## 1. Introduction

[2] The mean age of stratospheric air is the average time for an air parcel to travel from a source region in the troposphere (or near the tropopause) to a sample region in the stratosphere [*Hall and Plumb*, 1994]. The mean age is a fundamental transport time scale that has been widely used in stratospheric transport studies, particularly in the evaluation of chemical transport models and chemistry-climate models (CCMs) [*Hall et al.*, 1999a, 1999b; *Eyring et al.*, 2006]. However, the mean age only contains partial information of transit time scales. The complete information is included in the age spectrum, i.e., a probability distribution function of all the possible transit times since an air parcel had last contact with the tropospheric boundary source region [*Hall and Plumb*, 1994; *Waugh and Hall*, 2002]. Many studies have shown that the age spectrum is more useful than the mean age in diagnosing transport characteristics, e.g., the relative importance of different transport pathways into the lower stratosphere [*Andrews et al.*, 2001a; *Bönisch et al.*, 2009], the seasonal variations of stratospheric transport [*Andrews et al.*, 1999, 2001b; *Reithmeier et al.*, 2008; *Bönisch et al.*, 2009], and the horizontal recirculation rate in the tropical pipe region [*Strahan et al.*, 2009].

[3] The age spectrum is a kind of boundary propagator. By definition, the boundary propagator *G*(*r*,*t*∣Ω,*t*′) is a Green's function that solves the continuity equation for the mixing ratio of a conserved and passive tracer *χ*(*r*,*t*) [*Hall and Plumb*, 1994]. If the mixing ratio is uniform on the boundary source region Ω, this solution can be expressed by the following integration

where *r* is the sample region, *t*′ is the source time or the time the tracer had last contact with Ω, and *t* is the field time or the time the tracer is sampled at *r*. In many cases the boundary propagator is easier to interpret if it is rewritten as a function of the transit time *ξ* = *t* − *t*′, i.e.,

where *G*(*r*,*t*∣Ω,*t* − *ξ*)*d**ξ* represents the mass fraction of the air parcel at *r* and a specific field time *t* that was last in contact with Ω between *ξ* and *ξ* + *d**ξ* ago [*Waugh and Hall*, 2002; *Holzer et al.*, 2003; *Haine et al.*, 2008]. Here *G*(*r*,*t*∣Ω,*t* − *ξ*) is the age spectrum, which is also called the transit time distribution (TTD) in tropospheric and ocean transport literatures [e.g., *Holzer et al.*, 2003; *Haine et al.*, 2008]. In this paper we follow the stratospheric terminology and use age spectrum to refer *G*(*r*,*t*∣Ω,*t* − *ξ*).

[4] The age spectrum cannot be directly observed, and we rely almost solely on models to compute it. Several methods have been used to calculate the age spectrum, e.g., the Eulerian pulse tracer method [*Hall et al.*, 1999b], the Lagrangian trajectory method [*Schoeberl et al.*, 2003], and the Eulerian adjoint model method [*Haine et al.*, 2008]. The age spectrum can also be obtained empirically by fitting an assumed analytic form of the age spectrum with tracer measurements [e.g., *Andrews et al.*, 1999; *Waugh et al.*, 2003, 2004; *Schoeberl et al.*, 2005; *Khatiwala et al.*, 2009].

[5] The pulse tracer method has been used more commonly than other methods because it is the most direct approach and is easy to implement. A pulse of a conserved and passive tracer is placed in the boundary source region Ω at a specific source time t′ where it disperses throughout the interior volume. The time series of the mixing ratio of this tracer at any interior point r, which can be expressed mathematically as *G*(*r*,*t*′+*ξ*∣Ω,*t*′), represents the model's time-evolving response to a delta function boundary condition. *G*(*r*,*t*′+*ξ*∣Ω,*t*′) is called the boundary impulse response (BIR) [*Haine et al.*, 2008]. Thus the direct product of the pulse tracer method is not the age spectrum, but the BIR. In general the BIR *G*(*r*,*t*′+*ξ*∣Ω,*t*′) is not equal to the age spectrum *G*(*r*,*t*∣Ω,*t* − *ξ*).

[6] The relationship between the age spectrum and the BIR is illustrated in Figure 1a, which is an example of the boundary propagator map at 60°N and 420 K isentropic surface. The age spectrum and the BIR are perpendicular to each other in the boundary propagator map. The age spectrum is fixed in field time and increases toward older source time, i.e., a horizontal cut through the boundary propagator map from right to left. The BIR is fixed in source time and increases with field time, i.e., a vertical slice from bottom to top. For unsteady flow the boundary propagator is a function of both field time *t* and source time *t*′ and therefore the age spectrum and the BIR are not the same (Figure 1c). However, if the flow is steady, at least in a statistical sense, the boundary propagator is only a function of the transit time *ξ*, i.e., for any *t* and *t*′, *G*(*r*,*t*∣Ω,*t* − *ξ*) = *G*(*r*,*t*′+*ξ*∣Ω,*t*′); that is, in steady flow the age spectrum and the BIR are the same. Previous stratospheric pulse tracer age spectrum studies have made the assumption of stationarity to compute the age spectrum from the BIR [*Hall and Plumb*, 1994; *Hall et al.*, 1999a, 1999b; *Schoeberl et al.*, 2005]. These studies assumed steady flow in the sense that the seasonal variations of the BIR and age spectrum can be ignored and performed a single realization of the BIR as an approximation of the annual-mean or time-averaged age spectrum.

[7] The traditional stratospheric pulse tracer studies have greatly improved our understanding of the annual mean properties of the age spectrum [*Hall and Plumb*, 1994; *Waugh and Hall*, 2002; *Schoeberl et al.*, 2005], but their approach has disadvantages. By assuming steady flow and performing a single realization, their method cannot be used to investigate the seasonality of the stratospheric age spectra. Stratospheric transport has a strong seasonal cycle due to the seasonal variations of processes such as tropical upwelling, subtropical jets, and polar vortices [e.g., *Chen*, 1995; *Pan et al.*, 1997; *Rosenlof et al.*, 1997; *Ray et al.*, 1999; *Randel et al.*, 2001]. One would expect the stratospheric age spectra to have large seasonal variations. However, our knowledge of the age spectrum's seasonality is very limited. To date the only work that investigated the seasonal variations of stratospheric age spectra was done by *Reithmeier et al.* [2008] using a Lagrangian trajectory method. *Reithmeier et al.* [2008] found that the age spectra in the ECHAM4 general circulation model (GCM) have strong seasonal cycles, and that the shapes of the age spectra change significantly with latitudes. However, there are serious transport biases in ECHAM4. Specifically, the subtropical barrier is too weak and is located too far away from the equator compared to observations. These biases could be related to the limitations of the version of ECHAM4 used by *Reithmeier et al.* [2008], which has a very coarse horizontal (6°) and vertical (only 19 levels) resolution and a very low model top at 10 hPa. The model limitations and the poor transport performance cast doubts on the results of *Reithmeier et al.* [2008].

[8] Another concern about the traditional pulse tracer method is whether a single BIR is a good approximation of an annual-mean age spectrum. Because in reality the stationary assumption does not hold, the single BIR approach implies that the seasonality of stratospheric transport has small impact on the annual mean properties of the age spectrum. *Hall et al.* [1999b] found that the mean age of a single BIR agrees reasonably well with the annually averaged clock tracer mean age. This was used as evidence that the annual mean properties of the age spectrum could be well captured by a single BIR realization. But no previous studies actually investigated the seasonal change of the BIR and the differences between the age spectrum and the BIR in the stratosphere.

[9] The limits of the traditional stratospheric pulse tracer method can be addressed by performing an ensemble of time-dependent BIR simulations. *Holzer et al.* [2003] and *Haine et al.* [2008] described in detail a straightforward method to calculate the age spectra in unsteady flow using the pulse tracer. We will review their method in the next section. This method requires performing a large number of BIR experiments in different seasons and years to reconstruct the time-varying age spectra, and therefore it is computationally expensive.

[10] In this study we investigate the seasonal variations of the age spectra in the Goddard Earth Observing System Chemistry-Climate Model (GEOSCCM) using the pulse tracer method. We introduce an approach to significantly reduce the computational cost for calculating seasonally varying age spectra based on the method of *Holzer et al.* [2003] and *Haine et al.* [2008]. Our main purpose is to understand the seasonality of the stratospheric age spectra. Another purpose is to clarify the differences between the BIR and the age spectrum. Our work broadens the usage of the pulse tracer age spectra. These results will improve the understanding of the transport characteristics in the GEOSCCM, which has been shown to produce realistic stratospheric transport by various diagnostics [*Strahan et al.*, 2011]. Our results could also be used by empirical studies as guidance for age spectrum's seasonal variability.

[11] Our method for calculating the age spectra is described in section 2. We describe how to compute age spectra and BIRs in unsteady flow using a simplified version of the method of *Holzer et al.* [2003] and *Haine et al.* [2008]. We also briefly introduce the GEOSCCM and the simulations. In section 3 we discuss the seasonal and interannual variations of the BIRs. Seasonal variations of the age spectra are presented in section 4. Discussions and summary are given in section 5. All results presented in this paper are zonally and monthly averaged and then interpolated to the isentropic coordinate.