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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES, POPULATION BIOLOGYCorrection for “Aerosol–cloud–precipitation system as a predator-prey problem,” by Ilan Koren and Graham Feingold, which was firstpublished July 8, 2011; 10.1073/pnas.1101777108 (Proc Natl AcadSci USA 108:12227–12232).The authors note that on page 12229, right column, second full

paragraph, line 13, “c2 ’ 3·10−1 m−1” should instead appear as

“c2 ’ 3·104 m−1.”

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Aerosol–cloud–precipitation systemas a predator-prey problemIlan Korena,1 and Graham Feingoldb,1,2

aDepartment of Environmental Sciences, Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel; and bNational Oceanic and AtmosphericAdministration Earth System Research Laboratory, 325 Broadway, Boulder, CO 80523

Edited* by Inez Y. Fung, University of California, Berkeley, CA, and approved June 9, 2011 (received for review February 1, 2011)

We show that the aerosol–cloud–precipitation system exhibitscharacteristics of the predator-prey problem in the field of popu-lation dynamics. Both a detailed large eddy simulation of thedynamics and microphysics of a precipitating shallow boundarylayer cloud system and a simpler model built upon basic physicalprinciples, reproduce predator-prey behavior with rain acting asthe predator and cloud as the prey. The aerosol is shown tomodulate the predator-prey response. Steady-state solution tothe proposed model shows the known existence of bistability incloudiness. Three regimes are identified in the time-dependentsolutions: (i) the weakly precipitating regime where cloud andrain coexist in a quasi steady state; (ii) the moderately drizzlingregime where limit-cycle behavior in the cloud and rain fields isproduced; and (iii) the heavily precipitating clouds where collapseof the boundary layer is predicted. The manifestation of predator-prey behavior in the aerosol–cloud–precipitation system is afurther example of the self-organizing properties of the systemand suggests that exploiting principles of population dynamicsmay help reduce complex aerosol–cloud–rain interactions to amore tractable problem.

aerosol–cloud–climate interactions ∣ dynamical systems ∣ emergence ∣self-organization ∣ attractors

Shallow clouds in the Earth system are currently the subjectof a great deal of attention because of their importance for

climate predictability (1). These shallow clouds radiate in thelongwave at approximately the same temperature as the surfacebut reflect shortwave radiation back to space and therefore coolthe climate system. Moreover, the manner in which shallowclouds are represented in climate models has a significant effecton climate sensitivity. The effect of aerosol particles on theseclouds has also been identified as an important unknown (2).Aerosol particles are the nuclei upon which droplets are formedso that higher aerosol concentrations result in higher droplet con-centrations, and all else equal, smaller drops, more reflectiveclouds, and stronger cooling (3). Smaller droplets are also lessapt to collide and coalesce; this will tend to inhibit precipitation,possibly increasing cloud lifetime (4) and exerting an even stron-ger cooling effect. The feedbacks associated with aerosol effectson clouds and precipitation are myriad and complex [e.g., (5)],poorly quantified, and even more poorly represented in climatemodels.

The past decades have seen significant efforts to study shallowclouds and their interactions with aerosol, and much has beenlearned through intensive field campaigns (6, 7), remote sensing(8, 9), and modeling (10, 11). Fine-scale models that solve theNavier–Stokes equations on grids on the order of tens of metersand represent the coupled aerosol–cloud–precipitation systemare desirable because they address the processes at the appropri-ate spatial and temporal scales. Although these models exhibitcomplex responses to aerosol perturbations, it is becomingincreasingly apparent that the system prefers distinct “modes.”This preference raises the hope that the system is more predict-able than suggested by the very large number (order 107) ofdegrees of freedom that these models represent. Two concepts

are of particular note: (i) the concept of “buffering” (5) wherebyinternal processes buffer the system against strong perturbations,effectively reducing the number of degrees of freedom in thesystem; and (ii) self-organizing patterns in the form of mesoscalecellular convection (12) that constrain the system to specific con-vective modes (13). Process-level complexity therefore appearsto yield, in some cases, to a modicum of systemic predictability.

One example of these concepts is recent work which has shownthat open-cellular convection in precipitating boundary layersorganizes in such a way as to generate coherent precipitationwith a characteristic periodicity related to the size of the cells.In the open-cellular state, clouds appear as ring-like structuresthat are connected to adjacent cloudy rings, forming a lace-likepattern over distances of order 1,000 km (Fig. 1). Within the re-flective cloud walls, the buoyancy is positive, and clouds thickenuntil rain is formed. The rain consumes part of the cloud waterand once it falls below cloud base and evaporates, generates ne-gatively buoyant downdrafts that destroy the positively buoyantupdrafts that were responsible for its formation. The interactionbetween adjacent precipitation-driven outflows generates newpositively buoyant regions that are offset spatially from the earlierones. This manifestation of Le Châtelier’s principle, whereby rainreverses the dynamics that created it, appears as organized cellsthat form, dissipate, and reappear in an orchestrated manner(12). It is the coupling of cloudy/rainy cells in this self-organizedcloud field that is the source of the mesoscale periodicity in rain-rate. The existence of oscillations in open-cellular convectionmotivates us to explore whether oscillations exist in cloud-rainsystems more generally.

Approximately a century ago two researchers independentlyproposed a simple mathematical model to explain oscillatorybehavior in chemical reactions (14) and fish populations (15).In the intervening years various forms of the Lotka–Volterra(LV) equations have been applied extensively to natural systems,particularly in the biological sciences. The original equations aregiven as

dCdt

¼ Cða − bRÞ; dRdt

¼ RðeC − f Þ; [1]

where C is the population of prey, R is the population of preda-tors, and a, b, e, f are system-dependent constants. The prey(C) population grows exponentially in the absence of R, but isreduced by R preying on C. The predator (R) population dependsfor sustenance entirely on prey C and decreases exponentially inthe absence of C. To illustrate the system, numerical solution to

Author contributions: I.K. and G.F. designed research; I.K. and G.F. performed research;I.K. and G.F. analyzed model output; and I.K. and G.F. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.1I.K. and G.F. contributed equally to this work.2To whom correspondence should be addressed. E-mail: graham.feingold@noaa.gov.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1101777108/-/DCSupplemental.

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these equations is given in Fig. 2. The C population (e.g., rabbits)begins to grow, but upon representing a significant food source toR (e.g., foxes), C’s population diminishes. The fox populationgrows until it has overdepleted C and then decays. In the absenceof predation, C begins to recover, and so the cycle continues. Alimit-cycle (or phase diagram) view of the solution is shown inFig. 2B and is a useful means of tracking the trajectory of thetwo populations. Although the LVequations are highly simplifiedand somewhat unrealistic (e.g., the exponential growth in preyin the absence of predators), they prove instructive, and withmodification are very useful (16).

Consider now populations more pertinent to the aerosol–cloud–precipitation system. The system exhibits numerous exam-ples of coexistence of one or more components of the system.Examples include:

• Clouds grow in unstable conditions and therefore “consume”dynamical instability (17);

• Atmospheric aerosol particles are the nuclei upon which clouddroplets form so that aerosol and cloud must coexist. Super-clean aerosol environments do not support the existence ofstable cloud;

• Droplets consume water vapor and cannot thrive unlessdynamical forcing maintains a (super-)saturated vapor con-centration;

• Cloud droplets coalesce to form rain drops which consumecloud drops via collection. The formation of rain can spellthe demise of the cloud, or under some conditions, cloudsand rain may coexist. Because cloud drops form on aerosolparticles, rain that reaches the surface also indirectly consumesaerosol;

• In mixed-phase clouds, water and ice can coexist or, undersome conditions ice grows at the expense of liquid water viathe Bergeron–Findeisin process, resulting in complete glacia-tion and demise of the cloud.

We illustrate predator-prey behavior for a simple system com-prising two primary populations: cloud and rain. A third popula-tion, aerosol, mediates the interaction between these “species.”Because the aerosol directly affects cloud drop concentration Nd,we use Nd as a proxy for the aerosol. Consider model outputfrom large eddy simulation (LES) of a warm, marine stratocumu-lus cloud (11) which solves the Navier–Stokes equations coupledto detailed aerosol and cloud microphysics interactions andrepresents a high level of complexity at significant computationalexpense. Fig. 3 shows LES time series of liquid water path (LWP)(column-integrated liquid water content, gm−2) and rainrate R(mmd−1), which exhibit distinct predator-prey-like oscillations.The formation of rainfall sets in motion a negative feedback; raindepletes LWP and cloud depth H and slows down further rainformation. Cloud has to thicken again before rain can reform.The magnitude of the oscillations varies with time as a resultof LWP changing in response to various internal and externalforces. Thus the phase-space trace consists of a series of displacedanticlockwise loops as opposed to the recurring, superimposedloops in Fig. 2B. Nevertheless, in spite of the multiple processinteractions and feedbacks occurring in the system, the systemreveals a more simple emergent behavior (18) *.

Predictive ModelWe build a predictive cloud model that addresses the cloud-rainproblem using source and sink terms that are based on physicalprinciples, distillation of knowledge acquired from a number ofintensive airborne field campaigns, and empiricism from detailedmodels. The model is not designed to capture the details of thesystem, but rather its emergent behavior. Instead of solving bal-ance equations for both LWP (prey) and R (predator) as in Eq. 1,we capture the essential physics with one equation for clouddepth H plus a theoretically- (19) and empirically-based (7)diagnostic equation for R, with an appropriate delay function. His intimately related to LWP (Eq. 2), and as will be shown in theResults, has the benefit of producing a particularly simple analy-tical steady-state solution. In addition, a second balance equationis solved for Nd to account for its sometimes pivotal role in con-trolling R (12), and because fixing Nd would be overly restrictive.The proposed set of equations is just one of many alternatives,

Fig. 1. Satellite image showing a mesoscale cellular convective system in theAtlantic Ocean. The image, taken by the NASA MODIS imager on boardthe Terra satellite on July 21, 2004, shows bright, closed cellular regionsinterspersed with open cell regions where a lace-like pattern of brightcloudy filaments exist against the backdrop of the dark ocean. The scale isapproximately 1;300 km ðhorizontalÞ × 1;100 km ðverticalÞ.

A

B

Fig. 2. Solution to the Lotka–Volterra equations: (A) time series of predatorand prey; (B) limit-cycle view of (A).

*“emergent” behavior or “emergence” conveys the notion that system-wide patternsemerge from local interactions between elements that make up the system.

12228 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1101777108 Koren and Feingold

and has been chosen because it captures the basic physics in acompact and simple manner.

First we assume a cloud in which liquid water increases linearly(but not necessarily adiabatically) with H so that LWP varies as

LWP ¼Z

H

0

qðzÞdz ¼ c12H2; [2]

where q is the cloud water mixing ratio and c1 is a function ofcloud-base temperature and pressure; c1 ≃ 2 · 10−6 mmm−2 fora warm adiabatic cloud. The balance equation for H is writtenas

dHdt

¼ H0 −Hτ1

þ _Hrðt − TÞ: [3]

The first term on the left hand side represents an exponentialapproach of H to H0, with a characteristic time constant τ1, asa result of “dynamical forcing,” which we assume to include avariety of forcings including latent heating associated with phasechange, radiative flux divergence, entrainment-mixing, atmo-spheric instability, mesoscale forcing, etc. H0 represents the fullenvironmental potential for cloud development so that in theabsence of water sinks, H will approach the asymptotic limit ofH0 after several τ1. _Hrðt − TÞ represents loss of H as a result ofrain, a stochastic process that converts small cloud droplets toraindrops (a 10 order of magnitude increase in mass) in a rela-tively short period of time (T ≃ 15 min). The delay term (t − T)accounts for the time-dependence of rain production, which isa function of the state of the cloud some period of time priorto the current time step. Delay terms are common in populationdynamics; e.g., birth rates must account for maturation toadulthood.

To first order, rainrate R can be diagnosed from cloud depthHand Nd

R ¼ αH3N−1d [4]

based on theory (19) and observations (6, 7), whereα≃ 2 mmm−6 d−1 for warm stratocumulus. Thus cloud depth

(or more pertinently LWP) has a much stronger control overprecipitation than does Nd.

To represent the loss term for H we note that

dLWPdt

¼ −R; [5]

_Hr is then written as

_Hr ¼dHdt

¼ dHdLWP

dLWPdt

; [6]

and substituting Eqs. 2, 4, and 5 into 6 yields

_Hr ¼ −1

c1HR ¼ −

αH2

c1Nd; [7]

with Nd representing the cloud-mean value.Analogous to Eq. 3 we write the equation for Nd, including a

delayed loss term, as:

dNd

dt¼ N0 −Nd

τ2þ _Ndðt − TÞ: [8]

N0 can be considered to be the background concentration ofaerosol to which the system strives, and the last term representscoalescence processes. τ2 is the characteristic time constant forreplenishment of the aerosol and is of order τ1 or larger. Thecloud-mean loss term for Nd results from the conversion of cloudwater to rain water via drop collection and is calculated usinga simple expression based on detailed solution to the stochasticcollection equation (20):

_Nd ¼ −c2NdR [9]

with c2 ≃ 3 · 10−1 m−1.To simplify the formulation of the delay terms, we rely on

the fact that both H and Nd are continuous functions and assumethat there is an equivalent delay time T 0 that represents the meanvalue in the interval, and therefore replaces the integration overthe interval. _Hr in Eq. 7 is therefore expressed as αH2ðt − T 0Þ∕c1Ndðt − T 0Þ and Nd in Eq. 9 is calculated at t − T 0. Finally Ris diagnosed based on Eq. 4 as

RðtÞ ¼ αH3ðt − T 0ÞNdðt − T 0Þ : [10]

As will be shown below, the combination of Eq. 3 and the diag-nostic equation with delay Eq. 10 generates oscillating solutionsthat are equivalent to those produced by two coupled equations(e.g., Eq. 1). The addition of Eq. 8 allows for a more realisticevolution of the system because R is intimately related to Nd.The primary model (Eqs. 3, 8, and 10) are discretized and solvedusing a finite-difference scheme.

ResultsDynamic equations with variables characterized by mutual mod-ulation, and therefore the potential for first derivatives that areopposite in sign (21), produce solutions that have four basicmodes: (i) steady state, (ii) oscillations, (iii) chaotic behavior,and (iv) unstable solution. The steady-state regime, in whichsources and sinks of H are in balance, is amenable to analyticalsolution in our case and is explored below. Following that, someof the other modes of behavior are examined through time-dependent, numerical solution.

A

B

Fig. 3. Large eddy simulation of the aerosol–cloud–precipitation system:(A) time series of LWP and rainrate R over the course of 150 min;(B) limit-cycle display of (A). Loops in (B) evolve from lower left to upper leftin an anticlockwise sense.

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Analytical Steady-State Solutions. We start with an analyticalsolution to the equations that yields important insights. SettingdH∕dt ¼ 0 in Eq. 3 gives the steady-state solution to H in theform of a quadratic:

γτ1H2 þ NdH −NdH0 ¼ 0; [11]

where γ ¼ α∕c1. Because H ≥ 0, the only physical solution is

H ¼ ðN2d þ 4γτ1NdH0Þ12 −Nd

2γτ1: [12]

Fig. 4 shows contours of steady-state H in (Nd; H0) space forτ1 ¼ 60 min. Two regimes can be seen: (i) at Nd ≳ 30 cm−3, His almost entirely determined by the external dynamical forcingH0, except at higher H0 where Nd also plays a role; (ii) at lowNdð≲30 cm−3Þ, H decreases rapidly with decreasing Nd, with lit-tle influence by H0. As simple as this equation is, it already hintsat the potential for bifurcation of the aerosol–cloud–precipitationsystem between a nondrizzling solid cloud at highNd whose depthis determined by H0, and a drizzling cloud with depth stronglydependent on Nd (22). This result is not dependent on the valueof τ1; results for much larger τ1 (¼720 min) are qualitatively si-milar. The original idea of bistability emerged from a mixed-layermodel that included more complexity in the dynamics and micro-physics than the current model. Bistablity was also confirmed bymuch more complex LES (e.g., 12). It is therefore noteworthythat solution to Eq. 12 also points to this behavior.

Interestingly, Eq. 12 predicts a positive relationship betweenH and Nd, supporting earlier studies [e.g., (23)] that also pointto the cloud being able to achieve a deeper state under higheraerosol loadings. This result is still somewhat contentious, parti-cularly when considering the possibility of resulting feedbacks. Inaddition, at some point the larger H will likely fuel more rain (5)which will result in reduction in Nd (Eq. 12 considers fixed Nd).

Time-Dependent Solutions. For deeper analysis of the attractors ofthe system, the time-dependent model is run for a range of H0

and N0 until it achieves steady state. The simulations extendout to ∼7 d but steady state is usually achieved within hours.No attempt has been made to include a diurnal cycle of radiativeforcing in any of these, or subsequent simulations. Solutions toH, Nd, and R are displayed in Fig. 5. Under weakly precipitatingconditions (lowH0 and highN0),H is determined byH0, whereasfor more strongly precipitating conditions (higher H0 and lowerN0), H increases with both increasing H0 and N0. At small N0

(lower left),H is more strongly controlled byN0, in general agree-ment with the steady-state solution in Fig. 4. The figures differquantitatively because Fig. 4 is a solution to Eq. 12 alone whereasFig. 5 reflects the coupled solution to Eqs. 3 and 8 with 10. Thegray shaded area at upper left represents a region of parameter

space where stable solutions do not exist; strong precipitationresults in rapid demise of the cloud, which implies collapse ofthe boundary layer. Note that nowhere does H decrease in re-sponse to increasing N0, as suggested by some LES studies[e.g., (24)], however the latter response is a result of more effi-cient evaporation of cloud droplets and associated dynamicalfeedbacks–processes that are not represented by the currentmodel. Steady-state solutions to Nd and R are consistent withsolution to H; while Nd increases with increasing N0, it decreaseswith increasing H0, or increasing rainfall potential. Similarly,R is more strongly dependent on H0 than on N0 (as expectedfrom Eq. 4).

Weak R: damped oscillations to steady state.Fig. 6 shows an exampleof damped oscillations to steady state for one of the cases encom-passed by the parameter space in Fig. 5. The time series plotsshow how R develops in response to cloud thickening and peaksafter the maximum in H is reached. Nd is depleted by R anddepends on its source term to recharge. After a few cycles, thesolution approaches steady state. The limit cycles for (H; R)and (H; Nd) proceed in opposite sense. For the former, the cycleis anticlockwise whereas in the latter it is clockwise (the cycles ofH and Nd are approximately synchronous, although buildup ofNd precedes that of H). More generally, the sense of the rotationwill change depending on the phase differences between the(H; R) and (H; Nd) oscillations.

Moderate R: stable oscillations without steady state. Solutions tothe model equations tend to approach steady state more rapidly

Fig. 4. Steady-state solution to H (Eq. 12). Note the existence of tworegimes: The low Nd regime where H depends strongly on Nd , and thehigh Nd regime, where H is determined by H0.

A

B

C

Fig. 5. Time-dependent, steady-state solutions to (A) H, (B) R, and (C) Nd fora range of H0; N0. The gray shaded area represents strongly precipitatingparameter space where solutions are unstable and the boundary layer islikely to collapse. Input conditions: τ1 ¼ τ2 ¼ 60 min; delay time T 0 ¼ 10 min.

12230 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1101777108 Koren and Feingold

under weak precipitation (larger Nd) and tend to oscillate morevigorously in the presence of stronger precipitation (smaller Nd).In that sense Nd can be viewed as a damping parameter of thecloud-rain coupled oscillator. When Nd is large, the damping isstrong, and the system reaches steady state faster. When precipi-tation is sufficiently strong, the system may take some time toreach steady state (Fig. 7) and may even oscillate around a meanstate as in solution to the LV equations in Fig. 2. The probabilityof oscillation around a steady state increases as the delay timeT 0 increases, allowing the cloud to attain more significant depthbefore rain starts to develop.

Discussion and ConclusionsThere exist many examples of dynamical systems that, owingto their complexity, are not always tractable via the purelyreductionist approach. These systems do, however, benefit froma complementary “systems-based” approach (25) which seeks tocapture emergent behavior, as opposed to representing the de-tailed process interactions. The simple set of predator-prey-likeequations proposed here has been shown to mimic some aspectsof the emergent behavior of the aerosol–cloud–precipitationsystem revealed by detailed numerical simulation. The emer-gence takes the form of coupled oscillating cycles of cloud andrain, mediated by the aerosol. Thus the multitude of physical pro-cesses that interact in the cloud system reveal a pattern of rainpreying on cloud much like one species in the animate worldmight feed off another.

Coupled oscillators are commonly observed in chemical,biological, and physical systems (26–28) and also manifest them-selves in convective systems (12, 29). The existence of thisbehavior suggests that complex systems may be amenable torepresentation by a manageable number of parameters. In thecurrent case, the model captures qualitatively some modes ofbehavior of cloudy boundary layers using only five free para-meters: dynamical and aerosol replenishment parameters H0 andN0 [analogous to the “carrying capacity” parameters in modifiedLVequations; (16)] and their respective time constants τ1 and τ2;and delay time T 0. The first four parameters describe the externalforcings to the system while T 0 is determined by the internal

microphysical processes, i.e., the rate at which cloud water isconverted to rain water.

Solutions to the simple set of equations corroborate someearlier results and provide interesting insights. A steady-stateanalytical solution to the equation for H (Eq. 12) points to thepreviously described bifurcation of the system into two stablestates (22): one characterized by large Nd where H is determinedprimarily by dynamical forcing, and a second at lowNd whereH isdetermined by Nd. A similar pattern emerges from solutions tothe coupled equations, recorded when the system reaches steadystate, over a range of (H0; N0) (Fig. 5A). The fact that this bifur-cation, both observed in nature (open- vs. closed-cells in Fig. 1),and simulated by detailed LES, is captured by relatively simplemodels such as that used by ref. 22, or the even simpler preda-tor-prey model, is suggestive of emergence.

The time-dependent simulations of the equations (Figs. 6and 7) clearly reveal the predator-prey analogy to the aerosol–cloud–precipitation system. For a drizzling boundary layer themodel captures the coupled cloud-rain oscillations generatedby a much more complex LES (compare Fig. 3 with Fig. 6 or 7).Under conditions of weak rain, the system exhibits dampedoscillations to steady state (Fig. 6). The oscillations increasewith increasing R, and take longer to damp to steady state. Underrelatively strong drizzle and larger delay times, the dampingcomponent may disappear and the system reaches a state ofsteady oscillations in which the system traces out a region of(H; R) or (H; Nd) phase space (Fig. 7).

As in many other dynamic systems the model presented herehas a discrete number of preferred modes, as opposed to asmooth transition between states. For example, Figs. 5–7 showthat only a limited part of the phase space is occupied. Stableregions of parameter space, e.g., where precipitation is weak,tend to be robust. Simulations that randomly perturb H0 andN0 by �50% behave much like the unperturbed simulations inFigs. 6 and 7 (see SI Text, Fig. S1). In fact, the oscillating system(Fig. 7) experiences a stabilization in response to the perturba-tions (Fig. S2), provided the perturbations are not too strong,and do not persist for too long. This response is in accord with

Fig. 6. Damped oscillation to steady state. (A) time series (first 600 min);solid line: H; dashed line: R; red line: Nd ; (B) phase diagrams (for ∼7 d); solidline: (H;R); red line: (H; Nd ). Input conditions: H0 ¼ 530 m; N0 ¼ 180 cm−3,τ1 ¼ τ2 ¼ 60 min; T 0 ¼ 12 min.

Fig. 7. Oscillating, limit-cycle response of the H, R, Nd system. (A) time series(first 600 min); solid line: H; dashed line: R; red line: Nd ; (B) phase diagrams(for ∼7 d); solid line: (H;R); red line: (H; Nd ). Input conditions: H0 ¼ 670 m;N0 ¼ 515 cm−3, τ1 ¼ 80 min; τ2 ¼ 84 min; T 0 ¼ 21.5 min.

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the theory of self-organizing systems that are known to benefitfrom some degree of perturbation as the system is allowed toexplore a broader region of its attractor space (18). However,if perturbations are sufficiently strong, (e.g., when H0 is too largeand/or N0 too small), stable solutions cease to exist, indicatingthe potential for the system to migrate to a different state [e.g.,(12, 30, Fig. 1)]. It seems particularly pertinent to both theaerosol–cloud–precipitation system under discussion here, andmore generally for the climate system, to pose the following ques-tions: What are the preferred states, how robust are they, and dowe understand the mechanisms for transition between them?

The model presents an interesting variation on the well studiedtheme of predator-prey systems. First, cloud and rain water areboth the same “species” and differ only in terms of drop size andfall velocity. Unlike typical predator-prey systems, it is the prey(cloud) that spawns the predator (rain). Only once the first rain-drop embryos have been created can the traditional predator-prey behavior ensue, bringing to mind host-virus behavior wherethe virus cannot survive if it destroys the host.

Although rain production is primarily dependent on clouddepth (or LWP; Eq. 4), the aerosol is known to play a numberof interesting roles, some of which are highlighted here in thecontext of the predator-prey model. First, along with buoyancyand moisture, it is a basic “nutrient” source for the clouds. Aero-sol particles form the condensation nuclei for droplets, and theirabsence in sufficient concentrations does not support colloidallystable clouds. Second, the aerosol can be regarded as an immu-nizing agent in the consumption of cloud by rain: by suppressingrain production, it protects the cloud from the ravages of thepredator (rain), and as it becomes progressively more scarceas a result of wet-removal, acts to weaken the cloud (prey). Thischange in roles could mark the transition from a robust cloud-rain system to an unstable, runaway system where in the extreme,ultraclean state, clouds can no longer exist (Figs. 4 and 5, grayshaded area). An exception might exist when self-organization

helps maintain the cloud (e.g., 12, 30), but the current modelis not designed to simulate this. Third, because rain processesare less efficient at larger N0, aerosol perturbations provide thecloud a longer, undisturbed period to develop, and allow it toapproach its maximum potential H0. Depending on the magni-tude of T 0, this sequence of responses might even result in stron-ger rain under polluted conditions (5). The strong dependenceof R on H0 relative to N0 in Fig. 5B is a manifestation of thisresponse.

The solutions to the model clearly cannot represent the fullcomplexity of the detailed LES. In principle, the model couldperhaps, be designed to do so if the five parameters were time-dependent. This approach would be a nontrivial, perhaps futileexercise, requiring, amongst other things, connection betweenthe dynamical forcing (including a diurnal cycle of radiative for-cing) to H0, and aerosol sources to N0 with appropriate forcingtime scales. Moreover, the time-scale parameters are in realitycoupled to other variables. For example, the delay in rain forma-tion may be a function of Nd. Such attempts would run counterto the spirit of exploring underlying simplicity in the system andgaining insight into preferred modes, rather than attempting toreproduce the complexity of LES.

In conclusion, while future study of the aerosol–cloud–preci-pitation system must continue to pursue a process-oriented, re-ductionist approach that addresses detailed interactions betweenthe components (as in LES) the current work suggests it wouldbenefit from a parallel but integrated system-oriented approachthat yields the correct emergent behavior (25). Adoption ofsuch ideas might be particularly useful for representing aerosol–cloud–precipitation, and other subsystems, in climate models.

ACKNOWLEDGMENTS. This work was supported by NOAA’s Climate Goal andin part by the Israel Science Foundation (Grant # 1172/10) and the MinervaFoundation (780048). The satellite image (Fig. 1) is courtesy of the NASA/MODIS team.

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