a. Initial sounding

A thermodynamic sounding composited by the Global Energy and Water Cycle Experiment Cloud System Study (GCSS) boundary layer working group (available online at http://www.knmi.nl/samenw/rico) and based on soundings from the Rain in Cumulus over the Ocean (RICO) field experiment (Rauber et al. 2007) was used in Jiang et al. (2009). The composite sounding represents a relatively undisturbed trade wind cumulus regime with light precipitation. The area-averaged rainfall rate derived from radar data ranges from 1.2 to 7.2 mm day⁻¹ (Nuijens et al. 2009; Snodgrass et al. 2009), whereas the LES initialized using the composite sounding produced an area-averaged surface rainfall of 1 mm day⁻¹ under clean aerosol conditions (Jiang et al. 2009). The combination of the relatively stable initial sounding and the cancellation of the imposed large-scale radiative cooling and subsidence above 2250 m limits the growth of the cloud layer and inhibits precipitation.

To create an environment that is more conducive to convection and to understand how the precipitating population of clouds responds to variations in aerosol concentration, the composite sounding is modified as shown in Fig. 1. The model domain top is extended to 6 km, the initial potential temperature is 1.15 K km⁻¹ less stable than the composite profile between 2 and 4 km, and the initial total water mixing ratio r_t profile is moistened by 1.6 g kg⁻¹ between 2 and 4 km. These profiles are similar to the initial profiles observed on 19 January 2005 during the RICO campaign. The large-scale tendencies of potential temperature and moisture and the surface forcing used in this study are similar to those used in Jiang et al. (2009).

b. Numerical experiments

Five three-dimensional (3D) simulations were performed, as summarized in Table 1, with initial aerosol concentrations N_a of 100, 200, 300, 400, and 500 cm⁻³. The simulations were run for 12 h, but only the output from the last 9 h was analyzed. The domain size was 25.6 km × 25.6 km × 6 km with the horizontal grid spacing Δx = Δy = 100 m and the vertical grid spacing Δz = 40 m up to 4 km and vertically stretched thereafter with a stretch factor of 1.035. Lateral boundaries are cyclic in both the east–west and north–south directions. A time step of 2 s was used for all simulations. Note that the horizontal domain size meets the minimum benchmark for the development of deeper convection as suggested by Petch (2006), because it allows representation of the mesoscale. The simulations were performed on the National Oceanic and Atmospheric Administration (NOAA)/Earth System Research Laboratory’s high performance computer using 64 total 2.66-GHZ Intel Xeon processors.

c. Analysis of model output

a. Time series

To illustrate the temporal evolution of various cloud parameters derived from the model output, time series of LWP (averaged only over columns that have LWP > 20 g m⁻²), cloud fraction (the fraction of vertical columns that have LWC greater than 0.05 g m⁻³), domain-averaged surface precipitation rate R_sfc, and cloud-top maximum z_t,max and cloud-base minimum z_b,min are plotted in Fig. 2. The terms z_t,max and z_b,min are the highest and lowest cloud top and base in the domain, respectively, but their difference should not be interpreted as a proxy for cloud depth; rather, the domain-averaged cloud depth is on the order of 600 m. For clarity, only the N_a = 100, 300, and 500 cm⁻¹ results are plotted. The variables in the other two simulations (200 and 400 cm⁻¹) are generally intermediate to the others. Figure 2 shows that the clean simulation (100 cm⁻³) has the highest LWP, whereas the polluted simulation (500 cm⁻³) has the lowest LWP. The buildup of LWP in the clean simulation prior to each heavier R_sfc episode shows the strong correlation between LWP and R_sfc. The fairly distinct differences between LWP for clean and polluted cases prompted further in-depth analysis of these simulations. Profiles of relative humidity, equivalent potential temperature, and vapor fluxes (not shown) reveal the following interesting features: 1) In the clean simulation, the evaporation of precipitation establishes a cooler and moister subcloud layer; the moister air maintains a lower cloud base, and higher vapor flux at cloud base relative to the more polluted cases. 2) In the polluted simulations, the increased evaporation of liquid water near cloud top results in more moistening that promotes higher cloud tops relative to the clean simulation; nevertheless, maximum cloud-top heights are highest in the clean case (Fig. 2d). 3) On average, both cloud base and cloud top are shifted upward in the polluted case (not shown); thus, entrainment of environmental air tends to dilute the polluted clouds more than the clean clouds.

These results paint a somewhat different and more complex picture from what might have been expected based on precipitation losses (and associated stabilization) and preconditioning arguments alone. The rain produced by the clean clouds is insufficient to weaken convective development or significantly remove LWP. More effective preconditioning in the case of the polluted clouds is not strong enough to generate deeper clouds with higher LWP than the clean clouds.

Aerosol has the reverse effect on cloud fraction; the heavier R_sfc produced after 3.5 h in the 100 cm⁻³ simulation reduces cloud fraction significantly. On average, R_sfc is about 3 mm day⁻¹ in the 100 cm⁻³ simulation, only about 0.5 mm day⁻¹ in the 300 cm⁻³ simulation, and negligible in the most polluted (500 cm⁻³) simulation. The average R from the cleanest simulation falls well in the midrange of observations (1.2–7.2 mm day⁻¹; Nuijens et al. 2009; Snodgrass et al. 2009). The Z–R relationships are calculated from the modeled drop size distributions in two ways: (i) based on cloud- and lifetime-averaged Z and R for each individual cloud, where R ≥ 0.11 mm h⁻¹ (roughly equivalent to Z ≥ 7 dBZ), and (ii) based on individual cloudy columns where R is greater than 0.11 mm h⁻¹. The first method yields Z = 557R^2.35, and the second yields Z = 66.7R^1.62, compared to RICO observations Z = 88.7R^1.50 (Snodgrass et al. 2009) and Tropical Rainfall Measuring Mission (TRMM) Z–R relationship Z = 148R^1.55 (Nuijens et al. 2009). When comparing R converted from each Z–R relationship at Z = 10 dBZ, the precipitation rate in this study is 23% lower than that of Snodgrass et al. and 6% higher than that of Nuijens et al. for the first method and 32% higher than Snodgrass et al. and 76% higher than Nuijens et al. for the second method. Differences of this magnitude are not atypical for rain rates derived from empirical Z–R relationships.

The lowest cloud base is steady and located around 500 m over the time period shown. The highest cloud top fluctuates a fair amount, reflecting the variability in the strength of convection over the domain, but experiences a gradual increase from 3.5 to 4.5 km because of the imposed large-scale moistening above 3 km and drying below in the modeled environment. This added moisture promotes deeper cloud growth over time.

b. Cloud size, depth, and precipitation

The cloud size distribution n(a) of a cumulus population is defined as the number of clouds per area range (a; a + da). We only consider clouds larger than 300 m in diameter (defined as the square root of cloud area a). As shown in earlier studies, n(a) follows a negative power law (Fig. 3a), indicating a rapid decrease in the number of clouds with increasing size (e.g., Cahalan and Joseph 1989; Neggers et al. 2003; Zhao and Di Girolamo 2007; Koren et al. 2008). The size distributions are affected to some extent by N_a; the negative slope of the power law increases as N_a increases, as does the total number of clouds. Polluted cloud populations therefore have many more small clouds than clean cloud populations in accord with Xue and Feingold (2006) and Jiang et al. (2009). This same trend also holds for smaller clouds (<300-m diameter; not shown) However, for diameters greater than 1300 m, the clean simulations (100 and 200 cm⁻³) produce more large clouds.

Clouds with diameters of 500 m contribute very little to the cloud- and lifetime-averaged rainfall (Fig. 3b), in spite of their ubiquity (Fig. 3a). The value of R increases steeply as cloud diameter exceeds 500 m, and continues to increase to the maximum value at the largest cloud size. The largest contribution to R is also from the deepest clouds, in accord with observations (e.g., Short and Nakamura 2000) and with the fact that clouds tend to have a fairly constant aspect ratio (Benner and Curry 1998). The most polluted clouds (N_a > 300 cm⁻³) do not grow as deep or as large as the clean clouds; as expected (e.g., Warner 1968), they produce very little rain (Fig. 3c).

It has been proposed that, by delaying the rain process and allowing for more evaporation in the free troposphere, an increase in the aerosol may allow clouds to deepen, thus offsetting the precipitation suppression (Stevens and Seifert 2008). For the simulations with lower aerosol concentrations (N_a ≤ 300 cm⁻³), Fig. 3c shows that clouds do indeed grow slightly deeper with increasing N_a so that some offsetting of precipitation suppression is likely occurring. Interestingly, the deepening effect does not exist at N_a > 300 cm⁻³ so that other factors are clearly controlling cloud depth. It should be noted that there are only one or two clouds at these high cloud depths, so a general conclusion should not be drawn based on the small sample statistics.

Caution should be taken when comparing these results with observations made during RICO. First, the range of aerosol conditions during RICO was much narrower (50–200 cm⁻³; e.g., Hudson and Mishra 2007); second, the meteorological conditions were highly variable (e.g., Nuijens et al. 2009) and included confounding (positive) correlations between meteorological factors such as wind speed, cloud depth, and aerosol loading.

c. Frequency of occurrence of cloud lifetime

The frequency of occurrence of cloud lifetime is plotted for three simulations based on tracking thousands of individual clouds as discussed in section 2c(1) (Fig. 4). A geometrically increasing bin size is used to reduce the sampling “noise” at longer lifetimes where the number of samples is small. The general trend in the distribution is very similar among the three simulations (Fig. 4a). The frequencies in the other two simulations (200 and 400 cm⁻¹) are generally intermediate to the others and are not shown for clarity. The progressive increase in the number of clouds in association with higher N_a (Table 1, column 3) is reflected in the higher occurrences in Fig. 4a. The majority of clouds have lifetimes of 30 min or shorter. Only a few tens of clouds have lifetimes longer than 60 min.

Figure 4b relates cloud lifetime to cloud area, and shows the average cloud lifetime within each size bin. Clouds with sizes larger than 2000 m (diameter) are placed in the last bin to improve the sample statistics. On average there is a monotonic increase in cloud lifetime with increasing cloud size. For clouds with diameters less than 500 m and lifetime shorter than 30 min, there is no clear aerosol effect on lifetime, in accord with Jiang et al. (2006), but for the subset of large clouds, polluted clouds tend to live longer than clean clouds. Thus, although the cleaner conditions produce the largest and deepest clouds (Figs. 2d, 3a,c), the precipitation associated with these clouds has the effect of reducing cloud lifetime relative to their nonprecipitating counterparts.

d. Relationship between R, LWP, N_d, and cloud lifetime

For stratocumulus clouds, the precipitation rate R is often assumed to scale with LWP (or the cloud depth) and N_d as follows:

View Expanded

Equation (1) has been derived from observations (e.g., Pawlowska and Brenguier 2003; Comstock et al. 2004; vanZanten et al. 2005) and modeling (e.g., Geoffroy et al. 2008; Feingold and Siebert 2009) of stratocumulus clouds. The suitability of Eq. (1) to describe precipitation in cumulus clouds has not been thoroughly evaluated, except for a preliminary study by Mechem and Kogan (2008). This is now investigated using the model-generated population of cumulus clouds. At this point, we note that aerosol effects on R are expressed via N_d to avoid addressing the sometimes complex relationship between N_d and N_a.

Figure 5 shows a scatterplot of R_sfc versus LWP for three different initial N_a. As expected, R_sfc is positively correlated with LWP and negatively correlated with N_a (or N_d); R_sfc is negligible for the simulations initialized with N_a of 400 and 500 cm⁻³. For simplicity, it is now assumed that Eq. (1) also holds true for cumulus clouds. In the subsequent analysis, only clouds with R > 0.5 mm day⁻¹ and cloud area and lifetime-averaged LWP > 100 g m⁻² (precipitating clouds) are included in the analysis. Thus, the number of clouds (Table 2, column 6) is much smaller than that listed in Table 1 (column 3). The coefficients of α and β are derived from multivariate linear regression fits to the logarithms of the model output of R, LWP, and N_d, and R_reg (the regression-generated R) is calculated using Eq. (1) and compared to the model R. The various values of precipitation rate from R_sfc, R_zb, and R (where the overbar indicates an average over cloud depth) are all evaluated for their performance.

Equation (1) is further amended to incorporate cloud lifetime as follows:

View Expanded

where , and t_c denotes cloud lifetime. Similar to the tests of Eq. (1), and are also evaluated. Associated α, β, and γ values from Eqs. (1) and (2) regressions are listed in Table 2. The validation and comparison results are shown in Fig. 6. Because the differences in the α and β values are small among R, R_zb, and R_sfc regressions, only the R_zb and R_sfc and the I_zb and I_sfc are presented in Fig. 6. Figures 6a,b show significant scattering between the model-generated R_zb or R_sfc and their respective regression fits R_reg. The best fit using the multiple linear-fit method is plotted as a solid line for each case. Strong correlations are shown with correlation coefficients r ranging from 0.84 to 0.88 for the R regression and from 0.95 to 0.97 for the I regression (Table 2, column 5). The regression performs significantly better for the time-integrated rain amount I (Figs. 6c,d).

The α and β values (Table 2) are very similar, regardless of whether the regression is performed using R, R_zb, or R_sfc, except that the number of samples is less at the surface, reflecting that evaporation in the subcloud layer has reduced the number of samples that meet the cloud criterion. The average α (Table 2, rows 1–3) is 2.45, and β is −1.15. The corresponding parameters derived for stratocumulus clouds are 1.5 and −1.0 (vanZanten et al. 2005), 1.6 and −0.67 (Feingold and Siebert 2009), 1.75 and −1.75 (Comstock et al. 2004), and 2.0 and −1.0 (Pawlowska and Brenguier 2003) for α and β, respectively. The sensitivity to LWP is stronger in this study than in published studies of stratocumulus clouds, whereas the sensitivity to N_d falls in the middle of those of stratocumulus clouds. Including t_c in the regression in Eq. (2) reduces both the α₂ and β₂ (absolute) values slightly (Table 2, rows 4–6). It should be noted that, for stratocumulus, significant differences in regression coefficients can emanate from the definition of the physical parameters (e.g., how R and height are calculated) and measurement biases (in situ versus remote sensing; Brenguier and Wood 2009). In addition, the absolute magnitude of LWP is much higher in some of the deeper cumulus clouds, so small changes in α could a have significant impact.

Parameterizations of the form of Eqs. (1) and (2) assume that LWP, N_d, and t_c are independent, whereas we have already seen that this is not the case. Correlation coefficients r between LWP and t_c, N_d and t_c, and LWP and N_d are 0.52, 0.19, and 0.26, respectively (calculated from the same set of data used to derive the α₂ and β₂ values). The higher correlation coefficient between LWP and lifetime confirms that higher LWP is generally related to longer cloud lifetime. These correlations indicate that the parameters in Eqs. (1) and (2) are not independent, and their primary advantage is that they are directly measurable.

e. Precipitation susceptibility

Now that the relationship between R, LWP, and N_d has been established, other quantities such as precipitation susceptibility can be estimated. Precipitation susceptibility S_o (Feingold and Siebert 2009; Sorooshian et al. 2009, 2010) is a useful metric in that it represents the sensitivity of precipitation to changes in N_d,

View Expanded

Substituting R from Eq. (1), one has

View Expanded

where dln(LWP)/dln(N_d) represents the slope of a plot of LWP versus N_d on a log–log scale and is referred to as “slope” in Table 3. To derive S_o from the model output, data was binned at irregular increments in LWP_max to get similar numbers of samples in each bin. To be consistent, N_d,max and R_max are also used in the regression. The reason for choosing LWP_max over LWP is that LWP_max covers a wider range and R_max provides a larger number of samples than LWP and R provide. Note that calculations of S_o were also done using the cloud area- and lifetime-averaged LWP, N_d, and R, and a similar dependence of S_o with respect to LWP was obtained (not shown). Within each LWP_max bin, a multivariate linear regression fit is performed, and the associated variables (α, β, and slope) are derived (Table 3). The S_o values shown in Table 3 are calculated using Eq. (4) and can be derived from the negative slope of a plot of R versus N_d on a log–log scale with similar values (not shown). Values of S_o shown in Table 3 are statistically significant at the 95% level based on a one-tailed, t-statistics test. The correlation coefficients between log(R) and log(N_d) are in the range of 0.33–0.91 in each LWP bin.

As shown in Table 3, the differences in fit parameters between the calculations using R_sfc and R_zb are small for any given LWP_max interval. Evaporation in the subcloud layer reduces the total number of clouds in R_sfc from R_zb regressions by about 19% at the lowest three LWP_max bins, and the total number of clouds is similar in the higher LWP_max bins.

The S_o increases with LWP_max and reaches a maximum between 1100 g m⁻² < LWP_max < 1400 g m⁻² (Table 3, column 5) and decreases thereafter. The general trend in S_o is similar to that shown in Feingold and Siebert (2009) and Sorooshian et al. (2009, 2010), except that the LWP values are different because of different averaging methodologies. This behavior will be discussed further in section 4a.

f. Precipitation efficiency

The concept of precipitation susceptibility has direct links to the more commonly used metric of PE. In the case of PE, the accumulated R_sfc over the area that experiences rain is divided by the volume integral of the total amount of water condensed over the lifetime of the cloud (e.g., Flossmann and Pruppacher 1988),

View Expanded

where R_V = ∫R_sfc dA dt is the total volume of rainwater accumulated at the surface over the cloud lifetime, A is the area experiencing rain, and M_V is the total volume of condensed water accumulated over the cloud lifetime. PE therefore does not include a term related to drop concentration, although lifetime is inherently included in both the numerator and the denominator. To explore this further, one can perform regressions of the form

View Expanded

where Eq. (6) is a time- and volume-integrated form of Eq. (1). Figure 7a displays the model output using the same dataset that generated Fig. 6 and shows the strong correlation (r = 0.93) between the LES output and regression [Eq. (6)]. Table 2 shows that values of α₃ and β₃ are very similar to those from the regressions in Eqs. (1) and (2). Taking the natural logarithm of both sides of Eq. (6) and differentiating with respect to N_d, one can derive an expression for susceptibility analogous to Eq. (4) for time- and volume-integrated quantities.

Furthermore, substituting Eq. (6) into Eq. (5), PE can be calculated as

View Expanded

which has similar form to Eqs. (1) and (6) and now explicitly depends on N_d. Figure 7b shows a scatterplot of PE_LES calculated directly from Eq. (5) using LES output that implicitly includes the effects of N_d, as a function of PE_REG calculated based on the right-hand side of Eq. (7). The scattering is larger, and the correlation is weaker (r = 0.46) in Fig. 7b compared to Fig. 7a. Nevertheless the analysis points to the validity of the regression. Equation (7), with its explicit representation of N_d and its clear connections to S_o, is therefore proposed as an additional avenue for exploring PE.

PE is between 1% and 10%. These values are smaller than the 20%–30% range calculated based on the ratio of latent heat flux in the cloud layer to the surface value for the trade wind cumulus regime off the coast of Hawaii (Rauber et al. 1996) and in thunderstorm studies (Fankhauser 1988). PE is about 12% in simulations of a single warm cumulus cloud in a two-dimensional model (Flossmann and Pruppacher 1988) using the same definition for PE as given in Eq. (5). The rain intensity in that study was substantially higher, and the lifetime of the single cloud was 64 min.

Seifert and Stevens (2010) used a 1D model output to suggest a link between PE, autoconversion time scale, and cloud lifetime. We attempted a similar analysis based on the LES output but could not find a clear relationship between these parameters.

a. Nonmonotonic behavior of S_o

The behavior of S_o shown in Table 3 and Fig. 8a can be divided into three major regimes: small, intermediate, and large LWP_max. The relative insensitivity of S_o to LWP at the lowest LWP_max follows from the fact that small, short-lived clouds produce little precipitation regardless of the aerosol conditions. For LWP_max (100 g m⁻² < LWP_max < 1400 g m⁻²), S_o increases monotonically with increasing LWP_max as the precipitation process becomes more efficient and increases in N_d have a stronger potential to suppress collision–coalescence. At larger LWP_max (>1400 g m⁻²), S_o decreases with increasing LWP as the cloud switches from an autoconversion (N_d dependent) to an accretion (N_d independent) dominated process in accord with Wood et al. (2009). This qualitative behavior is also seen in the study of shallow cumulus clouds using satellite data (Sorooshian et al. 2009, 2010), in a parcel model (Feingold and Siebert 2009), and for LES of weakly precipitating cumulus (Sorooshian et al. 2010). The LWP values associated with inflection points and maxima remain uncertain and are likely related to cloud type. They are also highly sensitive to the methodology applied in calculating R, N_d, and LWP.

b. S_o controlling parameters

The S_o quantifies the sensitivity of precipitation to changes in aerosol. It identifies which types of clouds are most susceptible to aerosol changes. The dependence of S_o on LWP and other fundamental microphysical processes such as condensation, autoconversion, and accretion are now evaluated. Comparisons with other studies are made, as appropriate.

The original rationale for the choice of LWP as a controlling factor for S_o (Feingold and Siebert 2009) was that LWP can be viewed as a bulk cloud property that represents the macrophysical potential for the cloud to precipitate. The influence of microphysical properties then appears via N_d or N_a. This is analogous to albedo susceptibility, where LWP represents the macrophysical potential for a cloud to reflect solar radiation and N_d represents the microphysical influence. Recently, Wood et al. (2009) and Seifert and Stevens (2010) have explored similar susceptibility and precipitation efficiency ideas in terms of other controlling parameters such as the ratio of the condensation to rain time scales τ_cond/τ_driz and the ratio of accretion to autoconversion rates A_accr/A_auto. These ratios give a rough estimate of the water budget in clouds. The ratio τ_cond/τ_driz represents the time scale for generation of water through condensation relative to that for depletion through precipitation. Similarly, A_accr/A_auto estimates the rate at which precipitation depletes cloud water relative to the rate at which it can be produced by conversion of cloud water to rainwater.

To place our work in the context of these studies, we follow suit using the LES output as the source of these time scales and accretion rates. We note that, because of differences among the models, particularly with regard to the representation of the dynamics (LES versus simpler dynamical frameworks), quantitative comparison is not always feasible. Here, S_o is plotted as a function of τ_cond/τ_driz (Fig. 8b) and A_accr/A_auto in the corresponding LWP_max bins (Fig. 8c). The term is calculated as 1/r_l × (dr_l/dt) and τ_driz = r_l/(A_accr + A_auto), where r_l is the cloud liquid water mixing ratio. The terms A_auto and A_accr are directly calculated from the bin microphysics algorithms in the model, and they are processed and averaged in the same manner as described in section 2c(2). It is striking that S_o exhibits nonmonotonic response to all three variables. Closer examination reveals that the ratios of τ_cond/τ_driz and A_accr/A_auto are both positively correlated with LWP (Fig. 9). The correlation between A_accr/A_auto and LWP is stronger (r = 0.42) than that of τ_cond/τ_driz (r = 0.16). The stronger correlation between A_accr/A_auto and LWP along with the similar behavior of S_o in Figs. 8a,c suggests that the transition from an autoconversion-dominated regime to an accretion-dominated regime is well reflected in the LWP term. Intuitively, at low LWP, the ratio of accretion rates to autoconversion rates is small, because a cloud with low LWP has not generated many rain drops, and at high LWP the reverse is true. Similarly, a cloud in its formative stages will have small LWP and therefore small τ_cond/τ_driz, because microphysics is dominated by condensation and τ_driz is relatively large compared to τ_cond.

Also of note is that, whereas the LES results shown here, as well as those presented in Sorooshian et al. (2009), show a distinct maximum in S_o, the analogous study in Wood et al. (2009) shows S_o decreasing monotonically with increasing τ_cond/τ_driz or A_accr/A_auto. At this stage, it is unclear whether these differences are due to differences in model dynamics or to the parameter space explored by the different studies.

Figures 8 and 9 demonstrate the usefulness of LWP as a parameter controlling S_o and its relationship with more fundamental microphysical processes. From a pragmatic point of view, LWP is much easier to measure than τ_cond/τ_driz or A_accr/A_auto and therefore should also facilitate in situ and remote sensing evaluation of S_o (e.g., Sorooshian et al. 2009, 2010).

c. Generality of relationships

The generality of the relationships derived from the simulations is now evaluated by comparing results for the two different initial meteorological conditions shown in Fig. 1. The same regressions were performed using the model output based on the standard GCSS RICO case (solid lines in Fig. 1; Jiang et al. 2009). Table 4 summarizes the comparison. To ensure a sufficient number of samples, the R_max is used in the regression where R is the cloud-layer-averaged precipitation rate and R_max is the cloud-lifetime-averaged maximum R [for details, see section 2c(2)]. All data with LWP greater than 100 g m⁻² are included to produce the top two rows in Table 4. The β values for both these studies are close to 1, with those from Jiang et al. (2009) being somewhat lower. The value of α derived from Jiang et al. (2009) is about half of that in the current study. Based on the Table 3 breakdown of the dependence of β on LWP, it appears that the primary reason for the difference in β is due to the lower LWP regime in Jiang et al. (2009). Although α exhibits significant variability as a function of LWP in Table 3 across the entire LWP range, values of α are less than or equal to 1 in the lowest two LWP bins in Table 3. To test this hypothesis, we selected a subset of model output for cases with LWP in the range of 100–500 g m⁻² (bottom two rows in Table 4). Both this study and Jiang et al. (2009) now yield similar α and β values. This exercise provides us with confidence that the derived relationships are consistent when clouds of similar types are analyzed.

d. Thresholds for definition of precipitating clouds

The precipitating clouds have been defined throughout this study as R (averaged over cloudy region and lifetime for individual clouds) greater than 0.5 mm day⁻¹ and LWP greater than 100 g m⁻². Lower threshold values were tested using the regression of R_sfc shown in Table 2 as an example. The test results (Table 5) have shown that the correlation coefficients decrease slightly as the threshold values were lowered, but the regression parameters have been relatively insensitive to these choices.