful to examine more realistic clouds simulated with a large-eddy dynamic framework and explicit treatment of the cloud microphysical processes.
 The subgrid variability of cloud properties, which is important for highly nonlinear cloud processes such as droplet and precipitation formation, is also difficult to accurately parameterize in large-scale models. Since the spatial scales of cloud processes are typically much smaller than the model grid spacing employed in large-scale models, subgrid variability of these processes must be taken into account when there are nonlinear processes involved [Larson and Golaz, 2005]. In many GCMs such as the Community Atmosphere Model (CAM), a Gaussian normal subgrid distribution of vertical velocity is assumed for stratiform clouds, with the standard deviation (σw) calculated from the turbulent kinetic energy (TKE) [Ghan et al., 1997; Morrison and Gettelman, 2008]. This subgrid treatment is only applied to droplet nucleation and is not used in representations of other microphysical processes such as ice growth via the WBF process. In the latest version of CAM (i.e., CAM5), the subgrid variability of cloud properties is only considered for cloud liquid mixing ratio by assuming a Gamma distribution with a fixed variance, which is not consistent with a triangular probability distribution function (PDF) for the total water. For cloud ice and precipitation, no subgrid variability is considered. Further improvements to the representation of those cloud properties require better understanding of their variability on subgrid scales.
 A wide variety of functional forms have been used in the literature for the PDFs of total water or cloud liquid, including the exponential function [Bougeault, 1981], the Gamma function [Bougeault, 1982], and the Beta function [Klein et al., 2005; Tompkins, 2002]. One reason for this is that it is difficult to obtain generalized and accurate information from observations concerning variability down to small scales. Moreover, data from different observations suffer from various drawbacks. For example, aircraft or tethered balloon data suffer from undersampling problems [Tompkins, 2002]. Estimates produced from satellite retrievals have difficulties resolving cloud vertical structure [Cooper et al., 2006]. An alternate (complementary) characterization of PDFs can be produced from cloud-resolving models (CRMs) which can simulate cloud properties at spatial scales much closer to those of real clouds. In recent years, CRM simulations have been widely used to improve cloud parameterizations and the representation of subgrid variability for GCMs [Bretherton and Park, 2009; Larson et al., 2002; Tompkins, 2002; Xu and Randall, 1996].
 In this work, we use a cloud-resolving model with spectral-bin cloud microphysics [Fan et al., 2009a] for simulating Arctic SLMC and MLMC to improve our understanding of mixed-phase processes and subgrid variability of cloud properties in order to improve cloud parameterizations for the large-scale models.
2. Description of Phase Transformation in MPCs
 To help interpret our results later, we review some background information about phase transformation in MPCs. There are three possible regimes for the diffusional growth of liquid droplets and ice particles based on the in-cloud water vapor pressure (e) and the equilibrium vapor pressures over liquid water (es) and ice (ei) [Korolev and Mazin, 2003] (referred to as KM2003 hereinafter):
 1. e > es > ei: when the vapor pressure exceeds saturation for liquid and ice, both droplets and ice particles will grow simultaneously. This condition may occur in ascending mixed-phase clouds when updraft velocity exceeds a threshold value wth, which is given by
where η is a coefficient dependent on temperature T and pressure P, and Ni and are the number concentration and mean radius of ice particles, respectively [Korolev, 2008]. This expression is derived based on the equilibrium state of droplets (i.e., supersaturation with respect to liquid is zero) in a simple cloud parcel model framework with only diffusional growth. For non-spherical particles, ri is replaced by ri ci, where ci is a shape factor characterizing capacitance. Since the spectral-bin microphysics uses size-dependent ci for ice particles, i.e., ci = ci(r), [Khain et al., 2004], equation (1) should be generalized to
where = ri(r)ci(r)Ni(r)/Ni(r) is the “mean capacitance” and nr is the total number of size bins. This formula will be used later to calculate wth for our simulations.
 2. es > e > ei: when the vapor pressure exceeds ice saturation, but is below liquid saturation, droplets evaporate, whereas ice particles grow. The evolution of the mixed phase under this condition defines the WBF process. In mixed-phase clouds, the WBF process occurs when w*min < w < wth. Here w*min is the vertical velocity separating growth and sublimation of ice particles in the presence of liquid droplets. Based on the parcel model framework, when ice particles are at equilibrium (i.e., supersaturation with respect to ice is zero), w*min is given by
where χ is a coefficient depending on temperature T and pressure P, and Nw and are the number concentration and mean radius of droplets, respectively [Korolev, 2008]. Since droplets are generally spheres and the capacitance is a constant (1.0), no modification of the formula is necessary, but we need to calculate based on droplet size distribution in spectral-bin microphysics (SBM).
 3. When the vapor pressure is below both liquid and ice saturation (es > ei > e): both droplets and ice particles evaporate. Simultaneous evaporation of ice particles and liquid droplets may occur in downdrafts when w < w*min. Generally, based on the formula for w*min presented above, strong downdrafts are needed to make the cloud reach sub-saturation with respect to ice (shown later).
 In the latest release of CAM, i.e., CAM5, the two-moment cloud microphysics from Morrison and Gettelman  (referred to as MG08 hereafter) is used for stratiform clouds. The WBF process is taken into account by calculating ice depositional growth by assuming the in-cloud water vapor mixing ratio is saturated with respect to liquid, i.e., if liquid exists, vapor deposition onto ice depletes part or all of the liquid condensate [Gettelman et al., 2010]. Note that a maximum overlap between the liquid and ice fractions in a model grid-box is assumed in the current implementation, producing the largest possible depletion rate of liquid water via the WBF process. A similar microphysics and WBF treatment (i.e., MG08) were also implemented in the GFDL Atmospheric Model version3 (AM3) [Salzmann et al., 2010]. In the ECHAM5 (fifth-generation atmospheric general circulation model), the WBF process is parameterized such that a supercooled liquid cloud will glaciate completely within a single time step once a threshold IWC (0.5 mg kg−1) is exceeded [Lohmann et al., 2007]. Before the threshold IWC is reached, saturation with respect to liquid is assumed, which is similar to CAM5. These models provide examples of the treatment of WBF processes in large-scale models.
3. Experiment Setup and Case Simulations
3.1. Numerical Experiment Setup
 We employ a CRM referred to as the System for Atmospheric Modeling (SAM), coupled with spectral-bin microphysics (SBM) [Khain et al., 2004; Fan et al., 2009a], to simulate two typical types of MPC in the Arctic: SLMC and MLMC, which were observed during the U.S. Department of Energy Atmospheric Radiation Measurement (ARM) program's Indirect and Semi-Direct Aerosol Campaign (ISDAC) and Mixed-Phase Arctic Cloud Experiment Period A (MPACE_A) in April 2008 and October 2004, respectively. SAM employs the dynamical framework of a large-eddy simulation (LES) model [Khairoutdinov and Randall, 2003]. It solves the equations of motion using the anelastic approximation. The finite difference representation of the model equations uses the Arakawa C staggering, with stretched vertical and uniform horizontal grids. The advection and diffusion of momentum are of second-order accuracy. Advection of all scalar prognostic variables is done using a monotonic and positive-definite advection scheme and a damping layer is implemented in the upper third of the domain to reduce gravity wave reflection and buildup [Khairoutdinov and Randall, 2006; Khairoutdinov and Randall, 2003] for more details. The SBM is an explicit bin microphysical scheme [Khain et al., 2004] that has been coupled into SAM to investigate aerosol-cloud interactions [Fan et al., 2009a, 2009b]. Please refer to Fan et al. [2009a] for a more detailed description of SAM-SBM.
 Since the ice nucleation mechanisms in this type of MPC are still controversial [Fridlind et al., 2007; Fan et al., 2009a], for the ISDAC case (i.e., SLMC), we treat ice nucleation in a simple approach similar to the specification for the Surface Heat Budget of the Arctic Ocean (SHEBA) intercomparison [Morrison et al., 2011]. That is, when the total ice particle concentration falls below a specified ice nuclei concentration (NIN) and when ice supersaturation Si exceeds 5%, ice crystals nucleate at a rate required to keep the total ice concentration (Ni) at the value of NIN. The nucleation rate is therefore given by:
where Δt is the model time step. NIN is set to 0.4 L−1 for the ISDAC case based on aircraft measurements of ice particle concentration with size larger than 100 μm during 0:00–3:30 UTC, April 27, 2008 as shown in Figure 1a. For the MPACE_A case (i.e., MLMC), since clouds are much deeper and span a much larger temperature range, the above fixed ice formation may not be appropriate. Thus, we use the parameterization of Meyers et al. , which relates the number concentration of deposition and condensation-freezing ice nuclei (Nd) to Si:
where a = 0.639, and b = 12.96. Since the Meyers parameterization was derived from data for midlatitudes and it is known to over-predict ice crystal concentrations when applied to Arctic cases [Fan et al., 2009a; Prenni et al., 2007], we decrease Nd0 from the original value 1.0 L−1, to 0.1 L−1 to better match the observed ice particle concentrations for MPACE_A, which are around 0.114 L−1 on average based on the aircraft measurements for ice particles larger than 100 μm [Klein et al. 2007]. The new formed ice is put in the first bin for ice crystals of radius about 2.3 μm. Droplet freezing through the immersion mode is considered in both cases but it does not have any effect on the ISDAC case because the cloud temperatures are too warm (>−15°C).
 The SLMC that we simulate occurred about 100 km offshore from Barrow Alaska on 26–27 April 2008, one of the identified “golden day” cases during the ISDAC field campaign. This stratocumulus cloud had a very flat cloud top and base and was maintained mainly by mixing driven by cloud top radiative cooling [McFarquhar et al., 2011]. The simulated MLMC represents a case observed over Barrow during 5–8 October 2004. This cloud system was associated with a rather complex synoptic-scale flow and consisted of a number of distinct liquid layers with ice crystals falling between the liquid layers as indicated by aircraft and ground-based remote sensing measurements [Morrison et al., 2009].
 For the SLMC case, we use a bimodal dry lognormal aerosol size distribution with a standard deviation of 1.5 and 2.0, geometric mean diameters of 0.1 and 0.75 μm, and total number concentrations of 200.0 and 2.0 cm−3 for fine and coarse modes, respectively, based on the spectrum observed by the Passive Cavity Aerosol Spectrometer Probe (PCASP) from Flight 31 for the sub-cloud layers [Ovchinnikov et al., 2011]. For the MLMC, the bimodal lognormal size distribution is also used and it is the same as the MPACE Period B SLMC case [Fan et al., 2009a], as recommended by the intercomparison documents for this case [Klein et al., 2009]. Specifically, the fine and coarse modes have a standard deviation of 2.04 and 2.5, geometric mean diameters of 0.052 and 1.3 μm, and total number concentrations of 72.2 and 1.8 cm−3, respectively. For both cases, the major composition of ammonium bisulfate observed is assumed for droplet activation.
 The initial sounding for the ISDAC SLMC case is based on radiosonde observations at Barrow, but the temperature and moisture profiles are modified in the lower troposphere to match more closely the structure of the boundary layer at the time and location of the flight. Vertical profiles of temperature, moisture, and horizontal wind components are presented in our companion manuscript Ovchinnikov et al. . According to observations and the European Center for Medium Range Weather Forecasts (ECMWF) reanalysis around Barrow, the magnitudes of the sensible and latent heat surface fluxes do not typically exceed 10 W m−2 during ISDAC. Given that the cloud layer is not dynamically coupled to the surface for this case, we set the sensible fluxes to be zero and the latent heat surface fluxes of 10 W m−2 for simplicity. Large-scale subsidence is computed to balance the assumed horizontal wind divergence of 10−4 s−1. The large-scale forcing tendencies for temperature and moisture are specified to compensate for the effect of the subsidence and radiative cooling, minimizing the drift from the initial profiles above the boundary layer [Ovchinnikov et al., 2011]. For the MPACE_A (MLMC), the initial conditions, large-scale forcings and surface latent and sensible fluxes follow the intercomparison specifications [Morrison et al., 2009], which were derived with the ARM variational analysis [Xie et al., 2006]. There is no liquid water profile assumed initially and ice is allowed to form from the start of the simulations for both cases.
 Simulations are run on a 3-dimensional (3-D) computational domain comprised of 128 × 128 horizontal grid points and 120 vertical grid points. Periodic lateral boundary conditions are used. For the ISDAC SLMC, we use horizontal and vertical grid spacing of 100 and 20 m, respectively, since the cloud layer is thin. The dynamic time step is 2 s. For the MPACE_A MLMC, the horizontal grid spacing is 200 m and the dynamic time step is 4 s with a shorter sub-time step used for diffusion growth. Since the cloud system is much deeper, the vertical grid spacing starts at 50 m for a few lower layers and then increases to 80 m for the rest of the layers. The simulation time is 12 h for the SLMC but 72 h for the MLMC. The radiation scheme is called every 3 min.
3.2. Comparison of Simulations With Observations
Figure 1a compares vertical profiles of simulated liquid water content (LWC; including cloud water and rainwater), ice water content (IWC; including all ice particles), droplet (Nw), and ice particle number concentrations (Ni) with aircraft observations from Flight 31 during 0:00–3:30 A.M. on April 27, 2008. The observed data are available in the ARM data archive. LWC and Nw data were measured from the forward scattering spectrometer probe (FSSP) sizing particles in the size range 3 to 47 μm. The measurements of LWC from the King probe are also plotted for intercomparison (red line). IWC and Ni shown in the figure were calculated from composite size distribution measured by the Optical Array Probe two-dimensional cloud probe (2DC) and precipitation probe (2DP). For the IWC calculations, the Brown and Francis  mass-diameter (m-D) conversion was used. During ISDAC, the 2DC on the Convair 580 had anti-shattering tips. The anti-shattering tips play an important role in mitigating shattering for the 2DC, so shattering does not seem to be significant. The interarrival algorithm for filtering out shattering events was also applied to process both 2DC and 2DP data. Particles smaller than 100 mm are not reliably observed and therefore disregarded from the analysis. The simulated LWC and Nw agree with the observed values very well. As stated in Section 3.1, the ice nucleation is constrained by the observed ice particle number concentration that is about 0.4 L−1 (fourth panel of Figure 1a). The modeled IWC in the cloud is close to the observed values, but is underpredicted by about 50% below the cloud. Sublimation is unlikely to be a major factor for this underprediction since the layer between 700 m (cloud base) down to 530 m is supersaturated with respect to ice. The underpredicted IWC could occur because of overestimated fall velocities for ice particles in the model with correspondingly reduced time available for ice mass growth. For the near constant ice number, smaller mass for individual ice particles translates directly into smaller IWC. However, the measured IWC has up to 100% uncertainty, so it is hard to conclude that the model performs poorly below the cloud.
Figure 1b compares the time series of simulated liquid water path (LWP) and ice water path (IWP) integrated above 400 m with those from aircraft measurements. The simulated LWP is in good agreement with the observations but the IWP is underpredicted, consistent with the underprediction of the IWC as shown in Figure 1a. We also compare the modeled PDFs of vertical velocity (w) and LWC with those constructed from 1-s observations around the height of 800 m from a flight leg at similar heights (Figure 1c). Note that for aircraft speed of around 100 m s−1 one-second samples represent a distance comparable to the model's horizontal grid size. The PDF of w from the simulation agrees well with the observations. The modeled PDF of LWC at 800 m has a similar shape to the observations but with a smaller mean. This is expected because the modeled LWC peaks around 880 m but the observed LWC peaks around 820 m as shown in Figure 1a. IWC is not compared because the 1s sample volume does not provide reliable information given the very low ice concentration. The ground-based retrievals are not compared with the model for this case because we found by examining the cloud structures and properties that clouds over Barrow are very different from those over the offshore.
 The observed hydrometeor size distribution (HSD) from the flight leg at the height of 800 m is also compared with the simulated HSD (including liquid and ice particles) at similar heights (Figure 1d). The HSD were calculated from 2DC and 2DP measurements. The modeled HSD agrees with the observations for particles of sizes of 100–500 μm, and overestimates the particles of sizes of >500 μm. Below 100 μm, there are no reliable measurements of ice particles.
Figure 2 compares simulated cloud with observations for the MLMC. The modeled cloud fraction in Figure 2a is computed using cloud ice mixing ratio greater than 10−4 g kg−1 as the threshold defining a cloudy cell [Morrison et al., 2009]. The observed cloud fraction is from Active Remotely Sensed Clouds Locations (ARSCL) data in Climate Model Best Estimate (CMBE) product of ARM. The model captures the vertical structure of the cloud fraction, except for overprediction near the ground and around cloud top. The reason for the overprediction of cloud fraction near the ground is unclear but the faster fall speeds of ice particles as mentioned for the ISDAC case could be a contributing factor. The overprediction around cloud top is likely due to the thermodynamic conditions (water vapor and temperature) that are favorable for too much cloud formation. As shown in Figures 2b and 2c, LWC and IWC around 4–5 km are greatly overpredicted compared with both aircraft and retrieved measurements. Overall, the modeled LWC and IWC are in better agreement with the retrieved data than the aircraft measurements, probably because the aircraft data only include a few short flights that cannot capture the cloud characteristics at the other times. The coefficients for the m-D relationship used to calculate IWC from measured size distributions were determined by a technique described by McFarquhar et al. . The simulations have three distinct mixed-phase layers located around 1, 2 and 4 km, different from the retrievals suggesting that the clouds are mixed-phase over the entire vertical profile. But it agrees with the aircraft observations, suggesting that the clouds consist of a number of distinct liquid layers (i.e., three) with ice crystals falling between the liquid layers [Morrison et al., 2009]. Since it is very challenging to retrieve vertical distributions of LWC using ground-based remote sensors [Shupe et al., 2008], the retrieved LWC profile is highly uncertain, especially for the multilayer clouds due to the dominance of large ice particles.
4.1. Single Layer Mixed-Phase Clouds
4.1.1. Spatial Variability
 A PDF approach employing the assumption of a particular functional form is widely used to represent the subgrid variability of quantities in large scale models. For example, a Gaussian normal distribution is generally assumed for the subgrid variability of the vertical velocity (w) in stratiform clouds. The standard deviation (σw) describes the width of the Gaussian normal distribution and is frequently connected to TKE [Ghan et al., 1997; Morrison and Gettelman, 2008] in large scale model parameterizations of cloud properties. Here we fit the simulated vertical velocity PDF from the CRM with a Gaussian normal distribution, to estimate σw. Figure 3a provides the w distribution averaged over the cloud layer at 850 m and Figure 3b shows the distribution averaged over the entire cloud layer. The PDF can be represented quite well with a Gaussian normal distribution with the σw of about 0.3 m s−1. Vertical variations of σw in cloud are very small, with values ranging from 0.32 to 0.44 m s−1. However, σw is highly scale-dependent and decreases with decreasing resolution: σw becomes 0.1 and 0.03 m s−1 with the horizontal grid spacing of 500 and 1000 m, respectively. A refinement of the model horizontal grid spacing from 100 m to 50 m increases σw to 0.47 m s−1 for this case. By averaging w from the 50 m resolution simulation to the 100 m scale we find a σw of 0.31 m s−1, very close to that with 100-m resolution (i.e., 0.3 m s−1), indicating the 100-m resolution resolves the major dynamic features that contribute to vertical motion. Note that the values of σ or variances for PDF distributions are highly case-dependent. This study is more concerned with the qualitative insights from CRM simulations rather than the quantitative results. Further refining the model scale may affect the values of σ or variances, but the qualitative results about the PDF functions and the relationships among the cloud properties would not be affected. Currently, in GCMs, σw is generally a function of TKE and various empirical formulas are used to connect them [Ghan et al., 1997; Morrison and Gettelman, 2008; Storelvmo et al., 2008]. Given that σw is sensitive to horizontal and vertical scales [Guo et al., 2008], those formulas used to connect σw with TKE or the parameterizations of w need to be scale-ware so that it can produce better σw at coarse resolutions in GCMs.
 As stated in Section 1, in CAM5 subgrid variability of cloud liquid is currently represented but subgrid variability of cloud ice and precipitation is neglected. This treatment is not consistent with the total water content (qt) in CAM5 where a triangular PDF is used for diagnosing the liquid cloud fraction. Development of a consistent representation of subgrid variability of total water, cloud liquid, cloud ice, and snow (precipitating ice) is desirable, and estimates can be provided from CRM simulations. As shown in Figure 4 for a cloud slab at 850 m, the PDF of cloud liquid is well represented by a Gaussian normal distribution, very different from the Gamma distribution currently assumed in CAM with a fixed relative variance of 1.0 (the red line in Figure 4a). The relative variance (σ2) is given by variance divided by the square of the mean value of a quantity. Note that σ2 is set to be 2.0 in the current CAM release, which does not make the fit better either for this case. The LWC PDF can be well represented with a Gaussian normal distribution with a standard deviation σ of 0.013 g kg−1 (dotted line in Figure 4a). It also can be treated fairly well with a Gamma distribution with a relative variance of 0.05 (blue line), much lower than of the fixed variance in CAM (the smallest value used is 1.0). The PDF of IWC is well represented with a Gamma distribution with relative variance of 0.44 (Figure 4b), suggesting that Gamma functions can be used to represent the subgrid variability of ice water. By looking at the layers at about 400 and 600 m where snow is abundant, we found the snow (i.e., aggregates) PDF can also be well represented by a Gamma function but with a very small relative variance of 0.03. For the total condensed water content (TWC), the PDF is the same as that of LWC (Figure 4c) since cloud ice is about an order of magnitude less than the liquid phase and contributes little to the total water. For qt, we also see a Gaussian normal distribution (Figure 4d) with a σ of 0.02 g kg−1, which supports the subgrid treatment employed by Caldwell et al. . Note that we do not see much change of the PDFs of these cloud properties as the cloud evolves in this single-layer boundary cloud case. The PDFs for LWC, IWC, TWC, and qt revealed above are less scale-dependent than the PDF of vertical velocity. As expected, the variances decrease as the model grid size increase to 500 and 1000 m, but the sensitivity is not as strong as seen in σw. For the 50 m resolution simulation the variances are the highest, i.e., σ for LWC is 0.014 g kg−1, and the relative variance for IWC is 0.55.
4.1.2. Regimes of Diffusional Growth of Cloud Particles
 The three theoretically possible regimes of diffusional growth in the mixed-phase clouds, i.e., both liquid and ice grow, ice grows while liquid evaporates (i.e., WBF), and both liquid and ice evaporate, are examined in our CRM simulations. A snapshot of a vertical section of the clouds and the different regimes is shown in Figure 5 (regimes are defined based on the inequalities for e, es, and ei discussed for equations (1) and (3)). Both liquid and ice grow (referred to as “Both-Grow” for short) in much of the mixed-phase cloud, mainly in updrafts (Figure 5a). On average, the Both-Grow regime occurs in 49% of the mixed-phase cloud volume (Table 1). The WBF regime occurs in 51% of the mixed-phase cloud, and the regime of both liquid and ice evaporation is negligible. The fractions of these regimes in the horizontal slab at 850 m are close to the averages as shown in Table 1. Therefore, in this single-layer cloud, the WBF process occurs in only about 50% of the mixed phase cloudy-points and in the other half both liquid and ice grow simultaneously, consistent with the conclusion from Korolev  that the WBF process can only occur in limited conditions.
Table 1. The Fraction of Regimes With WBF and Both-Grow in the Mixed-Phase Clouds
A slab at 850 m
Upper-layer (at 3.9–4.1 km)
Middle-layer (at 1.9–2.1 km)
Lower-layer (at 0.9–1.1 km)
 In KM2003, wth that separates the Both-Grow process from the WBF process and w*min that separates the WBF process from both evaporation process were derived from an idealized adiabatic mixed-phase cloud with consideration of only diffusion growth. How realistic is this simple way of distinguishing these regimes? To answer this question, we calculate wth and w*min using equations (2) and (3), and compare with the regimes from the model determined by the vapor pressure criteria. Figure 6a indicates that the majority of points in the WBF regime occur where w < wth and the majority of Both-Grow regime occurs where w > wth. There is only about 6% of the WBF regime where w > wth, indicating the wth calculated with the approximate formula in KM2003 can reliably separate the WBF and Both-Grow regimes based on the CRM results with the full microphysics. As shown in Figure 6b, the majority of the WBF regime has w much larger than the calculated w*min that separates the WBF process from evaporation of both droplets and ice particles (w in the considered cloud generally varies from −1 to 1 m s−1), indicating that much stronger downdrafts are needed to cause ice sublimation in mixed-phase clouds. Except for deep convection clouds, such strong downdrafts usually do not happen (the calculated w*min can be close to −100 m/s, which would be never reached). Thus, the regime of both liquid and ice evaporation is negligible for stratiform clouds.
 It is also found that the positive w only account for about 6% of the WBF regime, meaning that about 94% of the WBF regime occurs in the downdraft regions (Figure 6c). Based on the calculated wth, which is generally less than 0.01 m s−1, we know that a small updraft can easily disable the WBF process, indicated also by Korolev . Our results with the full microphysics considered indicate that the WBF process could occur in somewhat higher updraft velocity (up to 10 times higher than 0.01 m s−1). This difference may result from the more completely dynamic and microphysical process treatments in the SBM (see the following section on process rates).
 In terms of water budgets. the evaporation of liquid mass in the updrafts (with w > 0) only accounts for about 3% of the total evaporation, suggesting that the vast majority of evaporation occurs in the downdrafts. Similarly, the condensation of liquid mass from the 9% Both-Grow regime in the downdrafts only accounts for 4.5% of the total condensation. Therefore, the vast majority of condensation and evaporation occurs in the updrafts and downdrafts, respectively. Note that subsaturation with respect to liquid in the updrafts and supersaturation in the downdrafts could be caused by mixing and the delay in the response of thermodynamics to changes of microphysics. The vast majority of the grid points with liquid supersaturation in the downdrafts are located at cloud tops where liquid supersaturation could be produced by strong gradients in RH and temperature across cloud boundaries or possibly by numerical problems arising during advection [e.g., Stevens et al., 1996; Grabowski and Morrison, 2008]. There are many grid points in the mixed-phase clouds with liquid subsaturation in the weak updrafts that are not around cloud tops, which could be caused by mixing or the WBF process itself.
 With a Gaussian normal distribution for w, we can schematically represent the Both-Grow and WBF regimes as shown in Figure 3c. Since wth is zero or only a little above zero and w*min is very negative, the Both-Grow regime accounts for about half of the area in the PDF and the WBF process accounts for the other half. Therefore, the liquid condensation process only occurs where w > wth. However, most GCMs do not explicitly calculate this condensation process in the microphysics. Condensate formed by large-scale processes such as radiative cooling and large-scale ascent is diagnosed from the water vapor saturation adjustment related to cloud macrophysics (e.g., cloud fraction) and is assumed to be liquid initially. Then in the microphysical calculation, the WBF process is calculated over the entire cloudy volume by the ice depositional growth process, i.e., vapor deposition in the mixed-phase clouds depletes part or all of the liquid water and in-cloud water vapor is assumed to reach water saturation [Gettelman et al., 2010]. Because w*min is very negative, there is only a negligible fraction of clouds (see Figure 3) where ice and liquid evaporate. Given the way that stratiform clouds are parameterized in GCMs, it seems that connecting the WBF process with the subgrid variability of vertical velocity is not necessary.
 It is also interesting to consider the co-location of ice and liquid within a cloud. In CAM5, maximum overlap is assumed between the liquid and ice fractions, implying the largest possible depletion rate of the liquid water via the WBF process. Our CRM simulations indicate that the correlation between CRM grid values of cloud liquid and ice water content at a certain layer in the clouds is very low (about 0.1) (Figure 7a). The correlation of LWC and IWC from the in situ aircraft observations (spirals only) for this case in the layer between altitudes 800–850 m is low as well (Figure 7b). Although the sample volumes for the data point in Figure 7b are much smaller than the volume of the CRM grid box, distance over which aircraft collected 1-s samples is comparable to the horizontal grid size of 100 m. We also examined the spiral data from multiple flights for the MPACE Period B (MPACE_B, another SLMC case occurring in the fall of 2004) where more data are available. They show similar low correlations (not shown). From the observed and modeled spatial distributions (Figures 1, 5b and 5c), we also can see that liquid water maximum occurs around cloud top while ice water maximum is near cloud base, consistent with Shupe et al.  for retrievals of MPACE single layer clouds. The lack of correlation between LWC and IWC does not contradict the results of Shupe et al. , which only says that LWC and IWC co-locate in updraft regions. For the other SLMC case such as MPACE_B, we see similar features as shown in Figure 3 of Fan et al. [2009a]. The in situ observations of Korolev et al.  suggest as well that the MPCs tend to be composed of either mostly ice or liquid at spatial scales of order 100 m and less. Therefore, all these model and observational results suggest that large cloud liquid values are not typically collocated with large cloud ice values at the small scale. As shown in Figure 7c, there are many pure liquid portions in the domain at the layer of 850 m, where liquid and ice (defined by LWC > 10−5 kg kg−1 and IWC > 10−6 kg kg−1) are present occurring at scales less than 500 m similar to the observational results of [Korolev et al., 2003]. When averaging over the scale of more than 1 km, they would be mixed with ice and be viewed as mixed-phase. Korolev and Isaac  came to a similar conclusion that at scales larger than 5 km, ice and liquid are conditionally mixed (i.e., single-phase “ice” and “liquid” clusters are mixed). Therefore, in the large-scale models that generally have resolutions of a few, tens, or even hundreds of kilometers, the maximum overlap assumption for the mixed-phase clouds seems to be appropriate.
4.1.3. Process Rates
 Rates for microphysical processes are another aspect that needs to be improved in the GCMs. In the MPCs, the ice growth rates by deposition and riming determine cloud microphysical properties and may affect the cloud lifetime. In current GCMs, however, these processes are often treated in a simplified way and some are ignored. In CAM5, for example, the ice depositional growth is calculated explicitly by assuming liquid saturation within the clouds [Gettelman et al., 2010]. Ice particles are assumed to be spherical in shape, so that the shape factor ci is fixed at unity for all sizes of particles. The CRM simulations with full microphysical processes with consideration of the size- and shape- dependent ice particle properties allow us to look at these process rates closely.
 By examining the liquid supersaturation in the CRM simulation, we find that in the WBF regime 95% of the cloudy cells are within 99.5–100% of the saturation liquid vapor pressure and also about 95% of the cloudy cells are between 100 and 100.2% of saturation over liquid in the Both-Grow regime. Cloud parcel model simulations and in situ-measurements [Korolev and Isaac, 2006] have showed that the relative humidity in mixed phase clouds is close to the water saturation. Our CRM results with full microphysical processes and dynamics add further support to the assumption that ice particles grow through vapor deposition at liquid saturation in the mixed-phase regime in climate models.
 The relationship between the ice depositional growth rate and IWC in the MPCs from the CRM simulations is useful and can be implemented to constrain the microphysics in the climate models. As shown in Figure 8a, we see a perfect linear relationship between ice depositional growth rate and IWC for IWC larger than 2 × 10−3 g m−3. The linear equation is similar in both the WBF and Both-Grow regimes, with a slightly larger slope of the ice deposition rate with IWC in the Both-Grow regime (not shown), because of higher supersaturation resulting from higher vertical velocity. For the Maxwellian diffusion growth where the capacitance of ice particles is a constant, the ice depositional growth rate should be proportional to Ni. Such dependence suggests that the ice growth rate should be proportional to (IWC)1/3. However, if the capacitance in the ice growth equation is a function of the particle size, as in the SAM-SBM, then the dependence of the growth rate versus IWC will have a power different from 1/3. The relationship of the capacitance with the ice particle size used in the SBM [Khain et al., 2004] leads to the linear relationship between the ice depositional growth rate and IWC when IWC > 2 × 10−3 g m−3 as shown in Figure 8a. The turning-point around IWC of 2 × 10−3 g m−3 corresponds to a significant increase in capacitance around the maximum dimension of 200 μm set for snow in the SBM. Therefore, the relationship between the ice depositional growth rate and IWC strongly depends on the capacitance of ice particles, which is strongly shape- and size-dependent [Pruppacher and Klett, 1997]. Simple assumptions for the capacitance of ice particles in the bulk microphysical schemes that cloud and climate models use could lead to a large deviation on ice depositional growth. In GCMs, the large deviation would strongly affect the WBF process and change the transformation rate of liquid to ice, resulting in a large uncertainty in climate simulations.
 The process of riming also can affects the rate of liquid to ice transformation in the MPCs a lot, but it is highly uncertain. In the simulation with explicit microphysical processes, we find that the riming is also linear with IWC over a certain range of IWC (separating by IWC of about 3 × 10−6 kg m−3) for this simple SLMC case as shown in Figure 8b. This relationship is mainly determined by the collision efficiency between drop-ice collisions and we know that the collision efficiency is highly size- and temperature- dependent. For this simple SLMC case, it may be simplified with the linear relationships shown in Figure 8b. It is noted that the riming rate is much smaller than the deposition rate for the majority of the cloudy volume where IWC is less than 0.01 g m−3. High riming rates (>5 μg m−3 s−1) are found only for very low probability (about 12%). Therefore, deposition should be dominant in terms of ice growth, in agreement with CPI images showing predominantly un-rimed dendrites. Before we generalize the relationships from this simple SLMC, we will look at the more complicated MLMC case.
4.2. Multilayer Mixed-Phase Clouds
4.2.1. Spatial Variability
Figure 9 shows the subgrid variability of w in the MLMC. There are significant differences in the PDF of w among different cloud layers. Although they can be approximately fitted with a Gaussian normal distribution, the standard deviation σw varies from 0.06 m s−1 at the middle mixed-phase layer (around 2 km) to 0.52 m s−1 at the top mixed-phase layer (around 4 km). The σw values for the low (around 1 km) and middle layers are much lower than the minimum value of 0.3 m s−1 used in some global models [Storelvmo et al., 2008]. Therefore, a minimum standard deviation of 0.3 m s−1 may be reasonable for the SLMC, but is probably too high for the other mixed layers except the top layer in MLMC. Without cloud top radiative cooling and surface fluxes, it is expected that the middle mixed-phase layer has much lower σw (0.06 m s−1 as shown on the panel) and the PDF is much less skewed than at the top and bottom of mixed-phase layers, which makes the layer last for only a few hours. In contrast to the SLMC case, where σw does not vary much in the cloud, σw varies a lot in the MLMC because the multiple layers influence the cloud top radiative heating rates for each cloud. Therefore, it is inappropriate to limit σw with a minimum value appropriate for single clouds layers, which will be too large for the low and middle cloud layers. In the CRM, w is a resolved quantity and the main processes contributing to it such as cloud top radiative cooling and entrainment are much better resolved than in large-scale models. GCMs generally use a coarse vertical resolution that can lead to a poor representation of radiative cooling profile and cloud top entrainment. Since these processes influence the TKE that is used to determine σw, this may generate additional problems for driving the cloud processes in GCMs. To produce more realistic clouds, it is suggested that the formulation for σw or the parameterizations of w should be scale-aware to produce better σw for coarse resolutions.
 The PDFs of LWC, IWC, TWC and qt at the different mixed-phase layers are shown in Figure 10. Surprisingly, the assumed Gamma distribution in CAM (with a fixed relative variance of 1.0) produces a reasonable fit to the PDF of LWC from the CRM. Note that in the previous SLMC case, the CRM PDF disagrees with the variance choice currently used in CAM but would agree reasonably well with a choice of a much smaller variance. Nevertheless, the PDF of LWC can be represented with a Gamma distribution in both SLMC and MLMC. The variability of IWC is similar for different layers (i.e., numbers shown on the panels) and it can be approximately fitted with a Gamma function. For TWC, the PDFs in the different layers are similar to those of IWC except the boundary mixed-layer where LWC contributes more significantly. All the modeled PDFs of TWC can be reasonably fitted with Gamma distributions and the variances are relatively uniform across the different mixed phase layers. Therefore, in this MLMC case, Gamma functions can be used to represent the subgrid variability of cloud liquid, cloud ice and total condensed water. The fourth column of Figure 10 shows the PDFs of qt, which can be accurately represented with normal distributions everywhere like the SLMC. The standard deviation is very low, indicating a very small variability. We also examined the PDFs of those cloud quantities at the different cloud stages and find no significant changes in shape, but the variances change as the clouds develop, suggesting that the variances should be parameterized to evolve with the dynamic and microphysical processes instead of being fixed at a constant value. Connecting the variances of cloud properties with the vertical velocity, turbulence, and cloud microphysical processes could be a way to account for the changes of variance [Larson and Golaz, 2005].
4.2.2. Regime of Diffusional Growth of Cloud Particles
 In the MLMC, the Both-Grow regime is slightly larger than the previous SLMC case, occupying about 53% of the cloud volume on average (Table 1). Figure 11 presents the vertical cross-section of the MLMC and it shows there are three distinct mixed layers (Figure 11a). The fraction of the cloud with the WBF process decreases but the fraction with the Both-Grow increases with increasing altitude of the mixed phase layers. The fractions of the WBF and Both-Grow regimes in the lower mixed phase layer (around 1 km) are the same as the previous SLMC. Thus, in both SLMC and MLMC, the WBF process only occurs in about 50% of the mix-phase regime and in the other half both liquid and ice grow simultaneously. As discussed in the previous case, given that most GCMs do not explicitly calculate liquid condensation and ice depositional growth occurs anywhere in the mixed-phase, it is not necessary to treat the WBF process at the subgrid scale in GCMs.
 Like the SLMC case, the vertical velocity in the majority of the WBF regime is less than wth, which is calculated using the approximate formula in KM2003 with the modification to account for the size-dependent capacitances (Figure 12a). However, there is about 14% of the WBF regime with w > wth and about 15% of Both-Grow regime is of w < wth in this more dynamically complicated case. As noted in SLMC, it could be caused by mixing and the delay in the response of thermodynamics to the change of microphysics. The wth calculated by the formula of KM2003 in the WBF regime is around 0.1 m s−1, much larger than in the SLMC case (about 0.01 m s−1), but that value still represents a weak updraft. The simulated vertical velocity in this regime is up to 1.0 m s−1, greater than those in the SLMC, meaning the WBF process can occur in the stronger updrafts of the MLMC. Similar to the SLMC, the majority of WBF regime has vertical velocities w > w*min (Figure 12b). Therefore, the cloud fraction where evaporation of both droplets and ice particles occurs is also negligible in this case. As shown in Figure 12c, the vast majority of WBF occurs in the downdrafts and Both-Grow occurs in the updrafts. In other words, there is about 16% of WBF regime occurring in the updrafts and about 14% of the Both-Grow regime occurring in the downdrafts. However, the condensation of liquid mass in the downdrafts only accounts for about 4% of the total condensation, and the evaporation of liquid mass in the updrafts (with w > 0) also only accounts for 4% of the total evaporation. Therefore, in both SLMC and MLMC, the threshold vertical velocity of zero or calculated with the approximate formula of KM2003 is good enough to separate the WBF and Both-Grow processes.
 Similar to what we see in the SLMC, in the MLMC the correlation of LWC and IWC at a certain vertical layer is also very low (Figures 11b and 11c), e.g., the correlation coefficient is only 0.11 in the top layer. Like the previous case, the pure liquid portions are seen in the low mixed-phase layer but their scales are again less than 500 m (not shown). In the other mixed-phase layers of the MLMC, the pure liquid portions rarely exist at the scale of 100 m but pure ice portions often exist. Since large pure liquid portions and large pure ice portions do not co-exist in the domain, and there will be mixing by cloud motions between these regions, the maximum overlap assumption seems to be appropriate in large scale models, based on both the SLMC and MLMC cases. But a certain threshold for IWC to start the WBF process (similar to what was done by Lohmann et al. ) can be set to prevent the unrealistic conversion of liquid to ice in GCMs since glaciation should be extremely slow if IWC is very low (e.g., cloud-grids with IWC < 10−6 kg kg−1 are generally treated as liquid-only cells in CRMs, meaning glaciation would not occur or occur too slowly to matter).
4.2.3. Process Rates
 The relationship between ice depositional growth rate and IWC is also examined for MLMC in Figure 13a. Similar to the SLMC case, the ice depositional growth rate is generally proportional to IWC over a certain range of IWC in all three mixed-phase layers. The different increasing rate of depositional growth rate with IWC in the different layers is a result of different supersaturation and relationships of the capacitance with size for different habit of ice crystals, which are determined by temperature [Khain et al., 2004]. As we discussed for the SLMC, the relationship between the ice depositional growth rate and IWC strongly depends on the size- and shape-dependent capacitance for ice particles. Simple size-independent assumptions for the capacitance of ice particles could lead to large changes in ice depositional growth. In MG08, the assumed capacitance of spheres can be improved since ice particles are not generally spherical. The slopes of the linear relationships between depositional growth rate and IWC shown in Figures 8 and 13 could vary with different cases, depending on the variability of Ni. Observations show that there is considerable horizontal variability of Ni, [e.g., McFarquhar et al., 2007; Verlinde et al., 2007; McFarquhar et al., 2011]. The lack of scatter in the two cases of this study is probably due to the narrow temperature range of the mixed-phase cloud layers.
Figure 13b presents the relationship between the riming rate and IWC in different mixed-phase cloud layers. Except for the lower layer representing boundary layer mixed-phase clouds, there is no clear correlation between them. The poor correlation of the riming rate with IWC in this MLMC is probably because ice particle size distribution and temperature are very different from the previous thin SLMC case, which lead to different particle fall velocity, collision frequency and efficiency. The relationship between the riming rate and IWC is dependent on the collision efficiency of drop-ice collision which is strongly size- and temperature- dependent. We also have checked the correlations of the riming rate with the other cloud properties such as LWC and find no strong correlation. We can suggest no simple relationship for the riming rate that can be generalized and implemented into the GCMs. Well-constrained collision efficiency of drop-ice collisions for different particle sizes and temperatures would help to further constrain the riming process.
 We have also examined (but do not show) other SLMC cases in the Arctic: MPACE_B [Fan et al., 2009a] and SHEBA [Morrison et al., 2011]. The PDFs of cloud quantities, the mixed-phase regimes, and the relationships qualitatively agree with the ISDAC SLMC case. The results and relationships revealed in this study seem representative for the Arctic mixed-phase clouds, at least for boundary layer mixed-phase clouds. The changes of variance for PDFs between the cases reinforce our suggestion of a parameterized rather than constant variance.
5. Conclusions and Discussion
 Two types of typical mixed-phase clouds in the Arctic, single-layer boundary clouds and multilayer mixed-phase clouds, observed during the ISDAC and M-PACE field campaigns, respectively, have been simulated using a 3-D cloud-resolving model with size-resolved cloud microphysics. The modeled cloud properties generally agree with the aircraft and the surface retrieved measurements. We have examined cloud properties such as PDFs of vertical velocity, cloud liquid and ice water contents, as well as the relationships among the cloud properties within the mixed-phase layers and growth regimes of cloud particles, with the purpose of gaining insights to improve the representation of the mixed-phase processes in the GCMs.
 We find that, in both SLMC and MLMC, the WBF process occurs in about 50% of the mix-phase regime with the vast majority in the downdrafts, and in the other half of the cloud both liquid and ice grow simultaneously. This is consistent with the conclusion of Korolev  that the WBF process occurs only in a limited region of mix-phased clouds. It is found that the WBF process can occur in somewhat stronger updrafts than those calculated with the approximate formula of KM2003. But, since the vast majority of condensation and evaporation of liquid mass occurs in the updrafts and downdrafts, respectively, a threshold vertical velocity of zero or calculated with the approximate formula in KM2003 can be used to reasonably separate the WBF and Both-Grow processes for stratiform clouds. In GCMs, since ice depositional growth occurs anywhere in the mixed-phase and liquid growth cannot be explicitly calculated, it is not necessary to treat the WBF process at the subgrid scale in GCMs.
 The CRM results show that vertical velocity is well represented by a Gaussian normal distribution in both SLMC and MLMC, validating that choice of distribution function used in many GCMs. The mixed phase clouds develop very near the saturation vapor pressure for liquid in our CRM simulations both for SLMC and MLMC, validating that assumption frequently used in GCMs.
 The PDFs of total water, liquid water, cloud ice, and total condensed water are examined as well for both SLMC and MLMC. The Gamma distribution with a relative variance of 1.0 assumed in CAM5 does not accurately represent the subgrid variability of liquid water for the mixed-clouds in our simulations, especially for the SLMC. But the PDF of LWC still can be fitted with a Gamma function in the both cases with a different choice of variances. The PDFs of cloud ice and total condensed water can be represented by Gamma or normal functions. The PDF of the total water is well represented by a normal distribution everywhere with a very small variability, which supports the normal distribution used by Caldwell et al. .
 We found the relationship between the ice depositional growth rate and IWC strongly depends on the capacitance of ice particles. Changing the assumptions about the capacitance of ice particles in the bulk microphysical schemes in GCMs such as MG08 could lead to large changes in ice depositional growth. This change would much affect the WBF process and change the transformation rate of liquid to ice, resulting in a large uncertainty in climate simulations. Improved characterization of ice particle growth is imperative. The relationship of the riming rate with IWC is dependent on the collision efficiency of drop-ice collision which is strongly size- and temperature- dependent. We find no simple relationship associated with the riming rate that can be generalized and implemented into the GCMs. Well-constrained collision efficiency of drop-ice collisions for different particle sizes and temperatures would help to further investigate the riming process.
 Although the correlation between LWC and IWC is consistently low on the small scales (less than 500 m), for the larger scales appropriate to GCMs, the assumption that ice and liquid are co-located (the maximum overlap assumption) looks appropriate.
 Although the ability of the CRM with size-resolved cloud microphysics to simulate cloud properties comparable to observations suggest that its simulations can be used to improve understanding of cloud processes and to evaluate the representation of cloud properties and processes in GCMs, conclusions must be tempered by recognition of limitations in our current understanding in cloud microphysics such as collision efficiencies between different sizes of droplets and ice particles, and the size distribution of capacitance and fall velocity for ice and snow.
 Based on our findings, we have the following suggestions to improve cloud parameterizations in the GCMs:
 (1) Although the subgrid treatment of the WBF process is not necessary since most GCMs do not explicitly calculate liquid condensation in the microphysics and ice depositional growth can occur anywhere in the mixed-phase, we still recommend that the subgrid variability of vertical velocity be connected with that of total water. With this connection, cloud fraction can be better determined in the situation where the PDF of total water is significantly skewed. A simple connection can be made by considering the covariance between them. The joint PDF of w and total water that was explored in [Larson et al., 2002] would be a good approach, but cloud ice should be added into the joint PDF for the mixed-phase clouds. Also, other scalars such as cloud droplet number concentrations can be treated potentially at the subgrid scale which could improve the representation of aerosol indirect effects.
 (2) It is dangerous to limit σw with a minimum value of 0.3 m s−1, which is unrealistically large for cloud layers below the top layer. Cloud top radiative cooling and entrainment may not be well resolved in GCMs and this drives TKE, which is in turn used to determine σw. Therefore, σw may not be well represented in coarse horizontal and vertical resolutions of GCMs. In order to produce reasonable σw at coarse resolutions, formulations of σw or w parameterizations should be scale-aware. For example, Larson et al.  did some preliminary work on scale-aware PDF parameterizations and showed that w PDF is significantly improved at 16 km grid spacing.
 (3) We recommend introducing a consistent subgrid treatment of total water, cloud liquid and cloud ice in GCMs. A Gaussian normal distribution is recommended for the subgrid variability of total water and Gamma functions can be used to represent the subgrid variability of cloud liquid and ice (and even snow). To avoid the analytical challenge for linking the Gaussian normal distribution for total water with the Gamma PDF for cloud liquid and ice, the Gaussian PDF could be replaced by a Gamma function. We also recommend that variances with total water, vertical velocity, eddy diffusion, and cloud microphysical processes be connected instead of using fixed variance for subgrid treatments of cloud liquid and ice.
 Our next step is to implement some of the above suggestions to improve the CAM performance in simulating Arctic mixed-phase clouds.
 This study was supported by the U. S. Department of Energy (DOE) Office of Science Climate Change Modeling Program and Atmospheric Research Program (ASR). The Pacific Northwest National Laboratory (PNNL) is operated for the DOE by Battelle Memorial Institute under contract DE-AC06-76RLO 1830. The authors are grateful to Hugh Morrison at NCAR and Peter Caldwell at LLNL for useful discussions. We also thank Matthew Shupe at NOAA ESRL for providing the retrieved cloud data for MPACE.