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 -5 to +7.5 dBZ in a characteristic time of 333 s. Simulations of the collision and coalescence process, such as those described in the review published by Beard and Ochs (1993), require longer times for precipitation formation, unless giant nuclei (aerosols with diameters greater than 2 µm) are incorporated in the simulation.

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-1, they can contribute significantly to the broadening of the spectrum (Johnson, 1982; Feingold et al., 1999; Yin et al., 2000; Van Den Heever and Cotton, 2007).

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 µm droplets together with collector drops with twice the volume (12.6 µm radius). From this initial bimodal distribution, he found that 100 of the 12.6 µm droplets per cubic meter (a 10-6 fraction), will grow more rapidly than predicted by the continuous growth model, experiencing their first 10 coalescences after a time of approximately 5 min, while the time to undergo 10 collisions assuming continuous growth was about 33 min.

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 10-6 lucky droplets are expected to reach 50 µm 10 times faster than the average droplet. More recently, Wilkinson (2016) further advanced the model by using large deviation theory (Touchette, 2009). He derived the probability for the time T to undergo N collisions being a very small fraction of its mean value and showed that the timescale for the initiation of precipitation is smaller than the mean time for a single collision.

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 r=40 µm compared to the average droplets. They estimated that the luckiest 10-3 fraction will cross the size gap around 5 times faster, and the luckiest 10-5 fraction was around 11 times faster, in good agreement with the results obtained by Kostinksi and Shaw (2005) (about 6 and 9 times faster, respectively).

However, previous efforts in this direction were mainly focused on finding the distribution of times for N collisions (Telford, 1955; Kostinski and Shaw, 2005; Wilkinson, 2016), while we were concentrated on studying the lucky droplet size distribution to determine whether or not the runaway growth process due to collision–coalescence has started.

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.

In this section, the methodology of analysis described before is applied to a dataset of cloud droplet size distribution (2–50 µm) collected with a Droplet Measurement Technologies fog monitor (FM-120) installed on a hilltop in Are, Sweden. The FM-120 is a single-particle optical spectrometer (Spiegel et al., 2012) that derives size from light scattered from individual droplets that pass through a focused laser beam. The equivalent optical size ranges from 2 to 50 µm. The fog monitor sample volume has a cross-sectional area of 0.25 mm2 and a flow speed of 14 m s-1. The raw data consist of each droplet's radius and inter-arrival time (elapsed time since previous particle). More than 7 million droplets were processed over a period of 4 h.

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 cm3 in volume, corresponding to a cloud length of approximately 400 cm (∼0.3 s interval in the time series). The droplet distributions in each unit block are equivalent to the distributions obtained for each realization (for a fixed time) of the Monte Carlo algorithm described in the previous section, and each block can be interpreted as an independent realization of a stochastic process.

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.

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 P(Admixturex), where x is the sample size, is calculated using Monte Carlo simulations. This calculation uses a given number of consecutive blocks with a mixture of distributions. The simulations are carried out by randomly choosing Ntotal samples from the measurements (that consist of consecutive blocks) of size x, fitting the data to the distribution in Eq. (6), and determining if they do or do not follow that distribution. The decision is based on application of the Kolmogorov–Smirnov (K-S) goodness of fit test for a confidence level α=0.05. The experimental statistics for the K-S test can be obtained by arranging the data in ascending order (x1,x2,…,xn) and deriving the maximum difference between the rank statistics (i-1)/n and the theoretically calculated cumulative density function F(xi): 12Dn=max⁡1≤i≤nmax⁡F(xi)-i-1n,max⁡in-F(xi). If this value of Dn is smaller than a certain threshold value Dnα, we accept that the data obey the probability distribution under consideration, and the null hypothesis H0 cannot be rejected at a significance level α. The significance level α refers to the probability of the assumed distribution pattern being rejected. The limiting values of Dnα can be calculated from the K-S cumulative distribution (see Eqs. 5a and 5b). Tables with limiting values can be found in, e.g., Gnedenko (2017).

However, given that the parameters of the distribution F(x) were estimated from the observed data, theoretical limiting values provided by the K-S cannot be used. In this case, the limiting values Dnα are smaller than the case with known parameters and must be obtained via Monte Carlo simulations (see Appendix C for more details). Thus, the conditional probability can be calculated as follows: 13P(Admixturex)=N0/Ntotal, where N0 is the number of cases for which the null hypothesis H0) at α=0.05 cannot be rejected. However, what is really needed is the conditional probability P(xAdmixture), which is the probability that a sample has size x, given that the data (viewed as a time series of maxima for each block) in that sample follow a mixture of distributions. This probability can be calculated using Bayes' theorem from the following expression: 14P(xAdmixture)∝P(Admixturex)π(x). By writing this theorem in the form (14), we are assuming that the marginal likelihood is considered a normalization factor. Therefore, P(xAdmixture) can be computed using expression (14) and then normalized under the requirement that it is a probability mass function (pmf). In Eq. (14), the prior probability π(x) is assumed to have a uniform distribution. Thus, the expected value x can be calculated from the following expression: 15x=∑P(xAdmixture)x. Turning to a concrete example, Ntotal=100 samples with sizes x=100,200,…,1000 were randomly selected from the data, and the probability P(Admixturex) calculated following Eq. (13). The probability mass function P(xAdmixture) (pmf) was obtained by applying the procedure previously described and the expected value was found to be x=544 (about 163 s).

A thorough statistical analysis was conducted by fitting Mmax⁡ to the combined distribution in Eq. (6) for 100 samples with sizes at and below the average (100, 200, 300, …, 500) that were randomly selected from the entire dataset (49 647 blocks). For each random sample three null (H0) hypotheses were verified: (i) the sample comes from a mixture of distributions (Eq. 6), (ii) the sample comes from a Gumbel distribution or (iii) the sample comes from a Gaussian distribution. The three hypotheses were examined by the K-S method with limiting values calculated from Monte Carlo simulations (see Table C1).

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 H0 at α=0.05 was rejected for 13, 35 and 92 samples for the mixture, Gaussian and Gumbel distributions, respectively. For case 2 (sample size 200), the null hypothesis was rejected for 27, 58 and 96 samples. Using n=500 for the mixture of distributions (Eq. 6), the null hypothesis H0 was rejected for 50 samples. For the Gumbel distribution, the null hypothesis was rejected for all the samples (100) and the null hypothesis for the Gaussian distributions was rejected for 83 samples.

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 n=500, the largest droplet mass empirical distributions obtained for four different samples that are distributed following the mixture and the corresponding fit of Eq. (6).

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 η, defined in Eq. (8), takes values between η=+1, -1, which correspond to pure Gaussian and Gumbel distributions, and when -1<η<1 the system is in the pseudo-critical domain. In the disordered phase, fluctuations and correlations are negligible, there are only a few collision events, and Mmax⁡ is the largest part of randomly distributed droplets. In that case, the distribution of the mass of the largest droplets follow a Gumbel distribution. At later times in the evolution of the finite system, there are many collision events and Mmax⁡ is the result of the coalescence of other droplets. There is an additive process, the central limit theorem applies and the mass (or radius) of the largest droplets follows a Gaussian distribution.

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 η was calculated for 1000 samples of size n=500 selected randomly from the data. Figure 8 shows that for 90 % of the samples the ratio η lies in the interval [-0.9, 0.9], clearly indicating that samples are in the pseudo-critical region.

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.