Analysis of stratospheric transport from an observational
point of view is frequently realized by evaluation of the mean age of air values
from long-lived trace gases. However, this provides more insight into
general transport strength and less into its mechanism. Deriving complete
transit time distributions (age spectra) is desirable, but their deduction
from direct measurements is difficult. It is so far primarily based on model
work. This paper introduces a modified version of an inverse method to infer
age spectra from mixing ratios of short-lived trace gases and investigates
its basic principle in an idealized model simulation. For a full description
of transport seasonality the method includes an imposed seasonal cycle to
gain multimodal spectra. An ECHAM/MESSy Atmospheric
Chemistry (EMAC) model simulation is utilized for a general
proof of concept of the method and features an idealized dataset of 40
radioactive trace gases with different chemical lifetimes as well as 40
chemically inert pulsed trace gases to calculate pulse age spectra. It is
assessed whether the modified inverse method in combination with the
seasonal cycle can provide matching age spectra when chemistry is
well-known. Annual and seasonal mean inverse spectra are compared to pulse
spectra including first and second moments as well as the ratio between them
to assess the performance on these timescales. Results indicate that the
modified inverse age spectra match the annual and seasonal pulse age spectra
well on global scale beyond 1.5 years of mean age of air. The imposed seasonal
cycle emerges as a reliable tool to include transport seasonality in the age
spectra. Below 1.5 years of mean age of air, tropospheric influence intensifies
and breaks the assumption of single entry through the tropical tropopause,
leading to inaccurate spectra, in particular in the Northern Hemisphere. The
imposed seasonal cycle wrongly prescribes seasonal entry in this lower
region and does not lead to a better agreement between inverse and pulse age
spectra without further improvement. Tests with a focus on future application
to observational data imply that subsets of trace gases with 5 to 10 species
are sufficient for deriving well-matching age spectra. These subsets can also
compensate for an average uncertainty of up to

Stratospheric meridional circulation, referred to as Brewer–Dobson circulation (BDC), is a key process for the comprehension of air mass transport throughout the atmosphere. The spatial distributions and atmospheric lifetimes of various greenhouse gases and ozone-depleting substances, such as halocarbons, are strongly influenced by this large-scale motion (Butchart and Scaife, 2001; Solomon et al., 2010). Therefore, the BDC affects not only the chemical composition of the stratosphere but also the radiative budget of the complete atmosphere. The BDC is a combination of a residual mean circulation with net mass flux and eddy-induced isentropic bidirectional mixing (Plumb, 2002; Shepherd, 2007; Butchart, 2014). Air is transported mainly through the tropical tropopause and then advected to higher latitudes, where it eventually descends. Primary drivers are tropospheric planetary- and synoptic-scale Rossby waves that propagate upward and transfer their momentum by breaking in the extratropical middle stratosphere (Haynes et al., 1991; Holton et al., 1995). At the same time, this wave drag induces stirring processes that are especially enhanced in this “surf zone” (McIntyre and Palmer, 1984). It has been shown that the tropical upward mass flux has a distinct seasonal cycle with a maximum during northern hemispheric (NH) wintertime, when wave excitation is largest (Rosenlof and Holton, 1993; Rosenlof, 1995).

A problem that arises especially regarding observational investigation of
the stratospheric circulation is the impossibility of direct measurements of
the underlying dynamics, i.e., slow overturning circulation. However, a
suited tool for quantification, which can be derived from observations of
chemically very long-lived trace gases and directly compared to model
results, is the concept of mean age of air (AoA; Hall and Plumb, 1994;
Waugh and Hall, 2002). Mean AoA can be understood as the average period of
time that elapsed for an air parcel at any arbitrary location since passing
a certain reference point. Usually, the reference is either earth's surface
or the tropical tropopause layer. Mean AoA provides not only insight into
the current overall strength of the BDC but also allows for an
investigation of temporal changes. If the circulation intensity varies over
time, the value of mean AoA will also show this trend but will be inversely
proportional (Austin and Li, 2006). Different models predict an enhanced
stratospheric circulation indicated by a negative trend of mean AoA (Garcia
and Randel, 2008; Li et al., 2008; Oman et al., 2009; Shepherd and
McLandress, 2011) in response to strengthened wave drag by rising greenhouse
gas concentrations. On the other hand, sparse observationally derived mean
AoA from balloon-borne SF

For an even more thorough analysis of stratospheric transport, the usage of a full transit time distribution is of advantage. The interaction of mean residual transport and bidirectional mixing as well as the influence of shallow and deep branch is expressed in this distribution, which is also referred to as age spectrum (Hall and Plumb, 1994; Waugh and Hall, 2002). At any given point in the stratosphere, the age spectrum denotes the fraction of fluid elements in an air parcel that had a certain transit time from the reference point to the chosen location. It can be considered a probability density function (PDF), with the mean AoA being the first moment of this distribution. An important advantage is that changes in the different ranges of transit times of the BDC can be visualized with a comparison of age spectra at different points in time. Variations of the shallow branch influence the age spectrum mostly at short transit times (ca. 1 to 2 years – Birner and Bönisch, 2011), whereas changes in the deep branch influence mostly at long transit times (ca. 4 to 5 years – Birner and Bönisch, 2011). Additionally, the effect of aging by mixing (Garny et al., 2014) on the tail of the spectrum may also be assessed during such an analysis. In model experiments, the age spectrum can be gained via periodically occurring pulses of a chemically inert trace gas. For mean AoA, either a linearly increasing chemically inert trace gas or the mean of the age spectrum can be applied. In reality, only few very long-lived trace gases exhibit a linear trend in the first approximation and can be utilized to gain mean AoA (Andrews et al., 2001a). When deriving mean AoA from observations of very long-lived trace gases with non-linear increase, however, assumptions about the shape of the age spectrum are required, where mostly the pioneering work of Hall and Plumb (1994) is considered. The underlying age spectrum for this calculation is only an approximation, which might not be representative in all cases and could bias the inferred mean AoA. A directly inferred complete age spectrum may constitute an improvement and could give further insight into transport processes. This information might be extracted from mixing ratios of long- and short-lived trace gases in an air parcel, as for such species, residual transport, mixing and chemical depletion are inevitably associated with each other. The state of depletion of these gases provides an estimate of the elapsed transit time since passing the reference point. In addition, trace gases with varying chemical depletion are only sensitive to certain transport pathways and thus contribute mainly to the age spectrum for transit time ranges up to their respective chemical lifetime (Schoeberl et al., 2000, 2005). The complete age spectrum may then theoretically be derived as a combination of those pieces of information, provided that a sufficient number of distinct mixing ratios are known. Since the amount of air parcels with transit times larger than 10 years is likely to be low even in the uppermost part of the stratosphere, trace gases with lifetimes of up to 10 years should be sufficient for retrieving a meaningful age spectrum. That makes short-lived trace gases a suitable tool, as a variety of species with diverse lifetimes were frequently measured during past airborne research campaigns. Unfortunately, to the best of our knowledge, there are only few publications about possible techniques to finally convert the information of the mixing ratios of short-lived trace gases into stratospheric age spectra (Schoeberl et al., 2005; Ehhalt et al., 2007) and none that include seasonality in transport. When analyzing seasonal variation in stratospheric dynamics by inferring age spectra from observations, though, a proper consideration of the seasonal cycle is also required to achieve reliable results.

This paper presents an application and evaluation of a modified version of the method by Schoeberl et al. (2005; Schoeberl's method) with a reduced set of fit parameters and an imposed seasonal cycle to account for seasonality in stratospheric transport. The modified technique is applied as a proof of concept to an idealized simulation of the ECHAM/MESSy Atmospheric Chemistry (EMAC) model (Jöckel et al., 2006, 2010). Section 2 provides insight into the method and the model simulation. In Sect. 3, resulting age spectra and related quantities are analyzed and assessed with respect to future application to observational data. Finally, there is a summary, conclusion and outlook in Sect. 4.

A frequent basis for the estimation of age spectra, which is also utilized in
Schoeberl et al. (2005) and Ehhalt et al. (2007), is the mathematical
description of a mixing ratio

Ehhalt et al. (2007) used the formalism of Hall and Plumb (1994) to
derive a method where mixing ratios of short-lived trace gases are
correlated against those of very long-lived ones. This correlation is
identical to a vertical profile in a trace-gas-based coordinate system.
Technically, the method can also be applied using other vertical coordinates
such as altitude, pressure or potential temperature as long as there is an
exponential decay in the vertical space. In order for this to work, a
height-independent chemical lifetime has to be assumed, as Eq. (2) only then turns into a Laplace transform of
the age spectrum and makes for a constant vertical diffusion coefficient.
This is quite an elegant approach, since it allows for an analytical
solution of the equation. The study concluded that the resulting age spectra
from data between 30 and 35

There are, however, two points concerning the method's formulation that are
worthy of discussion. First, the vertical coordinate

When studying seasonal variability in transport processes, it was already
mentioned that the tropical upward mass flux has a distinct seasonal cycle,
with a clear maximum during NH wintertime and a minimum in NH summer
(Rosenlof, 1995). This cycle is also visible in age spectra, as recent
transport model studies have shown (e.g., Reithmeier et al., 2008; Li et
al., 2012a; Ploeger and Birner, 2016). The annual mean shape of the pulse
spectra is well approximated by an inverse Gaussian distribution with one
obvious mode. But age spectra for single seasons show several modes
representing the variable flux of mass into stratosphere during the
different seasons. This is an important feature, especially for species with
a seasonal cycle (e.g.,

A possible approach to derive a multimodal spectrum for a specific season is
to scale the age spectrum in Eq. (6)
appropriately. Rosenlof (1995) found that the strength of the tropical
mass flux into the stratosphere across the tropical tropopause at 70

Scaling constants and phase shift for each season.

The use of model data for a general proof of concept of the method's capabilities is very suitable, since the idealized setup yields the advantage of lifetimes being independent of transit time and transport pathways. Scientific focus in this study is on the method's potential to capture stratospheric transport with its underlying assumptions in a solely dynamical model setup. The model of choice is EMAC, being a potent and very well-performing chemistry climate model which has been used in many studies regarding stratospheric transport and chemistry.

The ECHAM/MESSy Atmospheric Chemistry (EMAC) model is a numerical chemistry
and climate simulation system that includes sub-models describing
tropospheric and middle atmosphere processes and their interaction with
oceans, land and human influences (Jöckel et al., 2010).
It uses the second version of the Modular Earth Submodel System (MESSy2) to link
multi-institutional computer codes. The core atmospheric model is the
fifth-generation European Centre Hamburg general circulation model
(Roeckner et al., 2006). For the present study we applied EMAC (ECHAM5
version 5.3.02, MESSy version 2.53.0) in the T42L90MA resolution, i.e., with
a spherical truncation of T42 (corresponding to a quadratic Gaussian grid of
approximately 2.8

In order to retrieve age spectra from the model itself, 40 chemically
completely inert trace gases are included that are released as a pulse every
3 months at the tropical surface between 12.5

Exemplary BIR map for the analyzed time span of the EMAC
simulation at 55

For the application of the inverse method, a further set of 40 trace gases
is included with prescribed constant lifetimes ranging from 1 month up to
118 months by steps of 3 months. This simplification allows for an
investigation of the method's basic principle by eliminating the variability
in chemical depletion. These trace gases are constantly released in the same
source region as the pulse tracers with a mixing ratio of 100 %. Figure 3
visualizes annual mean vertical profiles of five of these radioactive
tracers at 85, 55 and 10

Vertical profiles of five radioactive trace gases at 85

A problem for the comparison of the spectra is that all pulse spectra are
referred to earth's surface, whereas the inverse method uses the tropical
tropopause as a reference layer. When trying to refer inverse spectra to the
surface, results broaden significantly and do not match the pulse spectra.
This is likely a consequence of the solution given by Hall and Plumb
(1994) that uses one-dimensional diffusion to approximate transport above
the tropopause and might not be sufficient for application in the tropical
troposphere. Still, to make the spectra comparable, the annually averaged
mean AoA from the clock tracer at the tropical tropopause is derived for the
evaluated field time period. It is considered to be the mean transit time
from surface to the tropopause within the model. This quantity
(

Atmospheric pressure is in general not a suitable choice when investigating
transport processes and especially mixing. In the stratosphere,
bidirectional stirring occurs mainly parallel to isentropic surfaces, which
makes potential temperature the best choice for such a study. To include an
adequate description of all transport mechanisms, the local potential
temperature difference to the tropopause is calculated for every
stratospheric model grid point and used as vertical coordinate

The following section provides the results of the model study to evaluate
the performance of the inverse method. Annual and seasonal spectra are
presented for three different pressure levels (70, 10 and 140

Age spectra at 55

Figure 4 shows age spectra at 70

Same as Fig. 4, but at 55

Age spectra around the previously defined upper boundary (10

Same as Fig. 4, but at 55

Finally, age spectra for the lower stratosphere on the 140

Since the seasonal cycle leads to an improvement of the inverse spectra except for the lower stratosphere and no significant changes in the annual mean state, all of the following results of the inverse method will include the imposed cycle.

The percentage differences of the annual pulse and inverse mean AoA to the
clock tracer are given in Fig. 7. The pulse mean AoA (panel a) matches the
clock mean AoA very well except for the lower stratosphere. This is expected
for sharp peaks of the age spectrum, such as in Fig. 6, where
the temporal resolution of 3 months might not be sufficient and leads to
inaccurate mean AoA. The tail correction of the spectra as well as their
sole initialization in the tropics might also contribute to these
deviations. For lower pressures, however, the approximation of the tail
correction seems to be adequate with matching mean AoA values. The main
differences between pulse and clock mean AoA also appear to follow the
shaded area (panel b) of the 1.5-year threshold (see Sect. 2.3). If
globally averaged for the pulse spectra, this results in deviations of

Annually averaged deviation of pulse

Mean AoA as global cross section. Left column shows absolute annual mean values, and remaining four columns denote percentage seasonal difference to annual average. Data are from clock tracer (Clock), from pulse spectra (Pulse) and from inverse spectra including the imposed seasonal cycle (Inverse). The tropopause is indicated by the solid black line. White dashed area marks pressure levels below clock mean AoA of 1.5 years. The wavy structure is visible in all seasonal plots of the inverse method, which follow hypothetical isentropic surfaces. Since potential temperature relative to the tropopause is used as vertical coordinate for the inverse method, these structures are artefacts of the implementation and do not influence the validity of results.

Figure 8 shows mean AoA of the clock tracer and the pulse and
inverse spectra as a latitude–pressure cross section. The left column gives
the absolute annual average, and the remaining four columns give the percentage
deviations of all seasons from each annual mean. The seasonal patterns of
pulse spectra and clock tracer are matching very well with a few differences
in the lowermost stratosphere (especially DJF and JJA). This coincides with
the results of Fig. 7. The structures of mean AoA for the inverse method
match the pulse spectra and clock tracer in each season very well qualitatively above the dashed area. Only the amplitude of the seasonal changes seems
to be enhanced in direct comparison. This indicates that the seasonality
that could already be detected in Figs. 4 and 5 extends mostly to the global
scale. A clear boundary of those seasonal patterns is visible in the lower
stratosphere around the threshold of 1.5 years of clock mean AoA (see, in
particular, MAM or SON in the north). Below, the contours do not agree with
the pulse spectra and clock tracer in multiple seasons and spatial regions
and display an opposite sign to the pulse or clock tracer, especially in MAM
and SON in the north. These differences generally emerge particularly in the
Northern Hemisphere and are only visible to a small extent in the
Southern Hemisphere (e.g., DJF – polar below 100

The following analysis focuses on three fixed latitudes, since the results
presented in Fig. 8 have already shown a good performance of the inverse
method beyond 1.5 years of clock mean AoA in both hemispheres. 85,
55 and 10

Vertical profiles of age spectra width as a function of their mean
AoA at 85

The ratio of second to first moment

Absolute annual and seasonal ratio of moments as global cross section for pulse (top row) and inverse spectra (bottom row). Again, the white dashed area represents data below clock mean AoA of 1.5 years.

Influence of spectra's tail length on width and mean AoA for
pulse

The reason for the deviation from Hall and Plumb (1994) and Volk et al. (1997)
may not only be coarser resolution and lesser complexity of the
models compared to EMAC. The tail of the age spectrum is vital for a
mathematically and physically correct description of transport. The width of
annual mean pulse and inverse spectra with tail lengths of 10, 20, 50
and 300 years is shown as a function of mean AoA at 55

Results of the proof of concept presented above have shown that the inverse
method in combination with the imposed seasonal cycle is in principle
capable of deriving seasonal age spectra correctly, except for the region
below 1.5 years of mean AoA. However, when it comes to real atmospheric data,
application of the method becomes even more challenging. One of the most
critical factors is the chemical lifetime of the gases used in the
inversion which is strongly dependent on time and space, tainted with
seasonal and inter-annual variability and only known with limited accuracy.
The inverse method in its form postulated above (Eq. 11) can include variability in lifetimes, since
the lifetime

Radioactive trace gas subsets.

A further limitation of the method when applying it to observational data is the number of trace gases necessary for the inversion to work properly. In reality, it is impossible to find 40 trace gases with lifetimes ranging from 1 to 118 months evenly spaced in steps of 3 months. Considering data and measured species of modern airborne research campaigns, it is more likely to find a set of 10 trace gases at most, which span a range of 10 years in chemical lifetime. On this basis, three subsets of tracers with 10, 5 and 3 trace gas species are selected and shown in Table 2 together with their corresponding lifetimes to investigate the effects of reducing the number of trace gases and introducing uncertainty in the knowledge of the lifetime. Three trace gases are considered to be the minimum in order to constrain different parts of the age spectrum (e.g., left flank of the peak, right flank and tail). A reduction of trace gases removes, on the one hand, redundant information about transport and strongly diminishes the amount of data but, on the other hand, also increases the risk of errors during the inversion leading to wrong age spectra. This is especially precarious if chemical depletion is spatially varying and inaccurately estimated.

To test the influence of uncertainties in assumed lifetimes, a Monte Carlo
approach has been used in which chemical lifetimes are varied
pseudo-randomly. In this simulation the mean lower error margin is evaluated
by giving a pseudo-randomly selected number of lifetimes in a trace gas
subset a certain preset uncertainty

Annual mean age spectra from radioactive tracer subsets compared
to full set at 55

These results imply that a reduced set of trace gases with either 10
(optimum) or 5 (sufficient) species should be recommended in order to
retrieve age spectra from observations. Values in Table 3 and also the
general shape of the error spectra indicate that on average both of these
subsets can also compensate a chemical lifetime uncertainty of up to

Percentage deviations of spectra amplitudes and modal ages averaged over all three subsets presented in Fig. 12. The calculated values are relative to the spectra of each subset without errors in the knowledge of chemical lifetime.

This paper presents a modified version of the inverse method by Schoeberl
et al. (2005) to derive stratospheric age of air spectra from radioactive
trace gases with fewer free parameters. It introduces a formulation of an
imposed seasonal cycle in the spectra to approximate seasonality in
stratospheric transport. The development process always focuses on achieving
the best possible compromise between accuracy and practicability for
observational studies with very limited data. The resulting spectra are
evaluated in comparison to EMAC pulse spectra. The inverse method is applied
on a simplified set of 40 short-lived trace gases with globally constant
prescribed lifetimes as proof of concept. Resulting spectra are assessed as
well as their first and second moments and the ratio between them. This
comparison is conducted for annual mean state and for respective seasonal
variation. Data within the transition layer, the first 30

The modified inverse spectra match the pulse spectra well both on an annual and
seasonal timescale with its new set of reduced fit parameters for most
parts of the stratosphere. The imposed seasonal cycle improves the already
well-described intrinsic seasonality and reproduces seasonal variation in
the spectra correctly. Multiple peaks of the inverse age spectra at
55

The ratio of moments of the pulse and inverse spectra is compared to the
results of Hall and Plumb (1994). The ratio of moments of the pulse
spectra exceeds their values by a factor of 3 but show a similar spatial
structure. Since the results show that the ratio of moments undergoes
seasonal variability, an annual average value of 2.0 years for the lower
stratosphere in both hemispheres is promoted. This is larger than the values
implemented by Volk et al. (1997;

With respect to observational data, a set of 40 trace gases with lifetimes
ranging evenly spaced from 1 month to 10 years cannot be found in
reality. Tests with reduced numbers of trace gases show that subsets
consisting of 5 (sufficient) to 10 (optimal) chemically active trace gases
should be used in order to invert consistent and matching age spectra in the
northern midlatitudes. It is recommended that the species in these subsets
are selected so that their average local lifetimes along the transport
pathway cover a period of 10 years uniformly. The analysis shows moreover
that errors in the assumed chemical lifetime, which affect a random number
of trace gases in these subsets, can be compensated during the inversion to
some degree. On average an uncertainty in the knowledge of lifetimes up to

All data can be made accessible on request to the authors.

Deriving the reference ratio

MH wrote the paper, evaluated the model data and developed the ideas presented in this study in close collaboration with AE. FF and HG planned and performed the EMAC model simulation and implemented and tested the method of Schoeberl et al. (2000) mentioned in Sect. 2.1. FF, HG and AE contributed to the preparation of the paper in many discussions.

The authors declare that they have no conflict of interest.

This work is supported by the German Research Foundation (DFG) priority program 1294 (HALO) under the project number 316588118. The model simulations were performed on the HPC system Mistral of the German Climate Computation Center (DKRZ) supported by the German Federal Ministry of Education and Research (BMBF). The authors cordially thank Felix Plöger from Forschungszentrum Jülich and Harald Bönisch from the Karlsruhe Institute of Technology (KIT) for their help and the useful discussions regarding the research presented in this paper.

This paper was edited by Rolf Müller and reviewed by three anonymous referees.