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.
Introduction
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=40µm 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.
An overview of previous theoretical and experimental resultsResults for infinite systems in coagulation and percolation theory
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):
∂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 average concentration of droplets with mass xi at time t,
and K(i,j) is the coagulation kernel related to the probability of coalescence of
two drops of masses xi and xj. In Eq. (1), the first term on the
right-hand side describes the average rate of production of droplets of mass
xi due to coalescence between pairs of drops, whose masses add up to mass
xi, and the second term describes the average rate of depletion of
droplets with mass xi due to their collision and coalescence with other
droplets.
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 (K(i,j)=Cxixj), an attempt to
calculate the second moment of the droplet mass spectrum:
M2(t)=∑i=1∞xi2N(i,t),
leads to the result
3M2(t)=M2(t0)1-CM2(t0)t,4Tgel=CM2(t0)-1.
Thus, after t=Tgel, the second moment may become undefined, and the
total mass of the system starts to decrease (see Appendix A for more
details). This result applies to infinite (with negligible fluctuations and
correlations) coagulating systems in the thermodynamic limit, which is the
limit for a large number K of particles where the volume V is taken to grow in
proportion with the number of particles. Then, in the limit K,V→∞,K/V→N<∞. The infinite system interpretation of the sol–gel transition assumes the
presence of a gel phase (which is not predicted by the KCE equation) and
introduces an additional assumption as to whether or not the gel interacts
with the infinite size clusters that are not described by the KCE equation.
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 p of active bonds is set randomly
between sites. Then clusters of size s are defined as an ensemble of s-occupied
sites connected by active bonds. For a definite value of p=pc, a
macroscopic cluster appears, corresponding to the sol–gel transition.
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:
K1(z)=∑k=-∞∞-1ke-k2π2z/6,
or the equivalent expression:
K1(z)=6πz∑k=-∞∞e-3(2k+1)2/(2z).
Botet and Płoszajczak (2005) also found evidence (from numerical solutions
of the KCE equation) that, for multiplicative coalescence (with a collection
kernel proportional to the product of the masses), the largest cluster
follows the distribution in Eqs. (5a) and (5b) at the time of the phase transition. At
this point, a hypothesis is formed in which the results obtained in
percolation are extrapolated in order to find the probability distribution
of the largest (runaway) droplet at t=Tgel.
Some theoretical and experimental results for finite systems in coagulation theory and nuclear physics
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 N=512 monomers), that the distribution of the largest cluster in
the pseudo-critical region can be described as a mixture of the well-known
Gaussian and Gumbel distributions:
f(x,θ,μ1,β,μ2,σ)=θGumbel(x,μ1,β)+(1-θ)Gauss(x,μ2,σ).
In Eq. (6), the coefficients θ and (1-θ) are the mixture
weights (probabilities associated with each component). The individual
distributions Gumbel(x,μ1,β) and Gauss(x,μ2,σ)
are the mixture components.
The Gumbel distribution is one of the asymptotic distributions from extreme
value theory (EVT) and has the following form:
Gumbel(x,μ,β)=e-e-(x-μ)/β,
where μ is the position parameter and β the scale parameter. The
distribution in Eq. (6) has its origin in the fact that, for finite systems
in the pseudo-critical zone, the system experiences large fluctuations and
the gel distribution is a combination of both distributions, a Gumbel and a
Gaussian (Gruyer et al., 2013). A similar result was obtained by Botet (2011)
using synthetic data from stochastic simulations, for collision
probabilities proportional to the product of the masses.
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).
Analysis of the largest droplet distribution obtained from synthetic dataResults for the product kernel (K(i,j)=Cxixj)
For synthetic data analysis, the empirical distributions of the largest
droplet mass (Mmax) were obtained from Monte Carlo simulations,
following Botet (2011). The species-accounting formulation (Laurenzi et al.,
2002) of the stochastic simulation algorithm (SSA) of Gillespie (1975), which
rigorously accounts for fluctuations and correlations in a coalescing system,
was used for the stochastic simulation in this work (see Appendix B).
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 r or mass in the simplest case)
and position. However, Gillespie's SSA works perfectly for our purposes
because, due to the finiteness of our systems, our simulations are performed
for small volumes with a small number of droplets (in the range 50–300 cm-3).
Our methodology of synthetic data analysis consists of generating
N-realizations (at each time step) using the algorithm of Gillespie. For each
realization, there is a certain distribution of droplets. The largest
droplet mass obtained from each distribution at each realization (for a
fixed time step) would be the distribution to be fitted to the distribution
in Eq. (6). Thus, the sample size would be equal to the number of realizations
of the Monte Carlo algorithm.
Simulations were performed for the product kernel
(K(i,j)=Cxixj), with an initial mono-disperse distribution of 100
droplets of 14 µm in radius (droplet mass 1.15×10-8 g) in
a cloud volume of 1 cm3, with C=5.49×1010 cm3 s-1.
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 C (C=5.49×1010 cm3 g-2 s-1) in the product kernel is the result of the
polynomial approximation K(x,y)=A+B(x+y)+Cxy (Long, 1974) of the
hydrodynamic collection kernel (Eq. 11).
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 (Mmax) both
around and far from the sol–gel transition time (Tgel), which was
calculated from Eq. (4) and found to be equal to 1378 s.
Panels (a)–(d) (histograms) show the largest droplet mass distributions calculated from
Monte Carlo simulations at four different times, for a system with an
initial mono-disperse distribution of 100 droplets of 14 µm in radius. The
solid line shows the fit using Eq. (6).
Figure 2 presents the time evolution of the coefficient θ, which
represents the mixing fraction in Eq. (6) for the time interval [500 s,
2000 s]. Despite the noisy behavior of the coefficient θ (due to the
finiteness of the system), there is a decreasing trend with time, showing
larger values of θ (∼0.65) for times close to 500 s
and values down to 0.2 at the end of the time interval. This figure
indicates that, although the largest droplet distribution is adequately
described by a mixture of Gaussian and Gumbel distributions, it has a larger
Gumbel component (see Eq. 6) during the early stages of the coagulation
process. As time progresses, the Gaussian contribution becomes more
important (smaller values of θ) in providing a better fit to the
largest droplet mass distribution.
Time evolution of the coefficient θ in Eq. (6), obtained for a simulation with the product kernel.
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 θ=0.5 at the infinite
system critical point, Tgel, found to be 1378 s from Eq. (4). However,
due to the finiteness of the system, the critical point corresponds
approximately to a value θ=0.35 (see Fig. 2).
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):
η=wGaussian-wGumbelwGaussian+wGumbel,
where wGumbel=θ and wGaussian=1-θ are the relative
weights of the Gumbel and Gaussian distributions, respectively (see Eq. 6).
By definition, η=+1,-1 corresponds to pure Gaussian and Gumbel
distributions. If -1<η<1, the system is in the pseudo-critical domain.
Alternatively, Botet (2011) estimates the limits of the pseudo-critical
region as the times when the largest droplet mass standard deviation
σ(Mmax) calculated from Eq. (9) is small.
σ(Mmax)=1Nr∑i=1Nr(Mmaxi-Mmax)2
In Eq. (9), Nr is the number of iterations of the stochastic simulation
algorithm of Gillespie (1975), Mmax the mass of the largest particle
and Mmax its ensemble mean over all
the realizations.
Even though the second moment of the distribution M2(τ) diverges
(see Eq. 3) for the infinite system, there is no divergence of the second
moment for a finite system (with no critical behavior). For that case, the
standard deviation for the largest particle mass (σ(Mmax)) is
expected to reach a maximum in the vicinity of Tgel=CM2(t0)-1. Moreover, computing the time evolution of the
normalized standard deviation σ(Mmax)/Mmax instead of σ(Mmax) yielded better results in
estimating Tgel in Inaba et al. (1999), Alfonso et al. (2008, 2010, 2013),
and Alfonso and Raga (2017).
Figure 3a shows the time evolution of σ(Mmax)/Mmax, as an example for the system defined at the beginning of
this section. Note that the maximum occurs at T=1315 s, close to
Tgel=1378 s calculated from Eq. (4), and the time when the maximum of
σ(Mmax)/Mmax occurs
is a reliable estimate of the sol–gel transition time for the corresponding
infinite system.
For the finite system, the normalized standard deviation σ(Mmax)/Mmax of the largest droplet
mass versus time (a). The initial number of droplets was set equal to
N=100 droplets of 14 µm in radius in a volume of 1 cm3.
Simulations were performed with the product kernel K(i,j)=Cxixj
(with C=5.49×1010 cm3 g-2 s-1), and Nr=1000
realizations of the stochastic algorithm were performed. The maximum value
of σ(Mmax)/Mmax is found to
be 1315 s (dashed vertical line) and is very close to the sol–gel
transition time (continuous vertical line) for the infinite system (1378 s). In panel (b) the small end of the pseudo-critical domain is estimated as
the time where σ(Mmax)=0.1σmax.
Botet (2011) defines σ=0.1σmax as the limits of the
pseudo-critical interval, which corresponds to tinf=0.37Tgel and
tsup=1.5Tgel (see Fig. 3b). While Eq. (8) could be used to
determine if a sample collected inside a cloud is in the pseudo-critical
region, Eq. (9) implies that the time evolution of σ(Mmax) is
needed, and therefore a practical application is only viable in the case of
synthetic data obtained from stochastic simulations or cloud droplet data
collected dynamically at different times or cloud levels.
Numerical results for turbulent conditions
In our simulations, turbulent effects were considered by implementing the
turbulence-induced collision enhancement factor PTurb(xi,xj) that is calculated in Pinsky et al. (2008) for a cumulonimbus with
dissipation rate ε=0.1 m2 s-3 and Reynolds number
Reλ=2×104 and for cloud droplets with radii
≤21µm. The turbulent collection kernel has the following form:
KTurb(xi,xj)=PTurb(xi,xj)Kg(xi,xj),
where Kg(xi,xj) is hydrodynamic kernel, which considers
collisions between droplets under pure gravity conditions and has the following form:
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 V(xi), which are functions of their
masses. In Eq. (10), E(ri) and E(rj) are the collection efficiencies calculated
according to Hall (1980).
Monte Carlo simulations were performed with an initial bi-modal distribution
(200 droplets of 10 µm in radius and 50 droplets of 12.6 µm) for
a cloud volume of 1 cm3.
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-3, with an average
of 146 cm-3 (see Fig. 6).
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 σ(Mmax)/Mmax is evaluated from 1000 realizations of the Monte Carlo
algorithm (see Fig. 4), which reaches its maximum at around 1815 s and
serves as an estimate for the sol–gel transition time for the infinite
system. Four empirical probability distributions were fitted to the combined
distribution (Eq. 6) for times in the vicinity of Tgel. The results are
displayed in Fig. 5a–d. Note also that for this case, the combined
distribution (Eq. 6) provides a good fit for the largest droplet mass.
Moreover, the coefficient θ decreases in time (check Fig. 5), in
concordance with the scenario described in Sect. 3.1.
Time evolution of the normalized standard deviation σ(Mmax)/Mmax of the largest droplet mass
versus time estimated from the Monte Carlo algorithm. The simulations were
performed for the turbulent hydrodynamic kernel with a bi-disperse initial
condition (200 droplets of 10 µm in radius and 50 droplets of 12.6 µm) in a volume of 1 cm3.
Panels (a)–(d) (histograms) show the simulated Mmax distributions in a system with an
initial bi-disperse distribution (200 droplets of 10 µm in radius and
50 droplets of 12.6 µm) at four different times. The
solid line shows the fit using Eq. (6). The simulations were performed for the turbulent hydrodynamic kernel.
Analysis of the largest droplet (gel) radius distribution from observations
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.
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 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):
Dn=max1≤i≤nmaxF(xi)-i-1n,maxin-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:
P(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:
P(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:
x=∑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.
For each sample size, the number of samples with the null hypothesis H0
rejected at α=0.05 for all the distributions.
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 p value of the
goodness of fit test and the parameter θ indicating the weight of
the Gumbel component.
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).
Discussion and conclusions
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.
Histogram of the ratio η=(wGaussian-wGumbel)/(wGaussian+wGumbel), which measures the distance to the critical
point.
Data availability
Access to the data used to produce the results discussed in this paper are
available from the first author upon request by email.
The sol–gel transition time Tgel is defined as the time when the second
moment M2(t)=M2(t0)1-CM2(t0)t becomes
infinite, then 1-CM2(t0)t=0 and Tgel=CM2(t0)-1. The equation for M2(t) (moment of order 2 with respect
to mass) can be found from the general equation for moment evolution that
was obtained by Drake (1972) from the continuous form of the kinetic
collection Eq. (1). It has the following form:
dMn(t)dt=12∫0∞∫0∞(x+y)n-xn-ynK(x,y)⋅N(x,t)N(y,t)dxdy.
In Eq. (7), K(x,y) is the collection kernel, N(x,t) is the average droplet concentration and
x is the droplet mass. If we consider the product kernel K(x,y)=C(xy) in Eq. (A1), then the equation for the second moment is
dM2(t)dt=C[M2(t)]2,
with the solution being M2(t)=M2(t0)1-CM2(t0)t.
After Tgel, a runaway droplet forms, and the kinetic collection
equation is no longer valid, since the assumption of a continuous
distribution breaks down. There is, in essence, a phase transition in the
system from a continuous distribution to a continuous distribution plus a
runaway droplet.
The Monte Carlo algorithm
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:
Initialization. Initialize the number of droplets in each species (the species
are defined as droplets of different sizes). There is a unique index μ for
each pair of droplets i,j that may collide. For a system with N species, n1,n2,…,nNμ∈NN+12. The set
μ defines the total collision space and is equal to
the total number of possible interactions.
Monte Carlo step. Determine the next collision to occur and the time to the
next collision. The next collision μ is calculated according to the
distribution Pμ=aμα, from the
inequality:∑ν=1μ-1aν<r2α≤∑ν=1μaν,where r2 is a uniformly distributed random number in the interval (0,1).
aμ is calculated from the following probabilities:
a(i,j)dt=V-1K(i,j)ninjdt≡Pr{two unlike particles i and j with populations (number of particles) ni
and nj will collide within the imminent time interval},
a(i,i)dt=V-1K(i,i)nini-12dt≡Pr{two particles of the same species i with
population (number of particles) ni collide within the imminent time
interval},
α=∑ν=1NN+12aν.
As the time to the next collision is exponentially distributed with
mean 1/α (Gillespie, 1975) and 1-r1=r1∗ is a uniformly distributed random number in the interval [0,1], the
time τ to the next collision can be calculated from the following expression:τ=1αln1r1∗.
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.
Procedure for estimating the limiting values for the
Kolmogorov–Smirnov goodness of fit test for distributions with unknown
parameters
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 p values can be obtained via Monte
Carlo simulations. First, the parameter vector ϕ^=θ^,μ^1,μ^2,β^,σ^ is estimated for a given sample of size n, and the
test statistics (Eq. 12) are calculated assuming that the sample is
distributed according to Fx;ϕ^,
returning a value of Dn. Next, a sample of size nFx;ϕ^ variates is generated, and the parameter
vector ϕ^1 is estimated. The test statistics are
calculated again assuming that the sample is distributed according
to Fx;ϕ^1. Such a
calculation was made for different sample sizes (n=100,200,…,500) 1000 times, and the distribution pattern of Dn was derived (see
Table C1). Thus, 5 % (for α=0.05) from the greater
side was taken as the estimated Dnα=0.05 limiting values. The
estimate of p value is calculated as the relative number of occasions in
which the test statistics are at least as large as Dn. The numerically
calculated K-S limiting values for the three distributions under analysis
(mixture, Gumbel and Gaussian) for α=0.05 are shown in Table C1.
As can be checked, the values are smaller than the case with known
parameters, which can be estimated (for α=0.05) as 1.36/n.
Estimated limiting values (for α=0.05) for the
Kolmogorov–Smirnov goodness of fit test for the three distributions.
Sample sizeK-S (estimated) limiting values (Dn) for α=0.05MixtureGaussianGumbel1000.07250.08730.08532000.04940.06240.06303000.04320.05170.04874000.03690.04610.04195000.03240.04140.0396Author contributions
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.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
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.
Financial support
This research has been supported by the Consejo Nacional de Ciencia y Tecnología (grant no. CB-284482.).
Review statement
This paper was edited by Sachin S. Gunthe and reviewed by two anonymous referees.
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