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Aerosol indirect effect from turbulence-inducedbroadening of cloud-droplet size distributionsKamal Kant Chandrakara,b, Will Cantrella,b, Kelken Changa,b, David Ciochettoa,b, Dennis Niedermeiera,b,c,Mikhail Ovchinnikovd, Raymond A. Shawa,b,1, and Fan Yanga,b

aDepartment of Physics, Michigan Technological University, Houghton, MI 49931; bAtmospheric Sciences Program, Michigan Technological University,Houghton, MI 49931; cLeibniz Institute for Tropospheric Research, 04318 Leipzig, Germany; and dAtmospheric Sciences and Global Change Division,Pacific Northwest National Laboratory, Richland, WA 99352

Edited by John H. Seinfeld, California Institute of Technology, Pasadena, CA, and approved October 14, 2016 (received for review August 1, 2016)

The influence of aerosol concentration on the cloud-droplet sizedistribution is investigated in a laboratory chamber that enablesturbulent cloud formation through moist convection. The exper-iments allow steady-state microphysics to be achieved, withaerosol input balanced by cloud-droplet growth and fallout. Asaerosol concentration is increased, the cloud-droplet mean diam-eter decreases, as expected, but the width of the size distribu-tion also decreases sharply. The aerosol input allows for cloudgeneration in the limiting regimes of fast microphysics (τc <τt)for high aerosol concentration, and slow microphysics (τc >τt)for low aerosol concentration; here, τc is the phase-relaxationtime and τt is the turbulence-correlation time. The increase inthe width of the droplet size distribution for the low aerosollimit is consistent with larger variability of supersaturation due tothe slow microphysical response. A stochastic differential equa-tion for supersaturation predicts that the standard deviation ofthe squared droplet radius should increase linearly with a systemtime scale defined as τ−1

s = τ−1c + τ

−1t , and the measurements

are in excellent agreement with this finding. The result under-scores the importance of droplet size dispersion for aerosol indi-rect effects: increasing aerosol concentration changes the albedoand suppresses precipitation formation not only through reduc-tion of the mean droplet diameter but also by narrowing of thedroplet size distribution due to reduced supersaturation fluctua-tions. Supersaturation fluctuations in the low aerosol/slow micro-physics limit are likely of leading importance for precipitationformation.

aerosol indirect effect | cloud-droplet size distribution |cloud–turbulence interactions

The optical properties of warm clouds depend on the dropletsize distribution and its moments such as number density and

effective radius, which, in turn, are influenced by the aerosol par-ticles that act as nuclei for the formation of cloud droplets (1, 2).Thus, aerosol indirect effects are considered among the largestuncertainties in climate response to changes in radiative forc-ing (3). This work addresses how the aerosol number concen-tration affects the cloud-droplet size distribution in a turbulentenvironment, which is relevant to both the aerosol first and sec-ond indirect effects (albedo and lifetime effects). The lifetimeeffect links the development of precipitation, and thus cloud life-time, to aerosol number concentration. The logic is that a higheraerosol concentration leads to smaller cloud droplets and nar-rower size distributions, and therefore suppression of the colli-sion and coalescence of droplets, thereby increasing cloud life-time and maintaining higher cloud liquid water content (4–7).The microphysical details of the transition from condensationgrowth to collision growth are not fully understood, however, andit is fair to say that the underlying mechanism of the second indi-rect effect is still a matter of active research (2, 8). Initiation ofprecipitation in warm clouds is favored when the average dropletsize is increased, but perhaps even more crucial is the sensitivityto an increase in the width of the droplet size distribution (9–11).

For decades, the cloud physics community has grappled with thetheoretical finding that cloud-droplet growth in a closed (adia-batic) parcel leads to a size distribution that becomes narrowerwith height above cloud base, tending to suppress the onsetof coalescence through differential gravitational sedimentation.Observations, in contrast, often reveal that cloud-droplet sizedistributions are relatively broad, and this result is usually inter-preted as a consequence of entrainment and mixing and sec-ondary activation (12, 13). Supersaturation fluctuations due toentrainment and droplet recycling have been postulated as amechanism for producing droplets larger than are able to be gen-erated in an adiabatic parcel (14–17). These approaches haveclose ties to the theory of stochastic condensation developeddecades ago and still being refined (18–20). In this paper, weseek insight into the mechanisms underlying the lifetime effectby asking the question, how do changes in aerosol concentrationinfluence the moments of the cloud-droplet size distribution thatare relevant to precipitation formation?

Cloud-droplet size dispersion is also directly relevant to thealbedo effect: for example, cloud albedo is enhanced not onlywhen the droplet number density n increases (for constant liq-uid water content) but also when size dispersion decreases withincreasing n (2, 21). Various contributions to this dispersioneffect have been identified, such as condensational narrowingand collisional broadening, which, respectively, tend to decreaseand increase albedo susceptibility (1, 7, 22). What other pro-cesses influence the dispersion? Here, we consider the conceptthat cloud-droplet growth occurs in a turbulent environment andthat the microphysical response to turbulent fluctuations dependson relative magnitudes of the characteristic time for droplets to

Significance

Atmospheric aerosol concentration is linked to cloud bright-ness and lifetime through the modulation of precipitation.Generally speaking, larger cloud droplets and wider-dropletsize distributions form precipitation more efficiently. We cre-ate steady-state clouds in the laboratory through a balanceof constant aerosol injection and cloud-droplet removal dueto settling. As aerosol concentration is decreased, the cloud-droplet mean diameter increases, as expected, but also thewidth of the size distribution increases sharply. Theory, sim-ulations, and measurements point to greater supersaturationvariability as the cause of this broadening in what can be con-sidered a low aerosol/slow microphysics limit.

Author contributions: K.K.C., W.C., K.C., D.C., D.N., and R.A.S. designed research; K.K.C.,K.C., D.C., D.N., M.O., and F.Y. performed research; M.O. and F.Y. performed simulations;K.K.C., W.C., R.A.S., and F.Y. analyzed data; and K.K.C. and R.A.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: rashaw@mtu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1612686113/-/DCSupplemental.

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adjust to varying humidity (τc) and the correlation time of theturbulence (τt ). The cloud response time τc is inversely propor-tional to droplet number density and the mean droplet diameter,so it stands to reason that aerosol properties can push a cloud intocontrasting regimes with τc < τt or τc > τt . For example, it hasbeen suggested (2), based on the conceptual picture of competingtime scales for turbulence and cloud response (23, 24), that vary-ing aerosol number density can shift a cloud into regimes wherethe nature of the cloud response to mixing and entrainment wouldbe distinct. Furthermore, recent field evidence seems to supportthe concept that limiting τc regimes influence the supersaturationvariability resulting from turbulence (25). We investigate theseconcepts using a cloud chamber capable of generating turbulentclouds with controlled aerosol input.

Experimental ApproachThe concept of the experiment is outlined in Fig. 1, showing aturbulent cloud generated through moist Rayleigh–Benard con-vection and the resulting isobaric mixing of warm and cool water-saturated air (26). Steady-state cloud properties are achievedwhen a constant rate of aerosol injection is balanced by cloud-droplet growth in the turbulent environment and subsequent

cool, humid

warm, humid

steadyaerosol

injec�on

clouddroplet

ac�va�on

droplet growthin turbulentenvironment

dropletsedimenta�on

turbulentconvec�on

lowaerosol

injec�on

highaerosol

injec�on

A B

Fig. 1. Overview of the experimental approach, showing two photographsof the turbulent cloud illuminated by a laser light sheet for two differ-ent aerosol-injection rates. A and B show low and high aerosol-injectionrates, respectively. A also schematically shows the microphysical steadystate achieved by steady aerosol-injection rate, followed by aerosol activa-tion, droplet growth in a turbulent environment, and eventual removal bydroplet sedimentation. The fluctuations in droplet number density are espe-cially visible in B. Turbulent convection transports energy and water vaporfrom the lower (warm) surface to the upper (cool) surface, both of whichare maintained at water saturation. The turbulent mixing of air from thetwo surfaces leads to a nearly uniform vertical profile of supersaturation.Note that the light sheet is seen from a side perspective and is ∼2 m indepth. The red dots at the bottom are from the crossing laser beams fromthe phase Doppler interferometer.

removal by gravitational settling. The resulting steady-state cloudcan be sampled for long times, allowing for high statistical sig-nificance, and the changes in cloud-droplet size distribution tovarying aerosol concentrations can be observed in detail. A cru-cial aspect of the experiment is that the clouds are generatedwith constant forcing: the usual confounding feedbacks that arealways present in the atmosphere are absent because the ther-modynamic and turbulent forcing of the cloud is externally con-trolled within the convection chamber. The two vertical imagesshown in Fig. 1 illustrate conditions achieved under two differentaerosol-injection rates.

Experiments are performed in the Pi Chamber (26) (≈ 5 m2

horizontal cross-section and 1 m high), with an unstable temper-ature gradient (26 ◦C bottom and 7 ◦C top boundary) appliedbetween top and bottom surface to create the turbulent environ-ment. All of the experiments start with moist Rayleigh–Benardconvection containing no aerosols (no cloud droplets) and areallowed to run for t > 10 h to ensure steady-state condition.Then, aerosol injection starts at a constant rate, and after a rel-atively brief transient (of order 10 min), cloud properties alsoreach steady-state conditions. Aerosol injection then continuesfor the next 10 − 12 h to ensure sufficiently long cloud andturbulent samples to minimize random sampling uncertainties.Aerosol inlet concentrations and background aerosol concen-tration inside the chamber during the experiment are measuredwith a TSI Scanning Mobility Particle Sizer. The aerosol injectionoccurs at a point and then aerosols are rapidly mixed by the tur-bulent flow; measurements at several points in the chamber con-firm that the aerosol concentration na varies by less than 10%.Cloud properties (n and d) are measured by a Dantec Dynam-ics phase Doppler interferometer at a point close to the super-saturation measurement. In steady state, the cloud-droplet num-ber density and mean diameter are observed to be stationary intime to within the sampling uncertainty. Other properties suchas temperature, water vapor concentration, turbulence intensityand dissipation rate, and interstitial aerosol concentration alsoare observed to be steady throughout the experiment due to theconstant forcing.

ResultsSteady-State Cloud Properties for Different Aerosol Concentrations.Fig. 2 shows the probability density of droplet diameter for fivedifferent aerosol input rates. The input rates na , measured cloud-droplet number density n , and mean diameter d are summa-rized in Table 1. The SD σd and relative dispersion σd/d fordroplet diameter are also included. With the decrease in theaerosol input and corresponding decrease in the cloud-dropletnumber concentration, mean (and mode) droplet diameter isobserved to increase. Intriguingly, the size distribution also isobserved to become broader and positively skewed, consistentwith the dispersion effect observed in naturally occurring clouds(7). The change in the mode diameter and the distribution widthwith aerosol-injection rate is very clear and reproducible, andit is especially intriguing that the pronounced large-droplet tailapproaches the drizzle range for the cleanest cases.

Origin of Droplet Size-Distribution Broadening. The appearance ofa large-droplet tail with reduced aerosol-injection rate is strik-ing, and begs for further interpretation. We consider here twohypotheses for the increasing breadth of the size distributionwith decreasing aerosol concentration: cloud-droplet collisionsand supersaturation fluctuations.

One possible explanation is that as the aerosol-injection rateis reduced, the mean cloud-droplet diameter increases andthereby leads to a higher droplet collision rate. As collisionsbecome more frequent a large-droplet tail appears. We assessthis hypothesis by estimating the mean collision time based onthe observed size distribution, by using a parcel model, and

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Fig. 2. Probability density functions (PDFs) for cloud-droplet diameterobserved under steady-state conditions for five aerosol-injection rates. PDFsare shown instead of size distributions to emphasize the change in shaperather than the change in number density. The aerosol-injection rates shownin the legend correspond to the aerosol source concentrations multiplied bythe injection flow rate and divided by the chamber volume to obtain vol-umetric sources within the Pi Chamber. Droplet size distributions are mea-sured with a Dantec phase Doppler interferometer, which measures in thesize range of d = 3 to 220 µm with a resolution of 1–2%.

finally through large eddy simulation (LES) of the convectionchamber, including water vapor and bin microphysics. The esti-mated collision times (Cloud-Droplet Collisions) are of order 1 hor greater. A typical droplet residence time from the chamberτres is estimated by observing the decay of n in the chamber whenthe aerosol-injection rate is turned off (for the lowest aerosolcase, which has an interstitial concentration of nearly zero). Wefind an exponential decay with time constant τres ≈ 9.7 min,much less than the estimated collision times. Detailed parcelmodel and LES calculations (Large-Eddy Simulations) confirmthis conclusion that the large droplet tail does not result fromdroplet collisions.

Supersaturation Fluctuations. Supersaturation s is derived fromtwo scalar quantities, temperature T and water-vapor-mixingratio qv , which themselves are advected and diffused within theturbulent atmosphere. Thus, in the absence of processes thatdeplete the supersaturation, such as phase transitions, s willexhibit fluctuations that are the hallmark of all turbulent flows(27). Indeed, under cloud-free conditions in the Pi Chamber (i.e.,no aerosol injection), the supersaturation can be calculated fromhigh-resolution, collocated measurements of qv and T , and theresulting PDF of fluctuations s ′ shown in Fig. 3 is observed to benearly Gaussian. We now explore whether these supersaturationfluctuations can account for the change in shape of the dropletdiameter PDF. From prior work (2, 14, 25), we anticipate thata cloud can exist in limiting cases of fast and slow microphysicalresponse compared with turbulent mixing and can be expressedas high and low Damkohler number, Da = τt/τc , where τt is theturbulence-correlation time and τc is the microphysical responsetime. In our experiments, the cloud can be pushed toward the fastor slow limits by changing the steady aerosol-injection rate. Themicrophysical response time is the phase-relaxation time, which

is related to cloud-droplet mean diameter and number density asτc ∝ (nd)

−1. If liquid water content w ∝nd3 is constrained to beconstant (e.g., due to constant thermodynamic forcing in the con-vection chamber), then it follows that τc ∝n−2/3 (i.e., increas-ing with decreasing cloud-droplet concentration and therefore,except for highly overseeded conditions, increasing with decreas-ing aerosol-injection rate). Values for the five experiments areshown in Table 1, and this trend is indeed observed to be valid.We can speculate that the supersaturation quickly adjusts to the“quasisteady” value for fast microphysics but that it spends moretime away from the quasisteady value for sluggish microphysics,thereby leading to greater variability in the local droplet conden-sation growth rates and a broader droplet size distribution. Toquantitatively evaluate this hypothesis for existence of the signif-icantly enhanced droplet size variability at low aerosol-injectionrate, we consider analytical solutions to a stochastic differentialequation for supersaturation, and we perform LESs of the tur-bulent moist convection with cloud represented using bin micro-physics (discussed in LES).

We extend the stochastic differential equation approach ofSardina et al. (28) to our experimental scenario. Droplet growthdue to diffusion can be written as dr2/dt = 2ξs , from which anexpression for the variability in r2 can be obtained:

dσ2r2

dt= 4ξs ′r2′ , [1]

where r , σr2 , and ξ are the radius of a droplet, rms r2 fluctuation,and condensation growth parameter, respectively. Overbar andprime indicate the mean and fluctuations. Thus, the challengebecomes finding an expression for the covariance of droplet sizeand supersaturation, s ′r2′ . To that end, we write a stochastic dif-ferential equation for the supersaturation:

ds(t) = s(t + dt)− s(t) =

[seq − s

τt− s

τc

]dt

+

(2σ2

s0dt

τt

)1/2

η(t). [2]

Here, supersaturation s(t) and Gaussian white noise η(t) areboth continuous random variables but are not differentiable (29).The first term on the right side of Eq. 2 represents a sourceterm due to moist convection that always tends to drive the sys-tem toward an equilibrium value of supersaturation seq . The sec-ond term represents the sink due to cloud-droplet growth, andthe last term is a random fluctuation term (Wiener process)representing variability due to turbulence [σ2

s0 and η(t) arethe supersaturation variance without cloud (cf., Fig. 3) anda Gaussian white noise term with mean of 0 and SD of 1,

Table 1. Time-averaged microphysical properties for the fivesteady-state clouds

na, cm−3 na,int ,min−1 cm−3 n, cm−3 d, µm σd , µm σd/d τc, s Da

1 2 21.3 16.6 6.7 0.40 58.1 0.032 10 76.9 15.1 5.7 0.38 17.6 0.114 36 201.2 12.7 4.5 0.35 8.0 0.2512 372 564.6 8.6 2.4 0.28 4.2 0.51,515 22,000 1,944.3 7.6 2.1 0.28 1.4 1.5

Shown are chamber-averaged aerosol-injection rate na, interstitialaerosol concentration na,int , cloud-droplet number density n, meancloud-droplet diameter d, SD of cloud-droplet diameter σd , calculatedphase-relaxation time τc, and calculated Damkohler number Da. Theturbulence-correlation time τt = 2 s is obtained from the measured velocityfluctuations.

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s'-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

PD

F

5

10

15

20

25

30

35

Measured DataGaussian Fit

Fig. 3. Supersaturation fluctuation distribution observed with no aerosolinjection and therefore no cloud droplets in the chamber. The fluctuationsare derived from measurements of water vapor mixing ratio using a LI-COR LI-7500A H2O analyzer and temperature using a resistance temperaturedetector. The absolute magnitude of s is susceptible to biases, but the fluctu-ations s′ are significant compared with instrument resolutions. The best-fitGaussian has a SD of 0.014.

respectively]. From this starting point, it is possible to obtaindifferential equations for supersaturation variance and fors ′r2′ : ds ′(t)2/dt =−2s ′(t)2/τs+2σ2

s0/τt , where τs = τcτt/(τc+

τt) is the combined time scale of the system and ds ′r2′/dt =

2ξs ′2 − s ′r2′/τs .Under steady-state conditions, expressions for s ′2 and s ′r2′

can then be obtained from the differential equations:

s ′2 =σ2s0τs

τt[3]

and

s ′r2′ = 2ξs ′2τs =2ξσ2

s0τ2s

τt. [4]

Combination of Eqs. 1 and 4 and integration with time givesthe expression for the width of a droplet size distribution,

σ2r2 =

8ξ2σ2s0τ

2s

τtτres . [5]

Here, τres is the time available for growth, which for theseexperiments is the typical droplet residence time in the cloudchamber. Droplets settle out of the chamber with a time scale of∼10 min, and, at least initially, we may assume that the lifetimeis independent of the aerosol-injection rate. Writing the globaltime scale as a function of Damkohler number, τs = τt/(1+Da),we obtain σr2 ∝ σs0/(1 + Da). Thus, the width of the cloud-droplet size distribution depends on the relative magnitudes ofthe turbulent-mixing time and the phase-relaxation time. Thelimits Da � 1 and Da � 1 therefore result in very differentbehavior, with the former “fast microphysics” scenario resultingin relatively narrow droplet size distributions and the latter “slow

microphysics” scenario leading to relatively wide droplet size dis-tributions. This result is qualitatively consistent with the trendsobserved in Fig. 2 and Table 1.

To test the linearity prediction, we plot the calculated σr2 ver-sus τs = τt/(1 + Da) in Fig. 4. Except for the point correspond-ing to the highest aerosol concentration, the linear dependencepredicted by Eq. 5 is excellent. In contrast, σr2 is plotted versusτc ∝ Da−1 in the inset, and the lack of collapse confirms thata simple linear dependence on Da does not describe the systembehavior.

The outlier point corresponds to the experiment with the high-est aerosol-injection rate and therefore the largest droplet num-ber density n . The aerosol-injection rate for that experiment istwo orders of magnitude larger than in the other experiments(cf., Table 1) and therefore provides insight into how the simplelinear prediction from Eq. 5 breaks down. First, it is possible thatinstrumental artifacts are influencing this data point: due to thevery high droplet concentration, n may be underestimated dueto coincidence effects; and due to the very small mean dropletdiameter, the width of the size distribution is getting closer tothe inherent size resolution of the measurement method. Sec-ond, the linear prediction σr2 ∝ τs depends on the assumptionof constant droplet residence time τres . When aerosol concen-tration and n are very large, however, we may anticipate that themean supersaturation in the chamber is suppressed due to strongcompetition between droplets, therefore resulting in a slow aver-age growth rate and an increase in the residence time. Thishypothesis is corroborated by the transient data, which resultsin τres ≈ 10 min for all cases except the highest aerosol case, forwhich τres is ∼4 times larger.

Fig. 4. Width of the r2 distribution versus the system time scale τs =

(τ−1c + τ−1

t )−1

. The theory predicts a linear dependence between σr2 and τs

that is observed for all runs except that with much higher aerosol-injectionrate. A linear fit to the four data points yields slope 88 µm2s−1. Uncertain-ties in the steady-state cloud properties calculated using the typical Poissonstatistic are very small because of the long data samples. Instead, uncertaintyis dominated by the turbulent fluctuations. Therefore, the uncertainty isestimated by dividing a steady-state data sample into smaller subsets and,for each subset, all of the physical quantities are calculated. Error bars in thefigure represent the maximum symmetric deviation of the individual quan-tity from the mean of all data subsets.

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The predicted slope for the linear dependence of σr2 on τs is√8ξσs0(τres/τt)

1/2. The parameter ξ = 109 µm2 s−1 is calcu-lated from the thermodynamic properties in the chamber, andτt = 2 s is obtained from the turbulence measurements. As dis-cussed already, the droplet lifetime τres ≈ 9.7 min. A measure-ment of σs0 is also needed, and from Eq. 3, it is evident thatthe measurement is obtained under cloud-free conditions: for nocloud, τs = τt such that s ′2 = σ2

s0 . As shown in Fig. 3, the PDFof s fluctuations is consistent with the Gaussian-distributed noiseassumption in Eq. 2, and has an observed width of σs0 = 1.4%.Using these values from directly measured properties in thecloud chamber, the slope is estimated to be 74 µm2 s−1. This pre-diction is compared with the fitted slope in Fig. 4 of 88 µm2 s−1.These agree to within about 20%, which is an encouraging resultgiven the uncertain estimate of τres and the challenge of measur-ing water vapor supersaturation necessary for estimation of σs0 .

LES. As a final check on this conclusion regarding the pro-nounced broadening of the cloud-droplet size distribution withdecreasing aerosol concentration, we simulated the Pi Chamberusing LES. The original model is the LES code (named Systemfor Atmospheric Modeling or SAM) (30), which has been widelyused to study warm and cold cloud processes in the atmosphere.We modified SAM to mimic the Rayleigh–Benard convection inthe Pi Chamber. Details of the model setup are included in theonline supplement.

Five cases were run with varying aerosol-injection rate so asto achieve Da = [0.033, 0.09, 0.18, 1.6, 2.8], approximately thesame range as in the experiments (cf., Table 1). The LES resultsprovide insight on at least three points from the preceding dis-cussion: (i) the cloud-droplet size distributions become signif-icantly broader as the aerosol-injection rate is decreased, withσd/d = [0.43, 0.42, 0.41, 0.34, 0.25], qualitatively consistent withthe experimental observations in Fig. 2 and Table 1; (ii) thecloud-droplet size distributions with and without accounting fordroplet collision and coalescence are nearly identical, confirm-ing that collisions cannot explain the broadening; and (iii) themean and SD of the supersaturation PDFs decrease with increas-ing aerosol-injection rate, consistent with the theoretical picturegiven by the stochastic differential equation for supersaturation(Eq. 2).

Atmospheric ImplicationsThe experiments described here provide strong evidence foraerosol-mediated broadening of cloud-droplet size distributionsin turbulent environments: clean clouds can reside in a slowmicrophysics regime in which supersaturation fluctuations arelarge and lead to broad size distributions, whereas pollutedclouds reside in a fast microphysics regime with small super-saturation fluctuations and narrow droplet size distributions.However, how general are the results, what are the limitingassumptions, and how relevant is this idealized laboratory resultto naturally occurring clouds in the atmosphere? As mentionedalready, strong fluctuations of scalars within turbulence is a ubiq-uitous and general problem (27). Indeed, it is known that super-saturation is a fluctuating quantity in the turbulent atmosphere(18, 31–33). For example, supersaturation variability in the atmo-sphere, without the influence of cloud-droplet activation andgrowth, can be estimated for one source, the rms vertical velocity:σs ≈ τt(ds/dz )σw . For warm clouds with ds/dz ≈ 5×10−4 m−1

(34), turbulence measurements suggest the ranges from σw ∼ 0.1to 1 ms−1 and τt ∼ 10 to 100 s (35), resulting in σs ∼ 10−4

to 10−2. Other sources of supersaturation fluctuations includeentrainment and isobaric mixing of cloud parcels with differenttemperatures. These estimates and similar ones in the literature(18, 25, 31, 32) confirm that the Pi Chamber experiment pro-duces fluctuations through isobaric mixing, with a σs within the

reasonable range for the atmosphere. Unlike the atmosphere,however, in the Pi Chamber σs0 is fixed, allowing the responseof droplets to varying aerosol conditions to be studied in iso-lation. In the atmosphere, feedbacks between the microphysi-cal evolution and σs may exist (e.g., due to varying radiativeflux divergence). In this context, the results presented here cer-tainly provide support for similar stochastic differential equationapproaches for cloud microphysics (9, 20, 28) and perhaps evenfor macroscopic cloud properties (36). It will be intriguing toinvestigate exactly how the current turbulence-induced aerosoleffect relates to that observed in closed parcel studies that con-sider wide ranges of aerosol concentrations (37, 38).

A key assumption in the derivation leading to the linearityprediction underlying Fig. 4 is that τc is a constant for a givenaerosol-injection rate. In reality, fluctuations of n and d (wherethe bar here would represent a local average) lead to a τc thatvaries in time and space (14). Thus, the theoretical approachgiven here only allows for a s ′2 that varies between the σ2

s and0, whereas fluctuations in the integral radius can lead to varianceof s greater than σ2

s . The variance of supersaturation is relevantnot only to the droplet size dispersion but also to the numberof activated aerosol particles (33). Therefore, the mean (qua-sisteady) supersaturation and s ′2 will together influence boththe droplet concentration and the shape of the size distribu-tion. How significant these effects are, their relative importance,and their relevance for atmospheric clouds, is a topic for furtherresearch.

The most central result is the influence of aerosols onbroadening of the droplet size distribution, known as the dis-persion effect. Our experiments explore the Da range 0.03–1.5(from lowest to highest aerosol-injection rate), corresponding toa transition from the very slow microphysics limit to the edgeof the fast microphysics regime. What range of Da exists inthe atmosphere? For the most typical conditions, τc ∼ 1 to 10 s(14, 18) and τt ∼ 10 to 100 s, suggesting that Da can vary fromorder 100 to 1, thereby residing primarily in the cloud-droplet-dominated, fast microphysics regime. Under very clean condi-tions, however, with droplet concentrations of order n ∼ 10 cm−3

and a corresponding τc ∼ 100s and Da ∼ 0.1, clouds can residein the turbulence-dominated, slow-microphysics regime. Ourresults suggest that precipitation formation would be stronglyenhanced due to the increase in droplet size distribution widthin such clean clouds (10). Even under relatively polluted con-ditions, the turbulence broadening mechanism can operate ascloud-processed air is entrained into clouds, thereby reducingthe droplet number density without fully evaporating the clouddroplets themselves (13). Perhaps most compelling is the specu-lation that the emergence of high aerosol loading resulting fromindustrialization has fundamentally altered the microphysics ofclouds and suppressed their ability to form precipitation.

Finally, we suggest that the results reported here can serveas the basis for physical-based representations of aerosol indi-rect effects, including the dispersion effect (1, 6, 7), in coarse-resolution models. Added aerosols tend to reduce the phase-relaxation time relative to the turbulence-correlation time andthereby suppress the broadening of the size distribution mak-ing development of precipitation difficult. The notion of Datogether with the stochastic differential equation for supersat-uration allows expressions for the relative dispersion σd/d tobe obtained and, in turn, can be coupled with expressions foronset of precipitation that depend on the relative dispersion(10, 22, 39). The results in Table 1 confirm that not only doesσd increase but σd/d does as well with reduction of aerosol con-centration. Following similar logic as used in obtaining Eq. 5,the result σr2/r2 =

√2τt/τresσs0/seq can be obtained. The

result is compelling because it is independent of Da , and indeedthe relative dispersion (cf., Table 1) is observed to be nearly,

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but not exactly, constant (the slight decrease is consistent withan increase in droplet lifetime with increasing aerosol concen-tration). In the atmosphere an increase or decrease in relativedispersion would depend on how the particle lifetime varies,which is related to precipitation development itself, and there-fore becomes a more complex scenario. Nevertheless, this workprovides a pathway for quantifying the central result: the widthof the droplet size distribution increases with decreasing cloud-droplet number density and, in turn, can be expected to enhancethe cloud albedo susceptibility directly (first indirect effect) and

indirectly by accelerating precipitation formation (second indi-rect effect).

ACKNOWLEDGMENTS. We thank G. Feingold and two anonymous reviewersfor helpful comments. This work was supported by National Science Founda-tion Grant AGS-1623429. D.N. thanks the Alexander von Humboldt Founda-tion for fellowship support. M.O. was supported, in part, by the US Depart-ment of Energy (DOE) Office of Science, under the Atmospheric SystemResearch Program. The Pacific Northwest National Laboratory is operatedby Battelle for the US DOE, under Contract DE-AC06-76RLO1830. Portageand Superior, high-performance computing clusters at Michigan Technolog-ical University, were used for the LES runs.

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