Transformation of the mass flux towards the particle from the
kinetic regime to the continuum regime is often described by the
Fuchs–Sutugin coefficient. Kinetic regime can be obtained as a limiting case
when only one term of the expansion of the Fuchs–Sutugin coefficient at small

The condensation sink (CS) is an important parameter for aerosol dynamics
quantifying the rate of vapour condensation onto an existing aerosol
population. The inverse of the CS has a clear physical meaning – being the
characteristic timescale for vapours to condense onto the surface of
existing aerosol. Due to the similarity between the processes of vapour
condensation on aerosol particles, and coagulation of the smallest particles
(monomers, dimers, and clusters) with the larger particles from Aitken and
accommodation modes, the CS proves useful for the quantification of a coagulation
sink

The average CS on new particle formation event days is
generally lower than that on nonevent days

For describing aerosol dynamics, we choose a modal approach. This approach
treats the whole aerosol population as a sum of modes, and the equations for
the first-order moments of the particle size distribution are obtained based
on the aerosol general dynamics equation. The number concentration, geometric
mean diameter and standard deviation of each mode can be calculated from the
moments

Instead of using the full general dynamics equation in this paper, we focus on one
physical process – aerosol growth by condensation – because of its importance
for atmospheric aerosol. The model developed is then tested against
atmospheric observations from a remote site in Hyytiälä (Finland), which
represents semi-clean boreal forest in the Northern Hemisphere. Typically
one can identify two or three modes with characteristic diameters less
than 200

Aerosol growth by condensation has been extensively investigated
theoretically

The novelty of the present work is that we obtain analytical formulas for the
CS and its time evolution in the range of intermediate Knudsen
numbers typical for atmospheric applications. Two regular approaches,
which can be found in the literature, either involve extensive calculations
starting with Boltzmann equations

Analysis of the timescales typical for the dynamics of CS and vapour concentration in the atmosphere, allows one to use a quasi-stationary approach for the vapour concentration. It also allows one to develop a simple model describing the coupled dynamics of the CS and the condensing vapour in the atmosphere, during the periods of aerosol growth by condensation inherent in the atmosphere.

The equation describing the growth of the aerosol population by condensation
is

In the kinetic regime

In the next order in

This function is shown in Fig.

One can introduce a limiting diameter as

Fuchs–Sutugin coefficient as compared to its one-term (kinetic) and
two-term (correction) expansions at small

The solution of the condensation equation obtained with the method of
characteristics for intermediate Knudsen numbers is

The solutions of the condensation Eqs. (

We next compare the kinetic limit solution and the “corrected” solution for
the constant growth rate of the particles

The mean free path for the molecules of sulfuric acid, which can be
considered as a typical low-volatility condensing vapour, is

In this Subsection we apply the correction to obtain analytical formulas for
the CS. The CS reflects the ability of vapours to condense on
the aerosol particles, and can be calculated from the formula

Assuming that the dynamics of an aerosol mode are described by solution (

Time evolution of the lognormal aerosol distribution: blue –
solution (

It can be seen that the CS grows in time as

Accounting for the correction valid for intermediate Knudsen numbers gives

If the parameters of the aerosol population do not depend on time, CS in the
kinetic regime can be calculated as follows:

The analogous formula defining CS for the intermediate Knudsen numbers is

Note that the term in brackets is similar to the limiting diameter in the formula
for the growth rate (

In order to demonstrate the influence of correction (

Formulas (

The ratio

We first evaluate formula (

Parameters of the lognormal distribution for different days.

To calculate the CS from Eq. (

The parameters of the particle size distributions on 27 March 2014 and 1 June
2008 are summarized in Table 1, and the examples of experimental data fitting
by the function (

The performance of the analytical formula (

We next aim to separate the errors introduced by the unsatisfactory
approximation of the particle number distribution and usage of Eq. (

CSs calculated using two approximations vs. CSs calculated using the
full FS formula. The magenta
line corresponds to

Overall, when considering the size range

One can obtain an estimate of the contribution of different modes to the CS using the
parameter map in Fig.

A diagram showing the condensation sink (CS) as a function of the
geometric mean diameter of the aerosol mode(

A coupled model of aerosol mode growing by condensation includes two
equations.

Equation of condensation:

Equation describing the time evolution of the vapour concentration:

The system is coupled in a sense, in that the
equation for the particle number distribution includes the dependence on the
vapour concentration through the growth rate as

In the following we consider, for simplicity, non-volatile vapours with

This formula is often used to get the proxies for vapour concentrations

Thus, the system of two differential equations can be reduced to a relatively
simple system of two algebraic equations:

Equation (

Equation (

Self-consistent dynamics of the system can be obtained from the simple
iterations of Eqs. (

This procedure can be readily extended for the two aerosol modes if the CS is
taken as the sum of the CSs calculated for each of the modes. The results of
the model calculations with two modes are shown in Fig.

The only free parameter in the system is the initial growth rate, taken to be
2.6 nm h

Even such a simple model gives quite reasonable predictions of the time
evolution of CS for the time characterized by continuous growth of aerosol
due to condensation. At nighttime the predicted values of the CS are higher,
but from Fig.

Next, we account for the decrease in the particle number concentration during
the nighttime in the simplest way, by assuming

The previous example illustrates that the growth due to condensation in the atmosphere can be captured by the model. Generally, aerosol modes do not exhibit these well-pronounced continuous dynamics, with the growth process rather often being interrupted due to either the changing air mass, precipitation or some other variable, which must be well understood and parameterized before these factors can be incorporated into the model. However, within the period where the meteorological conditions are more or less stable and low-volatility organic vapours are supplied, aerosol grows due to condensation and our model can be applied.

Finally, we comment on the choice of the initial diameter of the mode 20

We have obtained a solution for the condensation equation in the range of
intermediate Knudsen numbers (for particles with diameters up to

Based on this solution, we obtained the algebraic formulas describing the dynamics of the CS over time, assuming an initial lognormal particle number–size distribution. We tested the formulas against atmospheric observations for quasi-stationary conditions. For the typical parameters of aerosol modes in Hyytiälä (Finland), the correction resulted in a 5.5 % overestimation of the CS compared with the calculations using the full Fuchs–Sutugin formula. There is also an overall error due to the approximation of the data with a lognormal distribution, which varied and could be up to 50 % when the tail of the distribution corresponding to larger particles was not captured well. This error, however, did not exceed 15 % when two aerosol modes were considered.

We confirm the previous results by

Note that the difference between the CS in the kinetic regime and the CS in
the intermediate regime can be estimated from Fig.

The differential equation for the vapour concentration was coupled with the
equation for the evolution of the particle number distribution to obtain a
simple self-consistent model of CS dynamics in the atmosphere. For typical
atmospheric values of the CS, one can use a quasi-steady state solution for the
equation for the vapour concentration, in addition to the analytical formula for
CS. This model gives reasonable results for the dynamics of CSs
during the periods of pronounced aerosol growth by condensation for the
characteristic diameters of the mode

Note that in the framework of this model the characteristic diameter of each mode is permanently growing. In nature, in the case we considered, the diameter growth is likely to be interrupted by processes related to meteorology (e.g. morning and evening transitions in the boundary layer).

The model can be extended to investigate the dynamics of a cluster/nucleation
mode with a characteristic diameter of a few nanometres in the presence of a base
mode and a time-dependent vapour concentration. As we showed, these modes
will only have a negligibly small effect on the coupled dynamics of the base
mode and condensing vapours while the base mode will define the condensation
sink for the smallest particles. The simplest way to include the
cluster/nucleation mode is to add a general
dynamic equation with a nucleation term and a diffusion term into the system considered here

Data measured at the SMEAR II station (University of
Helsinki) are available on the following website:

The authors declare that they have no conflict of interest.

This work was supported by the Academy of Finland Centre of Excellence Programme (grant no. 307331) and the Academy of Finland professor grant awarded to Markku Kulmala (no. 302958). The results obtained are part of a project (ATM-GTP/ERC), which has received funding from the European Research Council (ERC), under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 742206). Edited by: Chak K. Chan Reviewed by: two anonymous referees