Stochasticity of the collisional growth of cloud droplets is
studied using the super-droplet method (SDM) of

Coalescence of hydrometeors is commonly modeled using the Smoluchowski
equation

Another limitation of the Smoluchowski equation is that it describes the
evolution only of the expected number of droplets of a given size. It does
not contain information about fluctuations around this number, which are
suspected to be crucial for precipitation onset

Moreover, although the Smoluchowski equation can be written for the discrete
number of droplets of a given size, it is more often used for droplet
concentrations. This adds an additional assumption that the coalescence
volume is large, somewhat in agreement with neglecting fluctuations in the
number of collisions and correlations in droplet numbers

A number of methods alternative to the Smoluchowski equation exist. They are
capable of addressing stochastic coalescence, but have some shortcomings that
make their use in large-scale cloud simulations impossible. The most accurate
one is direct numerical simulation (DNS). In DNS, trajectories of droplets
are simulated explicitly and collisions occur when they come into contact.
The downside of DNS is that it is computationally extremely demanding.
Running a large ensemble of simulations from which statistics could be
obtained would take a prohibitively long time. An alternative approach is to
use a master equation

Several Lagrangian methods have been developed to model cloud microphysics

In this section we present how collision–coalescence is handled in the
super-droplet method. Further information about the SDM can be found in

At the heart of the super-droplet method is the idea that many droplets with
same properties within a well-mixed volume can be represented by a single
computational entity, called the super-droplet (SD). As we are interested
only in droplet coalescence within a single cell, it is sufficient if SDs are
characterized by two parameters: radius

We will perform two types of simulations. In the “one-to-one” simulations,
all SDs have multiplicity

The second type of simulation, in which the number of SDs remains constant
(with rare exceptions), is closer to the ideas of

The goal of this section is to show that the “one-to-one” SDM is at the
same level of precision as the master equation. To this end, we calculate the
average droplet size distribution and the standard deviation of mass of the
largest droplet from an ensemble of “one-to-one” simulations. As a
reference, we use the results from

Figure

Mass of droplets per size bin at

Relative standard deviation of the mass of the largest droplet in
the system. Details of the SDM simulations are given in the caption of
Fig.

The “one-to-one” SDM with linear sampling is computationally more efficient
than solving the master equation directly, or using the SSA. It also puts no
constraints on the initial distribution of droplets. Therefore we can use the
SDM to predict gelation times for larger systems and more realistic initial
conditions. We use an initial droplet distribution that is exponential in
mass

Relative standard deviation of the mass of the largest droplet for
different cell sizes. Estimated from ensembles of

Mean

Fluctuations in time of conversion of cloud droplets to raindrops were
studied using direct numerical simulations by

Relative standard deviation of

In Fig.

To analyze the amplification of fluctuations in the “constant SD” method,
we plot the relative standard deviation of

Points depict the minimal, limiting value of the relative standard
deviation of

The Smoluchowski equation presents a mean-field description of the evolution
of the size spectrum. It is exact only in the thermodynamic limit
(

We compare the results of the “one-to-one” simulations with solutions of
the Smoluchowski equation for two cases – with fast and with slow rain
development. In both cases collision efficiencies for large droplets are
taken from

Rain content ratio

As in Fig.

Average and standard deviation of time (in s) for lucky realizations
to produce a single drop with

Next, we validate the Smoluchowski equation in a setup with slow rain
development. This time the initial droplet size distribution is below the
size gap, i.e., the range of radii for which both collisional and
condensational growths are slow. We use

In coalescence cells with

Mean concentration of raindrops from the same simulations as in
Fig.

There is a well-established idea that some droplets undergo a series of
unlikely collisions and grow much faster than an average droplet

We are interested in the time

There is a large variability in

Next we calculate the “luck factor”, i.e., how much faster the luckiest
droplets grow to

In the previous sections we have seen that the size of the coalescence cell
has a profound impact on the evolution of the system. In this section we
estimate the size of a cell that can be assumed to be well mixed. All methods
in which the probability of a collision of droplets depends only on the
instantaneous state of the cell and not on its history rely on the assumption
that the cell is well mixed. This includes the master equation, SSA, the SDM
as well as the Smoluchowski equation. The assumption that a cell is well
mixed is valid if

Rigorously, the characteristic time for coalescence is the mean time between
coalescence events, as in diffusion-limited chemical systems

Mean time until a system of

Droplets in the cell can be mixed through turbulence. Turbulence acts
similarly to diffusion and its characteristic time for mixing is

Another process that can mix droplets is sedimentation. It is difficult to
assess its timescale, because it strongly depends on droplet sizes. Droplets
of similar sizes are not mixed by sedimentation, but it is efficient at
mixing raindrops with cloud droplets. We can expect that it would prevent
depletion of cloud droplets in the surroundings of a rain droplet that was
observed for the smallest cells in Sects.

The super-droplet method can exactly represent stochastic coalescence in a
well-mixed volume. It was compared with the master equation approach (see
Sect.

The SDM was used to study stochastic coalescence for two initial droplet size
distributions – with small (

The additional rain–rain collisions do not affect results if droplets are
initially large. Then, collisions of cloud drops and raindrops and between
cloud droplets are frequent, so the increase in the rate of collisions
between raindrops is not important. The mean behavior of the system converges
to the Smoluchowski equation results with increasing cell size. Good
agreement with it is found for cells with

Another aspect of the slow-coalescence scenario is that in it, some lucky
droplets can grow much faster than average droplets. We found that a single
luckiest droplet out of 1000 grows 5 times faster than average and the
luckiest out of a 100 000, 11 times faster. These values are in good
agreement with the analytical estimation of

We estimate a well-mixed (with respect to coalescence) volume in the most
turbulent clouds to be only 1.5

Simulation code is available at

This study was financed by Poland's National Science Center “POLONEZ 1” grant 2015/19/P/ST10/02596 (this project has received funding from the European Union's Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement no. 665778) and Poland's National Science Center “HARMONIA 3” grant 2012/06/M/ST10/00434. Numerical simulations were carried out at the Cyfronet AGH computer center, accessed through the PLGrid portal. We are grateful to Wojciech W. Grabowski for fruitful discussions. Edited by: Ilona Riipinen Reviewed by: two anonymous referees