1. Introduction
The atmospheric component–Atmospheric Model, version 3 (AM3)–of the new Geophysical Fluid Dynamics Laboratory (GFDL) coupled general circulation model (CM3; Donner et al. 2011) includes a prognostic treatment of cloud drop number concentration. This is a significant departure from the previous version, AM2 (GFDL Global Atmosphere Model Development Team 2004), which separately prescribes cloud drop number for land and ocean grid points. The introduction of prognostic drop number was motivated by the desire to simulate the aerosol indirect effect, a major source of uncertainty in global climate models (e.g., Solomon et al. 2007).
Key to predicting cloud drop number is the representation of the activation process that converts aerosols serving as cloud condensation nuclei (CCN) into cloud droplets. At local cloud scales, aerosol activation is governed by the Köhler theory (e.g., Pruppacher and Klett 1997) and depends on aerosol mass, size distribution, chemical composition, and supersaturation. The supersaturation is itself controlled by cloud-scale vertical motions. General circulation models (GCMs) with their horizontal grid spacing of the order of 100 km are unable to resolve cloud-scale motions. As a result, additional assumptions regarding the subgrid variability are required to implement a local activation parameterization in a GCM grid box.
We describe the AM3 subgrid activation assumptions and contrast them to existing approaches. We also investigate the sensitivity of AM3 to these underlying assumptions. To limit the scope of this work, we solely focus on the subgrid variability aspect of the activation parameterization. We treat the local activation parameterization that links aerosols and vertical velocity to the nucleation of cloud droplet as given. In AM3, it is based on Ming et al. (2006) with modifications as described in Donner et al. (2011). While the present study focuses on subgrid activation assumptions, we note that explicit or implicit subgrid assumptions also impact the representation of other cloud processes in GCMs, such as radiative and microphysical impacts of vertical cloud overlap assumptions, and horizontal cloud condensate variability (e.g., Pincus and Klein 2000; Barker and Räisänen 2004, 2005; Pincus et al. 2006).
2. Background
For clarity, we distinguish between a local activation parameterization and a gridbox activation parameterization. We define a local parameterization as one that is applicable at cloud scale. It predicts the number of activated cloud drops based on aerosol properties and supersaturation following the Köhler theory. The dependence on supersaturation is often recast in terms of cloud updraft velocity as proposed by Twomey (1959). We define a gridbox activation parameterization as a local parameterization that has been modified for GCM application. We call “subgrid assumptions” the set of assumptions used to convert a local parameterization into a gridbox parameterization. Subgrid assumptions typically incorporate the effect of subgrid variability of vertical velocity.
A survey of the literature reveals that the treatment of the subgrid vertical velocity generally falls within one of two distinct approaches (see the appendix for a more in-depth review): (i) an explicit assumption about the shape of the vertical velocity probability density function (PDF) is made, and an integration of the local parameterization is performed to obtain the gridbox parameterization; or (ii) the existence of a characteristic vertical velocity is assumed. This velocity is, in general, different from the gridbox average velocity. The gridbox parameterization is obtained by substituting the velocity with the characteristic velocity.
3. AM3 subgrid nucleation assumptions
We now summarize the subgrid assumptions to implement the local activation parameterization of Ming et al. (2006) as a gridbox parameterization in AM3. A similar approach also applies to a version of AM3 that incorporates a two-moment bulk microphysics (Salzmann et al. 2010).
AM3 also includes the nucleation of cloud droplets in its shallow convective parameterization [Bretherton et al. (2004) with modifications described in Zhao et al. (2009) and Donner et al. (2011)]. Nucleation occurs at cloud base using the plume vertical velocity at that level. Cloud drops are transported upward in the plume. When liquid water is detrained, both mass and number are transferred to the large scale. Prediction of cloud drop number is not incorporated in AM3’s deep convective parameterization (Donner 1993).
4. Sensitivity experiments
Description of the REF and alternate model configurations (config.; S1, S2, and S3). 
Model configuration is similar to the fixed sea surface temperature (SST) simulation presented in Donner et al. (2011), except that interannual variability in boundary conditions and forcings are removed to enable shorter simulations. Interannual monthly SSTs are replaced with monthly climatologies for the period 1980–2000. Emissions of sulfate precursors, organic and black carbon, and greenhouse gas concentrations are held constant at their 1990 values. As in AM3, emissions are based on Lamarque et al. (2010). Impacts of explosive volcanoes are neglected. Results discussed in this section are based on 6-yr experiments following a 1-yr spinup except for some untuned experiments, which are based on 1-yr simulations.
Applied in isolation, formulation changes in Table 1 have a large impact on the predicted cloud drop number and radiation balance, as illustrated in Figs. 1 and 2, from 1-yr simulations. Radiative balance is affected mainly by the absorbed shortwave radiation. Lowering the minimum σw value from 0.7 to 0.2 m s−1 (REF → S1u) reduces the cloud droplet number and increases absorbed shortwave by making clouds less reflective. Allowing droplet nucleation to occur in all clouds rather than new clouds only (REF → S2u, S3u) increases cloud droplet number because nucleation occurs more frequently. (S1u, S2u, S3u refer to untuned experiments.) This in turn makes clouds more reflective and reduces absorbed shortwave. In light of radiation balance constraints, the relatively high minimum σw in the reference model is compatible with the assumption that nucleation only occurs in the newly formed clouds, while allowing drop nucleation in all clouds facilitates a lower minimum σw.
Impact of formulation changes listed in Table 1 on TOA shortwave absorbed and outgoing longwave radiative fluxes. Values shown are for untuned experiments (S1u, S2u, and S3u) and based on a 1-yr simulation.
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
Impact of retuning on cloud drop number concentration distributions. Box plots show 1st, 25th, median, 75th, and 99th percentiles. S1u, S2u, S3u refer to untuned experiments. S1, S2, and S3 have been retuned to have the same net radiation as the REF configuration.
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
It is interesting to note that these formulation changes alone have a radiative impact with a magnitude comparable to the radiative forcing resulting from changes in greenhouse gas concentrations between preindustrial (PI) and present-day (PD) conditions (Solomon et al. 2007). However, configurations S1u, S2u, and S3u could not be used for long-term coupled climate experiments, because the magnitude of their net top-of-the-atmosphere (TOA) radiation fluxes is unrealistically large. Therefore, as in commonly done in GCMs, these configurations are retuned to bring their energy balance in line with the reference configuration. This is accomplished by adjusting parameters from other model components whose exact values are uncertain.
The impact of the retuning process on cloud drop number concentration is illustrated in Fig. 2. The altered model parameters are listed in Table 2. They consist of the autoconversion threshold radius and the erosion constants in the large-scale cloud parameterization.
Model parameters modified in the retuning of alternate configurations (S1, S2, and S3) compared to REF, AM3, AM2, and AM2.1. See GFDL Global Atmosphere Model Development Team (2004) for a discussion of these parameters. AM2.1 refers to the atmospheric component of the coupled model CM2.1 (Delworth et al. 2006). Also shown are values of the TOA net radiation. AM3 and REF are the same model; they differ by the configuration of the simulations and therefore have different TOA radiation balance.
As surveyed by Rotstayn (2000), the autoconversion threshold is often used to tune GCM models. Values in the range of 4.5–7.5 μm are common, even though they are unrealistically small. The threshold radius of 10.6 μm in AM2 is comparable to values in cloud and mesoscale models. This value was reduced to 8 μm in the atmospheric component AM2.1 of the coupled model CM2.1 (Delworth et al. 2006). The value of the threshold radius in large-scale models, such as GCMs, is often reduced compared to finer-scale models to compensate for the neglect of in-cloud subgrid-scale variability (Pincus and Klein 2000; Larson et al. 2001) and the dispersion effect (increased breadth of the drop size distribution with increased number concentration; Rotstayn and Liu 2005).
The large-scale cloud scheme in AM3 solves prognostic equations for cloud amount, liquid, and ice (Tiedtke 1993). These equations include a sink term (“erosion term”) that governs the rate at which subgrid-scale mixing dissipates clouds in subsaturated grid cells. Tiedtke (1993) suggested a cloud erosion constant of 1 × 10−6 s−1. AM2 retained this value but also introduced larger values for grid boxes where either convection or vertical turbulence is active. As noted in GFDL Global Atmosphere Model Development Team (2004), the AM2 erosion of 4.7 × 10−6 s−1 under convective conditions might be about 40 times smaller than suggested by the analysis of large-eddy simulations (LESs) of trade wind cumulus clouds (Siebesma et al. 2003). Erosion constants in S2 and S3 are closer to the LES’s estimated value.
We emphasize that the specifics of the retuning process are subjective. Configurations listed in Table 2 are not unique. We also note that the distinction between formulation changes and retuning can be arbitrary. Here, we regard the modifications to the activation as formulation changes, and modifications to the cloud macrophysics (erosion scales) and microphysics (autoconversion threshold) as retuning. One could take an alternate view and regard the cloud macro- and microphysical changes as formulation changes and the modifications to the activation as part of the retuning process. Both reflect uncertainties in GCM parameterizations.
Basic simulation metrics are presented in Table 3. AM3 and REF are identical models but differ by their experiment setup (20 yr with varying boundary conditions versus 6 yr with climatological boundary conditions). AM3 results are from Donner et al. (2011). The REF configuration generally yields lower scores than AM3 owing to the different setup. REF is generally better than alternate configurations S1, S2, and S3, although scores often differ only very slightly.
Summary of root-mean-square error (rmse) and correlation (corr) between model configurations and various observations. Sea level pressure (SLP), and 200- and 850-hPa zonal wind (U200, U850) are compared to 40-yr European Centre for Medium-Range Weather Forecasts Re-Analyses (ERA-40s) (Uppala et al. 2005); precipitation is compared to GPCP v2 (Adler et al. 2003); shortwave and longwave cloud forcings (SWCF, LWCF, respectively) are compared to CERES EBAF (Loeb et al. 2009). All comparisons are based on annual means.
Additional simulation characteristics are shown in Fig. 3 in the form of annual zonal averages. Despite significant differences in the formulation of cloud drop activation and associated cloud retuning, basic simulation characteristics of these four configurations exhibit remarkably little difference, except for column integrated in-cloud drop number. Indeed, for most fields, intermodel differences are much smaller than differences between model and observations.
Comparisons of zonal mean model results and observations. Results from the four model configurations (REF, S1, S2, and S3) are represented by the colored curves. Observations are from (a)–(d) Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced And Filled (EBAF) and Earth Radiation Budget Experiment (ERBE) (Harrison et al. 1990); (e) Global Precipitation Climatology Project, version 2 (GPCP v2) and Climate Prediction Center Merged Analysis of Precipitation, version 2 (CMAP v2) (Xie and Arkin (1997); (f) ISCCP (Rossow and Schiffer 1999), and (g) Greenwald et al. (1993) and Weng and Grody (1994).
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
Figure 4 illustrates distributions of cloud drop effective radius and number concentration. Similar to Gettelman et al. (2008), distributions are shown for clouds between the surface and 100 hPa and with a minimum in-cloud water content of 2 ppmm. There are large differences between the four configurations. Effective radius distributions for REF and S1, which activate cloud drops only in new clouds, have more occurrences of clouds with low cloud droplet numbers and correspondingly large effective radii. Number distribution peaks at low values for REF and S1, compared to a distinct peak around 15 cm−3 for S2 and S3, where activation is allowed to occur in all clouds. From Fig. 4, it is clear that nucleating drops in all or new clouds only has a large impact on global cloud drop distributions. The minimum σw value has a comparatively smaller impact (REF versus S1), as does the choice of integrating over a PDF or using a characteristic velocity (S2 versus S3).
Distributions of tropospheric in-cloud (a) drop effective radius and (b) number concentration for the four model configurations.
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
It is also interesting to contrast Fig. 4 with Figs. 5a and 5c in Gettelman et al. (2008). The range of drop effective radius is larger in the four configurations tested here, especially for REF and S1. Drop number concentrations tend to be smaller; Gettelman et al. (2008) report a higher frequency of clouds with drop concentrations above 100 cm−3.
Based on the results presented in this section, it would be tempting to conclude that the various activation formulations and related cloud tuning have a decisive impact only on cloud drop number concentration. As we demonstrate in the next section, this is, however, not the case.
5. Impact on RFP and Cess climate sensitivity
We perform additional sensitivity experiments with the four configurations described in the previous section.
Figure 5 illustrates zonal-mean differences between PD and PI conditions for the net absorbed shortwave radiation and liquid water path. Although the zonal mean radiation and liquid water path fields are smooth under both PD and PI conditions, PD − PI differences are rather noisy. Lohmann et al. (2010) also observed that annual zonal means of RFP tend to be noisy. It is because the PD − PI signal is relatively small compared to model interannual variability. Extending simulations beyond 6 yr would probably reduce the noise level. Nonetheless, one can see that statistically significant differences occur in the northern latitudes, between 15°N and 60°N, presumably because of a larger anthropogenic aerosol signal. Differences in shortwave and liquid water path are consistent with each other. More cloud water results in less shortwave absorption. Overall, S1 tends to exhibit the largest PD − PI changes, while S3 the weakest.
Zonal mean differences in (a) TOA shortwave absorbed flux and (b) liquid water path between experiments with PD greenhouse gases and emissions, and PI conditions. Differences are relatively small and as a result noisy. Thick lines indicate where the PD − PI differences are statistically significant at the 95% confidence level based on a Student’s t test.
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
Global numbers for total clear-sky and all-sky anthropogenic RFP, Cess sensitivity, and temperature change index are listed in Table 4. Differences between RFPclr are relatively small and not statistically significant at the 95% confidence level based on a Student’s t test. However, all-sky RFP values exhibit large differences between the four configurations, with a range slightly larger than ±50% from the reference configuration. Differences are statistically significant between REF and S1 and S3 but not between REF and S2 because of large interannual variability. Compared to RFP, the variation in Cess climate sensitivity λ is small. All alternate configurations have a sensitivity that is within 10% of the reference. The sensitivity of the reference configuration (0.73 K W−1 m2) is larger than the sensitivity of 0.58 K W−1 m2 for AM2. Both values are larger than sensitivities for the Community Atmosphere Model, version 3 (CAM3) with a semi-Lagrangian dynamical core (0.41 K W−1 m2), the CAM3 with an Eulerian dynamical core (0.54 K W−1 m2), and the “superparameterized” version of the CAM (SP-CAM) (0.41 K W−1 m2) as reported by Wyant et al. (2006).
Summary of PD − PI clear-sky radiative flux perturbation (RFPclr), all-sky radiative flux perturbation (RFP), Cess climate sensitivity λ, and surface temperature change index ΔTs,i ≡ RFP × λ.
The surface temperature change index, ΔTs,i, is 0.66 K for REF (AM3), which is larger than the PD − PI temperature change of 0.32 K (1980–2000 minus 1880–1920 global average) in a five-member ensemble of the coupled model CM3 (2011). For AM2, RFP is estimated to be 1.44 W m−2 and therefore ΔTs,i ≈ 0.84 K, compared to a warming of 0.66 K in a five-member ensemble of CM2.1 (Knutson et al. 2006). Possible hypotheses for the smaller surface temperature change of the coupled model include the lack of equilibrium due to ocean thermal inertia, neglect of explosive volcanoes in our climatological configurations, and—to a lesser degree—model internal variability and simulation lengths. Similar differences have also been observed for the earth when comparing equilibrium temperature change with actual global mean surface temperature change (Schwartz et al. 2010).
RFP values in Table 4 are also plotted in Fig. 6 as a function of the autoconversion threshold selected in the retuning process. Quite remarkably, the four configurations are almost perfectly aligned along a straight line. To investigate whether this is simply a coincidence or whether there is a real correlation between the autoconversion threshold and RFP, we perform two additional experiments. REF− is identical to REF but with the threshold reduced from 8.2 to 6.2 μm, and REF+ is with the threshold increased to 10.2 μm. REF− and REF+ are subject to large TOA energy imbalances and therefore could not be used for coupled climate experiments. RFP values for these two additional configurations are shown in Fig. 7. Again, there is an almost perfect linear correlation between the autoconversion threshold and RFP. Moreover, the linear fit for REF, REF+, REF− (solid line in Fig. 7) is almost identical to the linear fit for REF, S1, S2, and S3 (dashed line in Figs. 6 and 7).
PD − PI RFP for the four different configurations plotted as a function of the autoconversion threshold. Dashed line is a linear fit. Error bars indicate standard deviation calculated as 
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
As in Fig. 6, but for REF and two additional experiments, REF+ and REF−, that only differ by the value of the autoconversion threshold. Thin solid line is a linear fit; dashed line is the fit from Fig. 6.
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
RFP includes the impact of greenhouse gases, and aerosol direct and indirect effects. The autoconversion threshold does not directly affect greenhouse gases or the aerosol direct effect. Although changes in cloud liquid water content could have a small impact on aerosol wet deposition, the correlation between autoconversion threshold and RFP is most likely caused by a modulation of the strength of the indirect effect. This is corroborated by the fact that RFPclr exhibits only small and statistically insignificant differences between the four configurations. The amplitude of the modulation of the indirect effect is of the order of 1 W m−2 for the range of threshold values tested here (Table 4). This range approaches the range of −1.8 to −0.3 W m−2 for the radiative forcing from the first indirect effect reported by Solomon et al. (2007). Although not unexpected, this is significant given that the autoconversion process is only one of several physical processes that impact the indirect effect. For example, Storelvmo et al. (2009) reported that only changing diagnostic relationships between aerosol mass and cloud drop number can also lead to a large modulation of the strength of the indirect effect from −1.94 to −0.62 W m−2. Hoose et al. (2009) investigated the impact of imposing a minimum cloud droplet or aerosol concentrations in Community Atmosphere Model (CAM)-Oslo. They found a large sensitivity of the indirect effect (−1.88 ± 0.14 to −0.62 ± 0.17 W m−2) to the choice of imposed minimum concentrations. We note that AM3 does not prescribe any minimum cloud droplet or aerosol concentrations.
Examples of autoconversion formulation impact on the magnitude of the indirect effect have been documented before. Lohmann and Feichter (1997) performed sensitivity studies to evaluate the effect of sulfate aerosols on the indirect effect. In one of their tests, they found that by replacing the autoconversion parameterization alone, the magnitude of the indirect effect could change by 2.6 W m−2. Rotstayn (2000) found that introducing a new autoconversion parameterization increased the magnitude of the indirect effect by 0.6 W m−2 compared to the original scheme. In a subsequent work, Rotstayn and Liu (2005) incorporated an autoconversion parameterization that accounted for the dispersion effect. They found that this alternate autoconversion led to a reduction in the magnitude of the indirect effect.
Quaas et al. (2009) compared the aerosol indirect effect in 10 different GCMs against satellite observations. Among their findings, they noted that all models overestimate the variations of liquid water path with anthropogenic aerosol optical depth, with the strength of this relationship influenced by the autoconversion parameterization.
The RFP uncertainty range of nearly 1 W m−2 documented here is substantially larger than the recent study by Lohmann and Ferrachat (2010). They estimated the impact of parametric uncertainties on the aerosol indirect effect by varying four parameters often used to ensure radiation balance in ECHAM5: rate of rain formation by autoconversion, rate of snow formation by aggregation, inhomogeneity factor for ice clouds, and entrainment rate for deep convection. They found an uncertainty range of 25% (0.25 W m−2) for the total anthropogenic aerosol effect when considering all their experiments and a smaller range of 11% when considering only experiments that satisfied energy balance constraints.
PD − PI liquid water path difference for the four different configurations plotted as a function of the cube of the autoconversion threshold (note the cubic scale of the horizontal axis). Dashed line is a linear fit, indicating a cubic relationship between autoconversion threshold and liquid water path difference as suggested by Eq. (15).
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
Finally, Fig. 9 shows the surface temperature change index, ΔTs,i, inferred from RFP and climate sensitivity. Because RFP is much more sensitive to the autoconversion threshold than climate sensitivity, there is again a nearly linear correlation with the threshold radius. To predict a realistic temperature evolution from PI to PD conditions, GCMs must balance RFP and climate sensitivity, often resulting in an inverse correlation between forcing and sensitivity (Schwartz et al. 2007; Kiehl 2007; Knutti 2008). Among our alternate configurations, S1 has the smallest value of ΔTs,i at 0.28 K. It is likely that this configuration would not produce a reasonable temperature change in a fully coupled climate model over the PI–PD period.
Estimate of PD − PI surface temperature change index defined as ΔTs,i ≡ RFP × λ, where RFP is the radiative flux perturbation and λ is the Cess climate sensitivity.
Citation: Journal of Climate 24, 13; 10.1175/2010JCLI3945.1
6. Conclusions
Subgrid activation assumptions—such as how subgrid variability in vertical velocity is treated in the cloud drop nucleation process and how newly nucleated drops are incorporated in clouds—can differ significantly from one GCM to another. These assumptions have a large impact on the predicted cloud drop number. By changing liquid cloud properties, they affect the amount of absorbed shortwave radiation.
Because of their radiative impact, changes in the subgrid activation assumptions are, implicitly or explicitly, accompanied by changes in other aspects of the cloud parameterization to achieve a realistic radiation balance. As an illustration, we built three alternate configurations based on AM3 but with modified subgrid activation assumptions and cloud tuning.
Despite significant underlying formulation changes, impacts on present-day climatology are small between these configurations. It would be difficult to constrain the subgrid assumptions and cloud tuning with current observations. However, there is a very large impact on the total anthropogenic radiative flux perturbation (RFP) between present-day and preindustrial conditions. RFP varies by ±50% (0.42–1.41 W m−2) in our alternate configurations compared to the reference AM3 value of 0.91 W m−2. This range is attributable to a change in the magnitude of the aerosol indirect effect. In the extreme case, the indirect effect appears to counteract the majority of the greenhouse gases effect. The impact on anthropogenic RFP is not caused by the subgrid assumptions themselves but through the required cloud retuning—in particular, the retuning of the autoconversion threshold radius. Adjusting the autoconversion threshold radius for radiation balance is a common practice, but using a different tuning strategy could have resulted in a smaller RFP spread.
Although sensitivities of the indirect effect to details of autoconversion parameterizations have been documented before (e.g., Lohmann and Feichter 1997; Rotstayn 2000; Rotstayn and Liu 2005), we find a linear relationship between the autoconversion threshold radius and the RFP previously not documented. The sensitivity originates from the numerical implementation of the autoconversion, which uses a limiter to avoid over depletion of water. The RFP uncertainty range of nearly 1 W m−2 documented here is substantially larger than the sensitivity study of Lohmann and Ferrachat (2010) (0.25 W m−2 for all their experiments and 0.11 W m−2 for their experiments in energy balance).
To simulate a realistic temperature evolution from preindustrial to present day, a GCM must balance RFP and climate sensitivity. Climate sensitivity estimated from Cess-like experiments does not change substantially in our alternate configurations. This suggests that in the present model setup, the warming from preindustrial to present day would be strongly controlled by the RFP. Some of our alternate configurations would likely result in an unrealistic temperature evolution compared to observations.
Our results serve as a reminder that uncertainties in the formulation of individual components of a cloud parameterization can translate into significant uncertainties in the aerosol indirect effect and total anthropogenic RFP. Reducing these uncertainties will necessitate more realistic physical representation of cloud processes and better observational constraints.
Acknowledgments
We acknowledge the Global Atmospheric Model Development Team (GAMDT) for its considerable efforts leading to the creation of AM3. V. Naik is acknowledged for preparing emission input datasets. C. T. Gordon and S.-M. Fan are acknowledged for their suggestions on an early draft. We thank the two anonymous reviewers for their constructive comments on the submitted manuscript.
ERBE and CERES-EBAF data were obtained from the NASA Langley Research Center Atmospheric Science Data Center. GPCP and ISCCP data were obtained from NASA Goddard Space Flight Center. NOAA/NCEP CPC Merged Analysis of Precipitation (CMAP) data were obtained from the IRI/LDEO Climate Data Library. ERA-40 data were supplied by the European Centre for Medium-Range Weather Forecasts (ECMWF).
APPENDIX
Review of Existing Formulations
We review some specific examples of GCM formulations for computing the grid box number of nucleated drops,
a. Subgrid distribution of w with diagnostic 
In their assessment of the radiative impact of anthropogenic sulfate, Chuang et al. (1997) were probably the first to implement a PDF-based activation parameterization in a GCM. They assume a normal distribution as underlying w PDF. Its mean is given by the GCM gridbox vertical velocity, and its standard deviation is set to a constant value of 0.5 m s−1. This value is chosen based on observations from the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment. They also perform sensitivity tests with standard deviations of 0.25 and 0.75 m s−1. Chuang et al. (1997) diagnosed the cloud drop number with Eqs. (1) and (3) in the form 
b. Subgrid distribution of w with prognostic
The treatment of the nucleation source term in their prognostic equation of
The same methodology is also employed in subsequent studies with a different GCM [Model for Integrated Research on Atmospheric Global Exchanges (MIRAGE)] to estimate the radiative forcing from anthropogenic sulfate aerosols (Ghan et al. 2001a) and the aerosol indirect effect (Ghan et al. 2001b). The lower bound on the standard deviation is increased to 0.2 m s−1. In an updated version of MIRAGE (Easter et al. 2004), cloud drop number is also predicted for subgrid stratiform and convective clouds in addition to large-scale stratiform clouds. Activation in subgrid stratiform clouds is treated similarly to large-scale stratiform clouds. Activation in convective clouds occurs at cloud base using a fixed convective velocity of 1 m s−1.
GCM implementations of subgrid w variability typically assume that the PDF follows a normal distribution. Studies have shown that normal distributions do not represent shallow cumulus convection well because they are unskewed. Distributions constructed from a mixture of two Gaussians can more faithfully describe the subgrid PDFs in boundary layer clouds (e.g., Larson et al. 2002). A turbulence cloud parameterization based on such a PDF has been developed (Golaz et al. 2002, 2007) and has recently been extended to the prediction of cloud drop number in the single-column version of the GFDL GCM (Guo et al. 2010).
c. Characteristic vertical velocity 
Ming et al. (2007) implemented a prognostic cloud drop number equation in a modified version of AM2. For convective clouds, the characteristic vertical velocity is chosen to be 
Recently, Morales and Nenes (2010) proposed a characteristic vertical velocity framework for activation that provides a droplet number concentration characteristic of the value averaged over the PDF of vertical velocity. They derived an analytical formulation for the characteristic velocity for microphysical processes that can be approximated with a power law of the velocity and a normal distribution of the subgrid velocity. They also evaluated the impact of droplet variability effects on autoconversion and effective radius.
Finally, we also note that to predict realistic cloud drop numbers, some mesoscale modeling studies also include subgrid velocity variability using a characteristic velocity derived from the TKE (e.g., Morrison and Pinto 2005; Morrison et al. 2008; Ivanova and Leighton 2008).
d. Validation studies
A few studies have attempted to validate some of the formulations discussed above against observational data and high-resolution models.
Lin and Leaitch (1997) were probably the first to investigate the application of local activation parameterizations (Ghan et al. 1993; Abdul-Razzak et al. 1998) against observations collected during 14 flights from the 1993 North Atlantic Regional Experiment (NARE). They found that an integration over a normal w distribution with zero mean and observed variance led to an underestimate of the cloud drop number compared to observations. They obtained better agreement by first estimating the maximum cloud drop number, 
Peng et al. (2005) compared observed cloud drop number for 11 marine stratus cloud cases with simulations from a parcel model. The observational data were collected during two field experiments in the North Atlantic Ocean [NARE and the Radiation, Aerosol and Cloud Experiment (RACE) in 1995]. Simulated cloud drop numbers were calculated using a characteristic velocity as well as a subgrid distribution. They estimated the maximum cloud drop number, 
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