The CONTRAIL data were obtained by collecting air samples once a month from April 2012 to December 2016 at longitudinal intervals of 10 or 15∘ along individual flight tracks at around 11 km altitude between France and Japan and partially between Russia and Japan. The period and longitudinal locations of this analysis were selected based on the CONTRAIL measurements, for which air sampling in the ExUTLS was usually carried out over Siberia. To identify the origins of ExUTLS air masses, kinematic backward trajectories are calculated for 10 years following the method of Inai (2018). Trajectories are initialized at uniformly distributed grid points (5.0∘ longitude × 2.5∘ latitude) within 45–80∘ N and 0–140∘ E at geopotential heights of 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16 km (Fig. 1). Initializations are made at 00:00 UTC on the 5th, 15th, and 25th of every month from January 2012 to December 2016, and use meteorological conditions prescribed by the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim dataset (1.5∘×1.5∘ horizontal resolution, 6-hourly temporal resolution, and 37 pressure levels; Dee et al., 2011). Although trajectories could be released at the exact CONTRAIL measurement locations and times, the grating initialization is employed because this study attempts to obtain uniform spatiotemporal tracer distributions as well as their transports by capitalizing on the CONTRAIL measurements. An example of the results is provided in Fig. 2, which shows where particles located as shown in Fig. 1 at 00:00 Z on 15 January 2015 were located 361 days prior (i.e., 00:00 Z on 19 January 2014). Many particles ending up at altitudes greater than 13 km (orange dots) traveled from the stratosphere, above 18 km. However, many particles ending up at altitudes below 10 km (purple to blue-green dots) distribute below 15 km, typically in the troposphere. Although the accuracy of individual trajectories is limited by the long-term nature of the calculations, statistical features of air mass transport can be investigated using a large number of trajectories.

Trajectories obtained from each run are categorized into several groups (trjk; k=1 to kmax) with criteria (hereafter denoted crik) of potential temperature, latitude, potential vorticity, and geopotential height along each trajectory. In this analysis, kmax is set to 4, with k=1 for the stratosphere, k=2 for the tropical troposphere, k=3 for the midlatitude LT, and k=4 for the high-latitude LT. Criteria for each k are summarized in Table 1. The trajectories are also used to determine whether trajectories categorized as trjk=1 passed through the deep or shallow branch of the BDC (k=1 d or 1 s). These trajectories are classified as shallow branch if they cross 400 K but do not reach 30 hPa within 4 years and as deep branch if they exceed 30 hPa within 4 years, following the method of Lin et al. (2015). Trajectories were categorized as trjk, according to the first set of three continuous days along the trajectory that satisfied the crik. This resulted in all trajectories being categorized as k=1, 2, 3, or 4 within 10 years. Trajectories trjk are assumed to travel along unique paths from origin k to the initial position of the backward trajectory. Origin fractions of air parcels with origin k (hereafter denoted fk) are calculated as a function of equivalent latitude (∅eq), potential temperature (θ), and month (M) of their release. Denoting as Nk the number of trajectories, which are classified into trjk groups with distinct ∅eq, θ, and M, the origin fraction for origin k is given by 1fk=∑i=1Nkρtrjkini(i)⋅cos⁡∅trjkini(i)∑k=1kmax(∑i=1Nkρtrjkini(i)⋅cos⁡∅trjkini(i)), where ∅trjkini and ρtrjkini indicate the initial latitude and density of the individual backward trajectories, respectively. Note that ρtrjkini is calculated from the equation of state. Results of a sensitivity analysis indicate that the estimated origin fractions are independent of the resolution of the input meteorological data (see Appendix A).

Similar methods are used to estimate the AoA, which is calculated as the average elapsed time until a trajectory goes back to the troposphere where it satisfies whichever criterion, k=2, 3, or 4. Thus, the AoA definition used here differs from that of Hall and Plumb (1994), who defined AoA as the elapsed time an air parcel spends in the stratosphere after traveling across the tropopause. In our estimates of AoA, however, a small fraction of trajectories are still in the stratosphere at the end of the 10-year calculation. Figure 3 shows the percentage of such remaining trajectories estimated as a function of ∅eq, θ, and M of their release. The percentages are almost zero in the region where potential vorticity is <4 PVU, whereas they are generally nonzero in the region where potential vorticity is >4 PVU. However, even in this region the values are <2.5 %. Here, we define ε as the percentage of trajectories that are still in the stratosphere after the 10-year backward calculation as a function of ∅eq, θ, and M (Fig. 3). Then, the AoA (ΓTrj) is obtained using the elapsed time since each trajectory trjk(i) left its origin k (≡τk(i)) according to 2ΓTrj=∑k=24∑i=1NkγTT⋅τk(i)⋅ρtrjkinii⋅cos⁡∅trjkinii∑i=1Nkρtrjkinii⋅cos⁡∅trjkinii⋅1-ε+Γtail‾⋅ε, where Γtail‾ is the average AoA for air parcels remaining in the stratosphere longer than the maximum length of the trajectory calculation tf (=10 years), calculated as follows: 3Γtail‾=∫γTT⋅tf∞τ′⋅PDFτ′⋅exp⁡-b⋅τ′-γTT⋅tfdτ′∫γTT⋅tf∞PDFτ′⋅exp⁡-b⋅τ′-γTT⋅tfdτ′,=γTT⋅tf+1b, where PDF is the age probability distribution function or “age spectrum,” and b is the exponential decay parameter of the PDF, with its value (b=0.2038 yr-1) from Diallo et al. (2012). The decay parameter in the present analysis may differ from that used by Diallo et al. (2012) because of differences in the vertical trajectory calculations (i.e., kinematic in the present study and diabatic in their work). However, this difference is expected to have little impact on the results because ε is small, as shown in Fig. 3. The term γTT in Eq. (3) is a correction factor for τk and is required because previous studies (e.g., Inai, 2018) have found that the AoA estimated by trajectory analysis using ERA-Interim data is underestimated, particularly when using a kinematic treatment. Inai (2018) found that this underestimation corresponds to 70 % of the observed value in the midlatitude stratosphere. To address this underestimation, the AoA values calculated here are corrected by comparing with AoA derived from SF6 mixing ratios (ΓSF6) assuming a linear trend relative to the time series at Mauna Loa (https://www.esrl.noaa.gov/gmd/obop/mlo/, last access: 14 May 2019) of 0.33 ppt yr-1 (ppt is parts per trillion by mole, with similar definitions for ppm and ppb). When γTT is set at 1.5, ΓTrj agrees well with ΓSF6 (Fig. 4). Thus, a value of 1.5 was used for γTT in this study. The AoA for air masses originating in the stratosphere and those that passed through the deep and shallow branches of the BDC were evaluated using trjk=1, trjk=1d, and trjk=1s, respectively. Note that because this study performs a trajectory analysis using an objective reanalysis dataset, subgrid-scale processes, such as the sporadic injection of tropospheric air masses into the ExUTLS, cannot be explicitly reproduced. Thus, to remove the influence of such events, CONTRAIL data with CO mixing ratios higher than 80 ppb in the region above 340 K and north of 60∘ N equivalent latitude are not used in this comparison (the same criteria are applied to the comparison shown in Fig. 7 in Sect. 2.2.2).

Distributions of origin fractions in a ∅eq–θ cross section are shown in Fig. 9 for January together with the climatology of monthly average potential vorticity for the period 2012–2016 obtained from ERA-Interim. In winter, origin fractions of the stratosphere dominate regions north of 40∘ N and higher than 340 K in altitude. In particular, regions where the altitude and equivalent latitude are greater than 360 K and 50∘ N, respectively, are almost entirely occupied by stratospheric air masses. Furthermore, the stratospheric air mass traveling via the deep branch of the BDC occupies roughly 30 % of the regions where the potential vorticity exceeds ∼10 PVU (Fig. 9c). However, origin fractions of the tropical troposphere dominate regions of lower latitude and altitudes where the potential vorticity is less than ∼4 PVU. These origin fractions are >50 %, except in regions lower than 320 K in altitude. Air masses in regions lower than 310 K generally originate in the midlatitude LT with mixing fractions up to ∼70 %, with few air masses originating in the high-latitude LT.

The origin fractions for April are shown in Fig. 10. In spring, origin fractions of the stratosphere are similar to their winter values, and dominate regions north of 40∘ N and higher than 340 K in altitude. Origin fractions of the stratosphere via the shallow branch of the BDC become slightly smaller than during winter, and those of the deep stratosphere via the deep branch of the BDC increase instead. Tropical tropospheric air masses continue to dominate regions where the potential vorticity is less than ∼4 PVU at equivalent latitudes below 50∘ N, except for regions below 320 K where midlatitude air masses are present. Origin fractions of the high-latitude LT remain small during spring.

Estimated origin fractions for July are shown in Fig. 11. In summer, origin fractions of the stratosphere become less dominant. In particular, those originating in the deep branch of the BDC (Fig. 11c) are small over the whole ExUTLS. Stratospheric air masses, almost all of which originate in the shallow branch of the BDC, are generally distributed in a small region where the altitude and equivalent latitude are greater than ∼370 K and 40∘ N exceeding 50 % of the origin fraction. In contrast, there is expansion of the region in which the origin fractions of the tropical troposphere are dominant. In particular, nearly 80 % of the air masses in the region above 340 K and south of 40∘ N originate in the tropical troposphere. Only during this season do origin fractions of air masses originating in the high-latitude LT reach up to ∼70 %, but these are limited to a region below ∼320 K. Origin fractions of the midlatitude LT become smaller than during spring, but the region where they are higher than 30 % expands up to 340 K at all equivalent latitudes.

Origin fractions for October are shown in Fig. 12. During autumn, high origin fractions of the stratosphere broaden again in the region above 360 K. However, those originating in the deep branch of the BDC are small. Origin fractions of the high-latitude LT are suppressed, and the region where origin fractions of the tropical troposphere are higher than 50 % becomes larger than during summer and extends up to 80∘ N along 330–340 K potential temperature surfaces. In the region below 325 K, midlatitude LT air masses dominate. These seasonal results are compared with previous studies in Sect. 4.1. The robustness and limitations of our estimates are discussed in Sect. 4.4.

As described in Sect. 2.2, detrended mixing ratios of CH4, N2O, CO, SF6, and CO2 observed in the tropical troposphere, midlatitude LT, and high-latitude LT are assigned to their original mixing ratios for k=2, 3, and 4, respectively. For k=1, the original mixing ratios are estimated by Eqs. (5) and (6) using trajectories trjk=1 for passive tracers and APs, and Eq. (9) for chemically active species. Figure 13 shows the original mixing ratios of each species assigned to an individual trajectory according to Eq. (6). Note that these values for stratospheric air masses are estimated based on their final state, unlike the case for regions k=2, 3, and 4, for which the values correspond to their original state. Whereas CH4 and SF6 show seasonal variations and latitudinal gradients in the troposphere, N2O does not. In contrast to the troposphere, CH4 and N2O in stratospheric air masses show distinct seasonal variations but somewhat different phase, with a minimum in boreal summer and maximum in winter for CH4, and a minimum in boreal spring–summer and maximum in autumn–winter for N2O. SF6 mixing ratios are significantly smaller in stratospheric air masses than in the troposphere throughout the year, and show seasonal variations with a maximum in September and minimum in March. A potential reason why the seasonality in the stratosphere differs among CH4, N2O, and SF6 is discussed in Sect. 4.3, together with seasonal variations in γLossS (Fig. 8). For CO, there are large seasonal variations in high-latitude and midlatitude tropospheric air masses, but tropical tropospheric values show smaller seasonal variations. The CO mixing ratios for the stratosphere show little seasonal variability, and are less than ∼40 ppb throughout the year. For CO2, seasonal variations are largest in the high-latitude troposphere; mixing ratios in the stratosphere show relatively small seasonal variations, but with a phase that differs from that in the troposphere.

The estimated AoA of stratospheric air masses is shown in Fig. 13f. Stratospheric air masses transported via the deep branch of the BDC have an AoA exceeding 6 years, whereas those transported via the shallow branch have an AoA of 1–1.5 years. The average AoA among air masses originating in both branches shows a seasonal variation, with maximum values of ∼2.7 years in March and minimum values of ∼1.9 years in September, of almost opposite phase to that of SF6 mixing ratios. The relationship between the original composition of stratospheric air masses and their AoA is discussed in Sect. 4.2.

To identify the characteristics of seasonal variations in origin fractions and reconstructions at fixed locations, four regions are selected: mid-equivalent latitude upper (MU) ExUTLS (45∘ N, 370 K), high-equivalent latitude upper (HU) ExUTLS (75∘ N, 370 K), mid-equivalent latitude lower (ML) ExUTLS (45∘ N, 320 K), and high-equivalent latitude lower (HL) ExUTLS (75∘ N, 320 K). Figure 18 shows seasonal variations in the origin fractions of each origin evaluated at the four locations. In the MU ExUTLS, origin fractions of the tropical troposphere become high, exceeding 50 % during summer and autumn. Accompanying this increase, trajectories originating in the tropical troposphere over around Asia are strengthened. In the other seasons, origin fractions of the stratosphere dominate. In particular, those that traveled via the shallow branch of the BDC exceed 50 %. The origin fractions of the midlatitude and high-latitude LT are nearly zero throughout the year, with the exception of that for the midlatitude LT in autumn. In the HU ExUTLS, origin fractions of the stratosphere dominate and exceed 60 % throughout the year. Furthermore, origin fractions of air masses that traveled via the deep branch of the BDC exceed 20 % during the period from January to April, whereas tropical tropospheric air masses generally fail to reach this region during this period. In the ML ExUTLS, tropospheric origin fractions are dominant. In particular, those of the midlatitude troposphere exceed 50 % during summer and those of the high-latitude troposphere exceed 20 % during July and August. During winter and spring, however, tropical tropospheric air masses dominate. In the HL ExUTLS, origin fractions of the midlatitude and high-latitude LT are enhanced during summer. Origin fractions of the high-latitude LT are comparable to those in the ML ExUTLS, but smaller than those of the midlatitude LT in the HL ExUTLS. This can be explained by enhanced exchange at the bottom edge of the subtropical jet (i.e., along the 320–330 K surface for summer; e.g., Gettelman et al., 2011). As shown in Fig. 11d, enhanced origin fractions of the midlatitude LT are distributed along such isentropes. In winter, origin fractions of the tropical troposphere and stratosphere are roughly 50 % and 40 %, respectively.

In addition to seasonal variations in origin fractions, seasonal variations in the tracer mixing ratios in origin regions (Fig. 13) also affect chemical distributions in the ExUTLS. Figure 19 reveals that seasonal variations in the reconstructions for each species and the trajectory-estimated AoA in each of the four locations have patterns that differ because they are based on a superposition of the origin fractions shown in Fig. 18 with the original time series for k=1–4 of the individual tracers shown in Fig. 13. Note that the CONTRAIL data are plotted if the measurement was conducted within ±5∘ in equivalent latitude and within ±5 K in potential temperature of one of the four locations. This results in few plotted CONTRAIL observations in the ML and HL ExUTLS regions during summer and no observations in MU and HU ExUTLS regions from June to January. This is caused by the seasonality of the thermal and dynamical structures of the ExUTLS and fixed flight altitudes. Despite the sparse and nonuniform observational field, the spatiotemporal distributions of chemical species, together with the origin fractions of the original air masses, can be resolved. This ability is one of the important advantages of the current analysis. The mixing ratios of CH4 and N2O show modest seasonal variations in the lower ExUTLS, whereas they show large seasonal variations in the upper ExUTLS, with minima in spring and maxima in autumn. The minima in spring are due to the transport of stratospheric air masses via the deep branch of the BDC, which have low CH4 and N2O mixing ratios and also low AoA. This seasonal variation in chemical abundance for stratospheric air masses is discussed further in the next section. In contrast to CH4 and N2O, CO mixing ratios show smaller seasonal variations in the upper ExUTLS than in the lower ExUTLS, with the exception of the high mixing ratio in the upper ExUTLS in August. This can be explained by the transport of midlatitude LT air masses, which have higher CO mixing ratios than the other air masses, to the lower ExUTLS during summer. In addition, a large fraction of air masses reach the upper ExUTLS only during August. The seasonal characteristics of SF6 mixing ratios are similar to those of CH4 and N2O. The phase of seasonal variations in the upper ExUTLS is nearly synchronized with, but slightly precedes, those of CH4 and N2O, and more closely resembles the upside-down pattern of AoA variations (Fig. 19f). The phase of seasonal variations in CO2 mixing ratios in the lower ExUTLS is nearly synchronized between the ML and HL ExUTLS, with the largest amplitude being evident in the ML ExUTLS. The phase of CO2 variations in the upper ExUTLS is quite different from that in the lower ExUTLS, with maxima during summer–autumn. This seasonal variation in the upper ExUTLS is consistent with observational estimates by Hoor et al. (2004) and Strahan et al. (2007).

Seasonal variations in AoA evaluated at the four locations are shown in Fig. 19f. The phase of seasonal variations for the four locations is roughly synchronized, whereas the absolute values are clearly different. For example, AoA in the HU ExUTLS has a maximum of >2.5 years during spring and a minimum of ∼1.3 years during the end of summer, whereas in the ML ExUTLS the maximum is only ∼0.5 years and occurs during the period from winter to spring. The amplitude of AoA variations in the ExUTLS is likely related to air mass mixing from the stratosphere, particularly when this involves air masses that have been transported via the deep branch of the BDC. This point is discussed further in the next section, in relation to seasonal variations in chemical composition.

As discussed in the previous section, the distributions of CO and CO2 in the ExUTLS are strongly affected by tropospheric air masses because CO has a short chemical lifetime and CO2 shows large seasonal variations in the high-latitude and midlatitude LT. For CH4, N2O, and SF6, however, seasonal variations in origin fractions of the stratospheric air masses and in the compositions of the original air masses are considered to be essential factors in their spatiotemporal distributions in the ExUTLS. Here, we discuss seasonal variations in the composition of stratospheric air masses and how this affects chemical distributions via mixing with tropospheric air masses in the ExUTLS. Figure 20 shows the relationships between chemical abundances from CONTRAIL measurements and the AoA estimated from the trajectories and interpolated to each CONTRAIL measurement location, along with these relationships for each original air mass. The AoAs for stratospheric air masses are the same as those shown in Fig. 13f, whereas the AoA for the tropical, midlatitude, and high-latitude troposphere are set to zero. Thus, the denotations for the tropospheric air masses only move vertically in the cross sections according to their seasonal variations. Overall, the CONTRAIL measurements are roughly distributed on lines connecting the tropospheric and stratospheric air masses for all seasons and chemical compositions. This linear distribution suggests that dynamical mixing of tropospheric with stratospheric air masses shapes the chemical distributions of the ExUTLS. Such linear “mixing lines” also suggest that the mixing took place rapidly (i.e., at a timescale shorter than their chemical lifetimes) along an isentropic surface (Plumb, 2007, and references therein). A comparison of the distribution of CONTRAIL measurements with trends in the troposphere for SF6 shows that the CONTRAIL measurements are distributed along the lines of the sign-reversed trend. According to Engel et al. (2002) and Bönisch et al. (2009), the mixing ratios of CO2 and AoA do not correlate below a level of ∼3 years AoA because the propagated signal of the tropospheric seasonal cycle into the stratosphere is still detectable. In agreement with their results, the CONTRAIL CO2 measurements also converge to the sign-reversed trend with increasing AoA. However, for CH4 and N2O, measurements depart from the sign-reversed trends toward lower mixing ratios with increasing AoA. This deflection can be interpreted as being due to their stratospheric sinks, i.e., chemical destruction of CH4 and N2O in the stratosphere, with no such destruction of SF6 and CO2.

Both the AoA and chemical abundance of the original air masses from the stratosphere show seasonal variations that might be caused by seasonal variations in mass fluxes from the deep and shallow branches of the BDC. Figure 20f shows seasonal variations in AoA and the value that is calculated by integration of “age spectrum” (PDF) from 0 to tf for air masses originating in the stratosphere as well as those separately evaluated for air masses that have traveled via the deep and shallow branches of the BDC. As the PDFs are calculated with a weighting factor according to area and density, as in Eq. (1), their integrations reveal relative masses. Air masses originating in both the shallow and deep branches have minima in September and maxima in March. These in-phase seasonal variations enhance both the seasonal variations in the total origin fractions of the stratosphere and its average AoA.

Interesting cyclic structures appear in CH4 and N2O mixing ratios and their AoAs in stratospheric air masses. For example, the CH4 mixing ratio is ∼1750 ppb (AoA of ∼2.3 years) in winter, ∼1700 ppb (AoA of ∼2.6 years) in spring, ∼1650 ppb (AoA of ∼2.3 years) in summer, and again ∼1700 ppb (AoA of ∼2.0 years) in autumn. Thus, clockwise rotations are the result of this pattern. The same is true for stratospheric N2O and AoA. These rotations are formed by seasonal variations in AoA that are at a maximum in spring and a minimum in autumn, in combination with seasonal variations in the relative chemical loss rate along the AP (defined as γLossS and discussed in Sect. 2.2.2) that is at a maximum in winter and a minimum in summer. These π/2 phase-lagged seasonal variations result in rolling variations in the relationship between CH4 and N2O mixing ratios and AoA in stratospheric air masses. The seasonal variation in AoA is determined by the mixing of stratospheric air masses via the deep branch of the BDC (Fig. 20f). Although the detailed mechanism driving the seasonality of the chemical loss rate along AP is unknown, it likely involves the seasonal change of the relationship between AP and AoA as a possible mechanism from a dynamical viewpoint. Other candidate mechanisms from a chemical viewpoint are seasonal changes in the abundance of disrupting substance along the AP, or seasonal changes in the solar radiation intensity and sunlit time. Further discussion of this topic is included in the next section, together with the mechanism driving the π/2 phase-lagged variations, i.e., rolling relationship between CH4 and N2O mixing ratios and AoA in stratospheric air masses.

The abundance of N2O and CH4 in stratospheric air mass may be related to the fraction of air masses traveling via the deep and shallow branches. The relationship between the chemical abundance and mass fraction of the two branches is now considered. The current study estimates approximately 24 % and 14 % of air masses following the deep branch are of stratospheric origin in spring and autumn, respectively, and the AoA is estimated to be ∼6.4 years (Fig. 20f). Andrews et al. (2001) estimated the N2O mixing ratio in the midlatitude deep stratosphere to be ∼80 and <40 ppb, where the AoA is estimated from CO2 mixing ratio to be 5.5 and 6.0 years, respectively. As their estimates are normalized to 1997 tropospheric values, the quantitative difference in the baseline N2O mixing ratios may differ by ∼20 ppb from the present values. If we assume the N2O mixing ratio of air masses originating in the deep branch of the BDC is 60 ppb, and that air masses are mixed at ratios of 24 % and 14 % with air masses whose N2O mixing ratio is 330 ppb, such mixing leads to ∼265 and ∼290 ppb N2O mixing ratios, respectively. These values are up to ∼20 ppb lower than the N2O mixing ratios of ∼280 ppb in May and ∼310 ppb in November estimated for the original stratospheric air masses shown in Figs. 13b and 20b. The same arguments are valid for CH4 with respect to the relationship between stratospheric CH4 mixing ratios and AoA; i.e., CH4 mixing ratios are <600 ppb in regions where the AoA is >5.5 years, as estimated by Röckmann et al. (2011). These overestimations of N2O and CH4 mixing ratios for the original stratospheric air masses might be due to overestimation of the AoA. This possibility is discussed further in the next section.

To examine the mechanism that drives the rolling relationship between CH4 and N2O mixing ratios and AoA in stratospheric air masses (Fig. 20a and b), separately estimated PDFs for air masses transported from individual origins are considered. Figure 21 shows an example of PDFs estimated for January. Each PDF has a spectral peak corresponding to the most probable transit time (modal time). For January, the modal times for high-latitude LT, midlatitude LT, and tropical tropospheric air masses are <0.2 years, whereas that for stratospheric air masses is 1.0 year. The modal times for stratospheric air masses for a whole year are summarized in Table 2. These demonstrate seasonality with a maximum in winter, but remain at ∼0.6 years during other seasons.

The PDF for stratospheric air masses also has a long exponentially decaying tail, which leads to a longer mean age. For example, the mean age is calculated to be 2.5 years (i.e., 2.5 times larger than the modal time) for January (Fig. 21). If we consider chemically passive species with linear trends in the troposphere, the relative abundance depends only on the AoA and they should be linearly correlated with each other. Therefore, the mean age corresponds to the average mixing ratio. This can be confirmed by seasonal variations with linear correlations between SF6 and CO2 mixing ratios and AoA in stratospheric air masses (Fig. 20d and e). However, for chemically active species, in particular CH4 and N2O, their abundance in stratospheric air masses as well as the ExUTLS is controlled primarily by chemical loss processes, as seen in the comparison in Figs. 5 and 7. The chemical loss rate changes with season. Thus, the relationship between the relative abundance and AoA changes seasonally. The most influential air masses on the seasonal variation in average mixing ratios should be those that traveled over the modal time rather than the mean AoA because of the larger PDF. In other words, the seasonality of the average mixing ratio of chemically active species is most sensitive to seasonal variations in the air masses that have been transported along a “modal path” (MP) that corresponds to the modal time. This provides an explanation of the rolling relationship between CH4 and N2O mixing ratios and AoA in stratospheric air masses, as described in the next paragraph.

If we assume the transit time to be 1.0 year in winter and 0.6 years during other seasons, according to the modal time (Table 2), and if the typical season when stratospheric air masses are affected by chemical losses is assumed to be the middle of the transit time, the chemical loss processes of CH4 and N2O are primarily affected 0.5 and 0.3 years prior to ending up in the ExUTLS during winter and other seasons, respectively. The higher rates of chemical loss during May–August estimated for CH4 and N2O in Fig. 8 are caused primarily by chemical processes at the midpoint of a MP during January–April, when from the aspect of the seasonal variation in the BDC, the MP is expected to extend deeper in the stratosphere. The slight phase difference between CH4 and N2O in stratospheric air masses might reflect differences in their chemical loss mechanisms. The chemical loss of CH4 is controlled by reactions with OH, O(1D), and Cl, whereas that of N2O is controlled primary by photolysis and secondarily by reactions with O(1D). Therefore, the seasonality of CH4 is affected not only by seasonal variations in solar radiation that is a primary and direct factor for N2O loss, but also by OH abundance along a MP. Thus, the seasonal variations in CH4 and N2O mixing ratios in stratospheric air masses (Fig. 13a and b) leading to the rolling relationship with AoA (Fig. 20a and b) are interpreted as a combination of seasonally varying chemical loss rates on a transport timescale near the modal time and a path close to that of the MP.

As discussed above, a better approach might be to first model the chemical loss for active species based on the modal time and MP and evaluate the mixing ratio in stratospheric air masses. Then, the distribution can be reconstructed using the origin fraction of stratospheric air masses. However, the PDF might change depending on the ∅eq and θ of the trajectory releasing point, and it is difficult to obtain adequate estimates of the PDF without a sufficient number of trajectories for all bins. Such an approach will be the focus of future work. The use of modal time can result in smaller values for the correction factors for e-folding times (Fig. 8) because of smaller values for ΓTrj in Eq. (9). Such an adjustment will affect the correction factors for e-folding times; however, it will not significantly affect results presented here, particularly those related to the reconstructed distributions of the five trace gases.

This study provides a detailed explanation of seasonal variations in chemical distributions and transport in the ExUTLS from a dynamical standpoint using trajectory analysis in combination with aircraft measurements. Results suggest that the spatiotemporal distributions of CH4, N2O, SF6, and AoA in the ExUTLS are controlled primarily by air mass transport via the deep and shallow branches of the BDC and by their mixing with tropospheric air masses in the ExUTLS, whereas those of CO and CO2 are controlled largely by tropospheric air masses because CO has a short chemical lifetime and CO2 shows large seasonal variations in the midlatitude LT. However, some assumptions and limitations of the current study should be mentioned.

First, some uncertainty results from the use of ERA-Interim data in trajectory analyses. Trajectory results generally depend on the resolution of the input data. We performed sensitivity analyses to clarify this dependency in our origin fraction estimates (Appendix A). Results confirm that our estimates are independent of the resolution of the ERA-Interim data, at least as they relate to statistical characteristics. Furthermore, it is known that AoAs calculated from trajectory analyses using ERA-Interim data are somewhat young-biased. For example, these estimated AoA values are ∼30 % younger than those estimated from balloon-borne observations in the middle stratosphere, as demonstrated by Inai (2018). To address this issue, trajectory-based AoA values are uniformly corrected by a correction factor of 1.5 (determined with reference to the AoA obtained from SF6 mixing ratios) in this study. There is, however, a possibility that the bias differs with the meteorological region because different mechanisms drive the shallow and deep branches of the BDC (e.g., Birner and Bönisch, 2011). This is a possible cause of the inconsistent relationship between the abundance of N2O and CH4 in stratospheric air mass and the mass fraction of the air masses traveling via the deep and shallow branches of the BDC. If the AoA of air masses traveling via the deep branch is assumed to be ∼5 years, the N2O–AoA and CH4–AoA relationships approach those of Andrews et al. (2001) and Röckmann et al. (2011), respectively. Trajectory results also generally depend on the vertical condition, i.e., kinematic (employed by the current study) or diabatic (employed by, for example, Diallo et al., 2017). Previous studies suggest that using kinematic trajectories leads to a stronger dispersion and somewhat young bias in AoA estimates compared with using diabatic trajectories (e.g., Schoeberl et al., 2003; Diallo et al., 2012). Therefore, using diabatic trajectories in this analysis might result in a correction factor (γTT) of <1.5.

The second limitation is related to the criteria for the determination of air mass origin. These criteria may strongly affect origin fraction estimates and are thus expected to contribute to the uncertainty of this analysis, to some degree. A comprehensive sensitivity test to address this issue, focusing on in-mixing in the tropical tropopause layer (TTL), has been reported by Inai (2018), who found that the mixing fraction can vary by 40 % to 180 %, depending on the choice of criteria. Though the same test could be applied to the current study, the estimated origin fraction distributions are comparable to those estimated based on trace gas observations by the In-service Aircraft for a Global Observing System-Civil Aircraft for the Regular Investigation of the atmosphere Based on an Instrument Container (IAGOS-CARIBIC; Umezawa et al., 2015). Moreover, these estimates are indirectly validated by the CONTRAIL observations, through the reconstruction of the chemical distributions (as evident in Figs. 5d and e and 7). This agreement supports our criteria selection and suggests that our estimated origin fractions are not, at least, grossly wrong. However, the breakdown of stratospheric air masses is subject to the limitations described in the last part of Sect. 4.2. If the relative fraction of air masses traveling via the deep branch is 7 % smaller than the estimated values (i.e., if they were 17 % and 7 % in spring and autumn), the relationship between the abundance of N2O and CH4 in stratospheric air mass and the mass fraction of the air masses traveling via the deep and shallow branches also approaches that of Andrews et al. (2001) and Röckmann et al. (2011).

Another limitation may arise from the analysis methodology. The observed mixing ratios of CH4, N2O, CO, SF6, and CO2 are used after removing linear trends for each time series, and are considered to be a function of month and treated separately from the long-term trend. This treatment decreases the precision of observations if the observed values have nonlinear interannual variations, which is mainly of concern for CO2. Furthermore, the CONTRAIL measurements were conducted once a month. Thus, one observed value represents atmospheric conditions at a specific spatiotemporal point, whereas the analysis field has a coarser spatiotemporal resolution, corresponding to, at minimum, that of the grid scale of the ERA-Interim dataset. Such a mismatch in spatiotemporal resolution may contribute to the lack of agreement between the reconstructions and CONTRAIL measurements during summer, particularly for CO2 (Fig. 5e). However, uncertainties arising from these issues are minimized by the use of equivalent latitude and potential temperature, which are dynamically conserved quantities in the stratosphere. In the troposphere, which is more unstable, potential temperature and potential vorticity are not conserved, or are conserved only on much shorter timescales, because of diabatic motion. It should be noted that tracer uplift from the LT into the UT during summer (particularly for CO2, as discussed above) cannot be reduced with the coordinate system employed here. Though the current study covers only the ExUTLS over a longitudinal range from 0 to 140∘ E for comparison with the CONTRAIL measurements, the origin fractions and reconstructions are trial-evaluated over North America (Appendix A). Results confirm that the origin fractions are consistent between the two regions, and thus support the robustness of the current study. In this study, linear trends for CH4, N2O, SF6, and CO2 are assumed for the reconstruction. Although this is a simplified treatment, given the length of the analysis period, these trends are roughly constant over this time period with the exception of CH4, and the CH4 reconstructions are more strongly affected by chemical loss, as is evident in a comparison of Figs. 5a and 8a. In the reconstruction procedure described in Sect. 2.2.1, it was necessary to assign the average of tropical aircraft and midlatitude LT measurements to the original values for k=2 to prevent underestimations. As discussed in Sect. 2.2.1, one cause of this underestimation might be subgrid-scale tropospheric upward transport that, although common, cannot be accounted for in the trajectory analysis.

For the aircraft measurement data used as original mixing ratios for air masses originating in the tropical troposphere and midlatitude LT, particularly those collected by TU over the sea close to Japan, may contain a mixture of polluted and unpolluted air masses to some degree. On this point, the data have different implications from measurement data obtained by background monitoring sites, which are employed for the high-latitude LT. For example, the background CO and SF6 mixing ratios in the midlatitudes are comparable to those in the high-latitude troposphere (as confirmed, e.g., on NOAA/ESRL web sites; https://www.esrl.noaa.gov/gmd/ccgg/globalview/co/co_intro.html (last access: 14 May 2019), https://www.esrl.noaa.gov/gmd/hats/combined/SF6.html, last access: 14 May 2019), whereas those used for the midlatitude LT are significantly larger than those for high-latitude LT, except during winter. In the real atmosphere, the tracer distribution in the ExUTLS is determined not only by influx of background air masses, but also by that of polluted air masses. Therefore, it must take such polluted air masses into account to reconstruct plausible distribution of trace gases, i.e., CO and SF6, in the ExUTLS. Since their artificial sources are mostly distributed in midlatitude LT, we might have been able to reconstruct the tracer distribution, which agrees well with the CONTRAIL measurements. A more proper approach would be to assign background values with the addition of incremental values due to pollution assumed in each latitude region; such an approach will be the focus of our future work.