The fractional release factor (FRF) gives information on the amount of a halocarbon that is released at some point into the stratosphere from its source form to the inorganic form, which can harm the ozone layer through catalytic reactions. The quantity is of major importance because it directly affects the calculation of the ozone depletion potential (ODP). In this context time-independent values are needed which, in particular, should be independent of the trends in the tropospheric mixing ratios (tropospheric trends) of the respective halogenated trace gases. For a given atmospheric situation, such FRF values would represent a molecular property.

We analysed the temporal evolution of FRF from ECHAM/MESSy Atmospheric Chemistry (EMAC) model simulations for several halocarbons and nitrous oxide between 1965 and 2011 on different mean age levels and found that the widely used formulation of FRF yields highly time-dependent values. We show that this is caused by the way that the tropospheric trend is handled in the widely used calculation method of FRF.

Taking into account chemical loss in the calculation of stratospheric mixing ratios reduces the time dependence in FRFs. Therefore we implemented a loss term in the formulation of the FRF and applied the parameterization of a “mean arrival time” to our data set.

We find that the time dependence in the FRF can almost be compensated for by applying a new trend correction in the calculation of the FRF. We suggest that this new method should be used to calculate time-independent FRFs, which can then be used e.g. for the calculation of ODP.

Chlorine- and bromine-containing substances with anthropogenic sources have a
strong influence on ozone depletion in the stratosphere. The gases are
emitted in the troposphere, where many of them are nearly inert before they
enter the stratosphere at the tropical tropopause. In the stratosphere, many
of the molecules will be broken down photochemically and release halogen
radicals that intensify ozone destruction

The fraction of a halocarbon at some point in the stratosphere that is
released from the organic (source) gas into the inorganic (reactive) form is
quantified by its fractional release factor (FRF). The quantity was defined
by

When entering the stratosphere at the tropical tropopause, ozone depleting substances (ODS) have a FRF that is zero. As they follow the stratospheric circulation, the air parcels get distributed by different transport pathways and pass through their photochemical loss regions, where the molecules get dissociated. The FRF increases until it reaches the value of 1 when the ODS is completely depleted and all halogen atoms it contained have been released.

FRF thus describes the effectiveness with which a certain ODS is broken down
in the stratosphere. For the same time spent in the stratosphere, shorter-lived species will have higher FRF than longer-lived molecules. FRF are
therefore used in the calculation of the ozone depletion potential (ODP), a
quantity which describes how effective a certain chemical is at destroying
stratospheric ozone

For every tracer with changing tropospheric mixing ratios, we thus need to ensure that this trend does not affect the FRF values derived from stratospheric observations. The observed mixing ratio of a chemically active species (CAS) in the stratosphere is, however, influenced by its tropospheric trend and by chemical breakdown. Only the latter should contribute to the FRF. In the calculation of FRF the tropospheric trend thus needs to be taken into account and corrected for. As the different transit pathways contributing to the air parcel are associated with different transit times and different photochemical breakdowns, the complex interplay between transport, mixing and photochemistry needs to be described correctly for this purpose.

In recent years, inconsistencies between FRF values derived from independent
observations at different times were identified

In the current formulation for the calculation of FRF

In this paper, we first examine how strongly the FRF calculated using the
current formulation is influenced by the tropospheric trend, using model
calculations of FRF for some typical CAS. We show that the tropospheric trend
has a significant impact on FRF. We then present a new improved formulation
to calculate FRF which removes the impact of tropospheric trends much better.
In Sect.

The quantity of FRF was first introduced by

The substance-specific FRF can then be expressed in general by the following
equation:

In contrast to

In the first formulation of FRF suggested by

The concept of age of air (AOA) can be understood as follows: we consider a
stratospheric air parcel that consists of infinitesimal fluid elements. An
air parcel at some point in the stratosphere will consist of a nearly
infinite number of such fluid elements. When entering the stratosphere, the
fluid elements get distributed along different transport pathways. If we
consider an air parcel at some location

As the sum of the probabilities of all transit times must be unity, the
integral of

As noted above, the photochemical breakdown of a chemical compound increases
on average with the time the air parcel has spent in the stratosphere (

Inserting Eq. (

Equations (

The stratospheric mixing ratio

The ECHAM/MESSy Atmospheric Chemistry (EMAC) model is a numerical chemistry and climate simulation system that
includes submodels describing tropospheric and middle atmosphere processes
and their interaction with oceans, land and human influences

In this study we analyse a reference simulation performed by the Earth System
Chemistry integrated Modelling (ESCiMo) initiative

The model uses observed surface mixing ratios for boundary conditions that
were taken from the Advanced Global Atmospheric Gases Experiment (AGAGE,

An important point in the model set-up is the additional implementation of
idealized tracers with mixing ratios relaxed to

Idealized tracers with constant tropospheric mixing ratios have been
implemented for the halocarbons CFC-11 (CFCl

A detailed description of ECHAM/MESSy development cycle 2 can be found in

FRFs are often analysed as a function of mean age of air

To calculate FRFs according to the current formulation, we need to solve
Eq. (

In this study we use an inverse Gaussian function for the transit time
distribution

As an example, Fig.

Fractional release (

FRF calculated from the idealized tracers (without tropospheric
trends) of the EMAC model in the mid-latitudes between 32 and 51

This is a first hint that there is a time dependence in the current representation of the FRF.

There may be several reasons for this time dependency. On the one hand, changes in the stratospheric circulation or chemistry could cause changes in fractional release on a given age isosurface; on the other hand, it is possible that the tropospheric trend of the species has an impact on the derived fractional release factor.

In order to separate the two possible effects from each other, we make use of
the idealized tracers described in Sect.

Assuming that the age spectra for different locations with the same mean age are similar, we investigate changes in FRF in the model on age isosurfaces instead of on geographical coordinates. As mean age e.g. at a given location shows some variability with time, this is expected to lead to reduced variability.

We calculated the temporal evolution of zonal mean FRF values derived from
monthly mean data on the constant mean age of air surfaces

Temporal evolution of FRFs calculated by the current formulation.
Results for the realistic tracers are shown in colour. The results for the
idealized tracers (cf. Fig.

On older mean age of air surfaces we find higher FRF values, which is
reasonable, because older air has had more time to travel through the
photochemical loss regions than younger air. The value of FRF depends on the
species and their photolytic lifetimes. CFC-12 (CF

We notice a seasonality in FRF, which can be explained by seasonal variations
in transport, chemistry and mixing. These are stronger in the upper
stratosphere, due to shorter local lifetimes. Beside this, we can see that
the FRFs for idealized tracers only slightly vary with time. The increase in
FRF is of the order of about 5 % per decade, which is in agreement with

The temporal evolution of the FRF of the realistic tracers (with tropospheric
trends) is analysed on the same latitude band and AOA surfaces as for the
idealized tracers. The results are shown in Fig.

The coloured lines in Fig.

It is obvious from Fig.

The results differ depending on the magnitude and on the direction of the tropospheric trend.

For N

The situation is different if we consider the anthropogenically emitted
chlorofluorocarbons and methyl chloroform, which had strong positive trends
in the 1980s (growth rate of about 6 % for CFC-11 and CFC-12, 8.7 %
for methyl chloroform,

To sum up, our model experiments show that the tropospheric trend influences
the current FRF calculation and imposes a time dependence. If trends are
sufficiently small, as for N

As shown in Sect.

We consider the propagation of a CAS with solely tropospheric sources into
the stratosphere. Air parcels enter the stratosphere at the tropical
tropopause. In the stratosphere, the CAS gets distributed by the meridional
overturning circulation (Brewer–Dobson circulation), which includes residual
circulation and mixing in a similar way as for an inert tracer. In addition,
the CAS will also be chemically depleted by sunlight or radicals during the
transport. The mixing ratio of the CAS at a certain location in the
stratosphere is thus influenced by the temporal trend in the troposphere,
transport and mixing in the stratosphere, as well as loss processes. As in
Sect.

In general, a tracer's stratospheric mixing ratio

The mixing ratio of a chemically active substance at some point

All of the three functions depend on transit time

In the case of an inert tracer,

However, only if

As shown here, this formulation depends upon the assumption that fractional
release for all fluid elements reaching point

In order to derive a new formulation of FRF with better correction for
tropospheric trends, we again take a look at the loss term in Eq. (

Assuming that

The arrival time distribution

In general

From this equation we can now calculate the mixing ratio of a chemically
active tracer at any location and time in the stratosphere as long as the
tropospheric time series, the new average FRF

This can be done by simply rearranging Eq. (

We interpret

Temporal evolution of FRF calculated by the new formulation, taking
into account chemical loss. The results of the realistic tracers are shown in
colour on different age isosurfaces. The results of the idealized tracers are
shown in solid black lines, whereas the tropospheric trend is plotted in
dashed lines. We find much better agreement between idealized and realistic
tracers compared to the current formulation of FRF (cf.
Fig.

The entry mixing ratio in this new formulation

A complication is of course that the normalized arrival time distribution

Following

In the last section a new formulation of FRF has been derived which should be
able to correct the effect of tropospheric trends when calculating FRF. We
apply our new formulation Eq. (

To solve Eq. (

The result of the new calculation of FRF according to Eq. (

We clearly notice the improvement of the new calculation method. The tropospheric trend of the species is almost corrected for and FRF values for the idealized and realistic tracers show much better agreement.

In contrast to the current formulation (cf. Fig.

As we would expect, the fractional release of N

The largest change can be seen for methyl chloroform, which is the analysed
substance with the largest variation in the tropospheric trend. The realistic
tracer now approaches the idealized tracer and we can see the improvement
especially for the highest considered age isosurface (

To sum up, we conclude that including chemical loss in the calculation
reduces the time dependence of the FRF value substantially. The
parameterization of loss was adopted from

In this paper we presented a study on fractional release factors (FRFs) and their time dependence. We analysed the temporal evolution of FRFs between 1965 and 2011 for the halocarbons CFC-11, CFC-12 and methyl chloroform, as well as for nitrous oxide. FRF is often treated as a steady-state quantity, which is a necessary assumption to use it in the calculation of ODP and EESC. In the current formulation of FRF, the transit time distribution and the tropospheric time series of the substances are taken into account, but the coupling between trends, chemical loss and transit time distribution is not included.

For chemically active species, the fraction of the air with very long transit
times (the “tail” of the transit time distribution) will have passed the
chemical loss region and therefore only contributes very little to the
remaining organic fraction, but is to a large degree in the inorganic form.
On the other hand, the fraction of the air with short transit times will be
to a large degree still in the form of the organic source gas, as it has not
been transported to the chemical loss region. This must be taken into account
when folding the transit time distribution with the tropospheric time series
to derive the fraction still residing in the organic (source) form. For this
we used an arrival time distribution, based on the concept and
parameterization suggested by

We applied the two FRF calculation methods (current and new) to EMAC model
data and studied the differences. For both methods we used exemplarily (but
without loss of generality) zonally averaged monthly mean stratospheric
mixing ratios in a latitude band between 32 and 51

A special feature of the used model simulation are the implemented idealized tracers with nearly constant tropospheric mixing ratios. We showed that the use of the new formulation of the propagation of chemically active species with tropospheric trends into the stratosphere results in FRF values, which are to a large degree independent of the tropospheric trend of the respective trace gas and thus give a quasi steady-state value of FRF. This is shown by a much better agreement with the FRF of the idealized tracers, which have no tropospheric trend.

In contrast, the classical approach yields FRF values that depend on
tropospheric trends, which change with time. This might be an explanation for
the discrepancies between FRF values deduced from observations at different
dates. The reason for the non-steady behaviour is obviously based on an
incomplete trend correction. In times of strong tropospheric trends, the
realistic tracers deviate most from the idealized tracers. On the other hand,
the FRF of the realistic N

This may lead to discrepancies in FRFs derived during different time periods.
Such differences in FRF have been observed between the work of

We also acknowledge that the new formulation is less intuitive than the
formulation used by

To include chemical loss in the transit time distribution, we applied the
parameterization described by

The newly calculated FRF values fit well to the results of the idealized steady-state tracers and the influence of the tropospheric trend can almost completely be corrected. This is remarkable, because we have to keep in mind that the parameterization was derived from a completely different and independent 2-D model and that we used the same shape parameters as for the classical age spectrum.

Our new method produces FRF values which are far less dependent on
tropospheric trends. In the case of small tropospheric trends the results
will converge with those using the current formulation and also with those
for idealized tracers without any trends. On the other hand, more model
information is needed for the derivation of the FRF values, as
species-dependent arrival time distributions need to be applied. The
parameterization given by

We suggest using the new formulation and reassessing former FRF data. Especially FRF values calculated from observations at times of strong tropospheric trends will profit from the new calculation method. Many fully halogenated CFCs showed strong trends prior to 1990, while many HCFCs still show very strong positive trends. This implies that FRF values currently used for HCFCs are likely to be underestimated, which would lead to an underestimation of their ODP values.

We suggest that this new method should be refined by calculating the arrival time distributions in state-of-the-art models and deriving parameterizations from these models. These new methods should be tested by including idealized tracers in the same models and subsequently be applied to observations which have been used to derive FRF values. Using these new FRF values, a reassessment of ODP values for halogenated source gases and also a re-evaluation of temporal trends of EESC are necessary.

The Modular Earth Submodel System (MESSy) is continuously further developed and applied by a consortium of institutions.
The usage of MESSy and access to the source code is licensed to all affiliates of institutions which are members of the
MESSy Consortium. Institutions can become a member of the MESSy Consortium by signing the MESSy Memorandum of Understanding.
More information can be found on the MESSy Consortium website (

The authors declare that they have no conflict of interest.

This work was supported by DFG Research Unit 1095 (SHARP) under project numbers EM367/9-1 and EN367/9-2. We thank all partners of the Earth System Chemistry integrated Modelling (ESCiMo) initiative for their support. The model simulations have been performed at the German Climate Computing Centre (DKRZ) through support from the Bundesministerium für Bildung und Forschung (BMBF). DKRZ and its scientific steering committee are gratefully acknowledged for providing the HPC and data archiving resources for this ESCiMo consortial project. Edited by: J.-U. Grooß Reviewed by: two anonymous referees