Convective Self-Aggregation as a Cold-Pool Driven Critical Phenomenon
CConvective self-aggregation as a cold pool driven criticalphenomenon
Jan O. Haerter Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100Copenhagen, DenmarkFebruary 11, 2020
Author affiliation : Blegdamsvej 17, 2100 Copenhagen, Denmark.
Corresponding author : Jan O. Haerter; [email protected]
Convective self-aggregation is when thunderstorm clouds cluster over a constant temperaturesurface in radiative convective equilibrium. Self-aggregation was implicated in the Madden-Julian Oscillation and hurricanes. Yet, numerical simulations succeed or fail at producingself-aggregation, depending on modeling choices. Common explanations for self-aggregationinvoke radiative effects, acting to concentrate moisture in a sub-domain. Interaction betweencold pools, caused by rain evaporation, drives reorganization of boundary layer moisture andtriggers new updrafts. We propose a simple model for aggregation by cold pool interac-tion, assuming a local number density ρ (r) of precipitation cells, and that interaction scalesquadratically with ρ (r). Our model mimics global energy constraints by limiting further cellproduction when many cells are present. The phase diagram shows a continuous phase tran-sition between a continuum and an aggregated state. Strong cold pool-cold pool interactiongives a uniform convective phase, while weak interaction yields few and independent cells.Segregation results for intermediate interaction strength. Introduction
Drawing a link between cloud and precipitation processes, and statistical mechanics is intriguing: similar tothe characteristics of critical phenomena, cloud fields often show long-ranged correlations and near-fractalscaling.
2, 3
The possibility that the cloud field might have scaling properties is appealing because this mightimply universal behavior, where only few features of the small scale interactions are relevant in capturingemergent organization. Invoking non-equilibrium statistical physics, a relation to the sand pile model hasbeen drawn, which coins the notion of self organized criticality (SOC). Loosely speaking, precipitation couldbe ascribed characteristics of an avalanche, when moisture, brought into a domain of interest, is abruptlyreleased after a critical mixing ratio is exceeded.
6, 7
A particularly appealing candidate for critical behavior is the radiative convective equilibrium frame-work. Radiative convective equilibrium requires overall constant and homogeneous boundary conditions,e.g. constant surface temperature and moisture as well as insolation, and that the energy fluxes entering andleaving the system be equal. Under such conditions, humidity perturbations have repeatedly been shownto grow over time, in a process termed convective self-aggregation.
Explanations have in common thata feedback must exist, by which already moist regions grow moister, while dry regions become even drier.One possible mechanism is that cloudy regions lose less heat through long-wave radiation than do cloud-freeregions. In this process, subsidence would build up over the cloud-free regions, leading to low-level diver-gence there and suppression of cloud and precipitation. In some observational work, however, no strongsensitivity of the radiative budget at the top of the atmosphere to self-aggregation has been found. a r X i v : . [ n li n . AO ] F e b ere, we use equilibrium statistical physics only as an analogy, but follow a dynamical systems approach.Taking large-scale organization as an emergent aspect from the small scales, we test whether interactions be-tween convective updrafts can give rise to system-scale organization. Cold pools, resulting from evaporativecooling under precipitating convective clouds, spread as gravity currents away from the center of the precipi-tation cell. At the scale of ∼ km , cold pool gust fronts have been shown to stimulate the initiation of newconvective updrafts. Cold pools can reach maximal radii of r max ∼ km over sea surfaces, andtravel at an initial radial velocity of ∼ m s − , which is however gradually reduced in the course of theirlifetimes.
23, 25, 31, 32
It is increasingly appreciated that correlations exist between the locations of presentprecipitation cells and those of subsequent ones, likely mediated through cold pool activity
26, 27, 30, 33, 34 andthat precipitation can widen the distribution function of boundary layer buoyancy. In recent work,
30, 34 itwas shown that the collision between cold pools is the dominant process of generating new convective events— activation by isolated gust fronts without collision may be of second order. Interestingly, strong cold poolactivity was stated to hamper the buildup of a self-aggregated state in simulations, while cold pools areindeed ubiquitous in regions of precipitating convection, in particular in cases of deep convection.We therefore ask, how much a convective cloud system can self-organize into an aggregated state, ifcold pool interactions are the relevant process leading to new convective cells? In recent conceptual andsimulation work, the detailed interactions between cold pools were considered. It was shown, by theanalysis of large-eddy simulations, that it is most often collisions between several colliding cold pool gustfronts, rather than individual gust fronts, that set off new convective updrafts. In describing the positions ofnew cells by a simplified geometrical model, positions were taken to be precisely defined, e.g. when growingcold pool gust fronts collided, the new cell was produced at the first point of intersection between the gustfronts.Here, we first relax the requirement that new cells must occur at precisely defined points. In other words,we take new cells to emerge near gust front collisions of existing cold pools, but allow the new cells to bedisplaced by a random distance of the order of the typical cold pool radius. Such displacement has indeedbeen mentioned in early work, where new cells emerge somewhat randomly, but generally along the lineof collision between two previous cold pool gust fronts.Such spatial randomness allows us to start from the assumption that cold pool interactions can be mod-eled by only the number density, rather than explicit positions, of cell centers in a given local environment.The continuum model we derive enables us to map out a phase diagram of aggregation. In a second step, weagain tighten the assumptions, and return to microscopic correlations, where new cells do have a precisely-defined position relative to the ones causing them. We contrast this refined model to the continuum model.Our results support the key conclusion that, in both types of models, collisions between cold pools canlead to convective self-aggregation while single cold pool triggering alone cannot. Radiative effects are onlyrequired in a domain-average sense, in order to constrain the total system energy flux. Results
Model
As our conceptual model aims to incorporate a global energy constraint, the heat entering the atmospherethrough the lower boundary (e.g., a tropical sea surface) should leave at the top. Our model considers thisenergy flux by assuming that heat is transported to the cloud layer through latent heating during cloudformation (Fig. 1a). It further assumes that heat within the cloud layer will equilibrate relatively fastwith the surroundings through comparably rapid gravity waves. All heat transported to the cloud layeris eventually radiated out to space by long-wave radiative emission. We simply mimic the global energyconstraint by assigning an equal energy flux ε to each convective cloud and refer to the total system energyflux as E .The model domain is a square of linear dimensions L x = L y = L and an effective elementary area a isrequired for each precipitation cell. a necessarily includes the precipitation cell itself, but also further effectsthat inhibit other precipitation cells from populating this area at the same time. This could be downburstsor strong temperature depressions caused near the precipitation cell. We additionally define a (generallylarger) area unit, a cp = π r max as the area that a cold pool gust front can cover, if its maximum radius is r max . As mentioned, r max ∼
10 —100 km , which is larger than the updraft shaft and downburst area, butsubstantially smaller than the domain area, that is a < a cp (cid:28) L . (1) loud level condensationheating rapid lateral redistribution of heatradiative emission tospace new updrafts triggered by cold pool interaction u n i f o r m S S T ε εε ε ε εL x ~ 1000 km L y ~ k m r ~ 100 kmρ( r )ρ( r ) Time ba Explicit two-cold pool interactionExplicit single-cold pool replicationStandard model (density based)existing cellnew cellρ = 3/9 r r+e x p s cde Figure 1:
Simple model for organization through cold pools. a , Schematic of cloud system,with vertical columns showing updrafts of energy flux (cid:15) each, cold pools (green ellipses at the surface),new cells created by cold pools (small columns), as well as the cloud level. b , Details of local coldpool interaction (lateral scale ∼ km ): schematic shows precipitation cell centers (red points) andtheir corresponding cold pools (green), where at a given timestep cold pools spread and form newcells near points of interference. ρ ( r ) is the number density of cells within the sub-domain shown.The process continues at the subsequent timestep, with cold pools again emanating from the newcell centers. c — e , Distinguishing processes: c , Purely density based formulation: the probability ofthe central lattice site becoming occupied is proportional to the density ρ ( r ) of occupied sites in itssurroundings. In the example shown, r max is chosen so that all eight neighboring sites contributeto the density, that is, ρ ( r ) = 1 /
3, since three neighboring sites are occupied (blue). d , Explicittwo-cold pool interaction: a new cell can be set off only if the lattice site r lies between any of theneighboring lattice sites. In the example shown, a new cell is possible, since blue cells lie both to theleft and right of the site r . The circles indicate possible cold pool gust fronts. e , Explicit single-coldpool replication: a new cell is possible within the neighborhood of an existing cell.3 he domain total number of precipitation cells is expressed as N . Specifically, listing all horizontalpositions (2D) of precipitation cells as c i we can write N ≡ (cid:90) r d r (cid:88) i δ ( r − c i ) , (2)where δ is the Dirac delta function and the integral is taken over the entire domain.The domain mean number density ρ ≡ N/N max with N max ≡ L /a the maximal number of cells. ρ ishence the probability of finding a given elementary area a occupied by a precipitation cell. The total energyflux then is E = N ε , but we will generally simply work in units of total number density ρ , as all our raincells have equal energy flux. Analogously to Eq. 2, at any given position r we also define a local numberdensity ρ ( r ) ≡ a a − cp (cid:90) r max r =0 (cid:90) πφ =0 drdφ (cid:88) i δ ( r − c i ) , (3)which specifies the number density of cold pools which may affect the point r (Fig. 1b,c), since the integralin Eq. 3 regarding r is taken within the range 0 < r < r max , that is, the maximal radius within which onecold pool can collide with another. Only to avoid possible confusion, we emphasize that the symbols ρ and ρ are here used as number densities, whereas, in other contexts, the symbol ρ sometimes denotes the densityof air within and surrounding cold pools.To describe the dynamics of ρ ( r ) we consider three dynamical processes: (i) spontaneous cell production;(ii) cell decay; (iii) cell interactions. • spontaneous cell production occurs for sufficient atmospheric instability and space for cells to emerge.To incorporate the former, the rate of spontaneous production is tied to the total energy flux bymaking the spontaneous production ∼ f sg (1 − ρ ). Lower rates of spontaneous production hence occurwhen the domain mean flux is already large. f sg ≥
23, 36 a limitationon the space available to new cells should be incorporated in the model. This is accomplished bythe additional factor 1 − ρ ( r ), which ensures that each vacant area a has the same probability ofexperiencing a spontaneous event. In total, spontaneous growth evolves as f sg (1 − ρ )(1 − ρ ( r )). • cell decay occurs for each cell present at an equal rate f d ≡ τ − d , where τ d >
23, 36 τ d is on the order of 1 hour ). Local cell density ρ ( r ) henceevolves as − f d ρ ( r ), leading to exponentially decaying cell populations if all other processes wereabsent. • cell interaction is due to cold pool processes. For all vacant areas in the vicinity of r , i.e. ∼ (1 − ρ ( r )),the cold pools emanating from all cells present within a radius r max around r can help instigate a newcell at r . We therefore introduce an effective probability p ≥ r . The cell production from cold pool interaction is then modeled by P ( ρ ) ρ ( r ) m (1 − ρ ( r )),where the probability P ( ρ ) = p (1 − ρ ) again warrants the total energy constraint. Large values of p mean, that cold pools are more efficient at generating new cells. It is important to understandthe meaning of the exponent m , which serves to distinguish cold pool processes resulting from oneor multiple cold pools: m = 1 describes a process where replication is proportional to the density ρ ( r ), that is, each cold pool replicates at a rate that is independent of the presence of others. m = 2describes processes that involve collisions, that is, the replication of one cold pool depends on thepresence of others in the surroundings. The general ability of cold pool collisions to trigger newconvective cells,
18, 23, 37, 38 and the explicit comparison between m = 1 and m = 2 suggest a crucialrole of collision effects between distinct cold pool gust fronts. Note that our key assumption, which will be relaxed later, is that cold pool interaction is a purely densitydependent process. Geometrical constraints and morphologies of cell organization are hence consideredhigher-order corrections. Such an assumption of a density field would be justified, when collisions betweencold pools are sufficiently noisy, meaning that new cells are never produced precisely at the position ofcollision. Hence, in our model the number of cold pools contributing to the production of a new cell is elevant, but not the detailed position of the cell produced — justifying a coarse-grained density field ρ ( r )near the point of collision.Piecing together the three processes above, the complete dynamical equation describing the evolution ofcell density ρ ( r , t ) is ddt ρ ( r , t ) = p (1 − ρ ) ρ ( r ) m (1 − ρ ( r )) + f sg (1 − ρ )(1 − ρ ( r )) − f d ρ ( r ) (4)= (1 − ρ )(1 − ρ ( r ))( p ρ ( r ) m + f sg ) − f d ρ ( r ) . (5)For subsequent use, we express all rates in units of f d . This is accomplished by dividing Eq. 5 through bythe precipitation frequency f d (inverse duration of a precipitation event). This amounts to the replacements p → p /f d , f sg → f sg /f d , f d →
1, leaving p and f sg as the remaining parameters. Time is now measuredin units of precipitation duration, and space in units of a (defined above). Furthermore, we define q ≡ − ρ and, from now on, drop the explicit reference to the argument r of ρ , taking the symbol ρ to always refer tolocal cell density, as opposed to ρ , which measures the system average density. We can then more compactlydefine the RHS of Eq. 5 as F q,p ,f sg ( ρ ) ≡ q (1 − ρ )( p ρ m + f sg ) − ρ . (6)To make the analogy to a critical phenomenon more explicit, we note that, as a polynomial in ρ , Eq. 6 canbe interpreted as the derivative of a potential, denoted V q,p ,f sg ( ρ ), where F q,p ,f sg ( ρ ) = − d V q,p ,f sg ( ρ ) /d ρ .For m = 2, V q,p ,f sg ( ρ ) becomes of fourth order in ρ and can exhibit two competing local mimima.A few comments are appropriate:(i) the growth limitation by the factor q = 1 − ρ is analogous to that for logistic growth in populationdynamics, where the term is interpreted as a total resource limitation; the term (1 − ρ ) has a similar effect,but encodes local space limitations. The factor (1 − ρ ) takes into account that two cold pools cannot bein the same location, which is reminiscent of the repelling force in a Van der Waals Gas of particles with afinite radius. Conversely, for m = 2 the generation of new cold pools by ρ m in Eq. 6 is promoted by havingtwo cold pools in each other’s vicinity, giving a competing effect between space limitation and growth.(ii) the factor p ρ m + f sg is crucial in describing the local dynamics of cell growth. For sufficiently lowdensity ρ , spontaneous cell production is dominant, while for large ρ , most new cells are produced by coldpool interactions;(iii) for bistable steady state solutions to Eq. 5, the exponent m should be larger than unity. To see this,consider that, for m = 1, F q,p ,f sg ( ρ ) = q (1 − ρ )( p ρ + f sg ) − ρ is quadratic in ρ and therefore can only havea single stable fixed point. More explicitly, F q,p ,f sg (1) = − F q,p ,f sg (0) = qf sg ≥
0. Hence, dependingon the choices of p and f sg , there is (a) either a unique stable fixed point at ρ ≤ ρ is evenunphysical) or (b) a unique stable fixed point at a density 0 < ρ <
1. For m = 2, which we will discuss indetail in the following, the steady state expression of Eq. 5 is of third order in ρ and bistability becomespossible for certain parameter combinations. Defining self-aggregation
In some studies, increases in the spatial variance of cloud or liquid water has been employed as a measureof self-aggregation. However, this measure would still allow for many disconnected cloud clusters. Here,we want to define self-aggregation as the state where, in the limit of t → ∞ , complete segregation of thedomain into a cloudy and a non-cloudy component takes places. To quantify this, we will subsequently usethe number densities of convective cells within the different phases. Case without spontaneous generation
Consider the case of f sg = 0, a situation where new cells are exclusively produced by cold pool collisions. Weare interested in the range of parameters, where a fully aggregated state can be stable. To make progress,we check for the stability of two bulk phases and the boundary between these bulk phases: the two bulkphases consist of the stable solutions to Eq. 5, where the density is either high or low. Using that f sg = 0,Eq. 5 simplifies to ˙ ρ = F q,p ( ρ ) , (7) F q,p ( ρ ) ≡ − ρ q p ( ρ − ρ + 1 / ( q p )) . (8)Looking for local solutions in dependence on q , apart from the trivial solution ρ = 0, we obtain ρ , = 12 ∓ (cid:18) − p q (cid:19) . (9) q. 9 offers physically meaningful solutions for p q ≥
4. Both ρ and ρ are stable fixed points, which wecheck by verifying that ∂F q,p /∂ρ | ρ = ρ , < q ≡ − ρ as a parameter. However, q should be obtained in such a way thatthe partitioning of the domain into regions of high and low density ( ρ and ρ , respectively) is stable. Onetherefore has to consider the interface between these two regions of densities ρ and ρ , respectively, anddemand that the density at the interface be constant (Fig. 2a). As an approximation, consider the interfaceto locally be a straight line boundary, that is, we neglect its curvature and consider that the interface issmooth and sharp, so that for one half plane ρ = ρ = 0, for the other, ρ = ρ , as given by Eq. 9. This straightline approximation is valid, as long as r max is far smaller than the radius of the aggregated sub-domain. Forfinite-size systems the curvature can generally not be neglected and should be treated as a correction. Wedescribe the effective density for any point at the interface as the average ρ ave ≡ ( ρ + ρ ) / ρ /
2, andfurther that ρ ave be constant, i.e. F q,p ( ρ ave ) = 0 , (10)or, equivalently, ρ = ρ / . (11)Using this together with Eq. 9 yields q = 1 − ρ = 92 p − , (12)and ρ = 0, ρ = 1 / ρ = 2 /
3. Hence, if the aggregated state exists, the total cloud-free area q is inverselyproportional to the interaction factor p and the density within the cloudy bulk area is independent of q and equal to 2 /
3. But for which p can aggregation be expected? Clearly, ρ = 1 − q , the average density,must lie between the densities of the two phases, hence, ρ < − q < ρ , (13)yielding 92 ≡ p ,min < p < p ,max ≡ . (14)Hence, the segregated phase is limited to intermediate values of p . For p ≤ p ,min , the domain isentirely cloud-free, that is, ρ = 1 − q = 0. For p ≥ p ,max , no long-lived cloud-free region will exist.However, transient density fluctuations are possible. Our solution predicts that, for p ≥ p ,max , ρ willfollow the uniform bulk solution (inserting ρ = ρ = 1 − q and f sg = 0 in Eq. 6) to F ( ρ ) = p (1 − ρ ) ρ − ρ = 0 , (15)and it is easy to check that for p ≥ p ,max there is only one stable, plausible, solution for ρ . Eq. 15gives ρ ( p ,max ) = 2 /
3, which matches that obtained from Eq. 12 for the segregated solution, hence ρ ( p ) iscontinuous at p ,max . We further check the slopes s < ≡ lim p → p ,maxp
≡ lim p → p ,maxp >p ,max ∂ρ∂p = 4243 , (17)(18)hence, s < > s > , signaling a continuous phase transition of ρ ( p ) at p ,max . We check this result by explicitsimulations (Fig. 3). Case with spontaneous generation
We now allow spontaneous generation ( f sg >
0) and work with the full Eq. 5, namely˙ ρ = F q,p ,f sg ( ρ ) , (19) F q,p ,f sg ( ρ ) ≡ − qp ρ + qp ρ − ρ (1 + qf sg ) + q f sg . (20)One obvious effect of f sg > ρ ≈ qf sg to linear order, making ρ positive. As a third order polynomial, the zeros of Eq. 20 are more complicated Fixed points for different models. a , Schematic exemplifying the effect of an infiniteboundary (approximately straight line boundary), vs a finite boundary, which then shows a pro-nounced curvature affecting the replication probability at the boundary. The red square indicates apoint at the boundary, which uses the density computed within the circled area, to determine a possi-ble update. b , Model with quadratic dependence on ρ , i. e. m = 2 in Eq. 5, where q = 9 / f sg + p )is chosen to ensure that ρ + ρ = 2 ρ , i.e., that an aggregated state is possible. Black, blue andgreen curves correspond to increasing values of f sg = { , . , . } , respectively, with fixed p = 5 . ρ = 1 − q . Note that the value of ρ increases with f sg and eventually departs from theallowed range ρ < ρ < ρ . c , Similar to (a) but for f sg = .
03 held fixed and p = { . , . , . } forcolors black, blue and green, respectively. Note that in this example only the intermediate value of p yields a self-aggregated state. algebraic expressions ( Details:
Supplement). However, to test for stability of the aggregated state we areagain mainly interested in fixed points, where ρ = ( ρ + ρ ) / q . Luckily, the condition on q becomes very simple, namely q = 92(9 f sg + p ) , (22)which yields Eq. 12 for f sg = 0, as it should. The resulting zeros are also simple, namely ρ / = 13 (cid:32) ∓ (cid:18) − f sg p (cid:19) / (cid:33) , (23) ρ = 1 / . We can quantify the contrast ∆ ρ ( p , f sg ) between the densities in the aggregated and cloud-sparse regime,as ∆ ρ ( p , f sg ) ≡ ρ − ρ = 23 (cid:18) − f sg p (cid:19) / , (24)which is sharp for large interaction strength p and small f sg .However, we have not yet established where this contrast is applicable, as we need the boundaries of theaggregated regime. To obtain this boundary, we again use the condition on q (Eq. 13), which together withEqs 22 and 23 yields the upper and lower lines of transitions p ( f sg ) (Eqs S4 and S5). While these are moreelaborate algebraic expressions, they are nonetheless closed and allow us to study the type of transitionoccurring at the boundaries of the aggregated regime.Evaluating the expressions for the boundary (Eqs S4,S5) together with Eq. 22, we first compare theprevious results for f sg = 0 (Fig. 3) with those of positive f sg (Fig. 4a). The primary effect of increased Cloud fraction vs. interaction. a , Cloud fraction ρ as a function of p in the casewithout spontaneous generation ( f sg = 0). Theoretical and simulation result shown as black line(green points). Red dotted lines indicate p ,min and p ,max , respectively, that is, the limits betweenwhich segregation can take place. Gray dashed line indicates the function ρ = 1 − / p (valid inthe segregated regime), to indicate the change of slope at p ,max . The slight discrepancy between thetheoretical and simulation results is likely due to the assumption of smooth and straight interfacesbetween the cloudy and cloud-free areas, as well as noise caused by finite size effects and latticediscretization. Simulations were carried out on a lattice of 300 ×
300 sites with periodic boundaryconditions using a Gillespie algorithm for all rates involved, several thousand system updates perparameter value were simulated before computing the average ρ . b , Plots indicating the spatialpattern of different steady states, cloudy (gray) and cloud-free regions (blue), for p = { , , , } ,hence cloud-free, aggregated with considerable cloud-free areas, aggregated but mostly cloudy, andfully cloudy. f sg is to increase total density ρ . However, for small values of f sg , the slope still changes abruptly whenincreasing p beyond a threshold. This continuous phase transition signals the entry to the aggregated state(Eqs S4 and S5). For sufficiently large f sg a change of slope as function of p is no longer observed.Why does a change of slope occur? Within the aggregated phase, an additional degree of freedomis activated, namely one where cells lump together to “aid” one another, thus leading to more favorableconditions for replication. At larger values of f sg the discontinuity vanishes — abrupt variations in stateare no longer possible. This is explained by f sg diluting free space so strongly that new aggregates willincessantly form all over the domain (compare: Fig. 2b).The figure also shows simulation results for the different parameter combinations. The contrast betweentheory and simulation is strongest for f sg = 0, but also the cases with spontaneous generation displaysystematic discrepancies. These are partially due to finite system-size effects, whereby at the transition tothe aggregated state very small clusters will initially form. These clusters suffer most from the curvatureeffect (Fig. 2a), hence, small clusters will enter a positive feedback loop of decay, where they become smallerand increase the detrimental curvature effect.Also for larger p , at the other end of the aggregated regime, a discontinuity occurs, which is however lessnoticeable. At this stage, the domain is so densely filled that interaction takes place all over, and clusteringis no longer possible. The slope in fact decreases when crossing the threshold (Eq. S5), a bunched-up statecould maintain higher activity, but it becomes statistically impossible to maintain such a segregated state.We summarize the findings for ρ ( p , f sg ) (Fig. 4b) and c ( p , f sg ) (Eq. 24, Fig. 4c).In summary, edges of the segregated phase within the two-dimensional phase diagram are generallycharacterized by continuous transitions in ρ . However, for a single combination of p and f sg this changeof slope should disappear, namely, when ρ = ρ = ρ = ρ ( compare : Fig. 2). This point is easy to obtain Phase diagram for a system with spontaneous generation. a , Cloud cover versusinteraction strength p . Similar to Fig. 3 but now allowing for finite spontaneous generation f sg ≥ L = 200, with simulations carried out as in Fig. 3). Curves from bottom to topcorrespond to f sg = { , . , . , / , . } , that is, for the first three we expect a change of slope asfunction of p , for the remaining two we expect smooth behavior. For orientation, the dotted linesindicate the values p ,min and p ,max for the case of f sg = 0. b , Surface plot showing domain averagecloud fraction ρ versus interaction strength p and spontaneous generation rate f sg . Color shadingfrom orange to blue indicates increasing values of ρ . Thin black lines are contour lines for severalvalues of ρ . Dotted line indicates the boundary between non-aggregated (continuum phase) andaggregated convection (clustered phase). Note the abrupt change of slope at the dotted line (mosteasily visible for p ≈ /
2. Note also the “triple point” at (81 / , / , / c , Heatmap of the contrast ∆ ρ ≡ ρ − ρ , between the densities within and outsidethe aggregated subdomain. Fig. S1 shows examples of spatial patterns. when using Eq. 22 and exploiting the equality of densities in Eq. 23. We thus obtain the “triple point” as f ∗ sg = 316 , (25) p ∗ = 8116 , (26) q ∗ = 2 / − ρ , (27)that is, a point that is characterized by a smooth transition between the continuum and the segregatedphases (marked in Fig. 4b,c). At this point, the contrast (Eq. 24 and Fig. 4c) vanishes. Including explicit spatial interaction.
As mentioned, our model has assumed some spatial displacement upon collision of cold pools, leading to aformulation where only the number density, not the explicit position of cells in the surroundings of r wasrelevant in determining interaction effects between cold pools (Fig. 1c). The details of how often and atwhich precise location cold pools trigger new convective cells is still an open research question. Literaturehowever states that cold pools reach typical maximal radii
23, 25, 31, 32 and that cold pool interactions oftenleave behind a new cell near the location of collision
18, 23, 30, 34, 37–39 — hence, requiring the new cell to liesomewhat in between the centers of the cold pools causing it.To qualitatively capture such geometric spatial effects, we now consider a refined process on a lattice,where each lattice site represents the area a ∗ , required by a precipitation cell and its cold pool. In practice,the value of a ∗ will be a result of a self-organization process, which can be understood as follows: neighboringprecipitation cells can instigate new cells, when their respective cold pools have each traveled more than aminimal distance, defined by the radius of the effective precipitation cell area a , but less than the maximaldistance r max a cold pool can travel. Typical distances, which define a ∗ , will lie at intermediate values,and their calculation is a non-trivial statistical mechanics problem, left to a future study. In the current ualitative discussion we take a ∗ to be a given system parameter.Given that the site at r is not occupied, and two neighboring, and spatially opposite sites are occupiedby existing cells, e.g., cells at r − e x and r + e x (Fig. 1d), a new cell can be produced at r by collisionof the cells. Since we only consider neighboring sites for the interaction, we are implicitly assuming that r max is on the order of a lattice site. Note that this type of interaction, where the resulting cell is alwaysproduced between existing cells, will, for geometrical reasons, be unable to allow boundaries between cloudyand cloud-free areas to grow in favor of the cloudy areas (compare: Fig. 2a). In fact, the boundary willgradually retreat and cloud-free “cavities” will tend to increase in size. A single cavity would then growlarger until it takes up the entire domain — no sustainable aggregation could take place. Such a run-awayeffect will not be physically plausible, since the convective instability would inevitably increase by surfaceheating and cloud-level radiative cooling. Eventually, other processes, e.g. the spontaneous ones discussedthroughout the text, or single-cold pool processes (Fig. 1e) would become active.We therefore now consider an alternate model, where, in addition to the two-cold pool process described,also single cold pool processes are incorporated (Fig. 1d,e), and the probability for a given vacant site tobe “infected” by any occupied neighbor is p s . We remind the reader that single cold-pool processes alonewere ruled out on theoretical grounds to give aggregation (discussion following Eq. 6). We here test thisconclusion numerically (Fig. S2a, gray curve). We again initialize our domain with a random seeding ofcells and allow the dynamics to evolve. Indeed, for p s (cid:38) . p = 0, the spreading from any occupied site toits eight nearest neighbors starts to outweigh decay, however, the spatial dynamics yields a random patternwithout aggregated clustering (not shown).We now set p = 10, allowing for two-cold pool interaction. Similar to simulations of convective self-aggregation, the initial effect is that small sub-regions become cleared from precipitation cells. Some ofthese cleared regions eventually expand and merge with other cleared region. Finally, only a single cloudypatch remains, which maintains its area indefinitely.Exploring the parameter p s systematically, we find that the domain is entirely cloud-free for smallvalues, and a continuous transition to an aggregated regime again exists, similar to the case of f sg = 0 forthe previous model (Fig. 3). With a further increase in p s , the area of the aggregated subregion furtherincreases. Finally, for very large p s , aggregation is no longer possible and a featureless cloudy regime isfound. For this high-density limit, we could however not detect any phase transition for the order parameter ρ . Rather, at large values of p s increasingly diffusive dynamics is found which smoothly leads to a randompattern in space. In summary, also with a stricter condition on the location of cells resulting from thecollisions, a phase transition to an aggregated regime is found, when two-cold pool interactions play asufficient role in generating the new cells. Discussion and Conclusion
Self-aggregation in convection has drawn substantial interest due to its potential applicability to large-scalestructures in tropical and subtropical cloud organization, most notably the Madden-Julian Oscillation inthe Indian and Pacific Ocean, as well as the buildup of hurricanes. Cold pool processes are now considereda crucial component in the interaction between convective clouds. However, their exact relation to self-aggregation has been unclear. This may, in part, be due to the large mix of effects, all of which cancontribute to the self-organization of the convective cloud field — most prominently the effect of strongerradiative cooling of cloud-free air masses, which has been implicated in the stabilization of a larger-scalecirculation pattern to reinforce aggregation. With so many effects contributing, the computational demand to map out the entire phase diagramof actual aggregation in fluid-dynamics models, is currently prohibitively large. Likely contributors, whichwould need to be explored, are the value of the sea surface temperature, the ventilation coefficient whichdescribes rain evaporation leading to cold pool formation, model grid resolution, which influences cold poolspreading and the sharpness of gust front boundaries, the radiation and cloud microphysics schemes, as wellas domain size and geometry.This study explored the possible implications of small-scale interactions between clouds. These interac-tions are active at the spatial scale of cold pool radii, which are typically on the order of tens of kilometers.The model imposed a large-scale energy constraint, by which the propensity of forming new precipitationevents is weakened when the large-scale energy budget is used up. Our finding is that, when perturbationsare weak, in other words, when the main cause of new convective events is the interaction between previous vents, mediated through their cold pools, aggregation is likely to occur. For larger perturbations a con-tinuum state would be reached, where the domain shows a rather featureless mix of more cloudy and lesscloudy “patchiness”.Notably, our cold pools have been constrained by a maximal radius r max , which sets a scale for coldpool interaction. Increasing r max would correspond to cold pools that can travel larger distances, beforetheir momentum decays. In that case, interactions would become very strong due to the larger value of ρ ( r )(Eq. 3), and a segregation into an aggregated and a cloudfree phase would no longer occur. In our model(Eq. 5), the effect of r max is captured in the parameter p , and increases in r max would translate to larger p . Hence, for sufficiently large r max , p > p ,max , and the aggregation regime in Fig. 4 would be left. Theseconsiderations are in line with previous findings from simulations, where self-aggregation was found to behampered by increased cold pool strength.In conclusion, we have here introduced a continuum model for convective cell spatial number densityas well as a discrete model with explicit spatial interaction, to qualitatively mimic the effect of cold poolinteraction within an energy flux constrained framework. When only single cold pools are allowed to setoff new convective updrafts, self-aggregation did not occur. Single cold pool processes display a diffusion-like dynamics, which does not constitute a basis for bistability. Conversely, independent of the model,the interaction between multiple cold pools can indeed give rise to sustained aggregation effects, whichself-organize from an initially random cloud distribution. Acknowledgments
JOH thanks S. J. Boeing, S. B. Nissen and K. Sneppen as well as the two anonymous reviewers for usefulcomments. JOH gratefully acknowledge funding by a grant (13168) from the VILLUM Foundation. Thisproject has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation program (grant agreement no. 771859). No new data were used inproducing this manuscript. eferences Yeomans JM (1992)
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