The impact of stochastic fluctuations in cloud droplet growth is a matter of broad interest, since stochastic effects are one of the possible explanations of how cloud droplets cross the size gap and form the raindrop embryos that trigger warm rain development in cumulus clouds. Most theoretical studies on this topic rely on the use of the kinetic collection equation, or the Gillespie stochastic simulation algorithm. However, the kinetic collection equation is a deterministic equation with no stochastic fluctuations. Moreover, the traditional calculations using the kinetic collection equation are not valid when the system undergoes a transition from a continuous distribution to a distribution plus a runaway raindrop embryo (known as the sol–gel transition). On the other hand, the stochastic simulation algorithm, although intrinsically stochastic, fails to adequately reproduce the large end of the droplet size distribution due to the huge number of realizations required. Therefore, the full stochastic description of cloud droplet growth must be obtained from the solution of the master equation for stochastic coalescence.

In this study the master equation is used to calculate the evolution of the droplet size distribution after the sol–gel transition. These calculations show that after the formation of the raindrop embryo, the expected droplet mass distribution strongly differs from the results obtained with the kinetic collection equation. Furthermore, the low-mass bins and bins from the gel fraction are strongly anticorrelated in the vicinity of the critical time, this being one of the possible explanations for the differences between the kinetic and stochastic approaches after the sol–gel transition. Calculations performed within the stochastic framework provide insight into the inability of explicit microphysics cloud models to explain the droplet spectral broadening observed in small, warm clouds.

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

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:

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

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

State space obtained from the initial condition

In a second step, the probabilities of all generated configurations are
updated according to the first order finite difference scheme (Alfonso,
2015):

The finite difference equation for

The time evolution of the probability of each state was calculated for the
product kernel

Time evolution of the probabilities of 4 of the 7 states for the
initial condition

After the calculation for each state is completed, the expected values for
each droplet mass can be found from the following relation (Alfonso, 2015):

Lushnikov (2004) demonstrated that right after the sol–gel transition, the
particle mass distribution splits into two parts: the
thermodynamically populated one with behavior described by the kinetic
collection equation, and a narrow peak with a mass very close to the gel
mass. For the infinite system described by the KCE
(Eq. 1) with kernel

For a finite system, the standard deviation (

This was explored in previous studies (Inaba, 1999; Alfonso et al., 2008,
2010, 2013), where

For the finite system, the relative standard deviation

The droplet mass spectrum at different times (

The evolution of a system with an initial monodisperse droplet size
distribution of

A discrete 40 bin grid was defined for our model. The mass for bin 1 is taken
to be the mass of a 17

The results for the droplet mass distribution are displayed in Fig. 4 at

Size distributions obtained from the stochastic master equation
(dashed lines) and the KCE (solid lines) at

To proceed further, the previous results are compared to the analytical
size distributions from the KCE (Eq. 1) calculated for the product kernel
with monodisperse initial conditions before (

The comparison of the droplet mass concentration (

Within the master equation approach, the expected value of the mass of the
largest droplet

Within the Monte Carlo stochastic approach (SSA), the expected mass of the
gel is the ensemble mean (

Expected gel mass calculated from the SSA, the master equation and the kinetic approach. Simulations were performed for the product kernel.

After the sol–gel transition, the mass of the gel can also be estimated by
using the infinite system approach from the following relation (Wetherill, 1990):

Collisions between droplets under pure gravity conditions are simulated with
the hydrodynamic kernel, which has the following expression:

For an infinite system modeled by the KCE (Eq. 1) with the hydrodynamic
kernel, the second moment of the mass distribution (

The sol–gel transition time can be estimated approximately by calculating the
time at which the time series of

Expected gel mass calculated from the SSA and the master equation. Simulations were performed for the hydrodynamic kernel.

Time evolution of the total liquid water content calculated from the analytical solution of the kinetic collection equation for the product kernel.

Time evolution of the relative standard deviation

The droplet mass spectrum at different times (

Comparison of the size distributions obtained from the stochastic
master equation (dashed lines) with that to the KCE (solid lines) at

Time evolution of the correlation coefficients for different bin
pairs

Although for the hydrodynamic kernel the critical time was longer than 20 min, we must emphasize that, in general, for concentrations larger than
30–40 cm

The evolution of a system with the initial bidisperse droplet size
distribution described in the previous section is calculated here using the
master equation (Eq. 2) with the initial condition

Before the sol–gel transition, the mass spectrum exponentially decreases with increasing drop radius for both the KCE and the master equation. After the sol–gel transition, there are two types of behavior in the droplet mass distribution of the master equation: (i) an exponential decay that resembles the KCE description, and (ii) a peak in the gel fraction of the distribution, in which the mass is calculated according to Eqs. (14) and (15). As can be observed in Fig. 9, there are substantial differences between the kinetic and the stochastic approaches, especially in the large end of the distribution after the critical time, with much higher values of the droplet mass concentration for the stochastic case.

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 (

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

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

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

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.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This study was funded by the grant “Estancias Sabáticas Nacionales” from CONACYT, in Mexico and it was completed during an academic visit at Centro de Ciencias de la Atmósfera, UNAM. L. Alfonso also thanks the Associate Program of the Abdus Salam International Center of Theoretical Physics (ICTP), in Trieste, for all the support provided for the completion of this paper during the summers of 2015 and 2016. Edited by: G. Feingold Reviewed by: two anonymous referees