Aerosol–cloud–precipitation system as a predator-prey problem

Abstract

We show that the aerosol–cloud–precipitation system exhibits characteristics of the predator-prey problem in the field of population dynamics. Both a detailed large eddy simulation of the dynamics and microphysics of a precipitating shallow boundary layer cloud system and a simpler model built upon basic physical principles, reproduce predator-prey behavior with rain acting as the predator and cloud as the prey. The aerosol is shown to modulate the predator-prey response. Steady-state solution to the proposed model shows the known existence of bistability in cloudiness. Three regimes are identified in the time-dependent solutions: (i) the weakly precipitating regime where cloud and rain coexist in a quasi steady state; (ii) the moderately drizzling regime where limit-cycle behavior in the cloud and rain fields is produced; and (iii) the heavily precipitating clouds where collapse of the boundary layer is predicted. The manifestation of predator-prey behavior in the aerosol–cloud–precipitation system is a further example of the self-organizing properties of the system and suggests that exploiting principles of population dynamics may help reduce complex aerosol–cloud–rain interactions to a more tractable problem.

Shallow clouds in the Earth system are currently the subject of a great deal of attention because of their importance for climate predictability (1). These shallow clouds radiate in the longwave at approximately the same temperature as the surface but reflect shortwave radiation back to space and therefore cool the climate system. Moreover, the manner in which shallow clouds are represented in climate models has a significant effect on climate sensitivity. The effect of aerosol particles on these clouds has also been identified as an important unknown (2). Aerosol particles are the nuclei upon which droplets are formed so that higher aerosol concentrations result in higher droplet concentrations, and all else equal, smaller drops, more reflective clouds, and stronger cooling (3). Smaller droplets are also less apt to collide and coalesce; this will tend to inhibit precipitation, possibly increasing cloud lifetime (4) and exerting an even stronger cooling effect. The feedbacks associated with aerosol effects on clouds and precipitation are myriad and complex [e.g., (5)], poorly quantified, and even more poorly represented in climate models.

The past decades have seen significant efforts to study shallow clouds and their interactions with aerosol, and much has been learned through intensive field campaigns (6, 7), remote sensing (8, 9), and modeling (10, 11). Fine-scale models that solve the Navier–Stokes equations on grids on the order of tens of meters and represent the coupled aerosol–cloud–precipitation system are desirable because they address the processes at the appropriate spatial and temporal scales. Although these models exhibit complex responses to aerosol perturbations, it is becoming increasingly apparent that the system prefers distinct “modes.” This preference raises the hope that the system is more predictable than suggested by the very large number (order 10⁷) of degrees of freedom that these models represent. Two concepts are of particular note: (i) the concept of “buffering” (5) whereby internal processes buffer the system against strong perturbations, effectively reducing the number of degrees of freedom in the system; and (ii) self-organizing patterns in the form of mesoscale cellular convection (12) that constrain the system to specific convective modes (13). Process-level complexity therefore appears to yield, in some cases, to a modicum of systemic predictability.

One example of these concepts is recent work which has shown that open-cellular convection in precipitating boundary layers organizes in such a way as to generate coherent precipitation with a characteristic periodicity related to the size of the cells. In the open-cellular state, clouds appear as ring-like structures that are connected to adjacent cloudy rings, forming a lace-like pattern over distances of order 1,000 km (Fig. 1). Within the reflective cloud walls, the buoyancy is positive, and clouds thicken until rain is formed. The rain consumes part of the cloud water and once it falls below cloud base and evaporates, generates negatively buoyant downdrafts that destroy the positively buoyant updrafts that were responsible for its formation. The interaction between adjacent precipitation-driven outflows generates new positively buoyant regions that are offset spatially from the earlier ones. This manifestation of Le Châtelier’s principle, whereby rain reverses the dynamics that created it, appears as organized cells that form, dissipate, and reappear in an orchestrated manner (12). It is the coupling of cloudy/rainy cells in this self-organized cloud field that is the source of the mesoscale periodicity in rainrate. The existence of oscillations in open-cellular convection motivates us to explore whether oscillations exist in cloud-rain systems more generally.

Fig. 1.

Satellite image showing a mesoscale cellular convective system in the Atlantic Ocean. The image, taken by the NASA MODIS imager on board the Terra satellite on July 21, 2004, shows bright, closed cellular regions interspersed with open cell regions where a lace-like pattern of bright cloudy filaments exist against the backdrop of the dark ocean. The scale is approximately 1,300 km (horizontal) × 1,100 km (vertical).

Approximately a century ago two researchers independently proposed a simple mathematical model to explain oscillatory behavior 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 are given as

[1]

where C is the population of prey, R is the population of predators, and a, b, e, f are system-dependent constants. The prey (C) population grows exponentially in the absence of R, but is reduced by R preying on C. The predator (R) population depends for sustenance entirely on prey C and decreases exponentially in the absence of C. To illustrate the system, numerical solution to these equations is given in Fig. 2. The C population (e.g., rabbits) begins to grow, but upon representing a significant food source to R (e.g., foxes), C’s population diminishes. The fox population grows until it has overdepleted C and then decays. In the absence of predation, C begins to recover, and so the cycle continues. A limit-cycle (or phase diagram) view of the solution is shown in Fig. 2B and is a useful means of tracking the trajectory of the two populations. Although the LV equations are highly simplified and somewhat unrealistic (e.g., the exponential growth in prey in the absence of predators), they prove instructive, and with modification are very useful (16).

Fig. 2.

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

Consider now populations more pertinent to the aerosol–cloud–precipitation system. The system exhibits numerous examples 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 cloud droplets form so that aerosol and cloud must coexist. Superclean aerosol environments do not support the existence of stable cloud;

• Droplets consume water vapor and cannot thrive unless dynamical forcing maintains a (super-)saturated vapor concentration;

• Cloud droplets coalesce to form rain drops which consume cloud drops via collection. The formation of rain can spell the demise of the cloud, or under some conditions, clouds and rain may coexist. Because cloud drops form on aerosol particles, rain that reaches the surface also indirectly consumes aerosol;

• In mixed-phase clouds, water and ice can coexist or, under some conditions ice grows at the expense of liquid water via the Bergeron–Findeisin process, resulting in complete glaciation and demise of the cloud.

We illustrate predator-prey behavior for a simple system comprising two primary populations: cloud and rain. A third population, aerosol, mediates the interaction between these “species.” Because the aerosol directly affects cloud drop concentration N_d, we use N_d as a proxy for the aerosol. Consider model output from large eddy simulation (LES) of a warm, marine stratocumulus cloud (11) which solves the Navier–Stokes equations coupled to detailed aerosol and cloud microphysics interactions and represents a high level of complexity at significant computational expense. Fig. 3 shows LES time series of liquid water path (LWP) (column-integrated liquid water content, g m^-2) and rainrate R (mm d^-1), which exhibit distinct predator-prey-like oscillations. The formation of rainfall sets in motion a negative feedback; rain depletes LWP and cloud depth H and slows down further rain formation. Cloud has to thicken again before rain can reform. The magnitude of the oscillations varies with time as a result of LWP changing in response to various internal and external forces. Thus the phase-space trace consists of a series of displaced anticlockwise loops as opposed to the recurring, superimposed loops in Fig. 2B. Nevertheless, in spite of the multiple process interactions and feedbacks occurring in the system, the system reveals a more simple emergent behavior (18) *.

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 left in an anticlockwise sense.

Predictive Model

We build a predictive cloud model that addresses the cloud-rain problem using source and sink terms that are based on physical principles, distillation of knowledge acquired from a number of intensive airborne field campaigns, and empiricism from detailed models. The model is not designed to capture the details of the system, but rather its emergent behavior. Instead of solving balance equations for both LWP (prey) and R (predator) as in Eq. 1, we capture the essential physics with one equation for cloud depth H plus a theoretically- (19) and empirically-based (7) diagnostic equation for R, with an appropriate delay function. H is intimately related to LWP (Eq. 2), and as will be shown in the Results, has the benefit of producing a particularly simple analytical steady-state solution. In addition, a second balance equation is solved for N_d to account for its sometimes pivotal role in controlling R (12), and because fixing N_d would be overly restrictive. The proposed set of equations is just one of many alternatives, and has been chosen because it captures the basic physics in a compact 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

[2]

where q is the cloud water mixing ratio and c₁ is a function of cloud-base temperature and pressure; c₁ ≃ 2·10^-6 mm m^-2 for a warm adiabatic cloud. The balance equation for H is written as

[3]

The first term on the left hand side represents an exponential approach of H to H₀, with a characteristic time constant τ₁, as a result of “dynamical forcing,” which we assume to include a variety of forcings including latent heating associated with phase change, radiative flux divergence, entrainment-mixing, atmospheric instability, mesoscale forcing, etc. H₀ represents the full environmental potential for cloud development so that in the absence of water sinks, H will approach the asymptotic limit of H₀ after several τ₁. represents loss of H as a result of rain, a stochastic process that converts small cloud droplets to raindrops (a 10 order of magnitude increase in mass) in a relatively short period of time (T ≃ 15 min). The delay term (t - T) accounts for the time-dependence of rain production, which is a function of the state of the cloud some period of time prior to the current time step. Delay terms are common in population dynamics; e.g., birth rates must account for maturation to adulthood.

To first order, rainrate R can be diagnosed from cloud depth H and N_d

[4]

based on theory (19) and observations (6, 7), where α ≃ 2 mm m^-6 d^-1 for warm stratocumulus. Thus cloud depth (or more pertinently LWP) has a much stronger control over precipitation than does N_d.

To represent the loss term for H we note that

[5]

is then written as

[6]

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

[7]

with N_d representing the cloud-mean value.

Analogous to Eq. 3 we write the equation for N_d, including a delayed loss term, as:

[8]

N₀ can be considered to be the background concentration of aerosol to which the system strives, and the last term represents coalescence processes. τ₂ is the characteristic time constant for replenishment of the aerosol and is of order τ₁ or larger. The cloud-mean loss term for N_d results from the conversion of cloud water to rain water via drop collection and is calculated using a simple expression based on detailed solution to the stochastic collection equation (20):

[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 N_d are continuous functions and assume that there is an equivalent delay time T^′ that represents the mean value in the interval, and therefore replaces the integration over the interval. in Eq. 7 is therefore expressed as αH²(t - T^′)/c₁N_d(t - T^′) and N_d in Eq. 9 is calculated at t - T^′. Finally R is diagnosed based on Eq. 4 as

[10]

As will be shown below, the combination of Eq. 3 and the diagnostic equation with delay Eq. 10 generates oscillating solutions that are equivalent to those produced by two coupled equations (e.g., Eq. 1). The addition of Eq. 8 allows for a more realistic evolution of the system because R is intimately related to N_d. The primary model (Eqs. 3, 8, and 10) are discretized and solved using a finite-difference scheme.

Results

Dynamic equations with variables characterized by mutual modulation, and therefore the potential for first derivatives that are opposite in sign (21), produce solutions that have four basic modes: (i) steady state, (ii) oscillations, (iii) chaotic behavior, and (iv) unstable solution. The steady-state regime, in which sources and sinks of H are in balance, is amenable to analytical solution in our case and is explored below. Following that, some of the other modes of behavior are examined through time-dependent, numerical solution.

Analytical Steady-State Solutions.

We start with an analytical solution to the equations that yields important insights. Setting dH/dt = 0 in Eq. 3 gives the steady-state solution to H in the form of a quadratic:

[11]

where γ = α/c₁. Because H≥0, the only physical solution is

[12]

Fig. 4 shows contours of steady-state H in (N_d; H₀) space for τ₁ = 60 min. Two regimes can be seen: (i) at N_d ≳ 30 cm^-3, H is almost entirely determined by the external dynamical forcing H₀, except at higher H₀ where N_d also plays a role; (ii) at low N_d( ≲ 30 cm^-3), H decreases rapidly with decreasing N_d, with little influence by H₀. As simple as this equation is, it already hints at the potential for bifurcation of the aerosol–cloud–precipitation system between a nondrizzling solid cloud at high N_d whose depth is determined by H₀, and a drizzling cloud with depth strongly dependent on N_d (22). This result is not dependent on the value of τ₁; results for much larger τ₁ (= 720 min) are qualitatively similar. The original idea of bistability emerged from a mixed-layer model that included more complexity in the dynamics and microphysics than the current model. Bistablity was also confirmed by much more complex LES (e.g., 12). It is therefore noteworthy that solution to Eq. 12 also points to this behavior.

Fig. 4.

Steady-state solution to H (Eq. 12). Note the existence of two regimes: The low N_d regime where H depends strongly on N_d, and the high N_d regime, where H is determined by H₀.

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

Time-Dependent Solutions.

For deeper analysis of the attractors of the system, the time-dependent model is run for a range of H₀ and N₀ until it achieves steady state. The simulations extend out to ∼7 d but steady state is usually achieved within hours. No attempt has been made to include a diurnal cycle of radiative forcing in any of these, or subsequent simulations. Solutions to H, N_d, and R are displayed in Fig. 5. Under weakly precipitating conditions (low H₀ and high N₀), H is determined by H₀, whereas for more strongly precipitating conditions (higher H₀ and lower N₀), H increases with both increasing H₀ and N₀. At small N₀ (lower left), H is more strongly controlled by N₀, in general agreement with the steady-state solution in Fig. 4. The figures differ quantitatively because Fig. 4 is a solution to Eq. 12 alone whereas Fig. 5 reflects the coupled solution to Eqs. 3 and 8 with 10. The gray shaded area at upper left represents a region of parameter space where stable solutions do not exist; strong precipitation results in rapid demise of the cloud, which implies collapse of the boundary layer. Note that nowhere does H decrease in response to increasing N₀, as suggested by some LES studies [e.g., (24)], however the latter response is a result of more efficient evaporation of cloud droplets and associated dynamical feedbacks–processes that are not represented by the current model. Steady-state solutions to N_d and R are consistent with solution to H; while N_d increases with increasing N₀, it decreases with increasing H₀, or increasing rainfall potential. Similarly, R is more strongly dependent on H₀ than on N₀ (as expected from Eq. 4).

Fig. 5.

Time-dependent, steady-state solutions to (A) H, (B) R, and (C) N_d for a range of H₀; N₀. The gray shaded area represents strongly precipitating parameter space where solutions are unstable and the boundary layer is likely to collapse. Input conditions: τ₁ = τ₂ = 60 min; delay time T^′ = 10 min.

Weak R: damped oscillations to steady state.

Fig. 6 shows an example of damped oscillations to steady state for one of the cases encompassed by the parameter space in Fig. 5. The time series plots show how R develops in response to cloud thickening and peaks after the maximum in H is reached. N_d is depleted by R and depends on its source term to recharge. After a few cycles, the solution approaches steady state. The limit cycles for (H; R) and (H; N_d) proceed in opposite sense. For the former, the cycle is anticlockwise whereas in the latter it is clockwise (the cycles of H and N_d are approximately synchronous, although buildup of N_d precedes that of H). More generally, the sense of the rotation will change depending on the phase differences between the (H; R) and (H; N_d) oscillations.

Fig. 6.

Damped oscillation to steady state. (A) time series (first 600 min); solid line: H; dashed line: R; red line: N_d; (B) phase diagrams (for ∼7 d); solid line: (H;R); red line: (H; N_d). Input conditions: H₀ = 530 m; N₀ = 180 cm^-3, τ₁ = τ₂ = 60 min; T^′ = 12 min.

Moderate R: stable oscillations without steady state.

Solutions to the model equations tend to approach steady state more rapidly under weak precipitation (larger N_d) and tend to oscillate more vigorously in the presence of stronger precipitation (smaller N_d). In that sense N_d can be viewed as a damping parameter of the cloud-rain coupled oscillator. When N_d is large, the damping is strong, and the system reaches steady state faster. When precipitation is sufficiently strong, the system may take some time to reach steady state (Fig. 7) and may even oscillate around a mean state as in solution to the LV equations in Fig. 2. The probability of oscillation around a steady state increases as the delay time T^′ increases, allowing the cloud to attain more significant depth before rain starts to develop.

Fig. 7.

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

Discussion and Conclusions

There exist many examples of dynamical systems that, owing to their complexity, are not always tractable via the purely reductionist approach. These systems do, however, benefit from a complementary “systems-based” approach (25) which seeks to capture emergent behavior, as opposed to representing the detailed process interactions. The simple set of predator-prey-like equations proposed here has been shown to mimic some aspects of the emergent behavior of the aerosol–cloud–precipitation system revealed by detailed numerical simulation. The emergence takes the form of coupled oscillating cycles of cloud and rain, mediated by the aerosol. Thus the multitude of physical processes that interact in the cloud system reveal a pattern of rain preying on cloud much like one species in the animate world might feed off another.

Coupled oscillators are commonly observed in chemical, biological, and physical systems (26–28) and also manifest themselves in convective systems (12, 29). The existence of this behavior suggests that complex systems may be amenable to representation by a manageable number of parameters. In the current case, the model captures qualitatively some modes of behavior of cloudy boundary layers using only five free parameters: dynamical and aerosol replenishment parameters H₀ and N₀ [analogous to the “carrying capacity” parameters in modified LV equations; (16)] and their respective time constants τ₁ and τ₂; and delay time T^′. The first four parameters describe the external forcings to the system while T^′ is determined by the internal microphysical processes, i.e., the rate at which cloud water is converted to rain water.

Solutions to the simple set of equations corroborate some earlier results and provide interesting insights. A steady-state analytical solution to the equation for H (Eq. 12) points to the previously described bifurcation of the system into two stable states (22): one characterized by large N_d where H is determined primarily by dynamical forcing, and a second at low N_d where H is determined by N_d. A similar pattern emerges from solutions to the coupled equations, recorded when the system reaches steady state, over a range of (H₀; N₀) (Fig. 5A). The fact that this bifurcation, both observed in nature (open- vs. closed-cells in Fig. 1), and simulated by detailed LES, is captured by relatively simple models such as that used by ref. 22, or the even simpler predator-prey model, is suggestive of emergence.

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

As in many other dynamic systems the model presented here has a discrete number of preferred modes, as opposed to a smooth transition between states. For example, Figs. 5–7 show that only a limited part of the phase space is occupied. Stable regions of parameter space, e.g., where precipitation is weak, tend to be robust. Simulations that randomly perturb H₀ and N₀ by ± 50% behave much like the unperturbed simulations in Figs. 6 and 7 (see SI Text, Fig. S1). In fact, the oscillating system (Fig. 7) experiences a stabilization in response to the perturbations (Fig. S2), provided the perturbations are not too strong, and do not persist for too long. This response is in accord with the theory of self-organizing systems that are known to benefit from some degree of perturbation as the system is allowed to explore a broader region of its attractor space (18). However, if perturbations are sufficiently strong, (e.g., when H₀ is too large and/or N₀ too small), stable solutions cease to exist, indicating the potential for the system to migrate to a different state [e.g., (12, 30, Fig. 1)]. It seems particularly pertinent to both the aerosol–cloud–precipitation system under discussion here, and more generally for the climate system, to pose the following questions: What are the preferred states, how robust are they, and do we understand the mechanisms for transition between them?

The model presents an interesting variation on the well studied theme of predator-prey systems. First, cloud and rain water are both the same “species” and differ only in terms of drop size and fall 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 where the virus cannot survive if it destroys the host.

Although rain production is primarily dependent on cloud depth (or LWP; Eq. 4), the aerosol is known to play a number of interesting roles, some of which are highlighted here in the context of the predator-prey model. First, along with buoyancy and moisture, it is a basic “nutrient” source for the clouds. Aerosol particles form the condensation nuclei for droplets, and their absence in sufficient concentrations does not support colloidally stable clouds. Second, the aerosol can be regarded as an immunizing agent in the consumption of cloud by rain: by suppressing rain production, it protects the cloud from the ravages of the predator (rain), and as it becomes progressively more scarce as a result of wet-removal, acts to weaken the cloud (prey). This change 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, gray shaded area). An exception might exist when self-organization helps maintain the cloud (e.g., 12, 30), but the current model is not designed to simulate this. Third, because rain processes are less efficient at larger N₀, aerosol perturbations provide the cloud a longer, undisturbed period to develop, and allow it to approach its maximum potential H₀. Depending on the magnitude of T^′, this sequence of responses might even result in stronger rain under polluted conditions (5). The strong dependence of R on H₀ relative to N₀ in Fig. 5B is a manifestation of this response.

The solutions to the model clearly cannot represent the full complexity of the detailed LES. In principle, the model could perhaps, be designed to do so if the five parameters were time-dependent. This approach would be a nontrivial, perhaps futile exercise, requiring, amongst other things, connection between the dynamical forcing (including a diurnal cycle of radiative forcing) to H₀, and aerosol sources to N₀ with appropriate forcing time scales. Moreover, the time-scale parameters are in reality coupled to other variables. For example, the delay in rain formation may be a function of N_d. Such attempts would run counter to the spirit of exploring underlying simplicity in the system and gaining insight into preferred modes, rather than attempting to reproduce the complexity of LES.

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