Although rain has been observed to form in warm cumulus clouds within about 20 min, calculations that represent condensation and coalescence accurately in such clouds have had difficulty producing rainfall in such a short time except via processes involving giant cloud condensation nuclei (with diameters larger than 2 µm). One of the possible origins of this discrepancy is the stochastic nature of the collision coalescence process that is not well reflected in current models that rely almost exclusively on the kinetic collection (or Smoluchowski) equation, hereafter referred to as KCE (Pruppacher and Klett, 1997): ∂N(i,t)∂t=12∑j=1i-1K(i-j,j)N(i-j)N(j)-N(i)∑j=1∞K(i,j)N(j), where N(i,t) is the concentration of droplets in bin iand K(i,j) is the collection kernel for droplets in bins i and j. Additionally, Eq. (1) fails to represent the droplet size distribution at the time at which raindrop embryos are formed (Alfonso et al., 2008), as there is a transition from a continuous distribution to a continuous distribution plus a massive raindrop embryo (i.e, the “runaway droplet”). At that point, the infinite system exhibits a sol–gel transition (also called gelation and in which the runaway droplet is labeled “gel”), the KCE breaks down and the total mass of the system calculated according to the KCE is no longer conserved.

One way to avoid this problem is to adopt the stochastic finite volume description of the coalescence process by using the stochastic simulation algorithm proposed by Gillespie (1975). The stochastic simulation algorithm (hereafter referred as SSA), correctly accounts for fluctuations and correlations, and has been used in cloud simulation studies with realistic collection kernels (Valioulis and List, 1984). However, the SSA has difficulties in accurately reproducing the large end of the droplet size distribution. This is due to the huge number of realizations required to obtain smooth behavior at the large end of the droplet size distribution (Alfonso, 2015). The alternative approach (within the stochastic framework) is to use the master equation: ∂P(n¯)∂t=∑i=1N∑j=i+1NK(i,j)(ni+1)(nj+1)×P(…,ni+1,…,nj+1,…,ni+j-1,…;t)+∑i=1N12K(i,i)(ni+2)(ni+1)×P(…,ni+2,…,n2i-1,…;t)-∑i=1N∑j=i+1NK(i,j)ninjP(n¯;t)-∑i=1N12K(i,i)ni(ni-1)P(n¯;t). The master Eq. (2) is a gain–loss equation for the probability of each state P(n¯). The sum of the first two terms is the gain due to transition from other states, and the sum of the last two terms is the loss due to transitions into other states. This formulation was introduced in the pioneer works of Marcus (1968) and Bayewitz et al. (1974), and was studied in detailed by Lushnikov (1978, 2004) and Tanaka and Nakazawa (1993). However, these studies only offer analytical results for a limited number of cases (with constant, sum and product kernels), for monodisperse initial conditions. Furthermore, most of these studies are limited to nongelling conditions and do not provide a coherent framework for the general case.

The exception are the methods developed by Lushnikov (2004) from the analytical solution of the master equation, and more recently by Matsoukas (2015), the later based on arguments from statistical physics. These methods, although also limited to very special cases (product kernel and monodisperse initial conditions), are capable of obtaining solutions in the post-gel regime. For example, in Lushnikov (1978, 2004), the coalescence process takes place in a system with a finite volume that includes a finite number of particles. Within this approach any losses of mass are, by definition, excluded. In the infinite system described by the KCE (Eq. 1), the coagulation process instantly transfers mass to the gel, while in the finite system the gel coalesces with smaller particles, decreasing their concentration – not instantly by rather in a finite time.

In order to study the droplet size distribution after the formation of raindrop embryos (sol–gel transition), for systems with kernels relevant to cloud physics and arbitrary initial conditions, we must rely on numerical methods that are capable of solving the master equation (Eq. 2). We can address this problem through a detailed comparison of the droplet size distributions obtained from the stochastic description for a finite system with the master equation (Eq. 2), and the deterministic approach for an infinite system by using the KCE (Eq. 1), using the numerical algorithm reported in Alfonso (2015). By the time the gel forms, certain differences are to be expected between the two approaches at the large end of the droplet size distribution.

This analysis of the sol–gel transition problem in the cloud physics context could provide an alternative explanation of the differences between modeled and observed droplet spectra in clouds. Several mechanisms have been proposed in the past (entrainment, presence of giant nuclei, supersaturation fluctuations, effects of air turbulence in concentration fluctuations and collision efficiencies, effects of film forming compounds on droplet growth), and a large amount of literature exists regarding the variety of mechanisms that may explain this disparity, but a conclusive answer is still absent. This study does not attempt to dispute any of the mechanisms already proposed, but to explore another mechanism that has not yet been widely considered in the mainstream literature.

The paper is organized as follows. Sect. 2 presents an overview of the numerical algorithm (following Alfonso, 2015). Numerical results (for the product and hydrodynamic kernels, respectively) with a detailed analysis of the method for calculating the sol–gel transition time and a comparison with averages calculated with the KCE are presented in Sects. 3 and 4. Finally, Sect. 5 presents a discussion of the limits of applicability of the KCE and an example of correlations in the critical region and conclusions.

The objective of this section is to present a description of the algorithm. A more detailed explanation of the method can be found in Alfonso (2015), and only a brief summary is presented here. The main idea of the algorithm consists of the numerical calculation of all states for a given initial configuration with probability P(n01, n02, …, n0N;0)=1, and the subsequent calculation of the temporal evolution of each state. The time evolution can be performed by considering that the only allowed transitions are of the form n¯1(+)→n¯1 if i≠j and n¯2(+)→n¯2 if i=j, where n¯1(+), n¯1 and n¯2(+), n¯2 are the state vectors: n¯1(+)=(n1,…,ni+1,…,nj+1,…,ni+j-1,…,nN)n¯1=(n1,…,ni,…,nj,…,ni+j,…,nN)n¯2(+)=(n1,…,ni+2,…,n2i-1,…,nN)n¯2=(n1,…,ni,…,n2i,…,nN). For a system consisting of N monomers at t=0, the number of possible configurations increases exponentially and can be approximated from the equation (Hall, 1967): R(N)≈14N3exp⁡π2N/31/2. For example, R(50)=217590 and R(100)=190569232.

The procedure is illustrated for a system with five monomers in the initial state, only for the purpose of demonstrating the method. As the system in this case has only six possible states, it is much easier to explain the details of the calculations. The six possible configurations generated from the initial state (5,0,0,0,0) are displayed in Fig. 1.

In a second step, the probabilities of all generated configurations are updated according to the first order finite difference scheme (Alfonso, 2015): Pn¯;t0+Δt=Pn¯;t0+Δt∑i=1N∑j=i+1NK(i,j)(ni+1)(nj+1)×P(…,ni+1,…,nj+1,…,ni+j-1,…;t0)+Δt∑i=1N12K(i,i)(ni+2)(ni+1)×P(…,ni+2,…,n2i-1,…;t0)-Δt∑i,j=1NK(i,j)ninjP(n¯;t0)-Δt∑i=1N12K(i,i)ni(ni-1)P(n¯;t0). From Eq. (5) it should be clear that the state probabilities Pn¯;t0+Δt at t=t0+Δt will increase if the states from which transitions are allowed have a nonzero probability at t=t0 (second and third terms in the right-hand side of Eq. 5), and will decrease due to collisions of particles from the same state at t=t0 (fourth term and fifth terms in the right-hand side of Eq. 5) if P(n¯;t0) is positive.

The finite difference equation for P(1,0,0,1,0) is presented to illustrate the method. From the generation scheme displayed in Fig. 1, note that the only allowed transitions to (1,0,0,1,0) are from the states (1,2,0,0,0) and (2,0,1,0,0). Consequently, at t=t0+Δt, P(1,0,0,1,0;t0+Δt) will increase if P(1,2,0,0,0;t0) and P(2,0,1,0,0;t0) are positive at t=t0. On the other hand, P(1,0,0,1,0;t0+Δt) will decrease due to collisions from particles within the same state at t=t0 if P(1,0,0,1,0;t0) is positive. Then, P(1,0,0,1,0;t0+Δt) is calculated from the following equation: P1,0,0,1,0;t0+Δt=P1,0,0,1,0;t0+Δt(1/2)K(2,2)(n2+2)(n2+1)×P(1,2,0,0,0;t0)+ΔtK(1,3)(n1+1)(n3+1)P(2,0,1,0,0;t0)-ΔtK(1,4)(n1)(n4)P(1,0,0,1,0;t0).

The time evolution of the probability of each state was calculated for the product kernel K(i,j)=Cxixj, considering C= 5.49 × 1010 cm3 g-2 s-1 following Long (1974) and for the initial condition P(5,0,0,0,0;0)=1. Due to the small number of droplets in the initial configuration (only 5), the simulated volume was set equal to 10-2 cm3, with an initial droplet radius of 17 µm. The time step was Δt=0.1 s. For this case, the time evolution of four of the seven configurations is displayed in Fig. 2.

After the calculation for each state is completed, the expected values for each droplet mass can be found from the following relation (Alfonso, 2015): nm=∑nnP(n,m;t), where the discrete probability mass function is calculated from the state probabilities following the expression: P(n,m;t)=∑Allstateswithnm=nPn1,n2,…,nm=n,…nN;t. The expected values nm calculated from Eq. (7) are the magnitudes that must be compared with the averages N(m,t) obtained from the KCE (Eq. 1).

Collisions between droplets under pure gravity conditions are simulated with the hydrodynamic kernel, which has the following expression: Kg(xi,xj)=π(ri+rj)2V(xi)-V(xj)E(ri,rj). The hydrodynamic kernel takes into account the fact that droplets with different masses (xi and xj and corresponding radii, ri and rj) have different terminal velocities, which are functions of their masses. In Eq. (18), E(ri, rj) are the collection efficiencies calculated according to Hall (1980).

In their pioneering studies using the stochastic framework, Marcus (1968) and Bayewitz et al. (1974) solved the stochastic master equation (Eq. 2) for a constant collection kernel and a monodisperse initial droplet distribution. The latter study revealed significant deviations from the KCE when there is a small number of droplets in the initial distribution (NTotal<50). In our work, we extended the results obtained by these authors by calculating the expected droplet size distributions for small systems within the stochastic framework, but by using collection kernels that are mass dependent and relevant to cloud physics (e.g. multiplicative and hydrodynamic). Our results confirm the findings of Bayewitz et al. (1974) that in systems of small populations the results of the kinetic deterministic equations approach may differ substantially from the stochastic means at the large end of the droplet size distribution.

The application of the KCE to coagulating systems also requires that the particles are well mixed (Bayewitz et al., 1974; Sampson and Ramkrishna, 1985), implying that every pair of droplets is always available for coagulation (Sampson and Ramkrishna, 1985).

Another important assumption is that the droplet population is sufficiently large for the existence of a droplet with particular properties to not be conditionally dependent on the existence or nonexistence of any other droplet. In other words, no correlations are assumed in the system, so that ninj=ninj.

The assumption that the system is sufficiently large is linked to the fact that the KCE is a deterministic equation that simulates only the mean values and gives an incomplete description of the coagulating system if fluctuations about the mean are very large (Ramkrishna and Borwanker, 1973). Since fluctuations are proportional to ∼1/NTotal, a large number of droplets is needed for the fluctuations to be small, and in fact, the larger the number of particles in a system, the smaller the fluctuations. This fact underscores the finite system description adopted in this work, as the collision–coalescence process is limited to pairs of droplets in close proximity to each other. The KCE is not expected to be accurate when the number of droplets or the volume of the system are small.

Additionally, the KCE can fail even if the number of droplets is large when a raindrop embryo forms. At that critical time, there is a transition from a continuous droplet distribution to a continuous distribution plus a raindrop embryo (or runaway droplet). This sol–gel transition is well known in other fields (e.g. astronomy), but has not been sufficiently explored in the context of cloud physics, where the gel would correspond to the raindrop embryo. This approach is developed in this paper through a detailed comparison of expected values calculated from the stochastic framework with averages obtained from the KCE for realistic collection kernels, before and after the sol–gel transition time.

The marked differences between these two approaches at the sol–gel transition can be related to the increase of correlations at the critical point, and that can happen even for a large number of particles in the initial distribution (Malyshkin and Goldman, 2001). When the sol–gel transition occurs, the occupation numbers niof all low-mass bins are strongly anticorrelated with bins from the gel fraction (calculated from Eq. 15). On the other hand, in the vicinity of critical time, the fluctuations for the finite system are larger, since the standard deviation of the mass of the largest droplet, σ(Smax⁡), has a maximum. As a consequence, the differences between the deterministic and stochastic descriptions become larger and divergent. To further analyze this problem, the time evolution of the correlation coefficients ρi,j=cov(ni,nj)Var(ni)Var(nj)=σninjσniσnj between the random variables n1 and ni from bins within the gel fraction (see Eq. 15) were calculated. In the simulations, the gel fraction at t=1310 s covers the interval bins from 29 to 58 µm, and narrows as time increases. The time evolution of the correlation coefficients for different bin pairs ρ1,20, ρ1,25, ρ1,30, ρ1,35 and ρ1,40 are displayed in Fig. 10, showing, in all cases, an increase in the magnitude of the correlation coefficients in the vicinity of the sol–gel transition time. Also in Fig. 10, correlations are quite large in magnitude and negative most of the time for ρ1,20, ρ1,25 and ρ1,30 as the mass of the gel increases through coalescence with droplets from low-mass bins. However, after 2500 s, pairs 1–20, 1–25 and 1–30 are positively correlated as the gel fraction narrows and droplets from bins as large as 20, 25 and 30 are also depleted by the gel. The correlation coefficients for all analyzed pairs have maxima between 1000 and 1500 s, in the vicinity of the critical time. The random variable n40 is always anticorrelated with n1, with values much higher than the other pairs after 1500 s, and increasing in magnitude until the end of the simulation, which reflects the fact that the gel actively grows by collecting smaller droplets. Thus, for realistic collection kernels, the mean values predicted by the KCE will be not accurate after the sol–gel transition. The stochastic approach captures the gel formation and evolution properly, with larger values of the expected droplet mass at the large end of the distribution.

In principle, this analysis could be performed by using the SSA, which is an alternative tool for the master equation formalism (Eq. 2). However, the number of realizations required to obtain smooth behavior at the large end in order to compare it with averages from the KCE, would be extremely large.

It is necessary to emphasize that our method (although it can be computationally expensive) works for any type of kernels, whereas the analytical techniques developed by Lushnikov (2004) and Matsoukas (2015) work only for very special cases.

The failure of the KCE to capture the gel formation could provide an explanation of the inability of explicit microphysics cloud models to explain the droplet spectral broadening observed in small, warm clouds. Therefore, even for large simulation cells, the use of the KCE is justified only in the absence of the sol–gel transition.

For the small-volume approach described in this paper, a model that considers the interaction between small coalescence volumes through sedimentation or other physical mechanisms for realistic collection kernels is needed. For a constant collection kernel, this theory was outlined by Merkulovich and Stepanov (1990, 1991) based on a scheme proposed by Nicolis and Prigogine (1977) for chemical reactions. Within this theory, the whole system is subdivided into spatially homogeneous subvolumes (coalescence cells) that interact through the diffusion process, and the coalescence events are permitted only between droplets from the same subvolume. As a result, we obtain a set of master equations for each subvolume. Although very complex, it could be a starting point for considering the interactions between small coalescence volumes through different physical mechanisms.