In recent papers (Alfonso et al., 2013; Alfonso and Raga, 2017) the sol–gel transition was proposed as a mechanism for the formation of large droplets required to trigger warm rain development in cumulus clouds. In the context of cloud physics, gelation can be interpreted as the formation of the “lucky droplet” that grows by accretion of smaller droplets at a much faster rate than the rest of the population and becomes the embryo for raindrops. However, all the results in this area have been theoretical or simulation studies. The aim of this paper is to find some observational evidence of gel formation in clouds by analyzing the distribution of the largest droplet at an early stage of cloud formation and to show that the mass of the gel (largest drop) is a mixture of a Gaussian distribution and a Gumbel distribution, in accordance with the pseudo-critical clustering scenario described in Gruyer et al. (2013) for nuclear multi-fragmentation.

A fundamental, ongoing problem in cloud physics is associated with the
discrepancy between the times modeled and observed for the formation of
precipitation in warm clouds. Observational studies show that precipitation
can develop in less than 20 min. For example, in Göke et al. (2007),
an analysis of radar observations in the framework of the Small Cumulus
Microphysics Study (SCMS), demonstrated that maritime clouds increased their
reflectivity from

Numerous mechanisms have been proposed to close the gap between observations and simulations. Some theories explain this phenomenon as an increase in collision efficiencies due to turbulence (Wang et al., 2008; Pinsky and Khain, 2004; Pinsky et al., 2007, 2008), turbulence-enhanced collision rate of cloud droplets (Falkovich and Pumir, 2007; Grabowski and Wang, 2013) or turbulent dispersion of cloud droplets (Sidin et al., 2009).

More recent papers (Onishi and Seifert, 2016; Li et al., 2017, 2018, and Chen et al., 2018) also investigated the effect of turbulence in early development of precipitation.

Other research points to the supersaturation fluctuations resulting from
homogeneous (Warner, 1969) and inhomogeneous mixing (Baker et al., 1980),
which allow some droplets to grow faster by condensation in areas with
larger supersaturation. Cooper (1989) found evidence of faster growth of the
larger droplets due to the variability that results from mixing and random
positioning of droplets during cloud formation. Shaw et al. (1998) explored
the possibility that vortex structures in a turbulent cloud cause variations
in droplet concentration and supersaturation (at the centimeter scale),
allowing droplets in areas of higher concentration to grow more rapidly.
Their calculations show an important widening of the spectrum from this
mechanism. Roach (1976) showed that the growth of larger droplets increases
due to radiative cooling at the top of stratiform clouds and the addition
of sulfate cloud condensation nuclei (CCN), activated as droplets as a result
of aqueous-phase chemical reactions (Zhang et al., 1999). In the same manner,
Feingold and Chuang (2002) proposed the theory that certain organic
compounds (film-forming compounds) can create a layer around droplets that
inhibits their growth, causing a fraction of droplets to grow under
conditions of higher supersaturation with the consequent widening of the
spectrum. The existence of giant CCN is another of the proposed mechanisms.
Even at concentrations as low as 1 L

More recently, the sol–gel transition has been proposed as a possible mechanism for the formation of embryonic drops that trigger the formation of precipitation (Alfonso et al., 2010, 2013). Although this phenomenon is not as well known in the field of cloud physics, the sol–gel transition (also known as “gelation” in English-language literature), has been widely studied in other fields of research to explain, for example, the formation of planets (Wetherill, 1990) and aerogels in aerosol physics (Lushnikov, 1978) or the emergence of giant components in percolation theory (Aldous, 1999).

In the framework of cloud physics, the sol–gel phenomenon can be interpreted as the formation of the lucky droplet that becomes the embryo for raindrops and is defined by a transition from a continuous system of small droplets, to another system with a continuous spectrum plus a giant drop (runaway droplet, embryonic drop, gel) that interacts with the system increasing its mass by accretion with the smallest drops.

Telford (1955) may be the first to propose the “lucky droplet” model for
collision–coalescence of cloud droplets. One of the novelties of Telford's
approach was to recognize the shortcomings of the “continuous growth
model” and took into account the statistical fluctuations inherent to the
collision–coalescence process and its discrete nature. He performed his
analysis for a cloud consisting of identical 10

The lucky droplet model was further developed by Kostinski and Shaw (2005),
who presented numerical evidence that their model can lead to a rapid
development of precipitation. Their analysis was based on the derivation of
the distribution of times for N collisions (which gave the result of an Erlang
distribution). They concluded that the 10

The results obtained by Kostinski and Shaw (2005) were tested by Dziekan and
Pawlowska (2017) by calculating the “luck factor”, i.e., how much faster the
luckiest droplets grow to

However, previous efforts in this direction were mainly focused on finding
the distribution of times for

Recent studies that address the sol–gel transition interpretation in cloud physics (Alfonso et al., 2013; Alfonso and Raga, 2017) analyze the problem from the theoretical and simulation point of view. The aim of the present work here is to find observational evidence of gel formation, taking as a reference recent studies in percolation theory (Botet and Płoszajczak, 2005) and nuclear physics (Botet et al., 2001; Gruyer et al., 2013), which can shed some light on the gel (largest droplet) size distribution during the initial stage of precipitation formation.

The paper is organized as follows: Sect. 2 presents an overview of previous results for both infinite and finite systems. An analysis of the largest droplet distribution from synthetic data obtained from Monte Carlo simulations (for the product and hydrodynamic kernels, respectively) is presented in Sect. 3. Sect. 4 is devoted to the analysis of experimental data. Finally, in Sect. 5 we discuss our results accompanied by the relevant conclusions.

The most commonly accepted approach to modeling the collision coalescence
process in cloud models with detailed microphysics relies upon the
Smoluchowski kinetic equation or kinetic collection equation (KCE),
governing the time evolution of the average number of particles. The
discrete form of this equation can be written as follows (Pruppacher and Klett,
1997):

However, the KCE may have a serious limitation in some cases (Lushnikov,
2004) and hence cannot accurately describe the coagulation process. The
limitation essentially lies in the fact that the coagulation equation
inevitably creates particles with infinite mass. For example, for a
multiplicative coagulation kernel (

The other scenario considers that coagulation takes place in a system with a finite number of monomers in a finite volume. This approach is based on the scheme developed by Markus (1968) and Bayewitz et al. (1974) and was studied by Lushnikov (1978, 2004), Tanaka and Nakazawa (1993, 1994), and Matsoukas (2015) by using analytical tools and more recently by Alfonso (2015) and Alfonso and Raga (2017) numerically. Within this approach there is no mass loss, and the phase transition is manifested in the emergence of a giant particle that contains a finite fraction of the total mass of the system. Solutions in the post-gel regime were obtained analytically by Lushnikov (2004) and Matsoukas (2015) and numerically by Alfonso and Raga (2017).

The sol–gel transition has been observed experimentally. For example,
aerogels in aerosol physics (Lushnikov et al., 1990) and in other
theoretical models, such as that of percolation (Botet and Płoszajczak,
2005; Kolb and Axelos, 1990), where there is a close analogy between
percolation and droplet coagulation. In bond percolation, each lattice
corresponds to a monomer, and a proportion

Recent results in percolation theory show that the largest cluster follows
the Gumbel distribution for subcritical percolation (Bazant, 2000) and, at
the critical point, follows the Kolmogorov–Smirnov (K-S) distribution (Botet
and Płoszajczak, 2005). The K-S distribution is the distribution of the
maximum value of the deviation between the experimental realization of a
random process and its theoretical cumulative distribution, and it has following the
cumulative distribution:

We will now consider some results obtained for finite systems in coagulation theory (Botet, 2011) and in nuclear physics (Gruyer et al., 2013). Unlike those in infinite systems, fluctuations and correlations in a finite system are not negligible.

We must emphasize that phase transitions cannot take place in a finite system. This is due to the fact that a phase transition is defined as a singularity in the free energy or any thermodynamic property of a system. For finite-sized systems, the free energy is proportional to the logarithm of a finite number of exponentials, which are always positive (Bhattacharjee, 2001). Consequently, those singularities are only possible within infinite systems by taking the thermodynamic limit. Thus, for finite systems, the notion of pseudo-critical region is introduced (which is the finite system equivalent of a sol–gel transition time).

Some interesting simulation and experimental results were obtained for these
systems in Botet (2011) for the Smoluchowski model (1) and in Gruyer et al. (2013) for nuclear multi-fragmentation. Botet et al. (2001) found, from
stochastic simulations of coagulation process with the product kernel (for a
system of

The Gumbel distribution is one of the asymptotic distributions from extreme
value theory (EVT) and has the following form:

The fundamental hypothesis of our work is that the gel mass (largest drop) in the initial phase of precipitation formation is distributed as a mixture of two asymptotic distributions: one Gumbel and one Gaussian, following the pseudo-critical clustering scenario described in Gruyer et al. (2013).

For synthetic data analysis, the empirical distributions of the largest
droplet mass (

The main difference between the Gillespie's SSA and other Monte Carlo
methods based on the simulation particles (SIPs) approach (like the super
droplet method developed by Shima et al., 2009) is that the Gillespie's
SSA involved the collision of only two physical particles (droplets in our
case) per MC cycle, while the approach based on SIP in each MC cycle
collides SIP (super droplets, for example), which represents multiple numbers
of droplets with the same attributes (radius

Our methodology of synthetic data analysis consists of generating

Simulations were performed for the product kernel
(

The product kernel is proportional to the product of the masses of the
colliding droplets. It is widely used because analytical solutions of the
KCE or Smoluchowski equation (Eq. 1) have been obtained for this kernel by
Golovin (1963), Scott (1968), Drake (1972), and Drake and Wright (1972). The
value of the constant

The empirical distribution of the maxima was obtained for 1000 realizations of the stochastic algorithm. There is no need for a larger number of realizations to get better statistics, since the number of realizations in our Monte Carlo algorithm must be equal to the sample size in the application of the block maxima (BM) approach (see the next section for more details). On the other hand, this number is not much bigger than the number of blocks in the data for which the largest droplet maxima was fitted to fog data.

Figure 1a–d present the largest droplet mass empirical distributions
obtained at four different times. Note that Eq. (6) provides a good fit for
the distribution of the mass of the largest droplet (

Panels

Figure 2 presents the time evolution of the coefficient

Time evolution of the coefficient

These findings are in accordance with Gruyer et al. (2013) and Botet (2011): at an early stage of coagulation development, correlations are negligible, and, consequently, the largest fragments can be considered independent random variables. Therefore, the probability distribution of the largest fragment is given by the limit theorem for extremal variables, which states that the maximum of sample-independent and identically distributed random variables can only converge in distribution in the form of one of three possible distributions: Gumbel, Fréchet or Weibull.

As the coagulation process continues, fluctuations and correlations between droplets increase and the system reaches a critical point (Alfonso and Raga, 2017). Where the largest droplets are no longer independent random variables, the limit theorem for extremal variables no longer applies, and the largest droplet distribution is no longer described by a Gumbel distribution. At later times, away from the pseudo-critical region, the Gaussian contribution is the most important part of the largest droplet mass distribution. This can be explained by the additive nature of the process at this stage (Botet, 2011; Gruyer et al., 2013; Clusel and Bertoin, 2008), and the central limit theorem applies.

In the intermediate zone (which can be defined as the pseudo-critical zone),
the distribution is well described by a mixture of Gumbel and Gaussian
distributions and the weights associated with each distribution are
comparable. It is expected that it can be observed that

We can find whether or not a system is in the pseudo-critical region by
defining the following ratio (Botet, 2011; Gruyer et al., 2013):

Alternatively, Botet (2011) estimates the limits of the pseudo-critical
region as the times when the largest droplet mass standard deviation

Even though the second moment of the distribution

Figure 3a shows the time evolution of

For the finite system, the normalized standard deviation

Botet (2011) defines

In our simulations, turbulent effects were considered by implementing the
turbulence-induced collision enhancement factor

Monte Carlo simulations were performed with an initial bi-modal distribution
(200 droplets of 10

As we want to perform simulations for small systems (with a small number of
particles) for which fluctuations and correlations are relevant, the number
of droplets per cubic centimeter used in the simulations are small. They are of the
same order of the droplet concentrations for each block obtained from
observations, which fluctuate between 0 and 392 cm

The empirical distribution for the largest droplet mass was generated by
extracting the maximum from the droplet distribution at each realization
for a fixed time step. Additionally, the ratio

Time evolution of the normalized standard deviation

Panels

In this section, the methodology of analysis described before is applied to
a dataset of cloud droplet size distribution (2–50

The block maxima (BM) approach in extreme value theory (EVT) was applied, which requires dividing the observation period into nonoverlapping periods of equal size and restricts attention to the maximum observation in each period (see Gumbel, 1958).

Following the BM approach, considering the sectional area and flow speed,
the time series was divided into consecutive unit blocks of 1 cm

The maximum (radius of the largest droplet) is recorded from each consecutive unit block in order to generate the distribution for comparison with the theoretical combined distribution described in Eq. (6). The sample size corresponds to the number of consecutive blocks in which the time series was divided, which in this case is 49 647 blocks, equivalent to about 4 h of data. Figure 6 displays the number of droplets in each block, which fluctuate between 0 and 392, with an average of 146. Since each block is considered a realization of a random process, the largest droplet radius series must be fitted to the combined distribution in Eq. (6) for samples with certain conditions of homogeneity.

Time series of the number of droplets per block, sampled at a hilltop in Are, Sweden.

The average sample size (number of unit blocks) for which the largest droplet maxima can be fitted to the combined distribution in Eq. (6) is then estimated. This expected value can be calculated from the following procedure.

The conditional probability

However, given that the parameters of the distribution

A thorough statistical analysis was conducted by fitting

The results for sample sizes 100, 200, 300, 400 and 500 are shown in Table 1. As an example, for case 1 (sample size 100) the null hypothesis

For each sample size, the number of samples with the null hypothesis

For four random samples that are distributed following the admixture
distribution (with sample size 500), observed (histogram) and fitted (solid
line) using Eq. (6). Also shown for each distribution are the

The results shown in Table 1 confirm that, for all sample sizes, the mixture
of distributions provides a better fit than the Gumbel and Gaussian
distributions, confirming the correctness of the choice of the mixture of
distributions (Eq. 6) for modeling the largest droplet radius. As an
example, Fig. 7a–d present, for a sample size of

An infinite system has two possible evolutionary phases: the ordered phase and the disordered or statistical phase. In the disordered phase there is a continuous droplet distribution and a near equality of the largest and second-largest mass. After the sol–gel transition, there is an ordered phase characterized by the existence of a giant macroscopic droplet (gel) coexisting with an ensemble of microscopic particles.

A finite system can be in the ordered, disordered and pseudo-critical
phases, according to the scenario described in Botet (2011) and Gruyer et
al. (2013). The ratio

In the pseudo-critical phase, the fluctuations and correlations are no
longer negligible and the distribution is of neither of the
asymptotic forms (Gumbel or Gaussian). In this case, the fit of the largest
droplet mass (gel), is a mixture of a Gumbel (disordered state) and Gaussian
(ordered state) distributions. As was demonstrated in the preceding section,
this combined distribution (Eq. 6) is a good approximation to the largest
droplet distribution (gel) in the pseudo-critical region. The fact that the
mixture of distributions provides a better fit than the Gumbel and Gaussian
distributions shows that the samples selected in our study are mainly in the
pseudo-critical phase. To confirm this fact, the ratio

We could show that the gel radius (largest droplet) is described as a mixture of the two asymptotic distributions because the effect of the collision–coalescence process was in some way isolated for the orographic cloud data analyzed in this report. A similar analysis could be performed for the early stage of a convective cloud formation, before some other processes, e.g., entrainment, mixing, turbulence or ice formation, could obscure the finite system pseudo-critical scenario, and the gel formation that is basically a consequence of the collision–coalescence process could no longer be observed.

In this work, the early stage of formation of a warm cloud is viewed in the context of critical phenomena theory and can be thought of as being in ordered, disordered or pseudo-critical phases. The disordered phase corresponds to a cloud with a droplet spectrum formed mainly by the cloud condensation nuclei activation process, with an almost random distribution of particles, and the distribution of the mass of the largest droplets is Gumbel. In the pseudo-critical phase a giant droplet (gel) locally coexists with a continuous ensemble of small droplets. As the system considered is finite, there is no sudden change from disordered to ordered phase (sol–gel transition), but instead there is a pseudo-critical phase in which fluctuations are important and the gel distributes according to Eq. (6). The analysis presented here of the largest droplet distribution provides useful insight into the early stages of cloud development in warm clouds. In follow-up studies, the analysis of cloud data at different times or distances from the cloud base would be helpful in identifying the pseudo-critical phase and tracking the transition from the disordered to the ordered phase dynamically.

Histogram of the ratio

Access to the data used to produce the results discussed in this paper are available from the first author upon request by email.

In this study, the species accounting formulation (Laurenzi et al., 2002) of
the stochastic simulation algorithm (SSA) of Gillespie (1975) was used for
the stochastic simulation. The steps below summarize the algorithm:

As the time to the next collision is exponentially distributed with
mean

Increase the time by the randomly generated time in Step 2. Change the numbers of species to reflect the execution of a collision.

If stopping criteria are not met, go to step 2.

When parameters of a distribution are estimated from the data, the limiting
values provided for the Kolmogorov–Smirnov criterion cannot be used. In this
case, approximate limiting values and

Estimated limiting values (for

LA performed the model runs of the Monte Carlo algorithm, performed the statistical analysis of synthetic and observational data, and wrote the paper. GBR and DB provided the observational data and edited and improved the paper. All authors contributed to developing the basic ideas and discussing the results.

The authors declare that they have no conflict of interest.

This study was funded by a grant from the Consejo Nacional de Ciencia y Tecnología of Mexico (SEP-Conacyt) CB-284482. We also thank the two anonymous reviewers for their helpful comments on our paper.

This research has been supported by the Consejo Nacional de Ciencia y Tecnología (grant no. CB-284482.).

This paper was edited by Sachin S. Gunthe and reviewed by two anonymous referees.