In this section, we give an overview of the CA for the tropical rain clusters use in this study. In the interest of clarity, the bulk of the mathematical details is presented in Appendix B. The rules of the CA are summarized in Table 1. The CA is defined on a square lattice with open boundaries. For lattice size Z, we consider the data resolution that was analysed in T17 (25 km satellite-estimated rain) and T21 (36 km numerical model simulation). While Z = 120 and Z = 96 was applied in T17 and T21 respectively, which were large enough to cover each of the tropical oceanic regions with deep convection, we extend the lattice size to Z = 512 for the CA here such that scaling up to very large rain clusters can be realized. The main prognostic non-dimensional variable at each lattice site, X, represents the stability of the column of atmosphere against moist convection over a homogeneous ocean surface. In this simple model, X is taken to be the difference between the boundary-layer (below ∼ 850 Pa) averaged specific moist static energy, h1, and the layer-averaged specific saturated moist static energy at the top of the troposphere (above ∼ 450 hPa), h3* (Fig. 1). By assuming that h1 and h3* are larger than the layer-averaged saturated specific moist static energy of the middle troposphere (denoted as h2; Eq. B1) at all times, deep convection (and consequently precipitation) only occurs at a lattice site j if Xj is positive (Eq. B2); no deep convection is triggered otherwise. In this model, X is bounded below at a value arbitrarily set to -1, after normalized by a maximum stability, Xo, that the atmosphere can attain. Large-scale forcing gradually destabilizes the atmosphere through surface fluxes and radiative cooling, represented by adding a predefined increment, d, to X at each driving cycle (Eq. 1). The driving timescale, tLS, is much longer than the convective relaxation timescale, ensuring separation of scales.

The CA operates in the following manner: at each outer or radiative time step t, X is increased by a value d at all lattice sites, representing the large-scale destabilization of the atmosphere due to the combined effects of latent heat fluxes in the boundary layer and net radiative cooling of the atmosphere (Eq. B3). The rate of change is 5.768 kJ kg⁻¹ d⁻¹ based on estimates from the literature (Appendix B1). For the unstable (toppling) sites with X > 0, deep convection is assumed to restore the local stability, represented in the model by relaxing their X immediately (after a microscopic or convective time step τ) back to some negative values. This relaxation represents the combined effects of convective transport and compensating downdrafts (Eqs. 2–4). These relaxed X values are drawn stochastically from a distribution g(x) estimated from published literature (see Appendix B2 for the estimation of g) to reflect the uncertainties introduce by various cloud processes during deep convection. Driving of all the model sites is only resumed after all toppling activities complete, after which all sites become stable again and the system as a whole is in equilibrium. Then the cluster statistics are analysed. As will be seen in the following, such separation of time scales (t and τ) is critical to the rain cluster size distributions simulated in the CA.

A raining (toppling) site in the model is assumed to interact with its neighbouring sites through two modes of horizontally propagating gravity waves excited by the latent heating profile at the site where the deep convection occurs. The first set of gravity wave, GW1, is excited at the mature stages of the deep convection when the maxima latent heating is at the mid-troposphere, while the second set of gravity wave, GW2, is excited at the later stage of the storm development when the heating maxima is higher up and presumably accompanied by evaporative cooling at the lower troposphere. These gravity waves are assumed to radiate horizontally away from the convective site with circular wavefront, with the phase speed of GW2 half of that of GW1 since the vertical wavelength of the earlier is taken to be about half of the latter (Fig. 2). GW1 acts to stabilize the atmosphere while the presence of GW2 acts to destabilize the atmosphere behind its wavefront. The effects of these gravity waves are represented in the CA by increasing X (destabilize) for the nearest-neighbours and decrease X (stabilize) for the next-nearest-neighbours of a toppling site (Eq. 5). The magnitude of the (de)stabilization of the (nearest-neighbour) next-nearest-neighbour is proportional to |ΔX| at the precipitating site with a tunable positive constant γ, which represents the degree of coupling between different locations by gravity wave dynamics (Appendix B3). Note that in this scheme of gravity wave effect, treatment is site-by-site and thus only near-field effect has been simulated, considering that the grid size in the lattice is around 30 km (as mentioned in Sect. 2.1). As such, the propagation speed of gravity waves has been considered in our model to a certain extent. In reality, convective cells form a cluster and collectively a strong gravity wave may propagate from the convective tower as a whole and influence the far field (i.e., longer range than the next nearest neighbour). We only simulate the near-field effect because that is more in accord SOC models, which generates long-range spatial correlation from short-range interaction. From another perspective, full-physics simulations in T21 using WRF still generated scaling exponents of cluster area and rain rate with deviations from those observed. Thus, we experiment with idealized, near-field interactions in the CA here.

Parallel update during the avalanche (when toppling activities in the CA are still happening) is enforced – all the sites excited (X > 0) at a microscopic time step, τ, will only be relaxed at the next microscopic time, τ + 1. Driving of the model resumes only after quiescence (X < 0 at all sites) ensuring the clear separation between the driving (synoptic) and relaxation (convective) time scale. Multiple relaxations or back-toppling, at a given site are permitted in the model. The rain clusters are identified at quiescence as the collection of sets of rainy sites connected with their nearest neighbours. During initialization of the model, randomly selected sites are driven by the value d defined above. The model is then analysed when it attains a stationary state based on the number of clusters in the model.

Originally, we envisaged that some non-trivial scaling relation between the cluster rain rate (taken as proportional to the avalanche intensity: I in Eq. (3) and summation over a cluster) and the cluster area would naturally emerge from the model. However, we find that the avalanche intensity from the CA is effectively proportional to the cluster area on the average, perhaps due to lack of representation of the detailed mesoscale convective system (MCS) circulation necessary to reproduce mesoscale cluster with area-dependent rain intensity as discussed in T21. As such the cluster total rain rate of a CA cluster after quiescence is computed by Eq. (6), as a product of the total avalanche intensity of the cluster with a cluster size-dependent coefficient k (in power-law form, with the exponent taken from the observational value from T17). In other words, Eq. (6) constrains the CA to reproduce the required observed scaling of E(R|a) for mesoscale clusters (T17).

Although the CA shares some common features to some known SOC models like the Olami–Feder–Christensen (OFC) model (Olami et al., 1992), namely the existence of a toppling threshold; homogeneous external drive; and the clear separation of slow driving and fast relaxation time scales, it is not an SOC model. To understand the reason why, we note that according to the Absorbing State Mechanism (ASM; Dickman et al., 1998; Vespignani and Zapperi, 1998), the CA can potentially exhibit SOC only if it has a vanishingly small dissipation coefficient, ϵ, defined in the case for the CA, as ratio of the net loss in X through its interaction with its neighbours per unit change in X, as a result of a single toppling. If we ignore, for simplicity, the lower bound constrain imposed on X, the total change in X at the twelve neighbours of a toppling site away from the boundaries (the four nearest-neighbours and eight next-nearest-neighbours of the toppling site) is γΔX - 4× 316γΔX = 14γΔX (Eq. 5). This means that ϵ = 1 + 14γ ≥ 1, since γ ≥ 0. As a model for tropical convection, this is not unreasonable since deep convection have a net stabilizing effect on the atmosphere. However, as discussed in detail in the next section, the CA still exhibits critical behaviour for unique value of d for a given γ.

After the steps described in Sect. 2.1–2.3, we analyse the cluster area and rain (based on the constraint we imposed) statistics after each radiative time step t, a snapshot has been shown in Fig. C1 (Appendix C) as an example. The cluster statistics are then used to generate the scaling results in the following.

Since CA depends on the two adjustable parameters, γ and d, we denote CA with the coupling constant γ, driven with a magnitude d, as CA_γ,d for brevity. For example, CA_0,d has the meaning of the CA with γ = 0, and with an instability of magnitude d added to each lattice sites at each driving cycle (Eq. 1).

The phase space for CA as a function of γ and d is represented schematically in Fig. 3. For 0 ≤ γ ≤ 2.5, the cluster distributions are found to follow the scaling ansatz similar in form to the 2-dimensional site-percolation (e.g., Christensen and Moloney, 2005) when d approaches a critical value (dc): 7fS(s)∝s-ζSGSsξSDSd→dc,a≫1,Z≫ξAs-ζSG̃SsZDSd→dc,a≫1,1≪Z≪ξA where fS(S) is the discrete probability distribution P(a) when S is A and fS(s) is the probability density function, f(r), when S is R; GS(x) and G̃S(x) are the scaling functions for the two different ξA regimes, but share the same generic property of approaching to some non-zero constants when x ≪ 1 and decays rapidly when x ≫ 1; ξS is the characteristic “length scale” of the cluster and the DS is a γ-dependent exponent for the distributions; dc a γ-dependent value of d where the CA is critical when d = dc; Z is the lattice size. When S is A, ξA has the meaning of a geometric length scale of the cluster (e.g., the averaged radius of gyration of the clusters; Cheraghalizadeh et al., 2024) and based on which separating the two scaling regimes in Eq. (7); DA, the cluster fractal dimension and ξADA, the characteristic cluster area. Similarly, ξRDR is the characteristic cluster total rain rate.

Furthermore, the characteristic scale scales as 8ξSDS∝d-dc-ηS,S={A,R}, where ηS is a γ-dependent exponent.

The motivation of the CA scaling ansatz can be most easily inferred from the case when γ = 0. For CA_0,d, there is no interaction amongst the lattice sites; whether a site become unstable (X > 0) after adding d units during its driving phase, depends only on its antecedent value of X. In its stationary state, the probability that a lattice site becomes unstable after being driven is equal to the ensemble mean of the fractional area coverage of the rain clusters, Ω. Hence CA_0,d can be mapped to the 2-dimensional site percolation on a square lattice of the same length, Z, with the probability of occupancy probability, p = Ω(d). This claim is supported by Fig. 4 where the cluster size distributions for CA_0,d being practically indistinguishable from the sample cluster size distributions of a 2-dimensional site-percolation with the estimated Ω of the CA_0,d chosen as p. For Fig. 4b, the drive d for the CA_0,d is set to approximately its critical value such that Ω(d=dc=0.2901) ≈ pc = 0.5928. This value of dc is determined from preliminary experiments based on a Z = 128 lattice.

For the standard site-percolation with occupancy probability p = pc, the system becomes critical, and the characteristic cluster length scale, ξA, diverges. At criticality the system transit to percolation, that is, an incipient infinitely large cluster emerges in the limit of an infinitely large domain, and P(a) ∝ a-ζA for a ≫ 1, with ζA = 187/91. For standard site-percolation, the characteristic cluster area, ξADA ∝ p-pc-ηA with ηA = 91/36 (Stauffer and Aharony, 1994). Since p = Ω(d) for CA_0,d, ξADA ∝ Ω(d)-Ωdc-ηA ≈ Ω′dc-ηAd-dc-ηA to a first approximation, Ω′dc (> 0) being the first order derivative of Ω evaluated at dc. On the other hand, when Z ≪ ξA, the distribution is finite-size scaling and the characteristic cluster area becomes proportional to ZDA instead. We will call the scaling ansatz when Z ≫ ξA simply as finite scaling to differentiate from finite-size scaling ansatz for clarity.

The finite scaling and finite-size scaling of CA_0,d cluster size distributions is demonstrated in Fig. 5. The d values are judiciously selected such that Z ≫ ξA(d) (Fig. 3a). As d approaches dc (∼ 0.2901), ξA increases rapidly and consequently the interval of cluster area where P(a) exhibits a power-law form becomes increasingly discernible. On the other hand, ξAd=0.2905 is much larger than Z ≤ 512, and Fig. 3b is suggestive of finite-size scaling of P(a) for Z ≤ 512.

Since the scaling function GA (G̃A) is identical in principle, across the different d(Z), the scaling ansatz (Eq. 7) can be tested directly using data collapse, i.e., plotting together the scaling functions inferred separately for the different distributions in Fig. 5a, b. The inferred scaling functions, G^A for the different d values are shown in Fig. 5c, where the reweighted probabilities, a-ζ^AP(a), are plotted against the x = a/d^c-d-η^A. Note that hatted symbols denote the estimates. The methods used to obtain these estimates are described in Appendix D. η^A = 2.681 ± 0.067 is slightly larger than the expected theoretical value of 91/36 = 2.528, while ζ^A = 1.914 ± 0.095 is in agreement with the theoretical value of 187/91 ≈ 2.055. The inferred scaling functions associated with the P(a) for the different d values collapse together well for x ≳ 10⁻². For small x (e.g., x ≲ 10⁻⁴ for the G^A with d = 0.28) finite scaling is no longer valid as the cluster distribution would then depends on the details of CA. As d approaches dc, ξA increases rapidly (Eq. 8). Therefore, the lower limit of x where the finite scaling breaks down decreases, evident from the decrease in the x where the overlap of the inferred scaling function beings as d approaches dc (compare the x where G^A with d = 0.28 collapse with the G^A with d = 0.27, and with d = 0.25 in Fig. 5c). Similar inferences for the scaling function G̃A is performed when the distribution is finite-size scaling (Z ≪ ξA), with d = 0.2905 as shown in Fig. 3d. Note that as the scaling ansatz in Eq. (7) is defined for finite clusters, the percolating clusters from the numerical simulation are excluded when sampling the clusters. Details of the methods used in estimating DA and ζA are in Appendix D. D^A = 1.945 ± 0.057, slightly larger than the theoretical value of 91/48 ≈ 1.896 while ζ^A = 2.084 ± 0.189 agrees with theoretical value of ζA.

The scaling characteristics for the CA_0,d cluster total cluster rain rate distributions are similar to that describe above for the cluster size distributions. Figure 6a, b shows the R-distributions when they are finite scaling and finite-size scaling respectively. Away from r ∼ 1, the increasingly discernable power-law form of f(r) for r > 1 when d approaches dc (Fig. 7a) and with increasing Z (Fig. 7b) are reminiscent to what is observed for P(a). The collapses of the distribution in the two scaling regimes are shown in Fig. 4c, d, using ζ^R = 1.724 ± 0.103, η^R = 3.742 ± 0.088, D^R = 2.596 ± 0.073 and dc = 0.2925 ± 0.0005 estimated using the same method used in obtaining the exponents for A (see Appendix D). The exponents for R are not independent from those of A due to the constrain E(R|a)αaβ imposed by the CA (Eq. 6 and Appendix A): ζA and ζR are related with β by the scaling relation (Eq. A4) as a result of the constrain (e.g., Pruessner, 2012). Estimate of β from the scaling relation ζ^A and ζ^R is 1.380 ± 0.338 which is consistent with the imposed value of 4/3. In addition, DR and DA area related as follows: The characteristic scale for R ∼ ZDR which scales as E(R|ZDA) ∼ ZβDA. Since the relation is true for all Z (1 ≪ Z ≪ ξA), then DR = βDA. Similarly, one can demonstrate through Eq. (8) that ηR = βηA, η^Rη^A = 1.3959 ± 0.0675 and D^RD^A = 1.3348 ± 0.0763 consistent with the imposed β = 4/3.

In qualitative terms, the non-interacting case (γ = 0) serves as a baseline for understanding the CA's behaviour. Here, each lattice site topples independently of its neighbours, and the only source of spatial organization is the geometric connectivity of toppled sites at quiescence. The fact that the cluster size distribution for CA_0,d is essentially indistinguishable from that of standard 2D site percolation demonstrates that this CA's toppling dynamics naturally generates a percolation-like critical state when the external drive d is tuned to a critical value. In this state, clusters of all sizes emerge, with the largest clusters becoming increasingly space-filling as d approaches dc. The power-law exponent ζA (for cluster area) matches the theoretical percolation value, confirming that the CA's connectivity rule (nearest-neighbour links between toppled sites) is sufficient to reproduce the universal cluster geometry of 2D percolation. This baseline case also establishes that the separation between slow driving and fast relaxation timescales is already sufficient to produce scale-invariant cluster statistics, even before introducing convective interactions.

The finite scaling ansatz holds as well for the distributions of CA_γ,d→dc for 0 < γ ≤ 2.5, albeit the following generalizations from CA_0,d→dc (Fig. 8 and Table 2): (1) dc and ηS are increasing functions of γ; (2) DS is a decreasing function of γ. The method of obtaining the scaling exponents and dc for each of γ tested are the same as the preceding discussion on CA_0,d (Appendix D). For example, Fig. 7c and d show reasonable data collapse for the two finite scaling regimes Z ≫ ξA and 1 ≪ Z ≪ ξA for CA_γ,d for the case when γ = 2, using estimates of ζS, ηS and DS and dc. Similar data collapse with their corresponding scaling exponents estimates are also obtained for γ = 0.5, 1, 1.5 and 2.5 (not shown).

dc increases with γ since the net instability being removed by a convection event (toppling) depends linearly on γ. Hence as γ increase, the magnitude of d needed to drive the model to criticality increases linearly as well (Fig. 8d). DA is a decreasing function of γ since the suppression of convection by GW1 becomes stronger with increasing γ. Hence, we might expect that the CA rain clusters to be increasingly riddled with “holes” and consequently, their fractal dimension which DA represents decreases. As expected, DR and ηR appears to be simply an amplification of DA and ηA respectively, since the imposed scaling of E(R|a) in the CA constrains the ratio of DR/DA and ηR/ηA to be β. Figure 8e shows that the expected relation is well reproduced in the CA for DR and DA, but not as well for γ ≥ 1.5 for ηR and ηA. The estimates of ζA, ζR and β for the CA also satisfy the scaling relation Eq. (A4) for all γ tested.

In contrast with ηS and DS, ζS appear to be insensitive to γ for γ ≤ 2.5, with an average value of 2.003 ± 0.058 for ζ^A, which agrees with the exponent for percolation (187/91) and 1.721 ± 0.067 for ζ^R. Whether the true ζA for the CA is identical to that of the 2-dimensional site-percolation (187/91) when γ > 0 remains to be explained. The site interactions due to the gravity waves would have introduced a degree of spatial correlation of X over the domain and hence, CA_γ>0,d would not have a one-to-one correspondence with the 2-dimensional site-percolation, as evident from the deviations of η^A and D^A from the percolation values with increasing γ. On the other hand, ζ^A for γ > 0 are the same across within their error estimates from our analysis (Table 2 and Fig. 6a). Although ζ^A for γ = 0.5, 2 and 2.5 is significantly different from the expected value for 2-dimensional percolation of 187/91, we note that for γ = 0, the point estimate for ζA is ∼ 0.055 less than 187/91 as well. The possible reason behind the discrepancy could be that although care has been taken in allowing a fairly large range of ξA and Z to probe the scaling functions in our present analysis, the number of clusters used (O(10⁷)) to probe scaling function for the larger domain sizes (Z = 128, 512) may still be insufficient, and contributes to the biases in the point estimates. The value of 187/91 would be within the prediction interval of all the ζ^A for γ > 0 if adjusted for similar magnitude of underestimates.

As d increases towards unity beyond dc, the percolating cluster becomes increasingly space-filling. Since the finite clusters can only exist in the gaps of the percolating cluster, the sizes of the finite clusters decrease rapidly with the increasing d. Consequently, on a probability plot such as those shown in Fig. 9, the frequency of clusters of size in the order of the domain size increases rapidly with d, while the characteristic size of the finite clusters decreases rapidly. When Z≫ξA(d), as the case for d = 0.42 and 0.6 for γ = 0 and γ = 2 respectively in the figure, there is a clear separation of scales in the cluster size distribution. When d is unity, there is only one trivial (percolating) cluster of a = Z2.

For increasing γ beyond 2.5, CA takes increasing time to reached stationarity and hence we do not have sufficient sample size to fully characterize the distribution. For γ > 4 the CA the domain averaged instability, X, is found to increase exponentially as the numerical simulation progresses, even with d ≪ 1. The reason behind such a behavior in the CA is because the amount of instability that can be removed at each of the lattice sites behind the GW1 wavefront is constrained by the minimum that X can attain (-1), while X behind the GW2 wavefront (the nearest neighbours of a relaxing site) is not bounded above (Eq. 5). Hence for large enough γ, the total change in X after a toppling can be positive which represents a deep convection rain processes that introduce net instability to the atmosphere instead of stabilizing the atmosphere. As such we regard such model configuration as non-physical and choose not to analyse the model in these large γ regime.

The β estimates analysed from the CA cluster samples (Table 2 and Fig. 9) are close to 4/3 as expected. The procedure of estimating β for each γ tested (γ ∈ {0,0.5,1,1.5,2,2.5}) is similar to that carried out for in T21. The only difference from T21 is that for the CA, b in the Eq. (2) is modelled as a linear function of 1/Z, following T17, as shown in Fig. 10a using the case when γ = 2 by way of an example. From Fig. 10b, β appears to be an increasing function of γ. This is because with increasing γ, the occurrence of multiple toppling at a given lattice site may have increased for larger clusters resulting in a slight dependence of the avalanche intensity with the cluster area.

Physically, the interacting case (γ > 0) introduces convective coupling between neighbouring sites through the stabilizing and destabilizing effects of gravity waves (GW1 and GW2). As γ increases, the spatial correlations in the instability field X become stronger: the destabilizing GW2 encourages convective triggering in the immediate vicinity of an existing event, while the stabilizing GW1 suppresses convection further away. This leads to two notable qualitative changes in the cluster statistics. First, the fractal dimension DA decreases with γ; clusters become more ramified and hole-ridden because the stabilizing far-field influence inhibits the filling of gaps within clusters. Second, the exponent ηA, which governs how rapidly cluster sizes grow as d approaches dc, increases with γ; the clusters become more sensitive to the precise tuning of the external drive.

Remarkably, despite these changes in the geometric details of the clusters, the exponent ζA remains robust to variations in γ, staying close to the 2D percolation value. This robustness suggests that ζA is governed by a topological constraint that is independent of the microscopic interaction rules: as long as clusters are defined by nearest-neighbour connectivity on a 2D lattice at criticality, the asymptotic power-law decay of the cluster size distribution falls into the percolation universality class. In contrast, the exponents describing the approach to criticality (ηS) and the geometric structure of clusters (DS) are sensitive to the details of the interactions, in other words they encode the “fingerprint” of how convection couples neighbouring sites. Thus, this CA provides a clear causal attribution: gravity-wave-mediated interactions shape the spatial structure and correlation length of rain clusters, but they do not alter the fundamental 2D percolation geometry that governs the asymptotic cluster size distribution.

Although the CA exhibits criticality, the ζA is different from the observed value of about 5/3 reported in T17. From the results in the preceding section, adjusting the value of γ does not change the ζA, although the other critical exponents, ηA and DA changes with γ. The robustness of the ζA lies perhaps more in the dynamics represented by the toppling mechanism in the CA; If we assume that only the destabilization in the vicinity of deep convection by GW2 is the main mechanism behind the rain clusters organization, the CA can be further simplified by excluding the effects of stabilization of the atmosphere by GW1 in Eq. (5). Following the discussion in Sect. 2, the dissipative coefficient, ϵ, for this simplified CA, call it sCA, would be ϵ = 1-34γ. By the ASM (Dickman et al., 1998; Vespignani and Zapperi, 1998), sCA may potentially exhibit SOC if γ → 4/3 (i.e., ϵ → 0) and d → 0⁺. In fact, sCA_γ,0⁺ (the subscripts having the same meaning as the preceding section) is similar to the OFC earthquake model (in which stress in the lattice sites is analogous to atmospheric instability in our CA and when stress is over a threshold a site topples; Olami et al., 1992), in terms of its rules for nearest-neighbour interactions, γ as a control of the level of non-conservation (i.e., not all toppling energy passes to neighbours), as well as how the model is driven. The OFC model is widely accepted to be an SOC when bulk conservation is enforced (i.e., dissipation ϵ = 0; Bröker and Grassberger, 1997; Chabanol and Hakim, 1997; De Carvalho and Prado, 2000). Indeed, preliminary survey with a set of small domain experiments indicates that sCA_4∕3,0⁺ is a SOC (Fig. 11), with ζA ≈ 1.35. We use the bulk driving scheme of the OFC model (Olami et al., 1992) to achieve the vanishing drive (d → 0⁺) for this set of experiment. When γ = 0, sCA_0,d also exhibits critically at d = dc ≈ 0.292 with ζA = 187/91 since it is equivalent to CA_0,d whose cluster size distribution is in turn, equivalent to the cluster size distribution of the 2-dimensional site-percolation. The behaviour of sCA_4∕3,0⁺ and sCA_0,d lead us to suspect that in general, for any γ ∈ (0, 4/3), there exist a d = dc(γ) where sCA_γ,dc exhibits criticality, and unlike the case for the CA, ζA could be a function of γ as well. We represent our hypothesis in Fig. 9c. Note that ϵ < 0 for γ > 4/3 and hence sCA cannot achieve statistical stationarity for γ > 4/3. We feel that the sCA is a cell-automata which may exhibit critical phenomena, and at the same time “tunable” via γ to reproduce the observed ζA ∼ 5/3. Indeed, some finitely driven cellular automata are reported to display varying power-law exponents in their event size distributions (Piegari et al., 2006).

Interestingly the ζA estimates from the clusters simulated by a tropical-band model in T21 over Indian Ocean and Atlantic Ocean are close to that of the CA clusters (cf. Table 1 of T21 and Table 1 here) which suggests that the modelled rain clusters may behave like percolating clusters, at least over the Indian Ocean and tropical Atlantic. Peters et al. (2009) analysis of observed rain clusters suggests a ζA close to the percolation value as well. However, more effort is needed to elucidate the reasons behind the discrepancy between the ζA estimates from observation reported by T17 and Peters et al. (2009) to ascertain which value is more representative of the actual exponent.

The distributions of the modelled rain cluster size, for example by regional climate model (RCM) in T21, for all tropical oceanic regions that include the spanning clusters in the data samples show a “bump” at the large cluster sizes (see Fig. 2 of T21). This is reminiscent of the CA in the supercritical region, where a domain-spanning cluster emerges. In the atmospheric context, such a spanning cluster should not be interpreted literally as a continuous precipitating entity, but rather as a large-scale, connected region of active convection (e.g., the ITCZ or a monsoon trough), within which individual convective cells are dynamically linked. The “bump” in the RCM distributions may therefore reflect the model's tendency to produce overly persistent and spatially extended convective organization compared to observations.

The construction method and results from our CA model have implications to some recent studies on tropical rain clusters statistics. While Haerter (2019) focused on cold-pool dynamics of convective self-aggregation, the stochastic model they constructed simulated the effect of gravity currents away from the centre of precipitation cell with scale ∼ 10–100 km, similar to the action of GW1 and GW2 in our model. However, cluster size statistics have not been explored in Haerter (2019). On the other hand, analysis of a range of dynamical models (from general circulation model to RCE models with different model settings) in Stephan and Stevens (2025) showed that large-scale vertical overturning (i.e., planetary and synoptic scale variability) is critical for rain cluster size statistics to follow power law. This might be related to the larger ζA over the Pacific Ocean compared with other oceanic regions obtained in T21. While such large-scale variability has not been represented in our CA model, our lattice cell interactions through GW1 and GW2 have provided an answer to one of the discussion questions raised by Stephan and Stevens (2025), namely, “Is there an important feedback from convection, i.e., is the generation of waves and vertical motions by condensation relevant for the relationship between atmospheric motions and the distribution of column water vapor?

Nevertheless, we have imposed the scaling relationship between cluster rain rate and size in our CA model (via β in Eq. 6). T21 has argued that β may be a more fundamental parameter than the individual scaling exponents for cluster rain and area (i.e., ζR and ζA) as β can be estimated through some fundamental mesoscale dynamical processes (convective heating, entrainment, updraught velocity). With only gravity wave effects implemented in our model, indeed we have obtained robustness of the ζR and ζA exponents under different coupling strength γ. Such robustness warrants further physical interpretation. The parameter γ controls the strength of gravity-wave-mediated coupling between convective sites. This leads to stronger spatial correlations in the instability field X, which manifests in the critical exponents that describe the spatial structure and correlation length of the clusters: the fractal dimension DA and ηA possessing sensitivity of the cluster sizes to the proximity of the critical drive. In contrast, ζA appears to be governed by a more fundamental topological constraint. The rain clusters defined by nearest-neighbour connectivity of toppled sites form a set of connected components on the lattice. Regardless of the details of the interaction rules that determine which sites topple, as long as the system is driven to a critical point characterized by a divergent correlation length, the cluster size distribution of such connected components falls into the universality class of two-dimensional percolation. In the CA, γ modifies the “microscopic” rules (the interaction strength) but does not change the fundamental fact that the clusters are nearest-neighbour-connected sets on a 2D lattice at a critical point, thus leaving ζA unchanged. The other exponents, ηS and DS, are “non-universal” in the sense that they describe the approach to criticality and the geometric details of the clusters, which are indeed sensitive to the specific interaction rules.

The fact that our simple CA, incorporating only gravity-wave-mediated interactions, yields a ζA near the 2-dimensional percolation value (187/91) rather than the observed ∼ 5/3 does not contradict the hypothesis that the observed scaling may depend on only a few fundamental physical constraints. On the contrary, it offers a valuable diagnostic attribution: it demonstrates that gravity wave dynamics alone drive the system towards a percolation universality class, and that reproducing the observed 5/3 exponent likely requires additional fundamental physical ingredients, such as cold-pool dynamics (Haerter, 2019), large-scale vertical overturning (Stephan and Stevens, 2025), or moisture–radiation feedback, that are not yet represented in the current model.

This highlights the distinctive level of explainability offered by reductionist CA models. Because each rule in the CA maps directly to a specific, well-defined physical process, any emergent change in the statistical exponents can be causally traced back to its underlying mechanism. In contrast, high-resolution dynamical models, while more realistic, often convolve numerous interacting processes, making it difficult to isolate which physical ingredient drives a particular statistical behaviour. The CA approach thus serves as a controlled testbed or sandbox for systematically testing which physical mechanisms are necessary and sufficient to reproduce observed rain cluster statistics.

A key direction for future research is to extend the CA framework with simple site coupling mechanism by incorporating additional candidate mechanisms (e.g., cold-pool spreading, horizontal moisture advection, or wind shear) and to examine whether they can shift ζA from the percolation value towards the observed 5/3, and whether the crossover behaviour from β = 4/3 to β = 1 in E(R|a) when a MCS grows in size can emerge naturally rather than being imposed. As a side note, Stephan and Stevens (2025) focused on comparing the scaling exponents of CWV “islands” cluster area (i.e., those above a certain water vapor threshold with triggering deep convection consideration) distributions with observations and high-resolution simulations (Li et al., 2022). CA models with CWV as the central ingredient, such as a few those we reviewed in the introduction, may be validated along that line of thought.