On the relationship between cloud contact time ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH: ATMOSPHERES, VOL. 118, 10,544–10,554, doi:10.1002/jgrd.50819, 2013

On the relationship between cloud contact time and precipitation susceptibility to aerosol Graham Feingold,1 Allison McComiskey,1,2 Daniel Rosenfeld,3 and Armin Sorooshian4,5 Received 13 June 2013; revised 12 August 2013; accepted 9 September 2013; published 24 September 2013.

[1] The extent to which the rain rate from shallow, liquid-phase clouds is microphysically influenced by aerosol, and therefore drop concentration Nd perturbations, is addressed through analysis of the precipitation susceptibility, So . Previously published work, based on both models and observations, disagrees on the qualitative behavior of So with respect to variables such as liquid water path L or the ratio between accretion and autoconversion rates. Two primary responses have emerged: (i) So decreases monotonically with increasing L and (ii) So increases with L, reaches a maximum, and decreases thereafter. Here we use a variety of modeling frameworks ranging from box models of (size-resolved) collision-coalescence, to trajectory ensembles based on large eddy simulation to explore the role of time available for collision-coalescence tc in determining the So response. The analysis shows that an increase in tc shifts the balance of rain production from autoconversion (a Nd -dependent process) to accretion (roughly independent of Nd ), all else (e.g., L) equal. Thus, with increasing cloud contact time, warm rain production becomes progressively less sensitive to aerosol, all else equal. When the time available for collision-coalescence is a limiting factor, So increases with increasing L whereas when there is ample time available, So decreases with increasing L. The analysis therefore explains the differences between extant studies in terms of an important precipitation-controlling parameter, namely the integrated liquid water history over the course of an air parcel’s contact with a cloud. Citation: Feingold, G., A. McComiskey, D. Rosenfeld, and A. Sorooshian (2013), On the relationship between cloud contact time and precipitation susceptibility to aerosol, J. Geophys. Res. Atmos., 118, 10,544–10,554, doi:10.1002/jgrd.50819.

1. Introduction [2] Precipitation formation in liquid (i.e., ice free) clouds continues to be a topic of great interest, particularly because of its implications for cloud cover and persistence, and climate forcing by clouds. Climate models typically parameterize rain production in terms of autoconversion, a process describing self-collection of small cloud droplets to form rain, and accretion, the collection of cloud droplets by raindrops. Autoconversion depends inversely on drop concentration Nd (Autoconversion / N–ˇ d ), but with some uncertainty in ˇ . Accretion is insensitive to Nd , and the relative importance of autoconversion and accretion is 1

NOAA Earth System Research Laboratory, Boulder, Colorado, USA. Cooperative Institute for Research in Environmental Sciences/NOAA Earth System Research Laboratory, Boulder, Colorado, USA. 3 Institute of Earth Science, Hebrew University, Jerusalem, Israel. 4 Department of Chemical and Environmental Engineering, University of Arizona, Tucson Arizona, USA. 5 Department of Atmospheric Sciences, University of Arizona, Tucson Arizona, USA. 2

Corresponding author: G. Feingold, NOAA Earth System Research Laboratory, 325 Broadway, Boulder, CO 80305, USA. ([email protected]) ©2013. American Geophysical Union. All Rights Reserved. 2169-897X/13/10.1002/jgrd.50819

intimately related to the potential for the aerosol to influence rain formation. [3] To quantify the potential role of aerosol in warm rain formation, one can apply the precipitation susceptibility metric So = –

dlnR dlnNd

(1)

where R is rain rate [Feingold and Siebert, 2009; Sorooshian et al., 2009]. The goal is then to establish the cloud parameters that control the magnitude of So . Although the denominator in equation (1) is drop concentration Nd , implicit in our discussion is the direct, although sublinear dependence of Nd on aerosol number concentration Na , Nd / Nca , where c 1. R has been shown both empirically [e.g., Pawlowska and Brenguier, 2003; Comstock et al., 2004; vanZanten et al., 2005] and theoretically [Kostinski, 2008] to have the form R L˛ Nd–ˇ

(2)

where ˛ 1.5. Sometimes these relationships are posed in terms of cloud depth H, but they can easily be related to equation (2) considering that L / H2 if cloud water increases linearly with height. From equations (1) and (2) we see that if equation (1) is a partial derivative at constant liquid water path L, then ˇ So . Therefore, L is an important parameter controlling So .

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FEINGOLD ET AL.: PRECIPITATION SUSCEPTIBILITY

Figure 1. Time series of (a) Nd , (b) –dNd /dt, (c) accretion/autoconversion, (d) R, and (e) Z based on box model calculations of collision-coalescence. Initially, autoconversion dominates rain production but with increasing time, more of the rain is produced by accretion. The dashed line in (Figure 1c) represents Autoconversion = Accretion. The dashed line in (Figure 1e) is for Z = –15 dBZ, a threshold often used to indicate the onset of drizzle. [4] Equations (1) and (2) have proven to be useful diagnostics but are somewhat dissatisfying in that they ignore an important parameter that influences rain formation, i.e., the time available for collision-coalescence tc . Cloud lifetime, an upper bound on tc in the case of isolated cumulus clouds,

has been explored in warm trade wind cumulus [Seifert and Stevens, 2010; Jiang et al., 2010]. Time scales for warm rain formation have been studied by Stephens and Haynes [2007] and Suzuki and Stephens [2009] using satellite-based radar. Cloud contact time, however, is a somewhat different concept and addresses the time that a parcel of air might spend in cloud on a typical trajectory through that cloud. Twomey [1976] identified the nonlinear dependence of the collision-coalescence process on liquid water content (LWC) and pointed to the dominance of pockets of high LWC, encountered for sufficiently long periods of time, in determining average coalescence rates. Using large eddy simulation (LES) with detailed cloud microphysics, Feingold et al. [1996] artificially manipulated in-cloud residence times in stratocumulus and showed how longer tc promotes rainfall production. Trajectory ensemble models that track a multitude of parcels through a host-model domain have also proven useful for analysis of tc [e.g., Stevens et al., 1996; Feingold et al., 1999]. [5] At small tc , autoconversion is the process responsible for the generation of a few collector drops. Once these larger drops have been formed, accretion takes over as the primary rain-producing mechanism. If clouds only live for short periods of time, and/or typical air-parcel in-cloud residence times are brief, autoconversion will tend to contribute more strongly to rain production. On the other hand, if there is ample time available for collision-coalescence then accretion will dominate. Because autoconversion depends on Nd but accretion does not, tc has the potential to affect So . [6] Previously published work disagrees on the qualitative behavior of So with respect to independent variables such as liquid water path L or the ratio between accretion and autoconversion rates. Two primary responses have emerged: (i) So decreases monotonically with increasing L [Wood et al., 2009; Terai et al., 2012] and (ii) So increases with L, reaches a maximum, and decreases thereafter [Feingold and Siebert, 2009; Sorooshian et al., 2009; Jiang et al., 2010]. (So analysis has also been applied to mixed phase clouds but the behavior is more complex [Seifert et al., 2012] and is not addressed here.) The response of So to L has important implications for the potential for aerosol to modify precipitation

Figure 2. Contour plots of radar reflectivity Z (solid) and the ratio of accretion to autoconversion (shading and dotted lines) in LWC-Nd parameter space based on box model calculations. (a) tc = 300 s and (b) tc = 600 s. With increasing tc , more of the domain is dominated by accretion. Note how Z and accretion/autoconversion are approximately parallel at large LWC and low Nd where drizzle is most active. 10,545


[7] The goal of this paper is to explore the role of tc in determining the qualitative response of So to an increase in L. As will be shown below, because tc determines the relative importance of accretion and autoconversion (all else equal), it may explain the discrepancy in So = f(L) evident in published work. It also provides further motivation for subgrid (climate model) representation of cloud life cycles.

2. Results [8] The results to be presented were derived from a number of simple modeling frameworks ranging from box models of collision-coalescence (size-resolved), to parcel models rising at constant velocity, and finally parcels driven along trajectories derived from LES of different cloud types. The models have been chosen because they isolate the primary precipitation-controlling parameters (L, Nd , and tc ); they do, however, neglect drop sedimentation, which could influence the results. Nevertheless, our earlier calculations in the fully coupled LES framework with the same microphysical scheme [Jiang et al., 2010] exhibit similar So responses and, therefore, the focus on simpler models is appropriate.

Figure 3. So versus LWC for two ranges of Nd , (a) 50 < Nd < 200 cm–3 and (b) 200 < Nd < 770 cm–3 , based on box model calculations of collision-coalescence. Line types correspond to different simulation times (in seconds) in both Figures 3a and 3b. rates and, in the case of climate models, for their representation of the lifetime of liquid cloud condensate. Models that rely strongly (inadvertently or advertently) on autoconversion to produce rain tend to have stronger aerosol-induced cloud lifetime effects [Quaas et al., 2009; Golaz et al., 2011; Gettelman et al., 2013]. A recent study [Wang et al., 2012] used satellite-based remote sensing to constrain an alternate form of So (precipitation susceptibility defined in terms of the probability of rainfall) and found much weaker cloud lifetime effects than those based on traditional climate models.

2.1. Box Model Calculations [9] First we solve the collision-coalescence equation in a simple box model using numerical solution to the stochastic collection equation [Tzivion et al., 1987]. The method predicts two moments (number and mass) in each of the 33 drop size bins covering the radius (r) range 1.5 m–2.5 mm, and has been evaluated for accuracy against analytical solutions. The bin drop size concentration is defined as n(r)dr where n(r) is the number of drops of radius r per unit diameter interval dr, and dr is based on a mass-doubling grid. Accretion and autoconversion rates, while not a standard model output, are easily diagnosed based on the bin-bin interactions. The initial drop sizeRdistributions are log normal, constrained by LWC (=4l /3 r3 n(r)dr) and Nd , with an assumed geometric standard deviation of 1.2. Results are somewhat sensitive to this initial choice of drop distribution. The time available for collision-coalescence is simply determined by the length of the simulation. [10] An illustrative time series of model output R is shown in Figure 1. Rain rate R is defined as 4/3 v(r)r3 n(r)dr over the full range of drop sizes, with v(r) the drop terminal velocity. Integrating over all drop sizes is appropriate to allow for the first stages of raindrop formation to be captured. (Note that integrating R only over larger drizzle drops immediately places the focus on the accretion process and the So calculations are quite different; see section 3.) Initially, autoconversion is the primary rain-producing process, but after 800 s, accretion begins to dominate. Note how the onset of the dominance of accretion (ratio greater than approximately 2) is correlated R with significant increase in (i) radar reflectivity Z = 64 n(r)r6 dr, expressed in units of 10 log10 Z (dBZ, with Z in mm6 m–3 ), and (ii) dlnNd /dt. To generalize these results, contour plots in Nd –LWC phase space are shown at two snapshots in time (300 s and 600 s; Figure 2). Contours of Z are superimposed on contours of accretion/autoconversion ratios. Two features are quite distinct: first, contours of Z are approximately parallel to those of accretion/autoconversion, particularly at higher LWC and

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Figure 4. (a) and (c): Contour plots of radar reflectivity Z (solid) and the ratio of accretion to autoconversion (dotted lines and shading) in L-Nd parameter space based on parcel model calculations. (b) and (d): Equivalent contour plots of R, mm d–1 (solid) and droplet effective radius re , m (shading and dotted line for re = 14 m). (Figures 4a and 4b): w = 0.5 m s–1 ; (Figures 4c and 4d) w = 1.0 m s–1 . With increasing w (decreasing tc ), more of the domain is dominated by autoconversion. For convenience, the equivalent cloud depth H scale is provided on top axes. lower Nd where drizzle is stronger, i.e., Z is a particularly good indicator of the importance of accretion; second, as time progresses and more rain is formed, more of the parameter space becomes accretion-dominated. The potential for time dependence of So becomes obvious. [11] The box model output is now converted to So and plotted as a function of LWC for different simulation times (Figure 3). Results are separated into 50 Nd 200 cm–3 and 200 Nd 770 cm–3 . For both plots, at short simulation times, So tends to increase with increasing LWC (henceforth “the ascending branch”), and only decreases at sufficiently high LWC. However, with progressively longer simulation times, “the descending branch” (dSo /dLWC < 0) starts to dominate; at long times (tc = 1800 s) it is only at very low LWC (and therefore negligible rain production) that there is evidence of the ascending branch. Note how for the larger values of Nd (Figure 3b), the maxima in So increase and shift to larger values of LWC. The higher maxima in So are indicative of a threshold Nd above which autoconversion is effectively suppressed. The shift to larger values of LWC follows since at higher Nd , higher LWC is required to produce rain. [12] Similar So calculations were performed at smaller ranges of Nd since during the transition from

autoconversion- to accretion-dominated rain production, the log R versus log Nd plots do not always follow a straight line (power law). For example, at the lower end of any prescribed Nd range, one might find an active rain production process characterized by low So and at the higher end of the range, where rain production is weak, by larger So . The two ranges were chosen to be illustrative of the general behavior. In all cases, however, the qualitative trends shown in Figures 3a and 3b are robust—namely, an increase from a baseline value of So of 2/3 to some peak value, followed by a steady decrease to 0. The value of 2/3 can be shown to be a consequence of v(r) for small cloud droplets (see Appendix A). More detailed analysis is performed in the following section. 2.2. Parcel Model Calculations [13] The adiabatic parcel model uses the same twomoment bin microphysical scheme applied to the collisioncoalescence equation but is extended to include droplet nucleation and condensation [e.g., Feingold et al., 1999; Sorooshian et al., 2009]. The calculations are performed at fixed updraft velocity w for a range of L and Nd . Because L is related to cloud depth H via L / H2 , specification of L amounts to specification of H, and at a fixed w, to

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Figure 5. Parcel model calculations of R versus Nd for different L (a) w = 0.5 m s–1 ; (b) w = 1.0 m s–1 . Note the degradation in the power law fit with increasing L particularly after the peak slope is reached, and beyond which accretion dominates collection. specification of tc . In the parcel model, Z and autoconversion and accretion rates are all calculated as mean values over the depth of the cloud layer. Figure 4 again shows how with decreasing w (increasing time), accretion becomes progressively more important. Nevertheless, for reasonable values of w, it is clear that significant parts of the parameter space can be dominated by autoconversion, and that cloud contact time tc (=H/w) may be an important factor in determining So .

[14] Before presenting So as a function of L (similar to that in Figure 3), the basic model output values of R and Nd , from which So is built, are shown for two values of w: 0.5 m s–1 (Figure 5a) and 1 m s–1 (Figure 5b). In both cases, it is clear that So only has well-defined power law behavior over the entire range of Nd (constant slope on a log-log plot) up until the point at which So reaches a maximum, i.e., only while autoconversion dominates. As in the case of the

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Figure 5. (continued) box model, initially, So is 2/3. During the transition to accretion-domination, dependence of R on Nd is more complex; at lower Nd (higher R), a reduction in the slopes (So ) is apparent, while there is a steepening of the slopes at higher Nd ; piecewise power law fits would be appropriate only over small ranges of Nd . [15] For this reason, So calculations are performed in different Na ranges for w = 1 m s–1 (Figure 6). With progressively increasing Na , there is an increase in the value of L associated with the maximum value of So,max because it takes more and more liquid water or cloud depth to produce precipitation via autoconversion. There is also an increase

in So,max with increasing Na (see also Figure 3) because at higher and higher Na , any addition of aerosol (and therefore Nd ) effectively shuts off autoconversion. Note, however, that high values of So are of no particular consequence if R itself is very low. With decreasing w (increasing tc ), So,max peaks move to smaller L and to lower absolute values but the qualitative behavior is the same (not shown). 2.3. Parcel Ensemble [16] The third modeling framework is the trajectory ensemble [e.g., Stevens et al., 1996] that uses an Eulerian host model (in this case an LES) to produce an ensemble

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R into path integrated LWC (= LWC(t) dt) bins, a variable perhaps more relevant for rain formation than L because parcels encounter a wide range of different conditions while in cloud, as opposed to simple vertical motion with well-described LWC(t). Figure 7b shows So as a function of R LWC(t) dt. In consideration of Figures 3 and 6, the results

Figure 6. So versus L for different ranges in Na based on adiabatic parcel model calculations for w = 1.0 m s–1 . The individual curves are for the geometrically increasing Na ranges, as indicated. Note how with increasing Na , maxima in So increase in value, and move progressively to higher L. of a relatively large number of parcel trajectories (in this case 500). During run time, the essential position and thermodynamic information is recorded for each trajectory, for a posteriori use. After the LES simulation is complete, each trajectory is used to drive a parcel model that experiences the same conditions encountered by the host model. Each trajectory experiences a different liquid water history and contact time, and there is no assumption of adiabaticity, unless undiluted cloud was predicted by the host model. The advantage of this modeling framework is that it synthesizes a very large number of calculations and allows one to capture tc and LWC(t) in a much more realistic manner than in the case of the box and constant updraft parcel models. Its disadvantage is that it does not allow communication amongst parcels, drop sedimentation, or feedbacks to dynamics, and therefore does not address the adjusting aerosol-cloud-precipitation system as does the LES. [17] The parcel trajectory derives from an LES of the FIRE-I stratocumulus case off the coast of California in 1987 [Stevens et al., 1996]. An analysis of the parcel in-cloud residence times and associated LWCs shows (Figure 7a) that most parcels spend only 15 min in cloud at typical LWCs of 0.15 g kg–1 , and only a very small fraction of the 500 trajectories spend up to 40 min in cloud. Note that these residence times are only accrued while a parcel is in cloud, if a parcel exits the cloud then the “clock” is reset, much as the collision-coalescence process would be reset in a natural cloud. Following the same methodology used in the simpler models, the trajectory ensemble (500 members) is now run for a range of aerosol concentrations Na (25 < Na < 1000 cm–3 ) which translates to a range of Nd depending on the w (and therefore supersaturation) encountered by any individual parcel. Results are aggregated and collected

Figure 7. (a) Joint probability distribution function of incloud residence time and average liquid water content for 500 parcels derived from large eddy simulation of a wellmixed stratocumulus-capped boundary layer. RAnalysis time is 1 h. (b) So versus path integrated LWC ( LWC(t) dt) for the ensemble of 500 parcel trajectories, each experiencing different tc and LWC(tc ). Results are divided into “All” (model output for the entire range of Na : 25 < Na < 1000 cm–3 ), “Low aerosol” (25 < Na < 250 cm–3 ), and “High aerosol” (300 < Na < 1000 cm–3 ). Because of limited in-cloud contact time and relatively low LWC, So R increases steadily with increasing LWC(t)dt indicating that autoconversion is responsible for an appreciable part of rain production. In the case of the high aerosol range, collision-coalescence is weak and So remains constant at approximately 2/3—the theoretically predicted value for cloud droplets in the Stokes regime.

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changes in Na . For the high aerosol range, So is approximately 2/3 and relatively constant, indicating the presence of very weak cloud drizzle with drops falling at velocities in the Stokes regime. The separation between low and high aerosol R ranges is congruent with the results in Figure 6 for low LWC(t)dt or L. [18] Results are now contrasted with two other cases. The first is a stratus case based on simulations of a North Atlantic Regional Experiment (NARE) case over a cold ocean. In this case the stratus cloud was decoupled from the surface. Parcels either stayed in the cloud for much of the hour long analysis period, or never entered the cloud at all (Figure 8a). Typical tc are therefore in excess of 50 min (for

Figure 8. As in Figure 7 but for LES of a decoupled North Atlantic stratus cloud layer. Analysis time is 1 h. Because the boundary layer is decoupled, some parcels spend no time in clouds (off axis) while others spend approximately 50 min (per hour) in cloud. The longer cloud-contact time at higher LWC allows for a more active collision-coalescence process. Maximum values of So for “Low aerosol” are therefore higher (compared to Figure 7) before dropping to baseline (cloud drizzle mode) levels of rain production. The “High aerosol” case experiences weak collision-coalescence and So never rises above the baseline value of about 2/3. are displayed for (i) all model output, (ii) a “Low aerosol” range (25 < Na < 250 cm–3 ), and (iii) a “High aerosol” range (300 < Na < 1000 cm–3 ). For “All,” only the ascending branch of So is evident, likely because of the well-mixed nature of the stratocumulus-capped boundary layer and the relatively short in-cloud residence times and exposure to relatively low LWC (Figure 7a). The implication is that for this case, autoconversion is an important rain-producing process and tc is a limiting factor in allowing a cloud with this R LWC(t)dt and H to produce its maximum R. For the low aerosol concentration range, So can be seen to be even higher because at these lower Na (and Nd ), autoconversion is able to generate precipitation and therefore to be influenced by

Figure 9. As in Figure 7 but for LES of a shallow continental cumulus case. Analysis time is 3 h. Cloud R fraction is low and so many parcels never enter cloud. LWC(t)dt values are similar in magnitude to Figure 8. Maximum valRues of So for the “Low aerosol” peak early, i.e., at low LWC(t)dt and then drop off to the cloud drizzle mode value. The “High aerosol” case experiences more significant collision-coalescence than R in Figures 7 and 8 and So rises to higher values once LWC(t)dt passes a certain threshold. R The descending branch starts to appear at the largest LWC(t) dt.

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the 1 h tracking period) at LWCs of about 0.2 g kg–1 . The So results (Figure 8b) again show the separation of All, High, and Low aerosol with similar trends as in Figure 7b. The larger values of path-integrated LWC enable a more active collision-coalescence process and therefore So is noticeably higher, and both ascending and descending branches are visible for “All” and “Low”. (Note changes in both x and y axis scales in Figure 8b compared to Figure 7b.) The results for the high aerosol range again show weak So . [19] Finally, a third case of shallow (liquid only) continental cumulus [Brown et al., 2002] complements Figures 7 and 8 by providing a view of So in individual clouds in a cloud field with approximately 15% cloud cover. Cloud bases are at about 1 km and maximum cloud tops at 3 km, cloud depths are highly variable. The low cloud cover is reflected in the relatively low number of counts in Figure 9 relative to Figures 7a and 8a. While tc tends to be small (order 10 min), there exists the potential for contact with significantly larger LWC (Figure 9), e.g., in the cores of deeper clouds. The path integrated LWC values are therefore of similar magnitude to those in the stratus case and So behavior follows suit (Figure 9). Note that in this case, collisioncoalescence is active enough that So does rise significantly in the high aerosol case. The low aerosol case peaks at lower path-integrated water contents, and at lower So values. Response for both the high and low aerosol ranges are in qualitative agreement with Figure 6.

3. Discussion and Conclusion [20] While the simplicity of the simulations shown here precludes categorical conclusion, the importance of tc seems indisputable. Cloud contact time plays an important role in determining whether rain production in warm, shallow clouds is dominated by accretion or by autoconversion, and therefore the extent to which the clouds are likely to be susceptible to aerosol perturbations. Seifert and Stevens [2010] showed with a 1-D kinematic model that precipitation efficiency exhibits threshold behavior when plotted as a function of the ratio of cloud lifetime to autoconversion time scale. They concluded that for a cloud of finite lifetime, ˇ would be smaller than one might expect from a purely microphysical point of view, due to a progressive increase in the importance of accretion as the cloud droplet size distribution ages. Jiang et al. [2010] performed analysis of an ensemble of individual clouds generated by a large eddy simulation of trade cumulus clouds. Although individual parcels were not analyzed, each cloud was tracked over its lifetime (an upper bound on tc ); So was calculated from an ensemble of clouds based on simulations with different aerosol concentrations. Accretion and autoconversion rates were also calculated. That study exhibited both ascending and descending branches of So (L) and So (accretion/autoconversion). Taken together, the large eddy simulations of Jiang et al. [2010] and the current study suggest that the relatively short cloud contact times in shallow clouds, be they due to the relatively short in-cloud residence times in well-mixed stratocumulus and/or short cloud lifetimes in the case of small cumulus, point to a significant role for autoconversion in producing rain, and therefore a potential susceptibility to changes in the aerosol. The extent to which this is true will clearly vary based on cloud regime and cloud characteristics (size/liquid

water/depth distribution and lifetime of clouds), boundary layer characteristics (e.g., depth, w, extent of vertical mixing), aerosol loading, and others. So will diminish with increasing cloud thickness, L, and parcel in-cloud residence time, and decreasing Nd . [21] As noted above, in this work R is defined over the full range of drop sizes to allow for the first stages of raindrop formation to be captured. Integrating R only over larger drizzle drops immediately places the focus on the accretion process and in this case So calculations exhibit a steady reduction from 2/3 to 1/3 to 1/6 to 0 as the v(r) dependence shifts from v(r) / r2 to r to r1/2 and finally r0 (as predicted by equation A8). Similar results were obtained by (R. Wood, personal communication, 2010). Given the sometimes short cloud contact times and therefore potentially important role of autoconversion in initiating the first raindrop embryos, So calculations based on R that includes all droplet sizes seems appropriate. The fact that the fall velocity determines the boundary behavior of So (2/3 at low L and progressive reduction to 0 after the So peak has been reached) is itself noteworthy. Scaling of cloud system responses with the terminal velocity of raindrops has also been noted in deep convective clouds [Parodi and Emanuel, 2009; Lee and Feingold, 2013]. [22] How might these results impact upon observations of So ? In clean marine conditions that readily produce drizzle, one might not expect to observe the ascending branch since observations would likely always include a significant accretion process. This would be particularly true if measurements of R were based on Z – R relationships. The chances of observing the ascending branch would likely increase under conditions of high Na and high L (Figure 6). They would also be more likely in cumulus clouds where contact times are relatively short (Figure 9b). [23] This work cautions against calculation of a single So over a large range of aerosol conditions (as represented here by Na ). When autoconversion dominates the rain formation process, power law behavior is distinct (equation (2), Figure 5). With increasing contributions from accretion, be they due to higher L, increasing time available for growth, or reduced Nd , the power law should only be used to calculate So over limited ranges of Na (or Nd ) (Figures 3, 5 and 6). With increasing Na , more liquid water is required to produce the same amount of precipitation. While this is no surprise, what is more interesting is that the maximum value of susceptibility, So,max is shown to increase with increasing Na , albeit over a limited L range. Below this L, R is small, and dominated by autoconversion (note the low baseline So = 2/3), above this L, R is more significant and dominated by accretion so that So rapidly approaches 0. The shift in the value of L at which So,max occurs to progressively larger L is indicative of a threshold behavior of precipitation formation [Kessler, 1969; Liu and Daum, 2004; Seifert and Stevens, 2010]. When the droplet effective radius re exceeds 14 m, there is a transition from a dominance of autoconversion to accretion (Figure 4). This re has been used as a rain initiation threshold [Rosenfeld and Gutman, 1994; Gerber, 1996; Freud and Rosenfeld, 2012; Rosenfeld et al., 2012]. [24] Climate models employ parameterizations of aerosol effects on precipitation to simulate aerosol effects on “cloud lifetime.” The idea is to parameterize potential influences of the aerosol on cloud condensate, fraction, and persistence.

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This is particularly challenging because spatial and temporal grids in climate models do not allow resolution of individual convective cloud elements, lifetime, and cloud contact times. The So or ˇ parameters are often used as a means of adjusting aerosol-precipitation interactions. Typical values of ˇ range from 0.5 to 2 [Quaas et al., 2009] and the choice of ˇ has significant influence on the aerosol cloud forcing [Golaz et al., 2011]. The inability of a climate model to capture convective cells severely restricts their ability to estimate ˇ , since clearly a single value of ˇ cannot represent the various cloud types and regimes. Ideally, ˇ could be built in a probabilistic fashion, regime by regime, taking into account typical probability distribution functions of L, Na , and tc based on large eddy simulation or cloud resolving models [see also Seifert and Stevens, 2010]. Until climate models are able to resolve clouds and convection, this approach may be worth pursuing. Alternatively, the multimodel framework [e.g., Wang et al., 2012] or new probability distribution function approaches to representing subgrid turbulence [Larson and Golaz, 2005] may prove useful.

[28] Simple dimensional analysis arguments yield ˛–ˇ =1

and ˛ = 1 + b/3;

Rh /

r3 n(r)v(r)dr

dz 0

Z

H

L/

(A1)

0

Z

1

r3 n(r)dr

ql / 0

Z Nd =

(A2)

1

n(r)dr

(A3)

0

where the integration is over all drop radii r. Proportionality constants in the above are immaterial for this discussion. The fall velocity relationship is represented by v(r) / rb

(A4)

where b takes on values of 2 (r < 30 m), 1 (40 m < r < 600 m), 0.5 (600 m < r < 2000 m), and 0 (r > 2000 m), i.e., b decreases with increasing drop size. [27] Based on equation (2), we assume –ˇ R q˛ l Nd

(A5)

and substitute the definitions of R and ql to show that 3 ˛ –ˇ Nr3+b m ' (Nrm ) N

(A6)

where rm is a characteristic drop radius for the size distribution.

r3 n(r)dr

(A10)

0

Z

H

Nh =

dz

1

n(r)dr.

(A11)

0

If ql / z, then for constant N, rm / z1/3 . If in addition we assume b invariant with height it can be shown that for Rh L˛ N–ˇ h ,

(A12)

equation (A8) is also valid. A3. Quadratic Change in Liquid Water With Height [30] Many clouds exhibit progressive entrainment with increasing height resulting in a quadratic form ql = a0 z2 +a1 z. Based on numerical integration of equations (A9)–(A11), and assuming b = 0.5

1

r3 n(r)v(r)dr

1

dz

0

Z

(A9)

0

Z

Appendix A: Theoretically Predicted So

R/

(A8)

A2. Linearly Increasing Liquid Water Cloud [29] Similar relationships can be derived for the cloud exhibiting a linear increase in ql with increasing height, which is typical of the stratocumulus clouds for which the empirical relationships have been derived. In this case, we consider Z H Z 1

Z

A1. Homogeneous Cloud [26] Consider the simple case of a homogeneous cloud for which there is no spatial variability in the drop size distribution (and therefore ql ). For this case,

ˇ = b/3,

i.e., the dependence of R on N is a function of the fall velocity regime and ˛ and ˇ are not independent. A progressive decrease in sensitivity of R to ql and N with increasing drop size is clear.

0

[25] Kostinski [2008] showed that there is a relationship between ˇ and drop fall velocity dependence on a drop of radius r (v(r)). Here we expand that analysis for clouds with different dependence of liquid water mixing ratio ql on height z. The analysis assumes a unimodal size distribution, which we recognize a priori is a crude approximation once drizzle becomes significant and size distributions become bimodal. Nevertheless, we believe it provides interesting insight.

(A7)

Rh NH1.80

(A13)

L NH1.70 ,

(A14)

yielding ˇ = 1/7 (as opposed to 1/6 for b = 0.5 and homogeneous or “linear” clouds) and, as before, ˛ – ˇ = 1. Similar relationships can be derived based on integration of the simple equations for other values of b. [31] Acknowledgments. This paper is dedicated to the memory of our dear friend, colleague and mentor, Shalva Tzivion (Tzitzvashvili), the primary architect of Tel Aviv University’s bin microphysical scheme (the TAU method of moments). G.F., A.M., and D.R. were funded by the DOE/ASR Program. G.F. and A.M. acknowledge support from the NOAA/NSF Climate Process Team (PI: V. Larson). A.S. acknowledges support from ONR grant N00014-10-1-0811. We thank R. Wood for useful discussions.

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