We introduce a framework to efficiently parameterise the aerosol water uptake
for mixtures of semi-volatile and non-volatile compounds, based on the
coefficient,

The most comprehensive description of aerosol composition and hygroscopic
growth is provided by models that calculate the full gas–liquid–solid
partitioning, i.e. the composition and state of the ion-pairs over the wide
range of temperatures and relative humidities from the surface in the tropics
to the winter polar stratosphere. Since thermodynamic equilibrium is the
final state of kinetic processes, many modelling approaches assume
equilibrium, which is reasonable if the atmospheric processes that lead toward it are fast compared to those that lead away from it

To calculate the multiphase partitioning, composition and associated water
uptake of multicomponent atmospheric aerosols, various equilibrium models
(EQMs) have been developed over the past decades including: EQUIL

These EQMs are often embedded in aerosol dynamical models (e.g.

To meet this challenge we introduce in Sect.

We introduce a mixed solution parameterisation framework to efficiently
calculate the aerosol water uptake for mixtures of semi-volatile and
non-volatile compounds with the constraint of using only one parameter, i.e.

Solving the multicomponent and multiphase partitioning analytically, by using a consistent set of equations,
based on one compound-specific single solute coefficient,

Breaking down the complexity of aerosol thermodynamics as much as possible, without the loss of crucial information
and critical numerical accuracy, by using chemical domains with a neutralisation order for all salt compounds listed in Table

Minimising the dependencies on the required thermodynamic data by using a pre-determined

Assuming

The relevant single solute equations (of M2012) are summarised in
Appendix

Thermodynamic data (Sect.

Chemical domains (introduced in Sect.

M2012 have detailed that a (unitless) single solute coefficient, i.e.

Table

To break down the complexity of aerosol thermodynamics as much as possible,
we minimise the number of chemical compounds and equilibrium reactions that
have to be considered. Following the original EQSAM approach

The domain definition (Table

To avoid the numerical minimisation of the Gibbs free energy, which is
required to obtain the equilibrium composition of mixed solutions

Anions:

Cations:

Neutralisation reaction order for Table

To solve the mixed solution framework we apply the NRO to balance
cation–anion pairs that have a non-zero ion–ion product. Within a chemical
domain (Table

Table

When the

For Reaction (

Equilibrium dissociation constants

The equilibrium dissociation constant of

Note that Reaction (

For the wet case, with RH above the compound RHD or mixed solution RHD (see
below), the situation is more complicated. In contrast to the gas–solid
partitioning described above, the gas–liquid equilibrium partitioning of,
e.g. gaseous ammonia,

Following SP2006 (their Sect. 10.4.3), Reaction (

For Reaction (

Here we express the product

At given

where

The

According to SP2006 (and references therein), Reaction (

Following the notation of SP2006 (see their Eq. 10.100), the equilibrium concentration (either in

To extend the calculation of the

To satisfy our key constraint (see Sect.

For Eq. (

Figure

Results of EQSAM4clim (red crosses) and ISORROPIA II (green squares)
for two idealised gas–liquid–solid partitioning examples: single solute
(binary) solution of pure

To analytically compute the equilibrium concentrations of the two
semi-volatile compounds,

With the initial (maximum) values of

With Eq. (

One can now solve with Eqs. (

Figure

To calculate the liquid–solid partitioning, we follow

For mixed solutions (two or more compounds and water), only the amount that
exists for

Results of EQSAM4clim (red crosses) and ISORROPIA II (green squares)
for the total aerosol water mass

However, comparing the water uptake calculation of EQSAM4clim with reference
calculations of, e.g. ISORROPIA II and E-AIM is somewhat precarious. The
reason is that for mixed solutions the calculated water mass mainly depends
on the threshold at which the mixture is considered to take up water. The
assumptions made to define the mixed solution

For ISORROPIA II, if the RH is within a mutual deliquescence

Here we follow the idea of a weighted mixed solution approach of ISORROPIA II, but we approximately solve the
liquid–solid partitioning by computing the weighting factor non-iteratively.
We compute the liquid–solid partitioning after solving the NRO (Sect.

with

where

always with a positive sign:

To solve the liquid–solid partitioning analytically, i.e. without iteration, we modify the approach of

The maximum value of

The concentration-weighted maximum

In case of mixed solutions, Eq. (

To adhere to our key-constraints (Sect.

Due to a lack of experimental data, we approximate

With the mixed solution molality,

where

Thus, with Eq. (

Mixed solution composition of

EQUISOLV II comparison – case 16. Bulk aerosol water mass as
a function of RH for different sulfate molar ratios, fixed for the entire RH
range (at constant

EQUISOLV II comparison – case 16. Bulk aerosol nitrate (ul),
ammonium (ur), total solid PM (ll), liquid

To calculate the mixed solution aerosol water uptake, the standard procedure
employs the widely used ZSR-mixing rule (see, e.g., SP2006, Eq. 10.98).
Assuming that solute concentrations are in equilibrium with the ambient air,
the total aerosol water mass,

Here we follow the standard procedure, while the liquid–solid partitioning
and the

In case the

Our mixed solution framework is independent of the total aerosol water mass because

Mixed solution

Finally, Eq. (

We apply our parameterisation using EQSAM4clim. EQSAM4clim is entirely based
on the mixed solution framework described in Sect.

To evaluate EQSAM4clim we compare the single solute and mixed solution
aerosol water uptake, as well as various other aerosol properties, against
different reference models using box and global modelling calculations at
various levels of complexity (see Table

fixed solute concentrations (9 cases): ISORROPIA II and E-AIM(see also Sect. S3.1 in the Supplement);

variable ammonia concentration: ISORROPIA II and SP2006(see also Sect. S3.2 in the Supplement);

variable solute concentrations (20 cases): ISORROPIA II and EQUISOLV II(see also Sect. S3.3 in the Supplement);

field observations (MINOS campaign, 184 cases): ISORROPIA II(see also Sect. S3.4 in the Supplement);

EMAC chemistry–climate model (year 2005): ISORROPIA II.

Selected results of each application case (1–5) are shown below, while the complete set of results are shown in the Supplement (Sect. S3). Throughout this work, all EQSAM4clim results will be primarily evaluated with respect to its ability to accurately simulate the water uptake of atmospheric aerosols, as this is a key process in climate modelling with our EMAC chemistry–climate model.

Observed and simulated total particulate matter

Figure

MINOS aerosol statistics (see Figs.

To further evaluate the aerosol water uptake calculations of EQSAM4clim for
variable concentrations, we first compare the mixed solution composition of

To scrutinise the differences between EQSAM4clim and ISORROPIA II, we further
evaluate 20 variable mixed solution cases, following the comparison presented
in

The comparison of total nitrate and aerosol ammonium (Fig.

To scrutinise further mineral-rich cases and to extend our model
inter-comparison to size-resolved aerosol observations, we further apply both
gas–aerosol partitioning schemes to (184) field measurements of the
Mediterranean INtensive Oxidant Study (MINOS) that were obtained during
a campaign in Crete in the period of 27 July to 25 August 2001

Residual gases

EMAC AOD vs. MODIS, MISR and AERONET (550

To extend the model inter-comparison of EQSAM4clim and ISORROPIA II to global
modelling applications, we use the atmospheric chemistry–climate model EMAC
in a set-up following

To evaluate the EMAC results, we compare the aerosol optical depth (AOD) to
three independent observational data sets, i.e. two satellites products,
i.e. MODIS (MODerate resolution Imaging Spectroradiometer) and MISR (Multi-angle Imaging Spectro-Radiometer), and one ground-based product, i.e. from the AErosol
RObotic NETwork (AERONET),

Figure

We have successfully extended the

The underlying research data is available on request.

Single solute molality as a function of water activity for several
electrolytes: (

The relation between solute mass fraction

List of names and abbreviations.

List of greek symbols.

List of symbols.

The solute molality is defined as the number of moles of solute per kilogram
of water, i.e.

Figure

The representation of water activity (M2012) relates

To break down the thermodynamics as much as possible, we use a simplified
representation of the

To express

To pre-determine

The RHD measurements at

To determine

The RHD

The parameterisation of solute molality,

Equation (

Growth factor of pure

Wet particle diameter,

We apply our new mixed solution parameterisation framework
(Sect.

To solve our mixed solution framework, we express

We solve

Step one:

Step two, repeated three times:

Figures

Besides significant computational speed-up, another advantage is that our
framework minimises the number of thermodynamic data that are typically
required, while it enables greater flexibility with respect to the extension
to other compounds, not considered in this evaluation. EQSAM4clim (v09) is
limited to the same salt compounds as ISORROPIA II, so that the single solute
parameter

EQSAM4clim has the advantage of being a short Fortran 90 (f90) code with approximately 850 lines, including comments (or 8 pages); see Fig. S1 in the Supplement for a sample. Figure S2.1 in the Supplement shows the flow chart of processes and operations; the computational algorithm is summarised in the Supplement (Sect. S2). For comparison, the gas–aerosol partitioning routine ISORROPIA II, also used in EMAC counts roughly 36 300 lines (or approx. 360 pages). For comparison, this is about one-third of the source code of the EMAC climate model core (ECHAM5.3.02), which has about 120 000 lines of f90 code (both including comments). Last but not least, due to its analytical structure the additional computational costs of EQSAM4clim are negligible for our climate applications, which will be detailed and presented separately.

RMSE – root mean square error between the model (

MBE – mean bias error between the model (

GFE – growth factorial error:

SS1 – skill score between the model (

PF2 is fraction of the number of points within a factor of 2 of the observations, PF10 is fraction of the number of points within a factor of 10 of the observations, and NPoints is the number of points used.

This work was supported the European Research Council under the European
Union's Seventh Framework Programme (FP7/2007-2013)–ERC grant agreement no.
226144 through the C8-Project. All EMAC simulations have been carried out on
the Cy-Tera Cluster, operated by the Cyprus Institute (CyI) and co-funded by
the European Regional Development Fund and the Republic of Cyprus through the
Research Promotion Foundation (Project Cy-Tera
NEA-Y