## 1. Introduction

The Brewer–Dobson circulation (BDC) is the slow meridional overturning circulation of the stratosphere, consisting of upwelling through the tropical tropopause, then poleward motion and downwelling through the midlatitudes and at the poles. This circulation is critical for the vertical transport of tracers such as ozone, volcanic aerosols, and chlorofluorocarbons (CFCs); for the temperature of the tropical tropopause and consequently the amount of water vapor in the stratosphere; and for stratosphere–troposphere exchange [e.g., Butchart (2014) and references therein]. Stratosphere-resolving climate models show a positive trend in the BDC—an increase in the tropical upwelling at a fixed pressure level—as a robust response to increasing greenhouse gases (Butchart et al. 2006; Hardiman et al. 2014). This increasing trend in the residual circulation, however, might better be described as a “lifting” trend, associated with the upward expansion of the tropopause (and entire tropospheric circulation) in response to global warming (Singh and O’Gorman 2012; Oberländer-Hayn et al. 2016). Reanalysis products are in qualitative agreement with the climate models, showing positive trends over the period 1979–2012, but with differing spatial structures for each individual product (Abalos et al. 2015). Satellite-derived temperature trends are also consistent with the model predictions (Fu et al. 2015).

The mean age of air (Hall and Plumb 1994; Waugh and Hall 2002) has been used as a metric for models’ abilities to reproduce the stratospheric circulation (e.g., Hall et al. 1999; Butchart et al. 2011). The apparent increase in the residual circulation has led to predictions that the mean age of air should decrease. Attempts to identify trends of decreasing age of air from observations of transient tracers in the stratosphere have found little evidence; in fact, age appears to be mostly increasing (Engel et al. 2009; Stiller et al. 2012; Haenel et al. 2015). However, for one thing, available data records are short enough—global satellite coverage of age tracers is available for less than a decade—that apparent trends could be indicative of interannual variability rather than of long-term trends (cf. Garcia et al. 2011). Moreover, mean age is a statistical average over many transport pathways (Hall and Plumb 1994), and at a given location it depends on mixing processes and not just mean advection (Waugh and Hall 2002; Garny et al. 2014; Ploeger et al. 2015a). Satellite observations of SF_{6} have been used to identify spatially inhomogeneous trends in age between 2002 and 2012 (Stiller et al. 2012; Haenel et al. 2015), and while these trends can be compared to model output, for which the contributions of advection and mixing can be isolated (Ploeger et al. 2015b), in reality they are difficult to disentangle (Ray et al. 2010).

There are certain aspects of the stratospheric age distribution that are dependent on the mean circulation alone. Using a “leaky tropical pipe” model, Neu and Plumb (1999) showed that, in steady state, the tropics minus midlatitude age difference on an isentrope depends only on the overturning mass flux and is independent of isentropic mixing, provided that diabatic mixing is negligible. This result has been used to assess transport in chemistry–climate models (Strahan et al. 2011). Here we present a generalization of this analysis. In section 2a, we show that the steady-state result of a simple and direct relationship between age gradient and overturning diabatic mass flux holds even in the absence of a “tropical pipe,” provided the isentropic age gradient is defined appropriately. For the more realistic case of an unsteady circulation, we show in section 2b that the result holds for the time average; in section 2c, the fully transient case is discussed. The accuracy of the theoretical predictions is demonstrated in section 3 using results from a simple general circulation model; the theory works well when applied to multiyear averages, though there are systematic discrepancies that appear to indicate a role for large-scale diabatic diffusion. Applications and limitations of the theory are discussed in section 4.

## 2. Age difference theory

*χ*follows the tracer continuity equation:

^{−1}) with a boundary condition of

*θ*) coordinates, we rewrite the full advection–diffusion operator as the divergence of a flux to obtain

### a. Steady state

*θ*surface and

*θ*surface. We define the mass-flux-weighted age of upwelling and downwelling air as

This relationship is essentially identical to that obtained by Neu and Plumb (1999) in their tropical pipe model, but the present approach avoids assumptions made in that model, other than steadiness and the neglect of diabatic diffusion (both of which will be addressed in the following sections). As discussed by those authors [and by Plumb (2002) and Waugh and Hall (2002)], (11) is remarkable and counterintuitive in that the gross isentropic age gradient is independent of isentropic mixing (except insofar as the mixing of potential vorticity drives the diabatic circulation) and depends only on the overturning mass flux through the *θ* surface; it is independent of path in the diabatic circulation. For a given mass flux, the age gradient is the same whether the circulation is deep or shallow.

The potential power of (11) lies in the fact that, unlike age itself,

### b. Time average

Although this derivation has been done for upwelling and downwelling regions, the time-average formulation does not require the two regions of the isentrope to be strictly upwelling or downwelling. As long as the isentrope is split into only two regions that together span the surface, any division will do. The overturning mass flux

### c. Time varying

If the time-derivative terms and the term involving fluctuations

Note that this differs from the time-average version of the theory, presented in section 2b. In the derivation of (30), the average of

## 3. Verification in a simple atmospheric GCM

### a. Model setup

To verify the theory, we evaluate the terms in (11), (17), and (26) in a simple atmospheric GCM with and without a seasonal cycle. The model is a version of the dynamical core developed at the Geophysical Fluid Dynamics Laboratory (GFDL). It is dry and hydrostatic, with radiation and convection replaced with Newtonian relaxation to a zonally symmetric equilibrium temperature profile. We use 40 hybrid vertical levels that are terrain following near the surface and transition to pressure levels by 115 hPa. Unlike previous studies using similar idealized models (e.g., Polvani and Kushner 2002; Kushner and Polvani 2006; Gerber and Polvani 2009; Gerber 2012; Sheshadri et al. 2015), the model solver is not pseudospectral. It is the finite-volume dynamical core used in the GFDL Atmospheric Model, version 3, (AM3; Donner et al. 2011), the atmospheric component of GFDL’s CMIP5 coupled climate model. The model utilizes a cubed-sphere grid (Putman and Lin 2007) with “C48” resolution, where there is a 48 × 48 grid on each side of the cube, and so roughly equivalent to a 2° × 2° resolution. Before analysis, all fields are interpolated to a regular latitude–longitude grid using code provided by GFDL.

In the troposphere, the equilibrium temperature profile is constant in time and similar to Held and Suarez (1994) with the addition of a hemispheric asymmetry in the equilibrium temperature gradient that creates a colder Northern Hemisphere [identical to Polvani and Kushner (2002)]. In the polar region (50°–90°N/S), the equilibrium temperature profile decreases linearly with height with a fixed lapse rate *γ*, which sets the strength of the stratospheric polar vortex. The stratospheric thermal relaxation time scale is 40 days. As an analog for the planetary-scale waves generated by land–sea contrast, flow over topography, and nonlinear interactions of synoptic-scale eddies, wave-2 topography is included in the Northern Hemisphere at the surface centered at 45°N as in Gerber and Polvani (2009). The Southern Hemisphere has no topography. As in Gerber (2012), a “clock” tracer is specified to increase linearly with time at all levels within the effective boundary layer of the Held and Suarez (1994) forcing (model levels where *p*/*p*_{s} ≥ 0.7) and is conserved otherwise, providing an age of air tracer.

The seasonally varying run has a seasonal cycle in the stratospheric equilibrium temperature profile following Kushner and Polvani (2006), with a 360-day year consisting of a constant summer polar temperature and sinusoidal variation of the winter polar temperature, so that equilibrium temperature in the polar vortex is minimized at winter solstice. The lapse rate is *γ* = 4 K km^{−1}, and the topography is 4 km high. With a lower stratosphere–troposphere transition, this topographic forcing and lapse rate were found by Sheshadri et al. (2015) to create the most realistic Northern and Southern Hemisphere–like seasonal behavior. The model is run until the age has equilibrated (27 yr) and then for another 50 yr, which provide the statistics for these results.

For the perpetual-solstice runs, the model is run in a variety of configurations, as described in Table 1. These four simulations correspond to those highlighted in detail in Figs. 1–3 of Gerber (2012), capturing two cases with an “older” stratosphere and two cases with a “younger” stratosphere. Note, however, that Gerber (2012) used a pseudospectral model and the age is sensitive to model numerics. All are run to equilibrium, at least 10 000 days, and the final 2000 days are averaged for the results presented here.

Summary of setup for the five runs used in this study. One run has a seasonal cycle as described in the text, and the others are perpetual solstice with varying stratospheric lapse rates *γ* (K km^{−1}) and wavenumber-2 topographic forcing *h* (km) in the one hemisphere only. The winter hemisphere in these perpetual-solstice runs is the same as the hemisphere with topography.

### b. Model seasonality

Figure 2a shows a 20-yr climatology of the residual vertical velocity at 53 hPa for the seasonally varying model run. Because of artifacts from the interpolation from the cubed-sphere grid and the high frequency of temporal variability, the field has been smoothed in time and latitude using a binomial filter of 2 weeks and 10°. Discontinuities are nevertheless visible in midlatitudes in both hemispheres. The seasonal cycle is barely evident; there is stronger polar downwelling in Northern Hemisphere winter/spring and weaker tropical upwelling during Southern Hemisphere winter. Figure 2b shows the climatology of the zonal-mean diabatic velocity

### c. Time-average results

We examine the time-average theory as described in section 2b. We calculate the terms in (17) for annual averages of 50 yr of the seasonally varying model run and for the average over the last 2000 days of the perpetual-solstice run with the same lapse rate and topography (runs 1 and 3 in Table 1). To demonstrate the flexibility of the definition of “upwelling” and “downwelling” regions, we have calculated

All of the points fall above the one-to-one line, a discrepancy consistent with the neglect of diabatic diffusion in the theory, as will be discussed in section 3f. The points on the 800-K isentrope are closest to the theory line, which is also consistent with diabatic diffusion. The results from the seasonally varying model agree as well with the theory as do the results from the perpetual-solstice run, demonstrating the success of the time-average theory in recovering the steady result.

### d. Time-varying results

Next we move on to the time-varying theory; consider Fig. 4. Figure 4a shows 3 yr of the age difference, and Fig. 4b shows the total mass divided by the mass flux for the same 3 yr of the seasonally varying model run. If the steady-state theory held instantaneously, these would be equal at all times. They are obviously not equal; in fact, their seasonal cycles are out of phase, with even negative values of age difference when there is polar diabatic upwelling of very old air associated with the final warming event each Southern Hemisphere spring.

We evaluate the time-derivative terms in (26), ^{−1} would make this term about 5% of the size of the total mass above the isentrope. The other term,

The average seasonal cycle over 20 yr of the model run for each of the terms in (26) is shown for three different levels in Fig. 5. At 400 K, shown in Fig. 5a, in the very low stratosphere, there is very little effect of the seasonal cycle. The product of the overturning strength and the age difference *M*. This discrepancy will be addressed in section 3f. The time-derivative terms are small. At 600 K, shown in Fig. 5b, the seasonal cycle is much more pronounced, and here the difference between

Because of the strong temporal variability, it is clear that the steady-state theory cannot be applied instantaneously. The contributions of the time-derivative terms are smaller upon long-term averaging, however. The magnitude of the annual average of these terms is shown as a percentage of the total mass above each isentrope in the solid lines in Fig. 6, and the standard deviation is shown in the shading. As we already observed from Fig. 5, the variability of

### e. Area-weighted averaging

This theory has been developed, by necessity, using mass-flux-weighted age of air. This precludes applying the theory directly to age data, because the circulation is unknown. We therefore try here to determine the bias introduced by using area-weighted age of air rather than mass-flux-weighted age of air. To do this, we define regions of upwelling and downwelling and take the area-weighted average of age in those regions. Because age of air data alone will not inform us as to the upwelling and downwelling regions, we also perform the averaging for the same “upwelling” and “downwelling” regions as for the time-average results (constant latitude bands). The bias introduced by this averaging in the seasonally varying model is shown at different levels in Fig. 8, where we show this bias as a fraction of the mass-flux-weighted age difference. The area-weighted age difference is smaller than the mass-flux-weighted age difference in all cases. Using the “true” upwelling and downwelling regions gives a bias of about 10%–15%, and the 40° band is similar. When the regions get farther from the “true” upwelling and downwelling, the bias becomes greater. In this model, the bias is quite consistent from year to year, as can be seen by the narrow range spanned by one standard deviation from the mean. This investigation demonstrates that we can qualitatively think about the age difference on an isentrope as a proxy for the diabatic circulation through that isentrope.

### f. The role of diabatic diffusion

*θ*in all latitudes and altitudes considered here. Since

*τ*

_{mixing}≈ 30 days. Using an average value for

## 4. Summary and conclusions

The theoretical developments in this paper have focused on extension of the simple relationship between the gross latitudinal age gradient on isentropes and the diabatic circulation, obtained by Neu and Plumb (1999) for the “leaky tropical pipe” model. Under their assumptions of steady state and no diabatic mixing, but without any “tropical pipe” construct, an essentially identical result follows. We then show that the result survives intact when applied to time averages of an unsteady situation but does not apply locally in time. The predicted age gradient is independent of isentropic mixing and of the structure of the circulation above the level in question.

Analysis of results from a simplified global model, in both perpetual solstice and fully seasonal configurations, shows that the time-averaged result holds quite well, although the predicted age difference overestimates the actual value by up to 15%, a fact that we ascribe to the neglect of large-scale diabatic mixing in the theory. Indeed, estimates of diabatic dispersion in the model are sufficient to account for the discrepancy.

The theory is, of necessity, formulated in entropy (potential temperature) coordinates, and consequently it is the diabatic circulation (rather than, say, the residual circulation) that is related to the latitudinal structure of age. While these two measures of the circulation can, at times (especially around the equinoxes), be very different, in the long-term average to which this theory applies they are essentially identical.

The relationship between age gradient and the circulation is straightforward, but in order to use age data to deduce the circulation there are some subtleties: in order to quantify the mean age difference, in principle one needs to know the geometry of the mean upwelling and downwelling regions and the spatial structure of the circulation (since, strictly, it is the mass-flux-weighted mean that is required). The theoretical result is unchanged if simpler regions (such as equatorward and poleward of, say, 30°) are used instead of those of upwelling/downwelling, but of course the mass flux involved is that within each chosen region, rather than the total overturning mass flux. When we test the agreement of the theory using an area-weighted age difference using the “true” upwelling and downwelling regions, this introduces a bias of 10%–15% underestimation of the magnitude of the age difference, with a smaller bias lower in the stratosphere. Using 40°S–40°N as the upwelling region and the rest of the globe as the downwelling region, which is a calculation that is possible entirely from data, we find that the area-weighted age difference has a similar underestimation of the mass-flux-weighted age difference as using the “true” regions. The area-weighted age difference can therefore be used to infer the circulation qualitatively, since the difference is only up to 15%. We caution that this bias estimate was performed with the simple idealized model, and, to get a quantitative estimate of the overturning circulation strength from data, a more realistic model is necessary and other factors must be considered.

Despite these caveats, these results offer an avenue for identifying trends in the circulation by seeking trends in age data, as done by Haenel et al. (2015) and Ploeger et al. (2015b). For one thing, the theory shows that it is the gross isentropic age difference, and not the age itself, that is related to the strength of the circulation. For another, these results show that annual averages (at least) are necessary to relate the strength of the circulation to the age, so good data coverage in space and time is necessary to eliminate the seasonal variability for which the theory is not applicable. It will require an accumulation of a long time series of global measurements to separate the long-term trends in the circulation from the short-term variability.

## Acknowledgments

We thank S.-J. Lin and Isaac Held for providing the GFDL AM3 core. This research was conducted with government support for ML under and awarded by the DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. Funding for AS was provided by a Junior Fellow award from the Simons Foundation. This work was also supported in part by the National Science Foundation Grants AGS-1547733 to MIT and AGS-1546585 to NYU.

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