GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-11-3261-2018A parameterisation for the co-condensation of semi-volatile organics into multiple aerosol particle modesParameterisation for the co-condensation of semi-volatile organicsCrooksMatthewmatthewcrooks@ntlworld.comConnollyPaulMcFiggansGordonhttps://orcid.org/0000-0002-3423-7896The School of Earth, Atmospheric and Environmental Science, The
University of Manchester, Oxford Road, Manchester, M13 9PL, UKMatthew Crooks (matthewcrooks@ntlworld.com)13August20181183261327819May201714July20175April201819April2018This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/11/3261/2018/gmd-11-3261-2018.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/11/3261/2018/gmd-11-3261-2018.pdf
A new parameterisation for the cloud droplet activation of multiple aerosol modes
is presented that includes the effects of the co-condensation of semi-volatile
organic compounds (SVOCs). The novel work comes from the dynamic condensation
parameterisation that approximates the partitioning of the SVOCs into the
condensed phase at cloud base. The dynamic condensation parameterisation
differs from equilibrium absorptive partitioning theory by calculating time-dependent condensed masses that depend on the updraft velocity. Additionally,
more mass is placed on smaller particles than at equilibrium, which is in
better agreement with parcel model simulations. All of the SVOCs with
saturation concentrations below 1×10-3µg m-3 are
assumed to partition into the condensed phase at cloud base, defined as
100 % relative humidity, and the dynamic condensation parameterisation is
used to distribute this mass between the different aerosol modes. An existing
cloud droplet activation scheme is then applied to the aerosol particles at
cloud base with modified size distributions and chemical composition to
account for the additional mass of the SVOCs. Parcel model simulations have
been performed to test the parameterisation with a range of aerosol size
distributions, composition, and updrafts. The results show excellent agreement
between the parameterisation and the parcel model and the inclusion of the
SVOCs does not degrade the performance of the underlying cloud droplet
activation scheme.
Introduction
Clouds make an important contribution to both weather and climate, so
understanding the complex physical processes involved in their formation and
continued existence is crucial for long-term weather and climate modelling.
The size and number concentration of cloud droplets can significantly alter a
cloud's albedo by changing the amount of reflected shortwave radiation and
absorbed longwave radiation . Cloud lifetime is also
tightly coupled to albedo and directly to cloud
droplet properties through the precipitation rate .
One of the most significant factors that influences cloud droplet number is
the properties of aerosol particles from which cloud droplets are formed by
the condensation of water under supersaturated conditions . An
increase in the number concentration of such cloud condensation nuclei (CCN) can
lead to an increase in cloud droplet number due to the higher abundance of
particles for water to condense onto. Conversely, in some situations, larger
particles, which typically activate at lower relative humidities, can deplete
available water and inhibit the activation of smaller particles within the
population . With such differing consequences resulting from
variations in aerosol particle properties, it is no surprise that the largest
cause of uncertainty in global mean radiative forcing is attributed to
aerosol–cloud interactions contributing an estimated -0.4
to -1.8 Wm-2.
An accurate representation of the cloud droplet activation process is
therefore of crucial importance. Global weather and climate models, however,
are not only restricted by our understanding of the microphysical processes,
involved but additionally by the computational expense required to model
them. Several cloud activation parameterisations have been developed to
predict cloud droplet number as a function of aerosol properties
. Despite an emphasis on computational
efficiency these parameterisations have been largely successful
. The most popular are and those based on
. The former was originally only tested up to a
mean radius of 0.1 µm, which is where it shows some deviation from parcel
model simulations. Both have been shown to perform well up to 250 nm at high
number concentrations with a tendency to over-predict at lower number
concentrations . The Fountoukis and Nenes parameterisation was
later extended to account for the kinetic limitations of larger droplets
. Under a wide parameter space the Fountoukis and Nenes
parameterisation, with the giant CCN extension at larger particle
sizes, is found to perform better than the Abdul-Razzak and Ghan
; consequently, this is the only activation
parameterisation studied in this paper.
The parameterisations mentioned above assume that the aerosol particles are
entirely involatile. Although it is common to make such an assumption for
primary emissions, regional- and global-scale studies of semi-volatile primary
organic aerosol have been carried out with mostly improved estimates of the
organic aerosol budget . Secondary organic aerosol
(SOA) is formed by the nucleation of new particles from organic vapours or by
the condensation of the oxidation products of precursor gases into the particle
phase. Subsequent particle-phase reactions age the condensed compounds to
produce compounds that are less volatile or functionally involatile. The
condensation of semi-volatile organic compounds (SVOCs) onto aerosol
particles increases their size, changes their chemical composition, and
consequently affects their ability to act as CCN. Depending on the
geographical location, between 5 and 90 % of the total aerosol mass can be
composed of organic material , with a significant but
uncertain proportion made up of SOA. Any realistic cloud activation scheme
must therefore include the effects of SVOCs in the formation of SOA.
Direct chemical and dynamical modelling of every organic species is not only
computationally impractical due to the many thousands of different organic
species present in the atmosphere but is rendered impossible by
only a small fraction of these having been identified . To
facilitate numerical modelling, large numbers of compounds are commonly
grouped together and are represented by fewer surrogate species with
effective chemical properties .
Equilibrium absorptive partitioning theory was introduced by to
calculate the equilibrium vapour and condensed phases of volatile compounds and
is often used in global models as a computationally efficient approximation
of the dynamically evolving vapour and condensed phases. The empirically derived
relation of was introduced as a method of treating multiple
organic species based on the results of two-compound experiments, and while it
benefits from its simplicity it has been found to be unrealistically
sensitive to changes in the concentration of the organic compounds
. The volatility basis set of allows many organic
species to be binned according to their saturation concentration. This was
later extended to include the condensation of water and also
introduces a new molar definition of the saturation concentration. A recent
advance has extended this molar-based approach to calculating the
equilibrium condensed concentrations across multiple aerosol modes of
different sizes and chemical composition when each particle contains a
non-volatile constituent. In some applications, this can also be applied to
particles that have previously nucleated from extremely low-volatility
compounds by approximating the resulting aerosol mass as involatile .
The work of proposes the only extension of the aforementioned
cloud activation schemes to include the effects of SVOCs. The
parameterisation assumes that the vapour and condensed phases of the SVOCs
are in equilibrium at cloud base, which is a reasonable approximation in all
but very low number concentrations of aerosol particles and high updrafts.
Equilibrium at cloud base is calculated using equilibrium absorptive
partitioning together with a log10 volatility basis set at a relative
humidity of 99.999 %. The condensed phase is assumed to have resulted from
the condensation onto a single mode of non-volatile particles with sizes
distributed according to a log-normal. The additional mass has the effect of
changing the median diameter and geometric standard deviation in such a way
as to preserve mass while keeping the arithmetic standard deviation constant.
The new particle size distribution and composition at cloud base is then
inserted into the and parameterisations in order to
calculate the number of cloud droplets.
Although the new parameterisation of was found to agree well with
a parcel model it does have the limitation of only applying to a single
aerosol particle mode, which is too restrictive for most atmospheric
situations. In this paper, we extend the parameterisation of to
the case of multiple aerosol modes, which is significantly more complicated.
While the SVOCs may be in bulk equilibrium at cloud base, as in the single
mode case, the timescale required for the condensed masses on each mode to
reach equilibrium is on the order of several hours. In many situations, the
aerosol particles may activate before the condensed phase of the SVOCs has
equilibrated between the different sizes of particles. The result is that the
multiple mode equilibrium partitioning of can miscalculate the
condensed masses achieved under the dynamic conditions experienced in cloud
droplet activation, especially at high updraft speeds. We present a new
parameterisation to calculate the condensed masses of SVOCs across multiple
aerosol modes during the rapid dynamic condensation induced by high relative
humidities near cloud base. These condensed masses at cloud base are then
used in the cloud activation scheme in an analogous way to the
equilibrium condensed masses in the single mode case . At low
updrafts there is more time for the condensed masses to equilibrate before
activation, resulting in the new dynamic parameterisation producing the same
results as using the multiple mode equilibrium theory. The new
parameterisation is found to significantly outperform the equilibrium model
at higher updrafts when compared to a dynamic parcel model.
Dynamic condensation parameterisation description
We describe here the model applied to approximate the partitioning of the
condensed masses of SVOCs at cloud base. For brevity we use the acronym DCP
for this dynamic condensation parameterisation. The condensation of the SVOCs
is based on the principle that if the vapour pressure exceeds the equilibrium
vapour pressure then there is net condensation and if it does not exceed equilibrium
then the organic compound undergoes net evaporation. During long-range
aerosol transport this can lead to the vapour and condensed phases
equilibrating as shown by the lower three inset boxes in Fig. . In the case of cloud droplet activation, the rapid
condensation of water can suppress the equilibrium partial pressure of the
organics to a negligible level, which is demonstrated in Sect. . This simplifies the condensation rate of the SVOCs to be
proportional to their partial pressure. In addition, it causes the organics
to undergo continuous condensation until their vapour phase is depleted. The
left three inset boxes in Fig. show this process. As
the aerosol particle rises in the atmosphere, water and the organics condense
onto the particle, reducing the mass of SVOCs in the vapour phase and
consequently reducing the condensation rate. In cloud, the condensed mass of
water increases drastically and causes all of the organics to condense into
the particle phase. In the following section, we justify this assumption
through an example before deriving the parameterisation for both monodisperse
and polydisperse aerosol populations.
Schematic of the two different condensation processes occurring in
the atmosphere with condensation rates of organics and water indicated by the
size of the green and blue arrows, respectively. The lower three inset boxes
show the equilibration between the vapour and condensed phases of the
semi-volatile organic compounds during large-scale aerosol transport. Gas-phase species change due to the air mass passing over anthropogenic and
biogenic emission sources, as well as oxidation and gas-phase reactions.
The left three boxes show the rapid depletion of the vapour phase of organics
that results from the relative humidity reaching and exceeding 100 %. Ground-based emissions sources of SVOCs do not contribute additional vapour mass to
the rising air parcel represented by the left three inset boxes.
Neglecting the equilibrium partial pressure
Ideality is assumed in order to calculate the equilibrium partial pressure of
both water and the organics. Consequently, the equilibrium saturation ratio
of both water and the organic compounds can be described by the mole fraction
of the condensing compounds multiplied by a Kelvin term. The condensed mass
of water in the mole fraction is calculated assuming that water condenses
sufficiently quickly to be perpetually in equilibrium. At the high relative
humidities experienced near cloud base, the denominator of the mole fraction
is dominated by the condensed water and consequently reduces the mole
fraction to negligible values in activated particles compared to the actual
saturation ratio.
To demonstrate the suppression in equilibrium partial pressure of the
organics we have run a parcel model with binned microphysics that
solves the dynamic condensation numerically. For this example, we have used
an aerosol population composed of three log-normal size distributions given by
the natural environmental conditions discussed in Sect. . We
used an SVOC mass loading of 27 µg m-3 and the vertical wind speed
is 2 m s-1.
Figure shows the ratio of the equilibrium
partial pressure of the three median diameters to the saturation partial
pressure as a function of relative humidity. On the second and third
modes (b, c), which activate into cloud drops, this ratio drops to below 0.2 and
0.1, respectively, as the relative humidity approaches 100 %. The first mode (a),
however, has a much larger value and actually increases above 1 near 100 % RH.
This is a result of this mode not activating, so it does not have a significant
quantity of water condensed on it to suppress the equilibrium partial pressure
of the organics.
The ratio of the equilibrium partial pressure, pieq, to the
partial pressure, pi, of each of the organic compounds on each of the
modes as a function of relative humidity. Each colour shows a different
volatility bin. On the first mode (a) the compounds with the lowest five C*
values look indistinguishable from each other and are all shown by the purple
line. Similarly, the six compounds with the lowest C* values in the second
and third plots (b, c) are shown by the blue line.
A monodisperse aerosol particle population
We first assume that the aerosol particle population is composed of N
particles of dry diameter Dd per cubic metre. The condensation rate of
the ith organic compound onto a particle is assumed to be proportional
to the difference between the partial pressure, pi, and the equilibrium
partial pressure, pieq, of the ith compound over the particle,
dyidt=αiDpi-pieq.
Here yi is the condensed mass on the particle and t is time. The
variable αi is defined as
αi=2πDv,iMiRT,
where Mi and Dv,i are the molecular weight and diffusivity of the
organic compound in air, respectively. The universal gas constant is denoted
R and T is the temperature. D is the wet diameter of the particle,
which is calculated assuming that the condensed water is in equilibrium at the
initial RH, from which the cloud droplet activation scheme is initiated. For
simplicity we set the temperature and pressure to their initial values in the
parameterisation, which is common in cloud droplet activation
parameterisations . Therefore, in the parameterisation the
variable αi becomes a constant parameter. The diameter, D, however,
varies with time as the SVOCs condense onto and evaporate off the particle.
This variation is highly non-linear and for simplicity we assume for the
parameterisation that setting D to the initial wet diameter is sufficiently
accurate an approximation.
The saturation ratio of the ith organic compound can be expressed as the
ratio of the mixing ratio, ri, to the saturation mixing ratio,
risat, or equivalently, the ratio of the partial pressure to the
saturation partial pressure, pisat. Equating these two definitions
gives
pipisat=ririsat.
For simplicity, we denote the ratio of the saturation partial pressure to the
saturation mixing ratio of the ith organic compound by βi so that
Eq. () simplifies to
pi=βiri.
The initial mixing ratio, ri0, corresponding to a condensed mass of
yi0 can be related to the initial partial pressure, pi0, through
Eq. (); namely, pi0=βiri0. The mixing ratio
subsequently decreases at the same rate as the increase in the total condensed
mass concentration of each compound. Hence,
ri=ri0-N(yi-yi0).
Substituting this into Eq. () yields
pi=pi0-Nβi(yi-yi0),.
Substituting Eq. () into () and neglecting the
equilibrium partial pressure produces
dyidt=αiDpi0+Nβiyi0-Nβiyi.
This can be integrated assuming constant diameter, temperature, and pressure
to give
yi=yi0+pi0Nβi1-e-αiDNβit.
Equation () represents an analytic approximation of the time
evolution of the condensed mass of the SVOCs on a single particle within a
monodisperse aerosol. The evolution of the total condensed mass per cubic
metre can be obtained by multiplying yi by N. Thus,
Yi=Nyi0+pi0βi1-e-αiDNβit.
All terms on the right-hand side of Eqs. () and
(), except t, are parameters that depend on the initial
conditions of the problem, such as the temperature, pressure, and initial
condensed mass of the organic compounds. The only time dependence is in the
exponential term.
In order to be applicable in atmospherically relevant situations, the
approximation in the previous section needs to be extended to polydisperse
aerosols. Suppose now that the particle population is composed of multiple
monodisperse populations of diameters Dd,j and number concentrations
Nj. In this case, we have an equation analogous to Eq. () for
each size of particle,
dyijdt=αiDjpi-pijeq.
We note that both the partial pressure and the parameter αi are
independent of size but the equilibrium partial pressure is dependent on
Dj, although, like in the monodisperse case, we neglect this.
The change in partial pressure resulting from condensation of the SVOCs is
proportional to the total condensed mass across all particles. Hence, the
evolution of the partial pressure, analogous to Eq. (), is given
by
pi=pi0-βi∑kNkyik-yik0,
where k is a dummy index used for the summation over j to distinguish
from the equations for the jth size of particle. The initial condensed
mass of the ith organic compound on a particle of size Dj is denoted
yij0. Substituting Eq. () into () and
neglecting the equilibrium partial pressure results in the equation
dyijdt=αiDjpi0-βi∑kNkyik-yik0.
We now multiply by Nj and sum over j, again using the dummy index kddt∑kNkyik=αiγpi0-βi∑kNkyik-yik0.
For simplicity, we have denoted
γ=∑kNkDk.
The total condensed mass of the ith compound across all particles is
denoted fi and can be expressed as
fi=∑kNkyik,
and similarly the initial total condensed mass is given by
fi0=∑kNkyik0.
Equation () can now be simplified as
dfidt=αiγpi0-βifi+βifi0.
This equation is qualitatively similar to Eq. () in the
monodisperse case, Eq. (), so the solution can be expressed in an
analogous way as
fi=fi0+pi0βi1-e-αiγβit.
To calculate the condensed mass on each of the individual particles we
substitute Eq. () into () to give
dyijdt=αiDjpi0e-αiγβit,
which can be integrated directly to produce
yij=yij0+Djpi0γβi1-e-αiγβit.
Equation () expresses the time evolution of the condensed mass of
each compound in a particle within a population composed of multiple
monodisperse modes. To obtain the total condensed mass per cubic metre on a
particular monodisperse mode, Yij, this expression can be multiplied by Nj,
Yij=Njyij0+NjDjpi0γβi1-e-αiγβit.
Polydisperse aerosol particle populations
Typically, atmospheric aerosol particles occur in a continuous range of
sizes. The continuous size distribution may be discretised into collections
of similarly sized particles to create an aerosol particle population that is
composed of multiple monodisperse modes and which approximates the continuous
size distribution. Alternatively, in many situations the continuous
distribution of particle sizes can be represented by one or more log-normal
size distributions as defined by the equation
dNdlnD=∑jNj2πlnσjexp-lnDDm,j2lnσj2.
Equation () denotes the number concentration of particles per
natural logarithm of the bin width. Here, Nj is the total particle number
concentration represented by the jth log-normal, and lnσj and
Dm,j are the geometric standard deviation and median diameter,
respectively. The advantage of representing a polydisperse particle
population in this way is that each log-normal size distribution can be
treated as a single mode by replacing the diameters, Dj, in Eq. ()
by the median diameters and multiplying by the total number of
particles in each mode. Hence, the condensed mass concentration on a log-normal
mode is approximated by the expression
Yij=Njyij0+NjDm,jpi0γβi1-e-αiγβit.
Fractional representation
As the condensed mass of an organic compound increases in the full
time-dependent equation, (Eq. ), the difference between the partial
pressure and the equilibrium partial pressure decreases and this slows the
rate of condensation. Eventually, the condensation rate approaches zero as
the condensed mass approaches the equilibrium value. In the parameterisation,
the equilibrium partial pressure has been neglected and this process does not
happen. The parameterisation, however, has been derived to be applicable
specifically when the relative humidity is close to 100 % for the purpose of
approximating cloud droplet activation; convergence on equilibrium in this
case is not directly of relevance. An additional and more important problem
arises in the high RH regime, however. In this case, the equilibrium vapour
partial pressure of the organic compounds is close to zero; the partial
pressure of an organic compound decreases as the condensed mass increases and
eventually becomes zero when all of the compound has entered the condensed
phase. Due to the approximations to the diameter, temperature, and pressure,
the partial pressure does not decrease at the correct rate and can reach zero
with either too little mass in the condensed phase or even calculate a
condensed mass that exceeds the total abundance of that compound. This
violates conservation of mass. To maintain mass within the system, we
introduce a fractional formulation that approximates what fraction of the
condensed mass exists in each mode as a function of time. To do this, we
divide the condensed masses given by Eq. () by the sum of Yij over
all particles,; thus,
Zij=Njyij0+NjDjpi0γβi1-e-αiγβit∑kNkyik0+NkDkpi0γβi1-e-αiγβit.
To calculate the distribution of the condensed mass at cloud base, the
fractional formulation, Eq. (), can be evaluated at cloud base and then
multiplied by the total abundance of SVOCs.
In the log-normal mode case, the fractional formulation takes the form
Zij=Njyij0+NjDmjpi0γβi1-e-αiγβit∑kNkyik0+NkDmkpi0γβi1-e-αiγβit.
Cloud drop activation parameterisationSingle aerosol mode
The parameterisation employed to calculate the number of cloud droplets
including the effects of SVOCs is a modification to that described in
. In this earlier work the system was seeded with a single
involatile mode whose particle sizes could be represented by one log-normal
size distribution. A description of the methodology is given here before the
theory is extended to the multiple mode case.
The initial temperature, pressure, and relative humidity are prescribed; in
this paper we use the values 293.15 K, 95 000 Pa, and 90 %, respectively. All
aerosol particles are assumed to contain an involatile constituent so that no
particle can evaporate completely. It is assumed that the SVOC vapours and
the involatile particles have coexisted for a sufficient time for the condensed
masses to be in equilibrium at 90 % RH, calculated using a molar-based
equilibrium absorptive partitioning theory . The additional mass
from the condensed SVOCs is added to the involatile mass and the new particle
sizes are assumed to follow a log-normal size distribution with the same
geometric standard deviation as the involatile particles but an increased
median diameter that is calculated to conserve mass. Further details are
given in Appendix and the original paper .
In the single mode case, the condensed masses of the semi-volatile organic
compounds are assumed to be in equilibrium with the vapour phase at cloud
base (99.999 % RH) and are calculated using equilibrium absorptive
partitioning theory. This additional aerosol mass is added to the initial
composite aerosol and a new median diameter and geometric standard deviation
of the dry aerosol size distribution are calculated which conserve mass and
maintain a constant arithmetic standard deviation, as defined by
SD=elnDm+12lnσeln2σ-1.
More details on deriving the size distribution at cloud base are given in
Appendix . These new aerosol size distribution parameters are
then input into a widely used parameterisation for cloud droplet activation
in the absence of SVOCs , which is already constructed to accept
multiple modes.
Multiple aerosol modes
In the multiple mode case we consider polydisperse aerosol particle
populations that can be represented by multiple log-normal size distributions.
As in the single mode case, we assume that the initial condensed masses of
each SVOC are in equilibrium with the vapour phase and these are calculated
using multiple mode equilibrium absorptive partitioning theory ().
The initial median diameter of each mode is calculated in the same way as in
the single mode case; assuming conservation of mass together with the same
geometric standard deviation as the involatile particle modes.
The bulk condensation of SVOCs into multiple log-normal modes is qualitatively
similar to the single log-normal case, and as such bulk equilibrium between a
condensed and vapour phase at cloud base is still achieved. It can take
several hours, however, for the condensation and evaporation of SVOCs between
the different modes to reach equilibrium. Therefore, it cannot be assumed
that the individual condensed masses at cloud base are in equilibrium even
though the assumption of bulk equilibrium still holds. Rather than
calculating bulk equilibrium using multiple mode, equilibrium partitioning
theory and summing over each mode, it is quicker and not significantly less
accurate to simply assume that all of the SVOCs are in the condensed phase
at cloud base. The parameterisation for the dynamic condensation of SVOCs,
described in the Sect. , can then be used to calculate how
this mass partitions between each aerosol particle mode.
An additional complication with using the DCP solution in the multiple mode
case, rather than equilibrium partitioning theory, is that there is a time
dependence and the solution changes depending on how long the particles
experience elevated RH values near cloud base before activating. This is
largely determined by the vertical updraft. The time, tcb, that it takes
for a parcel of air to reach 100 % RH from our initial value of 90 % assuming
a linear relative humidity, temperature, and pressure profile can be
calculated and is described in Appendix . The difference in
the condensed mass of the SVOCs only changes significantly in a dynamic model as
the RH approaches to 100 %. In addition, the DCP solution was derived under
the condition of constant temperature and pressure, as well as relative
humidities close to that at cloud base. Consequently, the DCP solution needs
to be evaluated at time that is shorter than tcb. We have found that
evaluating the DCP solution at the time it takes for the RH to increase from
99.9 to 100 % yields the best results at updrafts above about 1 m s-1,
although it is largely insensitive a t slower updrafts. Initial condensed
masses are still calculated using the initial RH. A review of this is
presented in Appendix and the Supplement.
Results
Total mass loadings of organics used in Sect. that are distributed between the volatility bins
using the anthropogenic volatility distribution given in Table .
Concentrations are given in µg m-3 at 293.15 K and 950 hPa.
The simulations are initiated at 95 % RH, a temperature of 293.15 K, and a
pressure of 95 000 Pa. The condensed concentrations of the SVOCs are assumed
to be in equilibrium with the vapour phase initially and the median diameter
increased to conserve mass accordingly. The geometric standard deviation of
the initial composite aerosol is the same as that of the involatile particle
mode.
Fraction of total aerosol particles that activate into cloud
droplets at a range of vertical updrafts and number concentrations. The solid
lines show the results from the parameterisation and the crosses show the
parcel model results, while the effect of the SVOCs is shown in green and
blue shows the analogous results without the organic vapours. The red
dashed lines show the effect of applying equilibrium absorptive partitioning
at cloud base and the dashed grey line shows the results of only including
the initial condensed concentration of SVOCs. The shape of the size
distribution of ammonium sulfate is shown in the lower right plots with
median diameters and geometric standard deviations written above. Number
concentrations used are specified above each plot.
We neglect the nucleation of new particles from the SVOC vapours as these are
unlikely to grow large enough to activate into cloud droplets. In addition,
the rapid growth of existing particles during the cloud droplet activation
process will induce significant condensation of the SVOCs that will act as
the dominant sink of the organic vapours. We further assume in the parcel
model simulations that the initial condensed SVOCs remain in the condensed
phase throughout the cloud activation process. As the relative humidity
increases monotonically up to the point of activation it is likely that the
further condensed water will act to increase the condensed mass of SVOCs
across all particles with minimal evaporation. The only particles that are
likely to undergo the evaporation of SVOCs are the smallest particles whose
condensed water is scavenged by the larger particles. These particles,
however, have only a small amount of condensed SVOCs compared to the mass of
the larger particles, especially near cloud base, and this additional mass
will have a limited effect on the activation of larger particles.
Same as Fig. but with a different size
distribution.
Figures to show the fraction of the
total number of aerosol particles that activate. The shape of the size
distributions of ammonium sulfate are shown in the lower right plots and the
total number concentration of particles is varied between the three
associated plots. The parameter values used in each plot are given above each
graph. To show the enhancement in cloud droplet number as a result of the
SVOCs, the fraction of activated drops both with and without condensing
vapours are shown in green and blue, respectively. The crosses show the
parcel model results, while the solid lines show the parameterisation. The
results from two alternative methods of including the SVOCs are also
compared. The first, shown by the dashed red line, applies multiple mode
equilibrium absorptive partitioning theory at cloud base to
distribute the SVOCs between the different modes, rather than the DCP. This
method is analogous to the original SVOC parameterisation of . The
second method includes the initial condensed concentration of SVOCs at the
temperature and pressure from which the activation scheme of is
applied, but without additional co-condensation of the remaining vapours, and
is shown by the dashed grey line.
Same as Fig. but with a different size
distribution.
Figure shows the result of having 6 times more aerosol
particles in the smaller mode than the larger. At all three particle number
concentrations, the inclusion of just the initial condensed mass of SVOCs
(dashed grey) produces little enhancement in CCN concentration compared to
the case when no organics are considered (blue line). The additional
co-condensation of SVOC vapours that occurs when the RH is above 95 % is
clearly important in this case. The assumption that the condensed mass of
SVOCs is in equilibrium across all modes at cloud base results in more
particles in the larger-sized mode activating at lower updrafts below 0.1 m s-1, but at higher updrafts the activated fraction is similar to the
case without co-condensation. The new DCP shows a pronounced enhancement
across all updrafts, shown by the green line. Below 1 m s-1 in the upper
left plot, the DCP captures the enhancement in CCN concentration calculated
in the parcel model well but at higher updrafts it over-predicts. It is worth
noting, however, that some of this over-prediction is attributable to the
activation scheme of , which also over-predicts the CCN
concentration even in the absence of SVOCs. At the higher particle number
concentration, shown in the top right plot, the DCP parameterisation is in
excellent agreement with the parcel model and outperforms the other models
significantly. All the parameterisations under-predict the CCN concentrations
at very high particle number concentrations, shown by the lower left plot.
The DCP parameterisation does outperform the others and is the only one that
predicts an enhancement by the SVOCs at higher updrafts, which is also seen
in the parcel model.
The size distribution used in Fig. has a larger portion
of the particle number concentration in the smaller mode than in Fig. . In this case the DCP parameterisation performs better
than in Fig. at lower number concentrations. The
application of equilibrium absorptive partitioning theory at cloud base
appears to agree with the parcel model at lower updrafts and predicts a
significant enhancement in CCN concentration compared to the case without
SVOCs. At higher updrafts, however, assuming equilibrium of the SVOCs at
cloud base activates a similar number of particles to the case with water only.
Neglecting the co-condensation of the organic vapours during activation
again produces little enhancement in CCN concentration. The new DCP performs
very well at the lower number concentrations used in the upper two plots
across all updrafts but, as was seen in Fig. , none of
the parameterisations predict a significant enhancement at high number
concentrations. Even in the absence of SVOCs, the parameterisation of
under-predicts compared to the parcel model and may indicate an
oversensitivity of the underlying cloud droplet activation scheme to high
number concentrations.
The size distribution used to generate Fig. has two
modes containing equal numbers of particles. The performance of all of the
parameterisations is qualitatively similar to that in Fig. .
The assumption of equilibrium of the SVOCs at cloud base predicts a large
enhancement in CCN from the larger-sized mode but has little effect on the
activation of the smaller particles. The DCP performs the best across all
updrafts at lower number concentrations, but at higher number concentrations
the parcel model predicts a significant enhancement in number of CCN that is
not captured by any of the parameterisations.
Activated fraction data from Figs. to
collated into one figure. The x axis shows the
activated fraction from the parameterisation and the y axis shows the
parcel model. The crosses show the results from the cloud droplet activation
scheme without SVOCs and the dots show the parameterisation with SVOCs.
Markers are coloured by the total number concentration of aerosol
particles.
Figure directly compares the activated fractions
from the DCP parameterisation against the parcel model from Figs. to . The crosses show just the
parameterisation without any SVOCs, while the dots show the effect
of the SVOCs using the DCP. The dots are coloured by total particle number
concentration and it is clear that the parameterisation under-predicts
compared to the parcel model at high number concentrations, as previously
discussed. The agreement at lower number concentrations, however, is largely
similar between the parameterisation with SVOCs and without, indicating that
the new parameterisation does not degrade the performance of the underlying
cloud droplet activation scheme.
Environmental variations
We now test the parameterisation for different environmental conditions by
varying the aerosol size distributions and volatility distribution. Different
aerosol size distributions are considered with parameter values taken from
, a study that has gathered together field data from multiple sites
across Europe to obtain typical measurements for four different environmental
conditions: natural, rural, near-city, and urban. Each environment varies in
proximity to major anthropogenic sources of pollution by distances of >50, 10–50, and 3–10 km
for the first three, respectively. The urban sites are
defined as having fewer than 2500 vehicles per day within a 50 m radius.
Table shows the summertime afternoon number concentration,
N, median diameter, Dm, and geometric standard deviation, lnσ,
from obtained by fitting three log-normal size distributions to the
data. It is clear from the values of N1 that there are significantly
higher concentrations of very small particles in the near-city and urban
environments than the natural and rural. This indicates that these small
particles are a result of the anthropogenic sources, most likely combustion
engines. Particles of less than 50 nm are typically created as a result of
particle nucleation during the initial cooling phase of vehicle emissions
() with 90 % of the number concentration
being in this nucleation mode (). This is exemplified in the
figures in Table with 85 and 95 % of the near-city and
urban number concentrations, respectively, being in the first two modes with
median diameters less than 50 nm.
Number concentration (cm-3), median diameter (nm), and
geometric standard deviation for the four cases studied in during
summertime afternoons.
is an accompanying paper to that further analyses the
aerosol properties in the different environments to provide aerosol
composition. We assume that the smaller two modes are composed of
carbonaceous aerosol particles. As much as 80 % of emitted black carbon is
hydrophobic () and will not contribute towards cloud droplet
activation. While in the atmosphere, however, these particles undergo an
ageing process involving oxidation, coating with sulfate, and SOA and
photochemical decomposition (), which
results in a hydrophilic composition. The degree to which these particles
have aged and their resulting chemical composition varies greatly with time
and location, and as such the possible values that can be assigned to the
material properties of these particles have a large variability.
In the current study of cloud droplet activation, assuming newly formed,
insoluble, hydrophobic, pure black carbon particles will not contribute to
the CCN number concentration and therefore will not provide an interesting
analysis. Instead, we attempt to choose parameter values that represent aged
hydrophilic particles. Molecular weights of several compounds found in
newly formed particulate matter from combustion range from 178–302 g mol-1
(). We choose a value of 200 g mol-1. The density of atmospheric carbonaceous aerosols has been found
to lie in the range 1–1.7 g cm-3 ()
and a value of 1.5 g cm-3 is used here. To simulate hydrophilic,
non-dissociative particles we choose a van't Hoff factor of 1; this is in line
with values found for levoglucosan ().
The larger mode in all four regions is modelled as composed of ammonium
sulfate, and this is chosen to represent all highly soluble compounds in a
single mode of particles that will act as effective CCN.
We use two different volatility distributions across the four sites; one
representing biogenic sources and the other anthropogenic sources. The former
will be used for the natural and rural sites and the latter for the near-city
and urban sites. The biogenic volatility distribution is taken from the 1DVBS
of , which uses the model of to distribute oxidation
products of α-pinene across nine volatility bins with
logC* values ranging from 10-5 to 103µg m-3,
separated by factors of 10. We assume for our modelling study that the molar-based C* can be obtained from the mass-based C* by dividing
by the molecular weight of the compounds in the volatility bin. The
volatility distribution is similar across all three sites in , with
little more than a rescaling in total concentration between them, so
without loss of generality, we take the values from Abisko in northern
Sweden; these values are given in Table . Biogenic SOA can
mostly be composed of compounds with molecular weights in the region of 130 g mol-1
() although much higher molecular weight
compounds have been identified (). A density of 1.4 g cm3 is
used and is taken from .
The anthropogenic volatility distribution is taken from , which is
derived from field measurements in Mexico City. Again, the volatility bins
are separated by orders of magnitude in logC* but range from
10-6 to 103µg m-3. We choose typical values in the
literature for hydrocarbons produced in vehicular combustion engines
(). A density of 1.25 g cm-3 and
a molecular weight of 200 g mol-1 are used, together with a van't Hoff
factor of 1.
Volatility distributions used to represent typical biogenic and
anthropogenic SVOC concentrations. These values are rescaled in order to
obtain the required organic mass fraction in the simulations. Concentrations
are given in ×10-2µg m-3.
Three concentrations of SVOCs are investigated and each is obtained by
rescaling the volatility distributions given in Table . Multiple
mode equilibrium partitioning is used to calculate the condensed masses at
90 % RH and the bulk organic mass fraction of the aerosol particles excluding
water is calculated. The volatility distribution is then rescaled until the
organic mass fraction is equal to 10, 50, and 90 %. Values of 10 and
50 % are chosen in line with upper and lower limits frequently encountered.
Organic mass fractions as high as 90 % have been measured
but not all of this will be attributed to condensed SVOCs. Our simulation
using a 90 % organic mass fraction of SVOCs is therefore used as an extreme
but still realistic scenario to see how the parameterisation performs under
a wide parameter space. Actual volatility distributions are given in Appendix
and we note that the 50 % organic mass fractions give total
SVOC mass loadings of about 27 µg m-3 for the natural and rural
sites and 37 µg m-3 for the near-city and urban environments. These
are in line with the total mass loadings measured in .
For comparison, we model each case with both the parcel model and the
parameterisation. The parcel model is initiated with equilibrium condensed
masses in the particle phase and new log-normal size distributions are
calculated assuming the same geometric standard deviation as given in
Table for particle modes with median diameters above 50 nm. For
smaller modes, we calculate a new geometric standard deviation that maintains
a constant arithmetic standard deviation. This has been found to be more
realistic under atmospheric timescales () although it does not
seem to have a significant effect on cloud droplet number. Initial condensed
masses are assumed to be involatile and the vapour phase is free to condense
with increasing altitude and condensed water.
Two further parameterisations are presented for comparison. The first assumes
that the initial condensed mass of SVOCs is in equilibrium but does not include
any additional condensation in the ascent to cloud base. This model is used
to demonstrate the importance of the co-condensation of SVOCs near cloud base on
cloud droplet number in addition to the effect of SOA that may be measured at
lower RH values. In order to justify the use of the new DCP we additionally
show the results from the parameterisation assuming multiple mode equilibrium
() at cloud base, which would be a direct analogy to the single
mode parameterisation (). This demonstrates the dynamic nature of
the condensation process near cloud base.
Activated fraction of aerosol particles using the natural
environment particle size distribution at a range of vertical updrafts. Three
different concentrations of SVOCs are used and are specified above each plot.
The crosses show the results from the parcel model and the solid lines show
the parameterisation both with (green) and without SVOCs (blue). The effect
of assuming multiple mode equilibrium at cloud base instead of the DCP is
shown in dashed red and the dashed grey shows the activated fraction
neglecting the co-condensation of SVOCs during the ascent up to cloud
base.
Figure compares the activated fraction of aerosol particles
between the parameterisations and the parcel model simulations for the
natural environment. The new parameterisation with the DCP is shown by the
solid lines with green showing the effect of SVOCs and blue showing
the analogous results without SVOCs. The analogous results from the parcel
model results are shown by the corresponding coloured crosses. As might be
expected, low concentrations of SVOCs have a limited effect on cloud droplet
number and this is demonstrated by the similarity between the green and blue
solutions in the upper left plot. This is true for both the parcel model and
the parameterisation, which are both also in good agreement for all wind
speeds.
With concentrations of SVOCs corresponding to an organic mass fraction of
50 % there is a pronounced increase in cloud droplet number as a result of
the SVOCs at wind speeds above about 0.5 m s-1. Above 1 m s-1
vertical updraft, an additional 10 % of the total number of particles
activate. This corresponds to as much as a 20 % relative increase in cloud
droplet number, or 200 cm-3, as a result of the SVOCs. Across the full
updraft range there is excellent agreement between the parcel model and the
parameterisation. If the parameterisation is used together with
multiple mode equilibrium absorptive partitioning at cloud base (red dashed),
however, there is a suppression in cloud droplet number compared to the cases
when there are no SVOCs. Similarly, if only the condensed SVOCs at 90 % RH
are taken into account (grey dashed) there is also a small suppression
between 0.2 and 2 m s-1, indicating that in this range the
co-condensation of SVOCs near cloud base is most important and creates a
narrower size distribution of particle sizes, which leads to an increase in
cloud droplet number.
In the lower left plot, corresponding to very high concentrations of SVOCs,
the parameterisation begins to deviate slightly from the parcel mode and does not pick up the suppression in cloud
droplet number between 0.2 and 1 m s-1. The agreement between the parcel model and the parameterisation is
overall still good considering the extreme abundance of SVOCs. The
discrepancies in the additional two parameterisations shown by the grey and
red dashed lines seen in the upper right plot are further demonstrated here
but to a much more severe level. In the cloud base equilibrium case, there is
a significant suppression in cloud droplet number of magnitude similar to the
enhancement due to the SVOCs seen in the DCP parameterisation and the parcel
model simulation. The co-condensation of SVOCs near cloud base is even more
important at higher concentrations and a significant suppression is seen
around 1 m s-1 vertical wind speed. This suppression occurs at much
higher vertical updrafts and to a much higher extent than is seen in the
parcel model.
Same as Fig. for the rural
environment.
Comparisons of cloud droplet number for the rural environment case between
the parcel model and the parameterisation are shown in Fig. . There is overall a smaller fraction of the particles
activating than in the natural environment. Due to there being twice as many
particles in the rural case, however, this translates into roughly the same
number concentration of CCN. Furthermore, at 10 m s-1 only
10 % of particles do not activate in the natural case and this
corresponds to the smallest mode. The smallest mode in the rural case has
roughly the same size distribution as in the natural case but contains half
of the total number of particles. These again do not activate to produce
the maximum activated fraction of 0.5 at high updrafts
The enhancement in the number of CCN reaches a maximum of about 300 cm-3 in
the 50 % organic mass fraction case at higher updrafts. This is a similar
relative increase of 20 % compared to the case without SVOCs that was seen in
the natural environment that had an analogous increase of 200 cm-3 and
a slightly lower number of CCN. The agreement between the parcel model and the
parameterisation is excellent with a very slight underestimate at lower
updrafts. The enhancement at higher updrafts is captured almost exactly. At
very high concentrations, however, the parcel model predicts very little
increase in the number of CCN from the SVOCs while the parameterisation shows a
noticeable enhancement. Overall, the agreement is still very good and the
errors from the new parameterisation with SVOCs are not larger than those
resulting from the Fountoukis and Nenes parameterisation without SVOCs.
Same as Fig. for the near-city
environment.
The importance of the DCP is again very prevalent in Fig. . The cloud base equilibrium parameterisation (dashed red)
significantly over-predicts the number of CCN at lower updrafts due to there
being more condensed mass on larger particles at equilibrium than is seen
under dynamic conditions. This increases the size of the largest particles
too much, allowing them to activate at lower supersaturations. The same effect
can be seen in Fig. below 0.2 m s-1. Conversely,
there is a less condensed mass of organics on smaller particles and this
increases the supersaturation required for activation. Therefore, at higher
updrafts, when smaller particles begin to activate in the parcel model, the
cloud base equilibrium parameterisation under-predicts the number of CCN.
Neglecting the co-condensation of SVOCs near cloud base produces similar
activated fractions to the cloud base equilibrium parameterisation. This is
due to a significant additional condensed mass on smaller particles occurring
from the co-condensation of SVOCs that creates a larger sink of water on
unactivated particles and suppresses the supersaturation.
Same as Fig. for the urban
environment.
Figure shows the results from the near-city case, which look
very similar to the rural environment. In this case, however, more particles
activate at higher updrafts but this corresponds to a lower activated
fraction. At very high updrafts there is a pronounced increase in the number of CCN
caused by the first and second modes beginning to activate. For a vertical
updraft of 10 m s-1, this produces an enhancement in cloud droplet
number of as much as 500 cm-3 compared to the case without SVOCs. Again,
the parameterisation without the DCP activates the first mode at much lower
vertical updrafts than the parcel model.
In the urban environment shown in Fig. , the SVOCs have
little effect at low concentrations, as was seen in the three other cases.
The SVOCs have little effect on cloud droplet number below an updraft of 1 m s-1 and this is likely to be the result of the fact that only the
largest mode activates here, and this mode contains a very small proportion
of the total number of aerosol particles. Above 1 m s-1, however, there
is a drastic increase in the number of CCN calculated in the parcel model and the
parameterisation with the DCP. The parcel model and the parameterisation are
both in excellent agreement, with both calculating an enhancement of 1000 cm-3 in the upper right plot. In the lower left plot, the
parameterisation predicts an enhancement of 1000 cm-3, which is lower
than the 1500 cm-3 from the parcel model. Overall, the agreement is
still very good. Above 1 m s-1, the additional two parameterisations
without the DCP predict a suppression in cloud droplet number resulting from
the SVOCs. As was previously discussed, this is a result of DCP partitioning
a significant amount of the condensed mass at cloud base onto the smaller
particles, which facilitates activation.
Scatter plot comparing the activated fractions from the parcel model
against the parameterisation for the four different environments: natural
(green dots), rural (green crosses), near-city (grey circles), and urban
(black dots). The parcel model data are the same as the green crosses in
Figs. to and the parameterisation data
have been interpolated onto the same wind speeds as the parcel model. The
1:1 line is shown by the grey dashed line.
All of the activated fraction data from Figs. to
are collated in Fig. . The x axis
shows the data from the parcel model and the y axis shows the
parameterisation with the DCP. As might be expected from Figs. to , the agreement in the
activated fraction is
very good with all data points lying close to the 1:1 line. The black dots
corresponding to the urban environment show a little deviation around an
activated fraction of 0.2, but these data points correspond to the highest
updrafts and the highest concentration of SVOCs, which are both extreme
cases.
Conclusions
A parameterisation of the cloud droplet activation process including the
effects of SVOCs is presented. The novel dynamic condensation
parameterisation (DCP) provides an analytic approximation of the
co-condensation of SVOCs near cloud base. In particular, it describes how the
condensed mass of SVOCs is distributed between different aerosol modes. This
is crucial for cloud droplet activation as it is important to predict the
change in particle sizes and chemical composition in order to ascertain the
critical supersaturation required for activation.
In the paper, we have presented results using equilibrium absorptive
partitioning theory to distribute the condensed SVOCs across the different
modes, and in general this approach places too much mass on larger modes
than is observed in the detailed parcel model. Consequently, the larger
particles can activate at lower supersaturations, as can the smaller
particles requiring higher supersaturations to activate. This can result in a
suppression in the concentration of CCN compared to the case without SVOCs in
simulations when the parcel model predicts an enhancement. The new DCP offers
much better agreement with the parcel model in these cases.
Four different environmental aerosol populations have been investigated:
natural, rural, near-city, and urban. In general, the aerosol particles become
smaller in size and higher in concentration in the more urbanised regions and
this reduces the fraction of particles that act as CCN. Up to 80–0 % of
particles activate in the natural environment compared to only 20–30 % in
the urban. The number of CCN in these two cases differs by only a factor of
2, however. In general, the inclusion of SVOCs produces an enhancement in
the concentration of CCN in all environments, with the only exception being a
suppression when the vertical updraft is around 0.5 m s-1 in the cases with a very
high concentration of SVOCs.
The most prominent enhancement in CCN concentrations is seen at higher
updrafts, resulting in a 20 % increase of 200 cm-3 in the natural
environment and a 50 % increase of 1000 cm-3 in the urban environment.
Significant underestimation in CCN concentrations of 50 % at higher updrafts
can result from either neglecting the co-condensation of SVOCs during cloud
droplet activation or assuming that the condensed phase of the organics reaches
equilibrium. Ignoring the dynamic condensation of SVOCs at cloud base at
lower updrafts can produce a gross overestimation in CCN concentrations by a
factor of 2.
MATLAB versions of the dynamic condensation parameterisation and its
associated cloud droplet activation scheme are available at
10.5281/zenodo.801398.
Change in size distribution parameters
In the parameterisation, we assume that non-volatile seed particles are
log-normally distributed. The additional aerosol mass from the condensed SVOCs
makes all particles larger, but different sizes of particles increase by
differing amounts. We assume that after the condensation of the SVOCs the
particle sizes are still log-normally distributed and describe in this
section how the median diameter and geometric standard deviation can be
calculated from the non-volatile particle size distribution parameters and
the mass of condensed organics. Depending on how close to equilibrium the
condensed masses of the SVOCs are between the different sizes of particles,
the standard deviation of the log-normal can vary. When close to equilibrium,
the size distribution is relatively wide with the larger particles increasing
in size by more than the small particles. This regime is described in
Appendix and is used to calculate the equilibrium size
distribution used at the start of the parcel model simulations and the
parameterisation. Under shorter timescales, the dynamic condensation process
of the SVOCs increase the diameter of the smaller particles by more than the
larger particles to produce a narrower size distribution. This scenario is
assumed at cloud base and is described in Appendix .
Initial size distribution
The median diameter and geometric standard deviation of the involatile seed
particles are denoted Dm and lnσm, respectively. For a total
number concentration of N, the volume of the involatile constituent is
Vm=Nπ6e3lnDm+92ln2σm.
The total volume of the composite aerosol at the initial 90 % relative
humidity is obtained by adding the total condensed mass of the SVOCs to
Vm. We denote the initial mass of the organics in the ith volatility
bin by mi0, which is calculated using equilibrium absorptive partitioning theory.
The total volume of the aerosol can then be calculated using
VT=Vm+∑imi0ρi,
where ρi is the density of the SVOCs in the ith volatility bin. To
calculate the initial median diameter of the composite aerosol, D0, we
assume a log-normal size distribution with a geometric standard deviation of
lnσm and eliminate Vm from Eq. () using () to give
Nπ6e3lnD0+92ln2σm=Nπ6e3lnDm+92ln2σm+∑kmi0ρi.
This equation can then be solved to find D0, which together with the
geometric standard deviation of lnσm defines the initial size
distribution of the particles including the condensed SVOCs.
Cloud base size distribution
We denote the new median diameter and geometric standard deviation of the dry
aerosol size distribution at cloud base as Dcb and lnσcb,
respectively. The expression for conservation of mass at cloud base,
analogous to Eq. (), is
Nπ6e3lnDcb+92ln2σcb=Nπ6e3lnDm+92ln2σm+∑kmiρi.
Here, mi is the condensed mass of the organic compounds in the
ith volatility bin at cloud base, which is calculated using the DCP. Both
Dcb and lnσcb are unknowns and an additional equation is
required in order to calculate their values. The arithmetic standard
deviation given by Eq. () is evaluated at 90 % RH and is the equated to
the analogous quantity at cloud base (100 % RH) to give
elnD0+12ln2σmeln2σm-1=elnDcb+12ln2σcbeln2σcb-1.
Together, Eqs. () and () form a set of
simultaneous equations for Dcb and lnσcb and can now be
solved to find the median diameter and geometric standard deviation of the
dry aerosol size distribution at cloud base.
Cloud base time
In order to calculate the time at which cloud base is reached and
consequently the time at which yij should be evaluated, we assume a
linear relative humidity profile with a gradient given by its initial value. We
further assume linear temperature and pressure profiles.
The water mixing ratio is defined as
r=ϵeP-e,
where e is the water partial pressure and ϵ=RaRv is
the ratio of the gas constants of dry air and water vapour, respectively.
Assuming hydrostatic balance gives the pressure gradient
dPdt=-PRaTgw.
Here, g is acceleration due to gravity and w is the updraft velocity. If
we evaluate this expression at the initial temperature, T0, and pressure,
P0, then the linear pressure profile can be written as
P=P0-P0RaT0gwt,
assuming constant updraft velocity.
Under subsaturated conditions there is little condensation of water, so
the water mixing ratio remains approximately constant. As a result the
release of latent heat from the water condensing is negligible compared to
the decrease in temperature caused by decreasing pressure with altitude.
The rate of change of temperature within a parcel of air can therefore be
expressed as
dTdt=RaTPcpdPdt,
where cp is the specific heat of air at a constant pressure. Substituting
in the pressure profile from Eq. () gives
dTdt=-gwcp.
This can be integrated to give a linear temperature profile of
T=T0-gwtcp.
The initial water mixing ratio can be calculated from the definition of
saturation ratio, S,
S=rrsat,
where rsat is the water mixing ratio at saturation and can be calculated
from Eq. () using the saturation vapour pressure. The initial water
mixing ratio is then defined as
r0=S0ϵesat(T0)P0-esat(T0),
where S0 is the initial saturation ratio and esat is the saturation
vapour pressure, which can be calculated using, for example, the Clausius–Clapeyron equation. To calculate a linear saturation ratio profile with time
we calculate S at some later time, δt, using the linear temperature
and pressure profiles of Eqs. () and (). We first
define the temperature and pressure at time δt by
T1=T0-gwδtcp,P1=P0-P0RaT0gwδt.
The water mixing ratio at time δt can then be calculated from
Eq. ():
r1=ϵe(T1)P1-e(T1).
Hence, the saturation ratio as a function of time can be approximated by
S=S0+tr1-r0rsatδt.
By defining cloud base as 100 % RH, the time to reach cloud base can then be
found by setting S=1 in Eq. () and rearranging to find t; thus,
tcb=(1-S0)rsatδtr1-r0.
Here, tcb is the approximate time that it takes for a parcel of air to
reach cloud base from the initial relative humidity. The DCP is derived to
apply only in the case when the relative humidity is close to 100 %, while
cloud droplet activation parameterisations are often initiated at 90 % RH.
This leads to a disparity in the time domain when the two parameterisations
are applicable. In the cloud droplet activation scheme that includes the
SVOCs, it is the cloud base size distribution that is needed and the DCP can
be used to calculate such a size distribution irrespective of the initial
relative humidity. The same linear RH profile that was used to derive
tcb can be used to derive approximate times to reach cloud base from any
initial relative humidity, which we define as tcb*. A discussion of the
best initial RH to use for the DCP is available in the Supplement
and is found to be 99.9 % so that the cloud base time used in the DCP is
tcb*=100-99.9100-RH0×tcb,
where RH0=S0×100%. This incorporates the fact that there is no
significant difference in equilibrium condensed mass until the relative
humidity approaches 100 % compared to 90 % RH as well as the fact that the
condensation rates derived in the DCP are only approximate.
Mass loadings
Stated here are the total mass concentrations used in the four different
environmental conditions under the three different mass loadings. The mass is
distributed between the volatility bins according to Table .
Total mass concentrations of the semi-volatile organic compounds
used in the four different environmental conditions. Values are in µg m-3.
The Supplement related to this article is available online at https://doi.org/10.5194/gmd-11-3261-2018-supplement.
The authors declare that they have no conflict of
interest.
Acknowledgements
The research leading to these results has received funding from the European
Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement
no. 603445 (BACCHUS) and the NERC-funded CCN-vol (NE/L007827/1) project.
Edited by: Klaus Gierens
Reviewed by: two anonymous referees
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