ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-17-3785-2017A new time-independent formulation of fractional releaseOstermöllerJenniferostermoeller@iau.uni-frankfurt.deBönischHaraldhttps://orcid.org/0000-0002-1004-0861JöckelPatrickhttps://orcid.org/0000-0002-8964-1394EngelAndreasan.engel@iau.uni-frankfurt.dehttps://orcid.org/0000-0003-0557-3935Institute for Atmospheric and Environmental Sciences, Goethe University Frankfurt, Frankfurt, GermanyInstitute of Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, GermanyDeutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Physik der Atmosphäre, Oberpfaffenhofen, GermanyJennifer Ostermöller (ostermoeller@iau.uni-frankfurt.de) and Andreas Engel (an.engel@iau.uni-frankfurt.de)20March20171763785379726August201616September201610February201710February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/17/3785/2017/acp-17-3785-2017.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/17/3785/2017/acp-17-3785-2017.pdf
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.
Introduction
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 as “the fraction of the halocarbon species x injected
into the stratosphere that has been dissociated”. It can be calculated by
comparing the original mixing ratio of a tracer that entered the stratosphere
to the mixing ratio that is observed at some point in the stratosphere. The
difference of this entry mixing ratio and the stratospheric mixing ratio is
equal to the amount of the species released due to photochemical breakdown.
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 . FRF should thus be specific for a
given molecule and a given atmospheric condition. If atmospheric conditions,
e.g. stratospheric dynamics or the actinic flux responsible for photochemical
degradation, change, FRF is expected to change. However, FRF should not be
dependent on the trend in the tropospheric mixing ratios of the chemical
compound (tropospheric trend) under otherwise unchanged atmospheric
conditions.
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 . This could either be caused by real changes in FRF, due to
changing atmospheric conditions, or by deficiencies in the way that the
tropospheric trends are taken into account in the calculation of FRF. The
latter is very likely, as data from different time periods are compared,
where trends differ not only in magnitude, but sometimes even in the
direction (positive/negative trend), suggesting possibly large impacts of the
way that tropospheric trends are considered in the calculation of FRF.
In the current formulation for the calculation of FRF ,
transport and mixing of chemically active species are treated in a similar
way as for chemically inert species, which are used to derive the mean age of
air. In brief this concept of mean age of the air
relies on the idea that different transport pathways (and associated transit
times) contribute to the chemical composition of an air parcel at a given
point in the stratosphere. Different transit times associated with these
different transport pathways have different probabilities, which are
described by a probability density function (pdf) also known as the age
spectrum. By folding the probability distribution for a certain entry time
into the stratosphere with the time series of the inert trace gas, its mixing
ratio at this point in the stratosphere can be derived, as long as there is
no chemical loss. The transit time distribution is called the age spectrum
G and the first moment (the arithmetic mean) is called the mean age
Γ. showed that this concept is only valid to
describe the propagation of inert tracers into the stratosphere. The
underlying reason is that air parcels which have already spent a lot of time
in the stratosphere will only contribute very little to the observed mixing
ratio of a compound which experiences photochemical loss, as a large fraction
of the molecules of this compound will not be in the organic form anymore.
Air parcels with long transit times thus need to be weighted less heavily
than air parcels with short transit times. Based on that finding,
introduced species-dependent arrival time distributions
(ATDs) to characterize the propagation of tracers undergoing chemical loss
and with changing tropospheric abundances.
used a 2-D model to calculate the species-dependent mean
arrival time Γ*, which is the first moment of the species-dependent
ATD. They showed that there is a large difference between Γ and
Γ* for CAS (in the case of an inert tracer Γ=Γ*) and
that by using Γ* instead of Γ it was possible to eliminate
differences between correlations of different species observed at different
times (and thus when tropospheric trends were different). Therefore it is
likely that the differences in the observed FRF could be influenced by the
current calculation method which is based on the mean age Γ and not on
the mean arrival time Γ*.
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. we present the classical and current
calculation methods of FRF. A description of the ECHAM/MESSy Atmospheric
Chemistry (EMAC) model and the simulations follows in Sect. .
We then calculate FRF with the model data (Sect. )
and show that the current calculation method yields time-dependent values. In
Sect. we derive a new formulation of FRF based on the
concept of arrival time distribution. In Sect. we show that
the new calculation method yields results with much reduced influence from
tropospheric trends. In the last section we discuss our results.
Calculation methods of FRF
The quantity of FRF was first introduced by as
“the fraction of the halocarbon species x injected into the stratosphere
that has been dissociated”. As explained above, FRF should be independent of
the tropospheric trend of the species, but is expected to change if
atmospheric conditions, especially stratospheric dynamics or photochemistry,
change. For the discussion of the different methods of calculating FRF, we
assume that stratospheric transport is stationary in time, i.e. that the
average circulation does not change with time. Under this assumption of
unchanged stratospheric transport, the fractional release factor f(r,t)
should not change with time, and thus be f(r), independent of t.
The substance-specific FRF can then be expressed in general by the following
equation:
f(r)=χentry(r,t)-χstrat(r,t)χentry(r,t).
Herein, χstrat(r,t) is the observed mixing ratio at the
location r and time t in the stratosphere. This is an observable
quantity, as it can be measured from balloon or aircraft samples or from
satellites. It is influenced by temporal trends in the troposphere,
stratospheric loss, and transport and mixing in the stratosphere.
χentry is the representative average entry mixing ratio of an
air parcel at location r and time t. Although both χstrat
and χentry depend on time, f should be a time-independent
quantity and thus only depend on the location in the stratosphere. The
representative average entry mixing ratio χentry should thus be
derived in a way that f is time-independent. To be consistent with previous
work , we will refer to this quantity as the
“entry mixing ratio” in the following.
In contrast to χstrat, χentry is not an observable
quantity, but serves as a reference to describe the original mixing ratio of
the CAS in the air parcel before photochemical breakdown. In the case of a
chemical compound which is in steady state between emissions into the
atmosphere and atmospheric loss, the tropospheric trend will be zero and
χentry will just be its tropospheric mixing ratio. However, if
the tropospheric mixing ratio of the trace gas changes with time,
χentry must be calculated based on assumptions about
stratospheric transport. As most ozone depleting substances are not in steady
state but have tropospheric trends, this needs to be taken into account in
calculating the entry mixing ratio χentry. It is through the
calculation of χentry that the time independence of FRF should
be achieved.
In the first formulation of FRF suggested by ,
the entry mixing ratio was calculated from the tropospheric time series by
estimating the time lag of the tracer mixing ratios between the troposphere
and the point r in the stratosphere based on mean age of air:
χentry=χtrop(t-Γ(r)),
where Γ(r) is the mean age of air, which is the mean time elapsed
since the entry of an air parcel at the tropical tropopause .
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 r in the stratosphere, it will
contain a mixture of fluid elements with longer and shorter transit times
t′ depending on the pathway they took. Note that in the following we will
use t to denote time, whereas transit time (i.e. the time of a fluid
element spent in the stratosphere) is denoted as t′. The distribution of
the probabilities of the different transit times is called the age spectrum.
It is denoted as G. Assuming that the average stratospheric transport is
stationary in time (i.e. no long-term changes in stratospheric dynamics), the
probability for a certain transit time t′ will only be a function of the
location in the stratosphere; thus, G=G(r,t′). In particular the age
spectrum will then only depend on the location r in the stratosphere and is
not a function of time t. For simplicity, we will make this assumption of
unchanged dynamics in the following, as FRF at a given location should be
unchanged as long as the stratospheric transport is unchanged.
As the sum of the probabilities of all transit times must be unity, the
integral of G(r,t′) over all possible transit times must be 1,
∫0∞G(r,t′)dt′=1,
and the arithmetic mean of the distribution can be calculated by its first
moment and is called the mean age of air:
Γ(r)=∫0∞t′G(r,t′)dt′.
Mean age is not a directly observable quantity, but it can be deduced from
observations of passive tracers like CO2 or SF6.
As noted above, the photochemical breakdown of a chemical compound increases
on average with the time the air parcel has spent in the stratosphere (t′).
FRFs thus show compact correlations with mean age of air Γ.
presented FRF as a function of mean age, including the age
spectrum in the calculation of χentry. In this formulation, the
entry mixing ratio for a certain mean age value is calculated by the
convolution of the tropospheric time series with the age spectrum
χentry(r,t)=∫0∞χtrop(t-t′)G(r,t′)dt′,
where χtrop(t-t′) represents the tropospheric mixing ratio at
time t-t′, and thus t′ before the date of observation. In this
representation of the entry mixing ratio the transport of the species to a
certain location in the stratosphere is represented by G. It takes into
account that several transit times and pathways are possible, which is an
improvement compared to the representation of the entry mixing ratio
according to Eq. (), where only a single transit
time is allowed for. Nevertheless, Eq. () is only valid for
chemically inert species and does not take account of chemical processes.
Inserting Eq. () into Eq. () yields
f(r)=∫0∞χtrop(t-t′)G(r,t′)dt′-χstrat(r,t)∫0∞χtrop(t-t′)G(r,t′)dt′.
Subsequently we will refer to Eq. () as the “current
formulation of FRF” as it has been used in and
.
Equations () and () respectively yield a single
fractional release factor f(r). In a similar way to mean age, this must be
interpreted as an average value, as of course the fractional release for the
fluid elements of an air parcel will differ depending in particular on the
time they have spent in the stratosphere, i.e. the transit time t′.
The stratospheric mixing ratio χstrat in Eqs. () and () can be deduced from observations or from model data, as
well as the tropospheric time series χtrop. In order to test how well
the current formulation can remove the effect of tropospheric trends in the
calculation of FRF, we analysed the temporal evolution of FRF using data from
the EMAC model. The EMAC model and the related simulations will be presented
in the next section and the time dependences of FRF calculated using Eq. () will be discussed in Sect. .
The EMAC model
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
. It uses the second version of the Modular Earth Submodel
System (MESSy2) to link multi-institutional computer codes. The core
atmospheric model is the 5th generation European Centre Hamburg general
circulation model (ECHAM5, ). For the present study we
applied EMAC (ECHAM5 version 5.3.02, MESSy version 2.51) in the T42L90MA
resolution, i.e. with a spherical truncation of T42 (corresponding to a
quadratic Gaussian grid of approximately 2.8 by 2.8∘ in latitude and
longitude) with 90 vertical hybrid pressure levels up to 0.01 hPa.
Simulations
In this study we analyse a reference simulation performed by the Earth System
Chemistry integrated Modelling (ESCiMo) initiative . The
RC1-base-07 simulation is a free-running hindcast simulation from 1950 to
2011. It is forced by prescribed sea surface temperatures (SSTs) and sea ice
concentrations (SICs) merged from satellite and in situ observations. The
initialization of the simulation starts in January 1950 and is followed by a
spin-up period of 10 years. Therefore we will analyse the data after 1965.
The model uses observed surface mixing ratios for boundary conditions that
were taken from the Advanced Global Atmospheric Gases Experiment (AGAGE,
http://agage.eas.gatech.edu) and the National Oceanic and Atmospheric
Administration/ Earth System Research Laboratory (NOAA/ESRL,
http://www.esrl.noaa.gov).
An important point in the model set-up is the additional implementation of
idealized tracers with mixing ratios relaxed to χtrop=1 ppt in
the lowest model layer above the surface. These idealized tracers have no
tropospheric trend, but the chemical kinetics in the stratosphere follow the
same mechanisms as for realistic tracers. However, there is no feedback of
these tracers into the chemistry, radiation or dynamics of the model. For all
tracers, the chemistry is controlled by the MECCA submodel (Module
Efficiently Calculating the Chemistry of the Atmosphere,
) and the photolysis rate coefficients are calculated by
the JVAL submodel .
Idealized tracers with constant tropospheric mixing ratios have been
implemented for the halocarbons CFC-11 (CFCl3), CFC-12
(CF2Cl2), methyl chloroform (CH3CCl3), Halon 1211
(CF2ClBr) and Halon 1301 (CF3Br), as well as for nitrous oxide
(N2O). These tracers have different lifetimes in the stratosphere:
CFC-12 and nitrous oxide are long lived with a similar stratospheric lifetime
of 95.5 and 116 years respectively . In contrast,
CFC-11 (CFCl3) and methyl chloroform (CH3CCl3) are shorter
lived, with stratospheric lifetimes of 57 and 37.7 years respectively
. The halons have stratospheric lifetimes of 33.5
(Halon 1211) and 73.5 years (Halon 1301) ().
A detailed description of ECHAM/MESSy development cycle 2 can be found in
, and references therein.
Time dependence of FRF in EMAC simulations
FRFs are often analysed as a function of mean age of air Γ. In EMAC, the age of air is calculated from a
diagnostic tracer. This tracer is linearly increasing in the lowest model
layer.
To calculate FRFs according to the current formulation, we need to solve
Eq. () and make some assumptions about the tropospheric
time series and the shape of the age spectrum. For the calculation of the
entry mixing ratio in the current FRF formulation (cf.
Eq. ), Eq. () is integrated 30 years back
in time. This is necessary to correct for the influence of the troposphere on
the stratosphere. The tropospheric time series before 1950 can be taken from
the RCP6.0 scenario . For most of the tracers
considered here, the mixing ratio before 1950 was close to zero, except for
the nearly linearly increasing tracer nitrous oxide (N2O). N2O has
increased very slowly and nearly constantly by 0.8 ppb yr-1 over the
past decades. The tropospheric mixing ratios of N2O before 1950 are
assumed to decrease by the same magnitude.
In this study we use an inverse Gaussian function for the transit time
distribution G with a constant ratio of
the squared width to mean age of Δ2/Γ=0.7 according to
and as used in previous studies . This parameterization can be used throughout most of the
stratosphere, but varies between stratospheric models .
As an example, Fig. shows the correlations of the FRF
of nitrous oxide and methyl chloroform with mean age of air using monthly
mean EMAC model data and the current calculation method (cf.
Eq. ). The correlations are compact but not
time-independent. Especially for methyl chloroform there are large
differences in the correlations, depending on the year.
Fractional release (f) as a function of mean age of air (AOA) in the
mid-latitudes between 32 and 51 ∘N for nitrous
oxide (a) and methyl chloroform (b) derived from monthly mean EMAC
model data. The FRF was calculated by the current formulation for different
dates. It can be observed that the correlations vary with time.
FRF calculated from the idealized tracers (without tropospheric
trends) of the EMAC model in the mid-latitudes between 32 and 51 ∘N.
The FRF is calculated on the 2- (purple), 3- (blue) and 4-year (green) age
isosurfaces. The absolute change in FRF per decade is noted in
parentheses.
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. . These tracers
have nearly constant mixing ratios of 1 ppt throughout the troposphere, but
in the stratosphere they experience the same transport and chemical depletion
mechanisms as the realistic tracers. The FRF of the idealized tracers can
easily be calculated by Eq. () with χentry= 1 ppt. FRF calculated by the idealized tracers gives a very
good proxy of a quasi steady-state value of FRF in the model.
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 Γ= 2, 3
and 4 years in the Northern Hemisphere mid-latitudes between 32 and
51 ∘N. In order to avoid possible spin-up effects, the analysis is
restricted to data after 1965. The temporal evolution of FRF calculated from
the idealized tracers is shown in Fig. .
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. ) are shown as black
lines for comparison. The related tropospheric trend of the species is
plotted in dashed lines over the entire range in order to compare the
magnitudes. There are obvious deviations between realistic and idealized
tracers that depend on the tropospheric trends of the species (see the text
for an explanation).
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 (CF2Cl2) and nitrous
oxide (N2O) are long lived (cf. Sect. ); even on the
4-year age isosurface, about half of the original amount remains in organic
form. In contrast, CFC-11 (CFCl3) and methyl chloroform
(CH3CCl3) are shorter lived. These species are largely depleted on
the 4-year age isosurface, with FRF values of around 0.8.
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
, who analysed changes in FRF in the Goddard Earth Observing
System Chemistry-Climate Model (GEOSCCM). These changes are consistent with
an acceleration of the Brewer–Dobson circulation due to climate change,
which is found in EMAC calculations, consistent with most other models
. A stronger circulation leads to a faster transport of
air parcels to their loss regions and thus to an increased FRF on a given
mean age level. Nevertheless, the FRF of the idealized tracers can be assumed
to be a good proxy for a quasi steady-state value in the model, as they are
not influenced by tropospheric trends.
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. show the results of the FRF
calculation for realistic tracers according to the current formulation. The
results of the idealized tracers are plotted in solid black lines and the
tropospheric trends are added by dashed black lines.
It is obvious from Fig. that the changes in FRF calculated
for the realistic tracers are much larger than for the idealized tracers. The
variation in the idealized tracers reflects the changes due to changing
chemistry and dynamics. As the only difference between the idealized and
realistic tracers is the tropospheric trend of the realistic tracers, the
larger variability of FRF for the realistic tracers must be due to the way
that the tropospheric trend is considered in the calculation of FRF according
to the current formulation.
The results differ depending on the magnitude and on the direction of the
tropospheric trend.
For N2O, which has a very small linear tropospheric trend of about only
0.2 % yr-1, the realistic and idealized tracers are in good
agreement, which means that the current formulation of FRF works well as long
as the trends are small.
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, ) and were phased out in the 1990s
due to the Montreal Protocol. For those tracers, the FRF is strongly
time-dependent and deviates systematically from the FRF of the idealized
tracers: in times of positive trends (before 1995), FRF is underestimated in
comparison to the idealized tracer. For methyl chloroform, whose positive
trend is followed by a strong negative trend from the middle of the 1990s, we
notice that the FRF is overestimated during the period of the negative trend
compared to the idealized tracer. The chlorofluorocarbons CFC-11 and CFC-12
have a much weaker negative trend from the mid-1990s than methyl chloroform
due to their longer stratospheric lifetimes. Here, the FRFs from the
realistic and idealized tracers are again in good agreement for the period
with small trends.
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 N2O or the CFCs in the 21st century, the
effect of the tropospheric trend is well removed. During periods of strong
positive trends in tropospheric mixing ratios, there is a low bias in the FRF
derived using Eq. () in comparison to the idealized
tracers. During periods of strong negative trends, as observed for
CH3CCl3 in the early 21st century, the FRF based on
Eq. () is overestimated. This time dependence could also
explain the differences between FRF values deduced from measurements at
different dates. If we for instance compare CFC-12 data on the 3-year
isosurface in 1980 and in 2000, there is an increase of about 50 % in the
FRF value (see Fig. ). The result of the calculation cannot
be regarded as a steady-state value and the possible change due to variations
in the stratospheric circulation cannot explain this magnitude of the
difference (see idealized tracers). Therefore, we conclude that it is caused
by an incomplete correction of tropospheric trends and develop a new
formulation of FRF in the following section.
A new formulation of FRF
As shown in Sect. , the currently used formulation
to derive FRF does not correct for tropospheric trends in a satisfactory
manner. In this section we will show a possible reason for and solution to
this issue.
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. , we again make the assumption of stationary
stratospheric transport, i.e. we derive a formulation of FRF which should be
constant as long as stratospheric transport (and radiation) is unchanged.
In general, a tracer's stratospheric mixing ratio χstrat(r,t)
can be formulated via its fractional release factor f if we consider that
it is the remaining fraction of the tracer which is not yet dissociated. This
fraction f will be a function of the transit pathway and the transit time
t′. For simplification, we assume that longer transit pathways will be
linked with more chemical loss and longer transit times; thus, we consider
f to be a function of t′ and location r only.
The mixing ratio of a chemically active substance at some point r in the
stratosphere at some time t, χstrat(r,t), can be calculated
by convoluting three functions: the tropospheric time series
χtrop(t-t′), the remaining fraction due to photochemical loss
(1-f(r,t′)), and the transit time distribution or age spectrum G(r,t′),
which is a function of transit time t′ and the location in the stratosphere
r. As explained in Sect. , G and f are not
functions of time t, as stratospheric transport is assumed to be stationary
in time.
All of the three functions depend on transit time t′:
χstrat(r,t)=∫0∞χtrop(t-t′)1-f(r,t′)G(r,t′)dt′.
Physically Eq. () states that the observed mixing ratio
of a CAS will be the sum over the mixing ratios of the individual fluid
elements with different transit times, different photochemical losses and
different original mixing ratios upon entry into the stratosphere. For short-lived species, the fluid elements with long transit times will contribute
very little to the observed mixing ratio in the stratosphere, as the original
content has been photochemically depleted. The tropospheric mixing ratio at
that time is thus not very relevant for the observed mixing ratio. Imagine
that a CAS has a decreasing trend in the troposphere and that its fractional
loss will be nearly complete after a transit time of 4 years. The observed
mixing ratio on the 4-year age isosurface will then be dominated by the short
fraction of the transit time distribution, whereas longer transit times must
be weighted less heavily. The probability density function describing how
strongly which transit time and thus the corresponding tropospheric mixing
ratio must be weighted should thus be different for species with different
chemical loss and in particular also different for species with chemical loss
than for species without chemical loss.
In the case of an inert tracer, f(r,t′)=0 for all possible transit times
and transport pathways. Thus the loss term (1-f(r,t′)) disappears and the
right-hand side of Eq. () reduces to
Eq. (). In this case the stratospheric mixing ratio
χstrat is identical to the entry value, as there is no chemical
breakdown.
However, only if f were constant for all fluid elements reaching point r
and independent of transit time t′, thus f=f(r) instead of f(r,t′),
could the factor (1-f(r)) be extracted from the integral, yielding
χstrat(r,t)=(1-f(r))∫0∞χtrop(t-t′)G(r,t′)dt′,
which can be rearranged to Eq. (), which is the form of
Eq. () used for the calculation of the fractional release factor
according to .
As shown here, this formulation depends upon the assumption that fractional
release for all fluid elements reaching point r is similar for all transit
times t′, which is clearly not a valid assumption.
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. (). (1-f(r,t′)) describes the loss as a function of
transit time t′. In general, the fraction of a species which has been
released from its source gas will depend both on the transit time t′ and
the transport pathway the air parcel has taken. However, on average f will
increase the longer an air parcel has stayed in the stratosphere, especially
the time spent in the loss region. For simplicity we therefore assume that
f will only be a function of the time spent in the stratosphere and not on
the pathway. The different fractional losses for different pathways are
ignored in this approach, following the “average lagrangian path” concept
proposed by .
Assuming that f will only depend on the transit time t′, we can define a
new loss weighted distribution function G*, which combines G with the
chemical loss term 1-f(r,t′):
G*(r,t′)≡1-f(r,t′)G(r,t′).
Following , we will refer to G* as the arrival time
distribution, as it represents the distribution of arrival times of
molecules, which have not been photochemically degraded.
The arrival time distribution G* is only normalized for inert tracers
without chemical loss. In this case, the loss term f(r,t′) disappears in
Eq. () and the arrival time distribution coincides with the
age spectrum G*=G.
In general G* satisfies the relation
∫0∞G*(r,t′)dt′≤1.
For this reason we define a normalized arrival time distribution
GN* by normalizing G*:
GN*(r,t′)=G*(r,t′)∫0∞G*(r,t′)dt′,
so that
∫0∞GN*(r,t′)dt′=1
with a corresponding mean arrival time Γ* that can be calculated from
the first moment of GN*:
Γ*(r)=∫0∞t′GN*(r,t′)dt′.
We now solve the integral over G*:
∫0∞G*(r,t′)dt′=∫0∞1-f(r,t′)G(r,t′)dt′=∫0∞G(r,t′)dt′-∫0∞f(r,t′)G(r,t′)dt′=1-f‾(r),
with f‾ being the first moment of the probability density
function of all fractional releases, and thus the arithmetic mean or average
fractional release
f‾(r)≡∫0∞f(r,t′)G(r,t′)dt′.
Replacing the integral in Eq. () with
Eq. ()
yields
GN*(r,t′)=G*(r,t′)(1-f‾(r)).
Solving Eq. () for G* and inserting this relation into
Eq. () yields
GN*(r,t′)1-f‾(r)=1-f(r,t′)G(r,t′),
and inserting it into Eq. () yields
χstrat(r,t)=∫0∞χtrop(t-t′)1-f‾(r)GN*(r,t′)dt′=1-f‾(r)∫0∞χtrop(t-t′)GN*(r,t′)dt′.
Note that in this formulation f‾(r) does not depend on transit time
t′ and can thus be extracted from the integral.
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 f‾ and the arrival
time distribution are known. The other way around it is possible to infer
steady-state FRFs f‾ from Eq. ().
This can be done by simply rearranging Eq. () and
solving for f‾:
f‾(r)=∫0∞χtrop(t-t′)GN*(r,t′)dt′-χstrat(r,t)∫0∞χtrop(t-t′)GN*(r,t′)dt′.
We interpret f‾ as the mean fractional release factor on a given
age isosurface. It corresponds to a quasi steady-state value. Of course, FRF
still depends on the mean age of air, which gives information on how long the
air parcel has already spent in the stratosphere. The new mean fractional
release factor f‾ should be independent of tropospheric trends
and is only expected to change if stratospheric transport or photochemistry
change. Equation () is similar to Eq. (),
suggested by , but G has been replaced by the normalized
arrival time distribution GN*. Note that for a species without a
tropospheric trend, Eqs. () and () will give
the identical result, as the integrals will yield the constant tropospheric
mixing ratios.
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
χentry(r,t)=∫0∞χtrop(t-t′)GN*(r,t′)dt′
now takes into account transport as well as chemical loss processes. Using
GN*(r,t′) instead of G results in a lesser weighting of the tail of
the transit time distribution, which is reasonable, especially for CAS with
short lifetimes. A shorter-lived species is almost completely depleted after
a transit time of e.g. 4 years; thus, this transit time t′ should not
contribute in the convolution with the tropospheric time series when
calculating the remaining organic fraction. For such shorter-lived species
the remaining amount in the original organic form is thus hardly influenced
by the tropospheric mixing ratios of air which entered a long time ago (the
“tail” of the age spectrum for an inert trace gas). The shorter-lived the
trace gas is, the more the weighting needs to be shifted to the short
fraction of the age spectrum. The arrival time distribution describes the
relevant weighting of the different transit times and is specific for each
trace gas.
A complication is of course that the normalized arrival time distribution
GN*(r,t′) needs to be known in order to solve Eq. (). This
arrival time distribution has been calculated from a 2-D chemical transport
model by .
Following , we call the first moment of this distribution
the “mean arrival time” Γ* which takes into account the chemical
loss of the species. A possible parameterization of Γ* was described
by . Γ* is a substance-specific quantity and depends
on mean age and the stratospheric lifetime of the tracers. In the following
section we test the new formulation of FRF f‾ given in
Eq. () by applying it to EMAC model data. We compare the results
to the current formulation of FRF f, based on Eq. ().
Results of the new formulation
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. () which takes into account
effects of chemical loss to the same data set as for the analysis of the
current FRF formulation presented in Sect. . This
means we examine the temporal evolution of FRF on the same latitude band and
age of air isosurfaces again using both idealized and realistic tracers.
To solve Eq. () it is necessary to find a good description of
GN*. We choose GN* to have the same shape as G, i.e. an inverse
Gaussian distribution but with the parameters Γ* (first moment) and
Δ* (second moment), so that GN*=G(Γ*,Δ*,t′). Like
for G we use a constant ratio of the squared width to mean age of
Δ*2/Γ*= 0.7 years.
derived a parameterization of a species-dependent mean
arrival time Γ* for a wide range of chemically active species from a
delta pulse emission calculation. Γ* can be calculated from the mean
age of air Γ and the mean stratospheric lifetime τ by a
parameterization scheme . Using Γ* instead of
Γ takes account of the chemical loss occurring on the transport
pathways. We computed Γ* for the considered species and applied it as
the first moment of our new arrival time distribution GN*.
The result of the new calculation of FRF according to Eq. () can
be seen in Fig. .
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. ), FRF is
slightly overestimated compared to the idealized tracer in times of positive
trends for CFC-12 (CF2Cl2) and methyl chloroform
(CH3CCl3). For CFC-11 (CFCl3), the FRF according to the new
formulation is somewhat underestimated on the 2- and 3-year age isosurfaces,
but fits the idealized tracer well on the 4-year age
isosurface. Furthermore, the FRF of methyl chloroform is underestimated when
tropospheric mixing ratios are declining. The reason for this feature is an
overly large correction between Γ and Γ*.
As we would expect, the fractional release of N2O is nearly unaffected
by the new calculation method, because of its small tropospheric trend. For
CFC-11 and CFC-12 there are still small deviations between the realistic and
idealized tracers, but the steady-state value is reached much earlier than in
the current formulation and overall the differences are much smaller. Indeed,
we do expect species- and age-dependent differences in the results, as the
same parameterization is used to derive Γ* from Γ for all mean
age values and different parameters are used for different species.
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 (Γ=4 years) in
comparison to the current formulation of FRF used in Fig. .
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 , who derived the
parameterization from a simple 2-D model. It could still be improved to
obtain an even better adaption to the idealized tracer. Besides this, we also
kept an inverse Gaussian distribution with a similar parameterization as for
mean age, which might not be the optimal choice for the new arrival time
distribution.
Summary and discussion
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 ∘N.
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 N2O tracer hardly differs from the idealized
tracer, because it has a very small tropospheric trend.
This may lead to discrepancies in FRFs derived during different time periods.
Such differences in FRF have been observed between the work of
and . The FRF values derived by
were lower than those derived by on the
3-year mean age isosurface. As the tropospheric trends were lower during the
observations used by , it is expected that the
re-calculation using our method should even increase the observed difference.
We therefore conclude that the calculation of mean age may be the reason for
the observed discrepancies, as suggested by .
We also acknowledge that the new formulation is less intuitive than the
formulation used by and . However, as we
have shown that the method used by and
yields values which are strongly influenced by the tropospheric trend, this
loss of intuitivity and the added dependence on model information are
necessary, as much more representative values are derived.
To include chemical loss in the transit time distribution, we applied the
parameterization described by . Using the new formulation of
the stratospheric mixing ratio (with loss), we constructed a new expression
of the FRF and validated it with EMAC data.
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 depends on the
stratospheric lifetime of the species. As fractional release also depends on
the lifetime, one may argue that there is a certain circular argumentation
involved. Indeed, if the assumption of stratospheric lifetime is very far
off, and tropospheric trends are large, then our new method will also fail in
correcting for the tropospheric trend. However, it should be noted that the
calculation is not extremely sensitive to the assumed lifetime. We
investigated the sensitivity for a CFC-12-like tracer with a linearly
increasing trend of 5 % yr-1. For an assumed steady-state FRF of
0.5 at a mean age of 4 years using our method, a value of 0.5 is found with a
deviation of 0.5 % for an uncertainty in the assumed lifetime of
20 %. Using the current method, ignoring the effect of chemical loss
would result in an FRF of 0.45, i.e. 10 % lower than the correct value.
The sensitivity to the assumed lifetime is thus rather small.
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 (http://www.messy-interface.org).
The data of the simulations described above will be made available in the Climate and Environmental Retrieval and Archive
(CERA) database at the German Climate Computing Centre (DKRZ; http://cera-www.dkrz.de/WDCC/ui/Index.jsp).
The corresponding digital object identifiers (doi) will be published on the MESSy consortium web page (http://www.messy-interface.org).
The authors declare that they have no conflict of
interest.
Acknowledgements
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
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