Condensational growth of cloud droplets due to supersaturation fluctuations
is investigated by solving the hydrodynamic and thermodynamic equations using
direct numerical simulations (DNS) with droplets being modeled as Lagrangian
particles. The supersaturation field is calculated directly by simulating the
temperature and water vapor fields instead of being treated as a passive
scalar. Thermodynamic feedbacks to the fields due to condensation are also
included for completeness. We find that the width of droplet size
distributions increases with time, which is contrary to the classical theory
without supersaturation fluctuations, where condensational growth leads to
progressively narrower size distributions. Nevertheless, in agreement with
earlier Lagrangian stochastic models of the condensational growth, the
standard deviation of the surface area of droplets increases as

The growth of cloud droplets is dominated by
two processes: condensation and collection.
Condensation of water vapor on active cloud condensation nuclei
is important in the size range from the activation size of aerosol particles to about
a radius of

Condensational growth due to supersaturation fluctuations
was first recognized by

Recent laboratory experiments and observations about cloud microphysics
also confirm the notion that supersaturation fluctuations may play
an important role in broadening the size distribution of cloud
droplets. The laboratory studies of

In an attempt to answer this question, we conduct 3-D DNS experiments
of condensational growth of cloud droplets, where turbulence, thermodynamics,
feedback from droplets to the fields via the condensation rate,
and buoyancy force are all included. The main aim is to investigate
how supersaturation fluctuations affect the droplet size
distribution.
We particularly focus on the time evolution of the size distribution

We now discuss the basic equations where we combine
the Eulerian description of the density (

The background airflow is almost incompressible and thus obeys the Boussinesq approximation.
Its density

List of constants for the thermodynamics: see text for explanations of symbols.

In addition to the Eulerian fields described in Sect.

Each superparticle is treated as a
Lagrangian point particle, where one solves for the particle position

The condensational growth of the particle radius

The initial values of the water vapor mixing ratio

Initially, 10

Summary of the simulations; see text for explanation of symbols.

We conduct high-resolution simulations (

Figure

Time-averaged kinetic-energy spectra of the turbulence gas flow for

Next we inspect the response of thermodynamics to turbulence.
In Fig.

Time series of the field quantities:

When changing

Comparison of the time evolution of droplet size distributions for
different

Our goal is to investigate the condensational growth of cloud
droplets due to supersaturation fluctuations.
Figure

Time evolution of

We further quantify the variance of the size distribution
by investigating the time evolution of the standard deviation of the droplet surface area

Comparing panels a and b of Fig.

Two comments are here in order.
First, we emphasize that we observe here

Condensational growth of cloud droplets due to supersaturation fluctuations
is investigated using DNS. Cloud droplets are tracked in a
Lagrangian framework,
where the momentum equation
for inertial particles are solved. The thermodynamic
equations governing the supersaturation field are solved simultaneously.
Feedback from cloud droplets onto

We observe that

The stochastic model developed by

In the present study, the simulation box is stationary,
which means that the volume is not exposed to cooling, as no mean updraft is considered.
Therefore, the condensational growth is
solely driven by supersaturation fluctuations. This is similar to the condensational
growth of cloud droplets in stratiform clouds, where the updraft velocity of
the parcel is close to zero

Entrainment of dry air is not considered here. It may
lead to rapid changes in the supersaturation fluctuations
and result in an even faster broadening of the size distribution

The source code used for the
simulations of this study, the Pencil Code (

XYL developed the idea, coded the module, performed the simulations, and wrote the manuscript. AB and NELH contributed to the development of the module and commented on the manuscript. GS contributed to the development of the idea and commented on the manuscript.

The authors declare that they have no conflict of interest.

We thank Wojtek Grabowski, Andrew Heymsfield, Gaetano Sardina, Igor Rogachevskii, and Dhrubaditya Mitra for stimulating discussions. This work was supported through the FRINATEK grant 231444 under the Research Council of Norway, SeRC, the Swedish Research Council grants 2012-5797 and 2013-03992, the University of Colorado through its support of the George Ellery Hale visiting faculty appointment, and the grant “Bottlenecks for particle growth in turbulent aerosols” from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048. The simulations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Royal Institute of Technology in Stockholm and Chalmers Centre for Computational Science and Engineering (C3SE). This work also benefited from computer resources made available through the Norwegian NOTUR program, under award NN9405K. The article processing charges for this open-access publication were covered by Stockholm University. Edited by: Ryan Sullivan Reviewed by: two anonymous referees