In this study, we analyse the model output of 10 state-of-the-art CCM simulations. All these simulations were conducted in the framework of the Chemistry-Climate Model Initiative phase 1 CCMI-1,. An overview on the model simulations that are used for analysis in this study and some aspects of their model setup is given in Table . Additionally, the name of the atmospheric model component is provided to demonstrate similarities between some of the models. This subset of models has been chosen on the basis of availability of the required data for the analyses in this study (AoA and residual circulation velocities).

Note that ACCESS and NIWA-UKCA, as well as EMAC-L47, EMAC-L90 and SOCOLv3 share the same atmospheric model component and that the two EMAC versions only differ in vertical resolution. ACCESS and NIWA-UKCA only differ by the fact that NIWA-UKCA is coupled to an ocean model and that the simulations are run on different platforms. The simulations we analyse are seamless simulations spanning the period 1960–2100, the so-called reference simulations REF-C2 (only r1i1p1 ensemble members). The simulations follow the A1 scenario for ozone-depleting substances and the RCP 6.0 scenario for other greenhouse gases, tropospheric ozone (O3) precursors, and aerosol and aerosol precursor emissions. For anthropogenic emissions, the recommendation was to use MACCity until 2000, followed by RCP 6.0 emissions. Out of the models used in this study, MRI, NIWA-UKCA, and WACCM have coupled an interactive ocean and sea ice module in these simulations for atmosphere-ocean interactions. In all other simulations, climate model fields (i.e. sea surface temperatures and sea ice concentrations) are imposed. A variety of different climate model data sets were used for this purpose e.g. HadISST1 or HadGEM2 data; for details see Table S1 in. More details on the simulation setups can be found in and in the citations given in Table .

The methodology of this study mostly follows the companion paper . A short description of the most important concepts used here is given in the following.

Stratospheric mean AoA is defined as the mean residence time of an air parcel in the stratosphere . In the CCMs, the AoA tracer is implemented as an inert tracer with a mixing ratio that linearly increases over time as lower boundary condition “clock tracer”. In some models, this lower boundary condition is global, in others only in the tropics, in NIWA-UKCA and ACCESS for example, AoA is kept at 0 in the boundary layer. AoA is then calculated as the time lag between the local mixing ratio at a certain grid point and the current mixing ratio at a reference point. This reference point, however, varies between the models (e.g. boundary layer, tropopause, 100 hPa), which could lead to inconsistencies in the AoA calculation. To avoid this, we subtract the mean AoA value of the respective model's tropical tropopause from the actual AoA value at all grid points. This means that in our analyses, AoA =0 at the tropical tropopause for all models. We perform this calculation for each time step (with monthly values) separately for the entire time period 1960–2100. Therefore, the AoA trend excludes any changes in tropospheric transport times due to the fact that the tropopause rises over time see.

The residual circulation transit time (RCTT) is the hypothetical age that air would have if it only followed the residual circulation, meaning that no processes such as eddy mixing or diffusion would come into play. These RCTTs are calculated using a concept described by , by calculating backward trajectories on the basis of the Transformed Eulerian Mean (TEM) meridional (v‾*) and vertical (w‾*) velocities (referred to as residual velocities, available in the CCMI-1 database for the chosen models), with a standard fourth-order Runge–Kutta integration. The RCTT is then the time that these backward trajectories require to reach the tropopause from their respective starting point in the stratosphere. For more details see also and .

The RCTT differs from AoA because of (resolved and unresolved) mixing. In the stratosphere, this is due to the mixing of air between branches and the in-mixing of air from the mid-latitudes into the tropical pipe, which leads to recirculation of old air around the BDC branches. In global model studies, this effect has been named aging by mixing (A_mix) and is interpreted as the difference between AoA and RCTT . However, it has to be kept in mind that the residual of AoA and RCTT does not only reflect this process alone, but actually includes resolved mixing as well as parameterized and numerical diffusion.

As a measure of the relative strength of mixing (independent of the residual circulation strength), we use the so-called mixing efficiency ϵ for analysis. ϵ is defined as the ratio of the mixing mass flux to the net residual mass flux between the tropics and the extratropics across the subtropical barrier. The net mass flux is the horizontal motion that is determined by mass continuity from vertical motion and corresponds to transport by v‾* . ϵ can be derived by means of the tropical leaky pipe (TLP) model . The TLP model is a one-dimensional transport model of the stratosphere that includes advection and horizontal mixing of air between the tropics and the extratropics. It assumes two columns of well-mixed air (a tropical and an extratropical column) and can be used to quantify the strength of mixing across the subtropical barrier. If we neglect vertical diffusion see, we can formulate an analytical solution for tropical and mid-latitude AoA. According to the TLP model, tropical AoA (AoAT) with altitude-dependent vertical velocity wT*(z) can thus be described as AoAT(z)=∫ztz1wT*(z′)dz′+ϵα(z)+1α(z)∫ztz1wT*(z′)dz′+H1wT*(z)-1wT*(zt)=RCTT(z)+ϵα(z)+1α(z)RCTT(z)+Tcorr(z) . Here, H denotes the scale height (7 km) and α(z) denotes the ratio of tropical to extratropical mass, which is approximated by α(z)=sin⁡(lat(w*(z)=0))-sin⁡(1-lat(w*(z)=0))2⋅sin⁡(90)-sin⁡(lat(w*(z)=0)), zt denotes the height of the tropical tropopause and Tcorr(z)=H1wT*(z)-1wT*(zt) the correction term for the altitude-dependency of the vertical residual velocity w* (an additional analytical solution term from horizontal advection; for details see ). AoA thus depends on the advective vertical velocity wT* (i.e. the residual velocity) and on the mixing efficiency ϵ (i.e. the amount of mixing between the tropics and the extratropics). Solving Eq. () for the mixing efficiency yields ϵ=AoAT(z)-RCTT(z)RCTT(z)+Tcorr(z)⋅α(z)+1α(z). Equation () shows that ϵ is approximately proportional to the relative increase in AoA due to mixing. For analysis, we calculate the 10 year running averages of ϵ to obtain climatologically representative values. Note that according to the concept of the TLP model, ϵ has to be viewed as a parameter valid for a given climate state in a certain model. Hence, it can vary between models and/or for different climate conditions. In the study by , the authors nevertheless found a constant ϵ for three different climate states in one model. The tropical profiles of w‾*, tropopause height and AoA provided for the TLP model are averaged over the latitudinal bands of the models' individual vertically averaged turnaround latitudes (which are also time-dependent and calculated from v* for consistency between the models; refer to the Supplement of ) and are interpolated to vertical coordinates relative to the tropopause height of each model. The mixing efficiency is then obtained by the TLP model's best fit to the CCM AoA profile. Here, the best fit is done for the altitude range from the tropopause to 32 km details for the calculation of the mixing efficiency are given in. According to the TLP formulation, aging by mixing (A_mix) is a function of the mixing efficiency and of the residual circulation strength: A_mix=AoA-RCTT=ϵ⋅α(z)+1α(z)⋅RCTT(z)+Tcorr(z). A_mix is proportional to the mixing efficiency, but inversely proportional to the vertical velocity. A higher mixing efficiency is, e.g. connected with more air parcels recirculating see, thereby increasing A_mix. But the vertical velocity also controls the speed of the air parcels that recirculate. Thus, the mixing efficiency has been shown to be a useful diagnostic tool, as it does not depend on the speed of recirculation e.g..

A multi-model comparison of stratospheric transport changes in the 21st century has been conducted before for the Stratosphere-Troposphere Processes and their Role in Climate (SPARC) report . In that study, the authors showed stratospheric mean AoA of 10 chemistry–climate model simulations that took part in the CCMVal-2 Chemistry-Climate Model Validation, project. To allow a direct comparison of the simulations that were analysed in with the simulations we use here, Fig.  presents the same depiction of AoA differences between the 2090s and the 1990s (ΔAoA) (a) at 50 hPa and (b) as the difference between the tropics and middle latitudes as in their Fig. 5.18.

The general structures of the AoA differences agree between the two model intercomparison projects. All models predict a decrease in mean AoA at 50 hPa and the smallest decrease in the tropics (Fig. a). A second minimum in decrease is found at 60∘ N/S and the greatest decrease is at 30∘ N/S and/or at the poles. The AoA behaviour in these regions of maximum change across the 21st century is investigated in detail by . They showed that these trends are related to the climatological AoA distribution, the upward shift of the pressure levels and the widening of the AoA isolines. In comparison with the CCMVal-2 models, the inter-model spread of the AoA difference is reduced in our results. However, it was the two UMUKCA (Unified Model/U. K. Chemistry Aerosol) models that led to the large spread in the SPARC report and are not part of our analysis. However, NIWA-UKCA and ACCESS are the direct successors to the UMUKCA models and these range in the lower end of AoA changes here. The ULAQ model has changed from a very large latitudinal ΔAoA amplitude to a rather small one from CCMVal-2 to CCMI-1. Other models that appear in both studies do not show large changes. The EMAC model was not included in the SPARC report. With a higher vertical resolution (47 to 90 levels in the vertical), the EMAC model tends to simulate larger ΔAoA and thereby detaches from the bulk of the other model simulations. Consistent with this, show that in the SOCOLv3 model, AoA also gets on average 1 year older when the model is run with 90 layers in the vertical (instead of 39) because of less vertical diffusion. All the statements above also count for Fig. b and the tropical to middle-latitude AoA differences with altitude (Fig. b).

To visualise AoA and its trends of the model simulations and their inter-model differences, we present the annual mean AoA data of the 10 CCMI-1 model simulations in Fig. . Moreover, the piecewise linear regression of AoA for the periods 1960–2000 and 2000–2100 is presented. These two periods were chosen because the year 2000 marks a change in stratospheric dynamics, which is due to the reversal in signs of ODS and ozone trends as a consequence of the Montreal Protocol see. We chose the 30 hPa pressure level and an average between 30 and 50∘ N, because the observation-based data from are from that region too. These data and their linear regression are included in the figure as well; they show a non-significant positive AoA trend of 0.15±0.18 years decade-1.

The absolute AoA values differ strongly among the models. For the first three decades of the simulations (i.e. the mean from 1960–1990) the values range between 5.52 years in the MRI model and 3.18 years in the ACCESS model simulation. This topic had already been discussed, for example in and in . Analysing the hindcast simulations of the CCMVal-2 and the CCMI-1 projects, showed that it is mainly the mixing rather than the residual circulation that causes the large AoA spread and that this is likely linked to the different resolutions of the model simulations.

All the model simulations show a clear negative trend over time across the first, as well as across the second period. The in situ measurement-based observations , in contrast, display a positive, but not significant, trend for the last couple of decades. A similar behaviour to the in situ measurements can also be found in satellite-based observations, for example, in , although for a shorter time series (2002–2012). A number of studies have investigated this discrepancy see e.g. and many reasons have been discussed to resolve this contradiction (e.g. effect of mixing in models, sparse sampling of observational data, differences in changes between deep and shallow BDC branches), but a viable explanation is still missing. In the present study, however, we do not discuss this issue any further, but rather focus on the analysis of the negative AoA trends in the model simulations. The models may agree in predicting a decrease in stratospheric AoA, but they do show large differences in the strength of this trend (σ(ΔAoA)=0.18 for μ(ΔAoA)=0.54; see below for ΔAoA). Also, the trends between the two periods (1960–2000 and 2000–2100) differ. Most models show a stronger trend between 1960 and 2000, only the two EMAC simulations have a stronger trend in the second period and in SOCOLv3 the trend almost remains constant. Note, however, that EMAC has a negative bias in ODSs, because the replacement products were not taken into account as F11 or F12 equivalents. This can possibly explain why the AoA trend is weaker between 1960 and 2000. Several studies e.g. have recently pointed out the importance of the role of ODSs for the trends in stratospheric dynamics. ODSs act as both, radiatively active greenhouse gases and chemically active gases controlling ozone depletion and recovery. During the period between 1960 and 2000, ODSs increase and thus act in concert with GHGs to cause an AoA decrease. Thereafter, however, ODSs decrease over time, which means that with respect to AoA trends, the ODS trend works against the trend in the continuously rising GHGs. This can possibly explain the weaker trend in the second period in most models. Due to this change in dynamical properties around the year 2000, an analysis of the trends across the entire period from 1960 to 2100 cannot be conducted without mixing-up various dynamical effects. The first period is relatively short for robust analyses of mixing trends and in the second period, two mechanisms work against each other, so that in most models, the trends are rather small. In the following, we therefore analyse the differences between the periods 1970–1990 and 2080–2100. Based on the sensitivity simulations in , which shows that the change in the slope in the year 2000 is due to ODSs, this allows the capturing of the GHG effect alone. At the end of the 21st century, ODS mixing ratios have declined to similar values as between 1970 and 1990 in the simulations. We do not start our investigation at 1960 due to the calculation method of RCTTs and ϵ, which are available only from 1970 onwards.

Table  provides an overview over mean AoA, ϵ and tropical upwelling (at 70 hPa) averaged between 1970 and 1990 (denoted as 1970) and between 2080 and 2100 (denoted as 2100) and their differences (Δ) for the 10 CCMI-1 REF-C2 model simulations. For consistency with ϵ and tropical upwelling (which are global values), we here averaged AoA globally (90∘ S–90∘ N) and over 10–100 hPa, which means that these AoA values are not comparable with those in Fig. . Table  will be used to explain and confirm some of the results throughout the paper.

Figure  shows the linear AoA changes, along with the ratio of the RCTT change to the AoA change (ΔRCTT / ΔAoA) for a model intercomparison of the zonal mean structure of the differences. Moreover, the contour lines show the climatologies of the 1970–1990 period of the simulations for AoA and for the RCTT / AoA ratio, respectively. The ΔRCTT / ΔAoA ratio provides the possibility to linearly separate ΔAoA into its contributions from residual transport and aging by mixing (A_mix). Specifically, values between 0.5 and 1 signify a domination of residual transport changes (reddish), while values between 0 and 0.5 mean that ΔAoA is controlled by A_mix variations (blueish). Values above 1 mean that the A_mix difference is positive and values below 0 mean that ΔRCTT is positive (assuming a negative ΔAoA).

In most models, a quite similar ΔAoA pattern can be seen. Relatively small ΔAoA dominates the tropical pipe where young air is prevalent and the differences increase with higher altitudes and latitudes. Several model simulations (ACCESS, CMAM, MRI, NIWA-UKCA), however, show their maximum ΔAoA in the lower stratospheric middle latitudes. This feature is connected with the climatological AoA gradient, the upward shift of the circulation and the decrease in RCTT and A_mix in . The two EMAC simulations show the largest ΔAoA throughout the stratosphere and the ULAQ model the weakest. Moreover, ULAQ is the only model that shows a pronounced hemispheric asymmetry, with stronger differences in the SH. These general ΔAoA patterns were also found in the multi-model study by who analysed 11 21st century model simulations that were performed for . Most models also agree in the ΔRCTT / ΔAoA pattern. Strong A_mix-dominated ΔAoA can be seen in the middle-latitude lower stratosphere and the residual circulation dominates ΔAoA mainly in the tropical pipe as well as in the downwelling branches in the high latitudes. Some model simulations hardly show any regions where the differences of residual transport accounts for less than 50 % of ΔAoA throughout the stratosphere, even in the upwelling and downwelling branches (EMACL47, EMACL90, GEOSCCM, SOCOLv3). Other models (GEOSCCM, ACCESS, NIWA-UKCA, WACCM) show distinct upwelling and downwelling branches. This indicates that ΔAoA is not influenced much by ΔA_mix in these particular regions. The NIWA-UKCA model shows a clear separation between aging by mixing and residual circulation dominated regions, and in the ULAQ model, the ΔA_mix is slightly positive in most parts of the stratosphere (i.e. ΔAoA / ΔRCTT >1). Altogether, these results suggest that changes in aging by mixing have a major impact on ΔAoA. This brings us to further analyse A_mix changes as well as the mixing efficiency and to discuss possible reasons for their variations.

Figure  suggests that most of the ΔAoA is determined by aging by mixing changes. However, A_mix itself depends on RCTT (see Eq. ) and is therefore not an independent measure for separation of the processes. AoA changes are commonly attributed to changes in the residual circulation e.g.. In the climatologies, the tropical upwelling as calculated via integration of v* between the turn-around latitudes; see e.g. in the 70 hPa pressure level has been shown to be a good measure for the strength of the residual circulation throughout the stratosphere see. To see if this relationship holds true for ΔAoA in our set of the CCMI model simulations, we present in Fig.  the inter-model correlations between ΔAoA and ΔRCTT, as well as between the ΔAoA and the differences in tropical upwelling in 70 hPa. These correlations are calculated across the 10 model simulations for each grid point separately. Note that for this, data from each model were interpolated to the resolution of the grid of the highest horizontal model grid resolution.

A high correlation between ΔRCTT and ΔAoA (Fig. a) can only be seen in the middle to high latitudes between around 70 and 10 hPa and low or no correlations in the other regions. In the upwelling region in the tropics this is particularly surprising since AoA should be controlled by the upwelling speed there. However, this connection apparently is not local. The correlations between ΔAoA and Δtropical upwelling (Fig. b) are high in the entire stratosphere above around 70–50 hPa. This reflects a clear connection between tropical upwelling and AoA, particularly in the deep branch. A stronger tropical upwelling generates a faster circulation and hence a decrease in AoA. The strengthening of tropical upwelling can be explained by an enhancement of the subtropical jet streams following from upper tropospheric warming and lower stratospheric cooling see e.g. and was linked with the upward shift of the tropopause by and . The rather patchy picture of the ΔRCTT vs. ΔAoA correlations suggests that the link between the residual circulation and AoA cannot be expected to be local, but rather of remote nature. It is mainly the ascent of air in the tropics that determines the ΔAoA-dependency on the residual circulation. The upwelling and downwelling regions in particular do not correlate well in Fig. a, although ΔAoA is clearly dominated by the changes in transport in those regions (see Fig. ).

showed that the climatological inter-model AoA spread in the CCMVal-2/CCMI-1 hindcast simulations can mostly be explained by model differences in mixing efficiency and only to some extent by differences in residual circulation. Hence, we now focus on the model trends (and differences between the two periods) in mixing efficiency and on the processes that are responsible for its changes. ϵ provides information on the models' relative strengths in mixing at the tropical barriers as a vertically integrated measure (see Sect.  for calculation method). Figure  shows the time series of the mixing efficiency of each model for the REF-C2 simulation period and, as in Fig. , the piecewise linear regressions for the time before and after the year 2000 are included. Note that due to the calculation of the 10-year moving average of ϵ, the first 10 years cannot be assessed here. For a more quantitative view, the Δϵ and the climatological ϵ of the two simulation periods are also presented in Table .

The climatological ϵ of the first 20 REF-C2 simulation years show similar values as the REF-C1 CCMI-1 reference hindcast simulations from 1960–2011; for details see values that were shown in . In this period, the main differences between the REF-C1 and the REF-C2 simulations are the sea surface temperatures and the sea ice distribution. Only MRI shows a somewhat higher mixing efficiency in the REF-C2 simulation (∼50 %). The multi-model mean of this climatological ϵ is 0.52%±31 %. This agrees well with the value (0.58%±32 %) that was found for the REF-C1 simulations (for ϵ calculated at the turnaround latitudes) by . Across the whole simulation period, the mixing efficiency decreases over time in most models. However, the sign of the trend often changes between the two periods. For example in MRI and in EMAC-L90, ϵ is at first almost constant and then it decreases; and in GEOSCCM and CMAM, ϵ at first decreases and in the later period it rises. Only the ULAQ model shows a positive trend in both periods. For the reasons we mentioned above (counteracting influence of GHGs and ODSs), we now again discuss the differences between the start and the end of the simulations to filter out the direct effect of ODSs on stratospheric dynamics. The ϵ values for the periods in question, as well as their differences are provided in Table . The ACCESS and the NIWA-UKCA model (the two models with the HadGEM atmospheric model component) show a slight positive Δϵ and the only model with a strong positive Δϵ is ULAQ. The fact that the mixing efficiency is not constant over time means that the absolute mixing strength does not change proportionally with the residual circulation. This is in contrast to the results of , who found a nearly constant ϵ when comparing three GCM simulations with different climate states. However, also make clear that it would not be surprising if ϵ varied over the course of climate change simulations, because the properties of the mixing barriers can also change over time. The mechanisms for the mixing changes are diagnosed using the potential vorticity gradient in Sect. . Next, we investigate if ΔAoA is also linked to variations in mixing efficiency, and not merely to residual circulation changes. For this, Fig.  shows the inter-model correlations (correlations across the 10 model simulations) between the local ΔAoA and the models' differences in mixing efficiency.

Figure  shows a clear link between ΔAoA and Δϵ above around 100–70 hPa, with correlation coefficients mostly ranging between 0.8 and 0.9. This reflects that a decline in mixing efficiency leads to a decrease in AoA, because there is less recirculation of air around the BDC. Only in the region around the Antarctic polar vortex is the correlation generally somewhat weaker. This may be due to changes in the polar vortex strength or position in the various models, which cannot be reflected in the (sub)tropical measure ϵ.

A_mix is connected with the mixing efficiency, but it also depends on the residual circulation. A higher or lower mixing efficiency causes more or less recirculation, which increases or decreases aging by mixing, respectively, and the RCTT controls the transit times of the air parcels that recirculate. Overall, we can again conclude that ΔAoA in climate models is connected with changes in both, mixing and residual transport. Most of the ΔAoA that is connected with residual transport can be attributed to tropical upwelling changes, at least for the deep BDC branch. This is caused by a climate change induced strengthening and/or upward shift of the subtropical jet streams see e.g.. The differences in mixing between future and past climate, however, have not been analysed in detail before. Therefore, we investigate the overall effect of mixing on the BDC changes in the remainder of the paper.

To quantify the impact that changes in mixing efficiency have on ΔAoA in the simulations, we again apply the TLP model, now by using w‾*, the tropopause height and a given ϵ to calculate the RCTTs and AoA (see Eq. ). We then compare the AoA and RCTT climatologies of the two periods 1970–1990 (1970) and 2080–2100 (2100). Figure  shows the tropical AoA and RCTT profiles (calculated with the TLP model) of these climatologies exemplarily for the EMAC-L90 model simulation. Moreover, two hypothetical AoA profiles (AoA′ and AoA*) are included in the figure (both for 2100). AoA′(2100) (gray line in Fig. ) displays the AoA climatology for 2100 with the A_mix fixed to the value of 1970, it is calculated by AoA′(2100)=A_mix(1970)+RCTT(2100)=AoA(1970)-(RCTT(1970)-RCTT(2100)). The difference between AoA(1970) and AoA′(2100) then yields the AoA change that is caused by the RCTT change only (without a change in A_mix). The AoA*(2100) (black line in Fig. ) profile displays the AoA climatology of the 2100 climate state by keeping the mixing efficiency constant at the value of 1970. We calculate this quantity by using ϵ(1970) in Eq. () for the AoAT(2100) calculation, thus it represents how much influence the change in ϵ has on ΔAoA (in contrast to the change in residual circulation only).

The RCTT and AoA climatologies of the EMAC-L90 simulation of both climate states show that RCTT explains about one-third of the AoA values. The difference (about two-thirds) is caused by aging by mixing (see Fig. ). This ratio had already been found in for the CCMVal-2 and CCMI-1 hindcast simulations. AoA′ reveals that for EMAC-L90 this ratio roughly also accounts for the differences between the periods. The AoA difference between 2100 and 1970 is subdivided by AoA′ into about one-third the share of RCTT difference and two-thirds of A_mix. This ratio can also be estimated from Fig.  for the EMAC-L90 simulation. However, as already stated above, the A_mix change also includes some RCTT change (see Eq. ), which implies that this is not a clear separation of the mechanisms.

The difference between AoA*(2100) and AoA(2100) reflects the impact of Δϵ on AoA. In the given example of the EMAC-L90 model, the fractional impact of the mixing efficiency change on the AoA difference is 29%±4% as calculated between 2.5 and 25 km above the tropopause. This value varies considerably among the 10 models, Table  provides an overview.

NIWA-UKCA, ACCESS and ULAQ show a negative fractional impact of mixing efficiency on AoA changes. These are the three models that also show a positive Δϵ (see Table ). In contrast to the other models, the mixing efficiency therefore leads to an AoA increase over time. The negative ΔRCTT therefore accounts for more than the entire negative ΔAoA to compensate for the effect of the ϵ change. With less than 5.5 % and 3.5 %, NIWA-UKCA and ACCESS have the lowest contribution of Δϵ on the AoA change and with up to 29 %, ULAQ and EMAC-L90 have the largest. The other models have a contribution of Δϵ on ΔAoA between 10 % and 29 %, the multi-model mean is 10.4 % (σ=18.3 %). It is reasonable that a large Δϵ leads to a high percentage of the ϵ-share on ΔAoA. The correlation coefficient between the two values is 0.83. This makes clear that the change in mixing efficiency does have a considerable impact on ΔAoA, as it could already be assumed from the high inter-model correlations between the two quantities (Fig. ). Due to the large model spread in Δϵ, however, this impact bears large uncertainties. Next, we quantify the impact of Δϵ on the model spread in ΔAoA.

To analyse the share of Δϵ on the AoA model spread, we again use the TLP model-calculated values of ϵ, α(z), Tcorr and RCTTT. These quantities are taken to calculate tropical AoA by AoAT=RCTTT+ϵ⋅α(z)+1α(z)⋅RCTTT+Tcorr,

as an average between 100 and 10 hPa. As above, this calculation is performed for each model for the two periods, so that ΔAoAT=AoAT(2100)-AoAT(1970). Using ϵ(1970) to calculate AoAT(2100) in Eq. () provides AoAT*(2100) and thus ΔAoAT*=AoAT*(2100)-AoAT(1970) provides the AoA difference without any changes in ϵ. Figure  displays the model distribution of ΔAoA with (blue) and without (red) a changing mixing efficiency.

Figure  clearly shows that when ϵ is kept constant, first, ΔAoA generally decreases (in absolute values) and second, the model spread of ΔAoA is considerably reduced (from 0.35 to 0.22 years). This reflects that the mixing efficiency changes lead to strong variations in AoA. When ϵ is held constant, for models with negative Δϵ, ΔAoA is reduced (in absolute values) (see for example EMAC) and ΔAoA is enhanced for models with positive Δϵ (see for example ULAQ). The model range decreases here because being the model with the largest ΔAoA, EMAC-L90 has a negative Δϵ and ULAQ, the model with the lowest ΔAoA has a positive Δϵ. A large or small (negative) Δϵ causes a large or small ΔAoA. From this analysis, we can conclude that |ΔAoA| generally increases through the changes in mixing efficiency (by 10.4 % as multi-model mean) and that Δϵ leads to a larger model spread in the AoA changes (Δϵ increases the ΔAoA model range by about 37 %).

Now, the question remains what the reasons for the changes in relative mixing strength are, or why ϵ changes in the course of climate change simulations. In the next section, we will explore an explanation for this by analysing various dynamical fields.

The relation between residual circulation and mixing determines the mixing efficiency and changes therein. To study possible dynamical reasons for the changes in mixing efficiency, the relation of wave driving, the residual circulation and mixing is analysed in the following based on the Transformed Eulerian Mean (TEM) momentum equation. This analysis is similar to that presented in , but here we use the quasi-geostrophic (QG) formulations on pressure levels (on which the CCMI data are available). We present and analyse multi-model mean (MMM) diagnostics to provide a general picture for the entire subset of CCMI-1 model simulations. Unless otherwise stated in the text, the individual models qualitatively agree fairly well in these diagnostics. The fields of the individual models can be found in the Supplement. Figure  shows the differences between the two periods (1970–1990 and 2080–2100) overlaid with the MMM climatologies of the first period of zonal winds, Eliassen–Palm flux divergence (EPfd), v*, the meridional potential vorticity gradient (∂PV /∂y), as well as Kyy and Kyy/|v*|. The calculation of Kyy and the meanings of these variables will be explained below. Due to data availability, not all models could be included in these MMMs. For SOCOL and ULAQ, the EPfd fields were not available, hence, for consistency, all MMMs are based on the remaining eight other models only.

The mechanism of the residual circulation increase in the lower stratosphere is well understood e.g.. It follows the upper tropospheric temperature increase that leads to an upward shift and increase of the zonal mean winds. This can be seen in Fig. a, which shows the MMM differences of the zonal mean wind between the two time periods. Subsequently, the critical layers that allow for wave propagation shift to higher levels. This becomes evident from Fig. b. There, a region of strongly negative ΔEPfd (enhanced wave dissipation) can be seen between about 100 and 50 hPa (with maxima around 70 hPa) and positive differences (less wave dissipation) can be found between around 200 and 150 hPa. The enhanced wave dissipation in the lower stratosphere leads to amplified poleward residual transport, which reflects in Δv* (Fig. c).

However, as shown in the sections above, transport changes in a future climate are not only due to this increase in residual transport, but also due to changes in eddy mixing. The EPfd (∇⋅F) not only drives the residual circulation, but it is also related to eddy mixing of potential vorticity q. Under the quasi-geostrophic (QG) approximation and neglecting parameterized (gravity) wave forcing, this can be formulated as ∇⋅F=v′q′‾, where v′ and q′ denote the deviation of their zonal means v‾ and q‾, respectively. Note that due to data availability, we show the full EPfd fields in Fig.  and not the QG EPfd. Note also, that strictly speaking this relation holds true on the beta plane , or approximately on isentropic surfaces, i.e. on Θ levels, on which mixing takes place. Therefore, the quantity may be somewhat distorted on the pressure levels presented here. This may lead to differences in the quantities, however, not qualitatively different conclusions in the following. Commonly, a flux–gradient relationship is assumed for the eddy PV flux, so that v′q′‾=-Kyy∂qy, with the diffusivity coefficient Kyy. ∂qy is the gradient of the QG PV see e.g. with ∂qy=∂q‾∂y=β-∂2u‾∂y2+∂f⋅∂Θ‾∂y/∂Θ‾0∂p∂p. Similarly to in , we calculate the diffusivity coefficient Kyy as Kyy=-∇⋅F∂qy. This relation states that horizontal eddy mixing is proportional to the EP flux divergence, and inversely proportional to the meridional PV gradient. Thus, a weak PV gradient indicates strong mixing, while a strong PV gradient acts as barrier to mixing, and thus mixing is suppressed.

The MMM differences show that next to the enhanced wave dissipation in most of the stratosphere (-∇⋅F increases); the PV gradient also increases from the subtropics to mid-latitudes (see Fig. d) due to the strengthened winds. Thus, Kyy tends to be enhanced due to increased wave dissipation, but reduced due to the increased PV gradient. As seen in Fig. e, the diffusivity changes are dominated by the increase in wave dissipation in the lower stratosphere (between around 100 to 50 hPa in both hemispheres) and by the decrease in wave dissipation below. This means that the vertical shift of EP flux convergence is reflected in ΔKyy, or in other words, that the region of strong mixing is shifted upward and the absolute strength of mixing increases in the lower stratosphere. At higher altitudes, the PV gradient contributes more strongly to Kyy, and in the SH mid-latitudes, the mixing strength Kyy even decreases despite enhanced wave dissipation due to the strongly enhanced PV gradient. In the NH, ΔKyy is smaller and mostly positive in the MMM. However, it should be noted that in the lower stratospheric subtropics in particular, which is the region of the climatological maxima of orographic gravity wave drag, the QG assumption can be misleading due to the neglect of parameterized wave drag.

The mixing efficiency derived from the AoA data represents the relative strength of mixing, i.e. the ratio of the mixing mass flux to the residual meridional mass flux. The mixing mass flux is proportional to the mixing velocity that can be expressed as vmix‾=Kyy/L for a given horizontal length scale L. Thus, the ratio of mixing vs. residual mass flux can be approximated as vmix‾v*‾=1LKyyv*‾. Given a constant length scale L, the mixing-to-mean-advection ratio is proportional to Kyy/|v*‾|. As shown above, residual transport as well as mixing increases in most regions (Fig. c and e). Figure f, however, shows that the ratio Kyy/|v*‾| decreases in large parts of the stratosphere. This means that mixing increases less strongly (or even decreases) than residual transport. Only in the (sub-)tropical lower to mid-stratosphere, the relative mixing strength increases. The changes in Kyy/|v*‾| mostly reflect the inverse change in the PV gradient, because the mixing diffusivity is inversely related to it. Note that if assuming v*‾=-f-1∇⋅F, the ratio Kyy/|v*‾| equals f/∂qy; i.e. the ratio is only related to the PV gradient. Changes in f/∂qy are overall similar to changes in Kyy/|v*‾| (not shown), except for some details that might be related to the simplification of the equations and/or to the neglect of parameterised wave drag (i.e. small scale gravity wave drag).

Overall, this analysis suggests that enhanced wave dissipation caused by the zonal mean wind increase; see amplifies the residual circulation as well as mixing, particularly in the lower stratosphere. However, the zonal mean wind changes also cause changes in the meridional PV gradient, and the stronger PV gradients act to inhibit mixing. Therefore, mixing increases less strongly than residual transport does, which explains the decrease in the mixing efficiency in the future (see Sect. ).

However, as shown in Sect. , the mixing efficiency does not decrease in all models. In Fig. , the mean ΔKyy/|v*‾| in the region of the subtropical barrier averaged between 20 and 40∘ S, and between 20 and 40∘ N is shown with altitude for the subset of eight CCMI-1 model simulations.

|Δ(Kyy/|v*|)| is larger in the SH compared to the NH, in particular at higher altitudes. This is due to the acceleration of the Antarctic polar vortex (Fig. a) that leads to an increase in ∂qy (Fig. d), thereby reducing effective mixing. In most models, Δ(Kyy/|v*|) is negative within these latitudes throughout the stratosphere in the SH as well as in the NH. Between 20 and 50 hPa there are only two exceptions, namely NIWA-UKCA and ACCESS in the NH, those two models that also have a slight positive Δϵ (see Table ). Moreover, the two EMAC model versions, which have the largest Δϵ also show the largest |ΔKyy/|v*||, in both hemispheres. Although we cannot find a clear inter-model correlation between Δϵ and Δ(Kyy/|v*|) here, these are strong indications of a link between the two quantities.

Hence, the inter-model differences in mixing efficiency changes are consistent with differences in the relative rate of change in mixing to residual transport. The differences in Kyy/|v*‾| changes between the models appear to be related to structural differences in zonal wind changes; i.e. the models with an increasing mixing efficiency (namely NIWA-UKCA and ACCESS) simulate zonal wind increases in the NH at higher latitudes than other models. As a consequence, the PV gradient also increases at higher latitudes, so that mixing increases more strongly in the region of the subtropical barrier (see figures in Supplement). Our approximations, as well as the limited number of models, do not warrant a robust conclusion from inter-model correlations. However, the results shown here suggest that the mixing efficiency changes, as well as the inter-model spread, are consistent with changes in the relative mixing strength due to changes in the background PV gradient.