Corresponding author: I. Koren, Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, Rehovot 76100, Israel. (firstname.lastname@example.org)
 Relative dispersion (ε), defined as the ratio between cloud droplet size distribution width (σ) and cloud droplet average radius (〈r〉), is a key factor used to parameterize various cloud processes in global circulation models (GCMs) and bulk microphysical scheme models (BSMs). Recent studies indicate that the impact of aerosol loading (N) and atmospheric thermodynamic conditions on ε are far from fully understood. Currently, a fixed value per hydrometeor type is used in most BSMs and GCMs, which imposes significant limitations on our ability to model and predict cloud processes and their impact on the environment, on regional to global scales. In this study, we use a detailed bin microphysics single cloud model to investigate the combined impact of atmospheric thermodynamic conditions and N on ε, in warm cumulus clouds. As initial conditions, we used different lapse-rates combined with 8 scenarios of aerosol loading, representing very clean (N = 25 cm−3) to heavily polluted (N = 1600 cm−3) conditions. Moreover, the results are analyzed per cloud evolutionary stage according to the dominance of microphysical processes. The use of this method indicated a different pattern of ε at each stage. Specifically, during the mature stage fitting of ε to rv is relatively resilient to changes in the environmental conditions. Such findings suggest a new view of the effect of aerosols on clouds, via changes in the cloud evolution patterns and a new approach to parameterization of ε based on rv, which can significantly improve the prediction of cloud processes by GCMs and BSMs.
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 Clouds play a key role in the Earth's climate, mainly through the radiation and water budgets [Dessler, 2010] and appear to impose the most complex and least understood effects on climate change assessments [Intergovernmental Panel on Climate Change, 2007; Stephens, 2005; Randall et al., 2007]. Therefore, reliable models are required in order to evaluate the impact of clouds on climatic processes at the global and regional scales. Due to computational time limitations this in turn requires the improvement of parameterizations which are currently widely used in bulk microphysics cloud resolving models (BSM), at regional to synoptic scales and in global circulation models (GCMs). One of the key factors used in GCMs cloud parameterization is the relative dispersion (ε), defined as the ratio between the standard deviation of the cloud droplets size distribution (σ) and the mean radius (〈r〉). Both σ and 〈r〉 are key variables because they determine important irradiative properties of clouds, such as reflectivity [Liu and Daum, 2000a, 2000b; Hansen and Travis, 1974; Slingo, 1989; Daum and Liu, 2003], directly by defining the effective surface area of light reflection [e.g., Slingo, 1990] or indirectly by controlling various microphysical processes, which have an impact on cloud properties, such as their dimensions and lifetime.
 In GCMs the relative dispersion is further used to relate the mean radius to the effective radius (re; equation (1)), using the parameter β = re/〈r〉 (expressed here for a gamma distribution) which is proportional to the relative dispersion (equation (2))
where L, N and rv are the (per volume) liquid water mass, number of droplets and the volumetric mean radius, respectively. Incorporation of equation (1) in GCMs is used commonly to investigate the first indirect effect which relates higher aerosol loading to smaller cloud droplets and higher cloud reflectivity [Twomey, 1977]. In bulk parameterization schemes the shape parameter, α, is used as a measure of the relative dispersion [Milibandt and Yau, 2005], while the relation between ε and α is given by
 The equations above show that the higher the ε values the larger is re and therefore result in weaker cloud reflectivity [Slingo, 1990].
 In GCMs the size distribution of cloud droplets is prescribed by fixing ε and in most bulk parameterization by fixing α, a priori. In the past, the relative dispersion was assumed to be invariant to the environmental ambient conditions. However, recently Liu and Daum  indicated by gathering measured data from different studies, that ε is sensitive and positively correlates with the level of aerosol loading (N), which was supported by modeling [Peng and Lohmann, 2003; Rotstayn and Liu, 2003] and theoretical studies [Liu et al., 2006]. Liu et al.  indicated that this so-called dispersion effect may account at least partially (42%) for significant discrepancies in the magnitude of the calculated indirect effect [Feingold et al., 2003; Rosenfeld and Feingold, 2003].
 However, the positive relationship between ε and N is in conflict with other studies which showed a negative relationship between the two parameters, based on modeling [Lu and Seinfeld, 2006] and observational studies [Miles et al., 2000; Yum and Hudson, 2005; Martins and Silva Dias, 2009; Ma et al., 2010]. This was explained by the broadening of the droplet spectra at lower Ns due to enhanced collision-coalescence and reduced condensational growth [Lu and Seinfeld, 2006; Martins and Silva Dias, 2009]. Accordingly, the reason for the contradicting results was attributed to the influence of different dynamical and microphysical factors on the size distribution of cloud droplets [e.g., Liu et al., 2008], with a dominating effect of updraft. A more uniform updraft velocity is associated with the narrowing of the size distribution of cloud droplets [Liu et al., 2006; Peng et al., 2007; Yum and Hudson, 2005; Martins and Silva Dias, 2009], while turbulent flow positively correlates with the degree of entrainment and the collision-coalescence processes, which in turn broadens the size distribution [Hsieh et al., 2009, and references therein]. Relative humidity influences the degree to which chemical hygroscopicity has an impact on cloud condensation nuclei (CCN) activation and hence dispersion [Martins and Silva Dias, 2009]. However, currently the impact of environmental parameters on ε and its relation to N is far from understood [Liu et al., 2008, and references therein]. In this paper we investigate the sensitivity of relative dispersion to different combinations of aerosol loading and meteorological conditions by employing a detailed bin microphysical model.
 In this study we used the Tel Aviv University axisymmetric non-hydrostatic numerical cloud model (TAU-CM) with a detailed cloud microphysical scheme [Tzivion et al., 1994; Reisin et al., 1996]. Only warm microphysical processes were included, accounting for nucleation of CCN, condensation and evaporation, collision-coalescence, binary break-up and sedimentation. The dynamical scheme of the model is described in detail in Tzivion et al. . The microphysical processes are solved using a multi-moment bin method [Tzivion et al., 1987], using a 34 drop bin radius size, ranging from 1.56 to 3.2*103μm. The critical values of super saturation for CCN activation at each grid point are calculated with the model for each bin size, using the Koehler equation [Pruppacher and Klett, 1997]. For simplicity, these calculations assumed that all CCN are pure sea salt aerosols (NaCl). The initial wetted particle size was based on a nucleation scheme which was proposed by Kogan , assuming that the initial droplet size formed on CCN with radii smaller than 0.12 μm was equal to the equilibrium radius at 100% RH. For larger CCN the initial radii were specified as less than the equilibrium radii at 100% RH.
 The different cloud simulations are based on three different sounding data of temperature and moisture from Bet Dagan, Israel (October 1st, 2006; 12Z sounding), Hilo, Hawaii (August 21th, 2007; 91285 PHTO Hilo Observations at 00Z) and Heraklion, Crete (May 6th, 2011; 16754 LGIR Heraklion (Airport) Observations at 00Z) meteorological stations, with few modifications to the temperature vertical profile to better enable convective development. In order to investigate the influence of environmental conditions, an additional four variations of the temperature lapse rate (γ) within the boundary layer were used for each profile (by 1 to 5%). In order to take into account the influence of aerosol loading in our analysis, each of the above mentioned 15 scenarios was simulated under 8 different aerosol loadings, ranging from 25 to 1600 #/cm−3 (120 simulations in total; see Table S1 in the auxiliary material). The domain size was 4000 and 5000 m in the radial and vertical dimensions, respectively, while vertical and radial resolutions were set to 50 m. Convection was initiated by introducing a bubble 1°C warmer than the environment at the lower boundary of the grid, for a one time step. The time step was 1 second and the total simulation time was 80 minutes.
 The analysis was focused on the evolution stage of microphysical properties along the core of the cloud. For this to be achieved, we separated the temporal evolution to a few distinct stages, based on the relative impact of different microphysical processes on L change in time (t), in the grid-cell (for more details see section 1 of Text S1 in the auxiliary material). During the “pre” stage, relatively fast growth in L occurs via nucleation of new droplets and condensational growth of small cloud droplets (〈r〉 < 10 μm), such that d(log (L))/dt > 0. During the “mature” stage condensation of larger cloud droplets (〈r〉 ≥ 10 μm) occurs, such that d(log (L))/dt is small and during the “post” stage collision-coalescence is linked with enhanced sedimentation, such that d(log (L))/dt < 0. The “post” stage was separated into three different stages, referred to as “post_1”, “post_2” and “post_3”. During “post_1”, collision-coalescence has similar or greater influence on the droplet's size distribution, as compared with condensation. During “post_2” enhanced sedimentation of cloud droplets, linked with collision-coalescence, occurred, while during “post_3” and “post_1”, which followed and preceded “post_2”, respectively, sedimentation was smaller. A value of 5E-6gr/kg/grid was used as a threshold for a significant sedimentation rate. All calculations included the whole range of cloud droplet sizes.
3. Results and Discussion
 The relative dispersion (ε) is defined as the ratio σ/〈r〉. For a given warm convective cloud, both 〈r〉 and σ depend on key factors related to the environmental conditions and to the dynamic and microphysical processes. The key factors that were initially considered in this study are aerosol loading (N), atmospheric stability (γ) and time (t), representing the cloud development, horizontal distance from the cloud center (d) and the vertical height (z).
 Preliminary results indicated a correlative response of 〈r〉 and σ to all factors listed above. However, the dependence on the horizontal position was less significant (as long as it is far enough from the boundaries of the cloud), as suggested by Rosenfeld and Lensky  and shown for shallow non-precipitating cumulus in Zhang et al. . As expected, 〈r〉 and σ were most sensitive to z (reflecting the relation re ∝ z1/3 [Liu and Hallett, 1997]). However, their ratio (ε) was shown to be less sensitive to the factors listed above (N, γ, and d) including changes in z (again, as long as the investigation took place far enough from the cloud boundaries, such that the role of entrainment was relatively insignificant).
Figure 1 presents the vertical profiles of 〈r〉, σ and ε for 4 Hawaiian clouds, representing four combinations of aerosol loading and temperature lapse rates (N = N2, N6 and γ = γ1, γ5), during the “mature” stage (t = 30 min) for cloud central column. A similar pattern and values of ε were obtained for each of the 4 N-γ combinations throughout most of the clouds' central columns (beside the upper third), indicating a strong correlation (r2 ≥ 0.92) between 〈r〉 and σ, yielding relatively small changes in ε (standard deviation ≤ 0.017). ε tends to decrease with height at the lower approximately third part of the central cloud column , due to a high condensational growth rate, and therefore, a relatively fast increase in 〈r〉 (i.e., maintaining a relatively low ε variance), in agreement with other studies [e.g., Politovich, 1993; Pawlowska et al., 2006]. ε increases dramatically in the upper approximately tenth part of the central cloud column, which we attribute to entrainment processes through the cloud top, involving evaporation of cloud droplets. As is further demonstrated below, ε was found to be relatively insensitive to N, γ and d and more sensitive to entrainment and the time dimension-reflecting cloud evolution stage. Therefore, in order to reduce the complexity we avoided entrainment zones and focused our analysis on the core of the clouds.
Figure 2 uses 40 simulations (5 lapse rates × 8 aerosol loadings), based on the Hawaiian sounding data. This is in order to investigate ε values obtained during the different cloud development stages (“pre, “mat”, post_1”, “post_2” and “post_3”), at 5 different height levels along the central cloud column, where entrainment is insignificant during the “mature” stage. ε is plotted vs. rv (which is widely used for microphysical parameterizations by applying equation (2)), indicating different ε patterns during each of the different stages. Figure 2 indicates relatively less dispersive and smaller ε values during the “mature” stage, reflecting relatively small dependency of ε on N, z and γ, during this stage. Toward the end of the “Pre” stage ε tends to decrease, reflecting narrowing by enhanced condensational growth. During “post_1” and “post_2”, ε increases due to enhanced collision-coalescence. During the “post_2” stage ε shows a bi-modal pattern. At the beginning of this stage a relatively small portion of the droplets grow by collision-coalescence leading, therefore, to a sharp increase in ε up to ∼6. This is then followed by narrowing of the distribution, toward the end of “post_2” (ε averages ∼0.64) when a larger portion of the droplets are involved in collision-coalescence, which is also linked to enhanced sedimentation. During “post_3” ε averages ∼0.48, since no significant microphysical process occurs further in the remnants of the cloud (Figure 2). Figure 3 presents the evolution of the average ε during the “pre”, “mature” and “post” stages, based on the 40 simulations using the Hawaiian sounding data, at 5 different height levels. Figure 3 further shows that during the “mature” stage ε tends to be most confined and relatively small, compared with the other stages (Figures 3a and 3c). We attribute this to the fact that at this stage, the droplets growth by condensation and collision-coalescence is relatively minimal (see Figure S1 in the auxiliary material). This is because part of the super saturation is already consumed at earlier stages and on the other hand the coalescence rate of droplets is still small. This fact is critical when considering the effect of aerosol loading (N) on the relative duration of cloud stages. In addition, the impact of entrainment during this stage on the inner cloud part is relatively small. Figure 3 also indicates a moderate and steady increase in ε during the “mature” stage, which is further investigated in Figure 4.
Figure 4 focuses on the ε values obtained during the “mature” stage, using the same conditions as in Figures 2 and 3 for the Hawaiian sounding data, as well as the sounding data from Heraklion, Crete and Beit Dagan, Israel (Figures 4b and 4c, respectively). In agreement with Figure 3, Figure 4 indicates an increase in ε for larger rv for all 3 investigated sounding data. Moreover, for all 120 simulations representing different combinations of N and γ (based on the 3 different profiles, and for each at 5 different height levels (z)), the ε values during the “mature” stage consistently show a linear correlation with rv, with a similar fit slope (averaged 1.48 ± 01) and intercept (averaged 0.24 ± 0.03). We attribute this to the decrease in condensational growth, together with the gradual increasing impact of collision-coalescence during the “mature” stage, which both act to broaden the size distribution. A slight but steady increase in time, in the contribution of the collision-coalescence processes to the droplet growth, explains this trend (see Figures S1a and S1b in the auxiliary material).
 The continuous increase of ε with rv during the “mature” stage together with the relatively small variance of ε per given rv (STD ≤ 0.15*ε), suggests that ε could be inferred from the value of rv during this stage. Since rv is normally resolved in GCM, the proposed relationship between rv and ε could be further used to improve the ε parameterization in future GCM simulations. Similarly, since α is used in BSMs (section 1) and based on the relationship between ε and rv, the above also suggests a more precise estimate of α and σ in future BSMs simulations. As previously stated, N, z and γ were all shown to affect the mean droplet radius. However, during the “mature” stage rv is dominantly influenced by the time evolution of the microphysical processes and monotonically increases during this stage together with ε, as can be seen in Figure 3b. Our results show that changes in N account for only ∼15–20% of the overall ε range during the “mature” stage. This suggests that determining ε directly by fitting it to rv is more accurate than determining ε based on N, as is widely done in current BSM.
4. Concluding Remarks
 The results shown here are the outcome of analyses of 3 different profiles of warm cumulus clouds and 40 variations of N and γ that were analyzed per profile. This study shows that the relative dispersion magnitude and variance depends heavily on the cloud evolutionary stage and offers a new method to estimate ε. It shows that during the mature stage most ε values range between 0.25–0.45 in agreement with several previous observational studies [e.g., Andreae et al., 2004; Pawlowska et al., 2006; Martin et al., 1994]. Furthermore, as is indicated by Figure 2, the sensitivity to changes in the location in the cloud (z), the atmospheric stability (γ) and the aerosol loading (N) during the “mature” stage are relatively insignificant, and can all be expressed as a linear function of rv: ε = a 〈rv〉 + b, where a = 1.48 ± 0.1 and b = 0.24 ± 0.03 (Figure 3). In the young (“pre”) stage the ε values were less consistent and higher, around 0. 51 ± 0.5 for all clouds, while during most of the “post” stages ε was highly dispersive, ranging 0.34–0.70 (“post_1”), 0.40–2.63 (“post_3”), and during “post_2” ε was bi-modal and highly variable (ranging 0.4–4.2), representing significant sedimentation.
 We suggest that trends in ε around the cloud core, can be explained by the balance between the two main growth processes that dictate the droplet size distribution (in the pre and mature stages). When either of them, condensation or collision-coalesce, leads to a fast change in rv, then ε significantly increases. During the mature stage, the relative importance of the collision-coalescence induced growth slowly increases, such that ε growth is relatively slow.
 The results shown here offer a new view on aerosol effects on clouds. In many cases analyzed here, an increase in N led to an increase in the duration of the mature stage. During this stage the cloud is at its peak reflectance. This adds another dimension to the cloud lifetime effect [Albrecht, 1989], namely aerosols prolonging the cloud lifetime, especially during the stage of maximum reflectance. This aspect is currently being studied further.
 The results presented may have an applicable use in improving the accuracy of estimations of cloud radiative forcing and aerosol effects on clouds when using bulk microphysical schemes and GCMs. By estimating the cloud evolutionary stage and calculating an initial ε and rv, one can iterate and update ε as a function of rv. The fact that a similar relationship between ε and 〈r〉 holds during the “mature” stage suggests a way to better estimate σ in future BSM (auxiliary material). In order to have our proposed methods applied in future BSM/GCM, computationally efficient algorithms should be developed to estimate the degree of cloud entrainment and determine the cloud evolutionary stage. We will address these issues in our future research.
 This work was supported in part by the Israel Science Foundation (grant 1172/10).
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.