Corresponding author: D. Demirhan Bari, Faculty of Aeronautics and Astronautics, Department of Meteorological Engineering, Istanbul Technical University, Istanbul 34469, Turkey. (email@example.com)
 The longitudinal variations in the time-mean transport by the Brewer-Dobson circulation are studied using a three-dimensional (3-D) residual circulation approach to analyze the effects on zonal asymmetries in stratospheric ozone (O3) and middle atmospheric water vapor (H2O). For January, the monthly mean residual winds, including both the Eulerian flow and the eddy-induced time-mean flow, were derived from general circulation model simulations with interactive chemistry (HAMMONIA), reanalysis (ERA-Interim), and satellite data (Aura/MLS). Extending the picture of the zonal mean two-dimensional Brewer-Dobson circulation, we find a 3-D circulation structure in relation to the zonal wave one in the middle atmosphere, including northward and downward residual winds over northern Europe/Asia with the downwelling directed toward the center of the polar vortex over northern Siberia, as well as southward and upward residual winds over the northern Pacific/Aleutians, and a pronounced cross-polar transport from Asia to North America in the middle stratosphere. The residual advection of O3 and H2O shows that the observed wave one patterns in O3 and H2O are produced by the zonal asymmetries in the residual mass transport in which Eulerian and eddy time-mean transports are largely counteracting. In comparison to observations, the model underestimates the effects of planetary waves but overestimates those of transient waves in configuring the stationary waves in O3 and H2O. Overall, the 3-D residual circulation approach provides a useful diagnostic for understanding regional differences in middle atmospheric trace gas distributions and for validating general circulation models with interactive chemistry.
 The stratospheric meridional mass circulation, also known as the Brewer-Dobson circulation (BDC), describes the time-mean transport of air masses from the tropics to midlatitudes and polar regions and is usually formulated in the zonal mean two-dimensional (2-D) framework of the transformed Eulerian mean (TEM) equations [e.g., Andrews and McIntyre, 1976; Holton, 1986; Randel et al., 1998]. The TEM equations include the effect of both the zonal mean Eulerian wind components and the zonal mean eddy fluxes on the time-mean transport, which leads to the approach of the 2-D residual circulation representing the zonal mean transport of important atmospheric constituents such as ozone (O3) and water vapor (H2O). However, the zonal mean approach excludes longitudinal variability in the residual circulation and its effect on the zonally asymmetric structures of O3 and H2O. Recently, Kinoshita et al.,  formulated an extended three-dimensional (3-D) residual circulation including the effect of time-mean 3-D eddy fluxes, based on earlier works of Hoskins et al. , Plumb [1985, 1986], and Trenberth . Alternatively, longitudinal variations in the transport of important trace gas constituents such as O3 can be examined in the framework of isentropic coordinates [Sato et al., 2009], but the 3-D residual approach provides a more detailed understanding of the wave-driving processes. In this paper, we apply the formulation of the 3-D residual circulation on different data sets in order to understand the time-mean 3-D tracer transport and its effect on the zonally asymmetric components of O3 and H2O in the stratosphere and mesosphere of the Northern Hemisphere. The main purpose of this paper is to identify the longitudinal variations in the 3-D residual circulation hidden by the 2-D residual approach, which leads to an extended picture of the 3-D BDC as a circulating 3-D conveyor belt. This also includes longitudinal variations of northward and southward tracer transports in relation to the wave one structure usually observed in the northern winter middle atmosphere. We use three different data sets, i.e., simulations of the general and chemistry circulation models—the Hamburg Model of the Neutral and Ionized Atmosphere (HAMMONIA) [e.g., Schmidt et al., 2010], the assimilated ERA-Interim data of the European Center for Medium-Range Weather Forecast (ECMWF) [Dee et al., 2011], and the Aura/MLS satellite data used for investigating the physics, dynamics, and chemistry of the middle atmosphere [e.g., Santee et al., 2005; Manney et al., 2007; Schwartz and co-authors, 2008]. A detailed comparison of the data sets is presented, revealing the influence of transport driven by quasi-geostrophically balanced planetary waves, and by transient synoptic-scale waves on the stationary wave patterns in O3 and H2O, demonstrating that the 3-D residual approach opens a new viewpoint for validating chemistry-climate model simulations and interpreting local changes in transport characteristics.
 Large-scale transport by ultralong planetary waves is one of the main factors determining the zonal asymmetries in total column ozone [e.g., Austin and Butchart, 1992; Hood and Zaff, 1995; Peters et al., 1996]. Sato et al.  show that observed zonal asymmetries in the vertical distribution of O3 in the Antarctic polar vortex result from longitudinal variations in the transport by the BDC. During winter, the zonal asymmetries in stratospheric O3 and in stratospheric and mesospheric H2O are characterized by a pronounced stationary wave pattern with zonal wave number 1 (wave 1), and a large fraction of these patterns is related to zonal asymmetries in tracer transports by the time-mean geostrophic flow [e.g., Gabriel et al., 2011a]. Zonal asymmetries in gravity wave activity contribute to zonal asymmetries in mesospheric geopotential height [Smith, 2003] and might also have an effect on the 3-D transport of mesospheric H2O. However, there is a strong gap in the understanding of the effect of total 3-D mass transport on the 3-D tracer distributions including the time-mean effects of the wave fluxes due to planetary and synoptic-scale wave activity. In this context, the approach of the 3-D residual circulation proposed by Kinoshita et al.  provides a suitable tool to investigate the time-mean tracer transport and related processes.
 There were several earlier works introducing a 3-D wave activity flux and a 3-D residual circulation by using time-means of the eddy fluxes instead of zonal mean eddy fluxes [e.g., Hoskins et al., 1983; Plumb, 1986; Trenberth, 1986]. These previous concepts differ in the way they formulate the transformation from Eulerian into residual components, and they are used mainly for the investigation of wave-mean flow interactions in the troposphere but not for exploring the 3-D transport in the middle atmosphere. Based on these earlier works, the recently introduced formulae of Kinoshita et al.  for the 3-D residual circulation is a comprehensive and consistent formulation describing the net 3-D mass flux and related tracer transports similar to the 2-D TEM formulation. A specific point of Kinoshita et al.  is the consideration of small-scale disturbances like gravity waves using a modified formulation of Miyahara , in addition to the large-scale disturbances due to planetary and synoptic-scale waves. In the study of Kinoshita et al. , the formulae presented are applied in a case study of Southern Hemispheric ozone transport processes using the results of a chemistry-climate model simulation. Hence, the concept is shown to be useful for diagnosing the contributions of the individual transport tendency terms to the configuration of stratospheric ozone during Southern Hemispheric spring.
 Another way to examine the 3-D structure of the BDC is given by the 3-D diabatic circulation in isentropic coordinates, which has been demonstrated by Callaghan and Salby  based on a 3-D primitive equation model with prescribed tropospheric wave forcing. They found a cross-polar flow in the upper stratosphere as well as a pronounced zonal asymmetry in the vertical diabatic flow component. Here, we show that the 3-D residual approach suggests a similar picture of the 3-D BDC based on different model data and model-independent observations, which demonstrate the robustness of the resulting picture of the 3-D BDC and provide confidence for further studies on the related net transport of O3 and H2O.
 Following the suggestions of Kinoshita et al. , we apply the approach of the 3-D residual circulation on different data sets to improve our understanding of the 3-D aspects of the time-mean transport of air masses and trace gas constituents in the northern middle atmosphere. In this paper, the 3-D residual circulation approach is applied for the northern winter stratospheric and mesospheric circulations. In section 2, the data sets are briefly described. We use long-term simulation with the HAMMONIA model for the years 2001 to 2006, ERA-Interim reanalysis for the years 2001 to 2010, and Aura/MLS data for the years 2005 to 2010. The periods of the HAMMONIA and Aura/MLS data do not match exactly due to the availability of the data. Therefore, we directly compare time-mean values derived from these data with those of ERA-Interim, which allows reasonable comparison of the circulation and transport patterns derived from HAMMONIA and Aura/MLS for altitude ranges above the upper limit of the ERA-Interim data. For the Aura/MLS data, the wind components are derived from temperature profiles based on a quasi-geostrophically approximated flow. We additionally use this procedure for both the HAMMONIA and the ERA-Interim data, providing both evidence in the comparability of the data sets and a separation in the effects of quasi-geostrophically balanced planetary waves and synoptic-scale transient as well as gravity wave activity on the residual transport. In section 3, the 3-D residual approach of Kinoshita et al.  is described, and section 4 examines the resulting 3-D residual circulation for the Northern Hemisphere winter as derived from the three different data sets. For orientation, section 5.1 describes the zonal asymmetries in O3 and H2O providing an overview of the zonal wave one structures in the stratosphere and mesosphere. In section 5.2, the time-means of the transport of ozone and water vapor by the residual wind components are examined elucidating the effect of the 3-D residual circulation on the observed spatial structure of the stationary waves patterns in O3 and H2O. The paper ends with a discussion and outlook in section 6, which also summarizes the main findings of this study.
 The characteristics of the zonal asymmetries in the residual circulation are investigated by using data for the years 2001 to 2006 of a long-term transient simulation with the general circulation and chemistry model HAMMONIA (data provided by H. Schmidt, Max-Planck-Institute for Meteorology, Hamburg), ECMWF reanalysis of 2001 to 2010 (ERA-Interim, provided by ECMWF, Reading), and Aura/MLS satellite data from 2005 to 2010 (provided by NASA). These data sets were compared to provide a reliable and validated insight of the time-mean 3-D residual circulation and associated tracer transports of the 2000s for both the stratosphere and mesosphere. The periods used for the HAMMONIA and Aura/MLS data do not match exactly because of the differences in data availability. Therefore, we analyzed the ERA-Interim for both periods separately, providing evidence that the effects of interannual variability on the 6 year averages are relatively small, i.e., that the derived residual circulation and transport patterns are representative for the 2000s. The investigations are focused on long-term January means of the Northern Hemisphere, representative of winter conditions, which is the period when stationary wave patterns are most pronounced.
 HAMMONIA is a general circulation and chemistry model that describes dynamics, physics, and chemistry interactively covering altitudes from the troposphere to the thermosphere [Schmidt et al., 2006, 2010]. The model includes the vertically extended dynamics and physics of the general atmospheric circulation model ECHAM5 [Roeckner et al., 2006; Manzini and co-authors, 2006] and the chemistry of the Model of Ozone and Related Tracers (MOZART3) chemical model [Kinnison et al., 2007]. Gravity wave breaking processes are parameterized following Hines (1997a, 1997b), with the uniform background gravity wave source strength specified by Giorgetta et al. . We use simulations with a horizontal resolution of ≈3.75° (triangular truncation at wave number 31) and 119 levels up to 250 km with a vertical resolution in the stratosphere of about 800 m. The HAMMONIA model is widely used to interpret ground-based and satellite-based observations, to estimate the effects of climate change in the middle atmosphere, and to investigate the involved physics, dynamics, and chemistry [Schmidt et al., 2006, 2010; Yuan et al., 2008; Offermann et al., 2009; Lossow et al., 2009, Gabriel et al., 2011b]. In this study, the results of a transient model run forced by AMIP-SST up to the year 2006 are used. A diagnosis is presented for the years 2001 to 2006 based on daily means of temperature, geopotential height, winds, ozone, and water vapor extracted from the model output.
 ERA-Interim is the latest ECMWF global atmospheric reanalysis data set starting from the year 1979. The ERA-Interim ozone analyses are improved compared to ERA-40, providing better agreement with independent observations. In particular, the ERA-Interim model uses an upgraded version of the ozone chemistry parameterization [Cariolle and Teyssendre, 2007], and ozone retrievals from a much larger number of satellite observations are included in the assimilation [Dragani, 2011]. There are seven different sensors from different satellites, which include, TOMS (on various satellites), SBUV (on various satellites), SCIAMACHY and MIPAS on Envisat, GOME on ERS-2, and OMI as well as MLS on EOS-Aura [Poli et al., 2010]. Our diagnosis is based on daily means that we derived from the ERA-Interim data provided on a 1.5 × 1.5 longitude-latitude grid and 37 standard pressure levels up to 1 hPa with time increments of 6 h.
 EOS-Aura/MLS satellite data is available at http://mls.jpl.nasa.gov/index-eos-mls.php (here, we used version 2). The data includes temperature, ozone, and water vapor profiles [Santee et al., 2005; Lambert et al., 2007; Manney et al., 2007; Froidevaux et al., 2008; Schwartz et al., 2008]. The Aura/MLS data provided sufficient resolution in time and space to derive daily mean profiles of stratospheric ozone, stratospheric and mesospheric temperatures, and water vapor. Similar to the procedure of Gabriel et al. [2011a], we sampled the irregularly spaced temperature, ozone, and water vapor profiles of a specific day to obtain a homogeneous data set of daily means on a 10 × 10 grid in longitude and latitude. In order to avoid small-scale perturbations, the data are smoothed by a 9 × 9 grid point filter. As described in Gabriel et al. [2011a], the zonal and meridional geostrophic wind components ug and vg are derived from the temperature field T (i.e., from geopotential height as determined by T via the hydrostatic equation), and the zonally asymmetric component of the daily mean vertical wind wb* is derived using a quasi-geostrophically balanced approach of a steady-state potential temperature equation. Generally, these approximations lead to weaker amplitudes in the winds in comparison to observations. For direct comparison, we therefore apply exactly the same procedure on the HAMMONIA and ERA-Interim data, resulting in quasi-geostrophically approximated data on a horizontal 10 × 10 grid denoted here by HAM(QG) and ERA(QG). Overall, this procedure provides a comparison of the flow fields of HAMMONIA and ERA-Interim with model-independent data, but important insight is also gained in the different roles of quasi-geostrophically balanced planetary waves and transient synoptic-scale as well as gravity wave fluxes configuring the 3-D residual circulation and transport.
3 3-D Residual Mean Circulation
 The theoretical formulae of time-mean (−) 3-D residual wind components developed by Kinoshita et al.  are given as:
where the prime (' ) denotes the deviation from the time-mean, and
is defined as the difference between the kinetic and potential perturbation energies. The subscript “res” indicates the residuals of zonal (u), meridional (v), and vertical (w) wind velocities. Partial derivatives with respect to longitude, latitude, and height (here, denoted as x, y, and z) are given as indices. In the formulations, ρ0 is the basic air density, Φ is the geopotential, and f is the Coriolis parameter. The third terms of the right-hand sides of equations (1) and (2) are the primary terms defining the eddy-induced time-mean transport, analogously to the 2-D TEM equations but based on monthly mean and not on zonal mean eddy heat fluxes. They have also been used in earlier works [e.g., Trenberth, 1986] and describe the effect of planetary and synoptic scale wave fluxes on the net mean mass transport. As described in detail by Kinoshita et al. , the equations include an additional term related to the effects of gravity wave activity, based on a modified formulation of Miyahara , which plays a significant role mainly in the upper mesosphere. The transformation satisfies the continuity equation, and the resulting 3-D residual circulation accords with the sum of the Eulerian-mean flow and the Stokes drift analogously to the 2-D residual circulation approach [Kinoshita et al., 2010].
 For this study, a slight change in calculating monthly mean eddy heat fluxes ( and ) is included compared to Kinoshita et al., 2010]: eddy temperature perturbations are used instead of geopotential perturbations, which is given by the hydrostatic approximation (H is the scale height, R is the gas constant). For all three data sets, the analysis is based on the monthly means () and the daily deviations (' ), i.e., the eddy heat flux terms are calculated based on the deviations of daily means from monthly means . Our investigation is focused on the residual circulation in the northern extratropics; therefore, we include total and quasi-geostrophic eddy flux terms only for latitudes φ > 10°N. Additionally, we use 1/f ≈ f/(f² + ε²) with ε² = 10−9 s−2 to avoid unrealistic values when f becomes small near the tropics.
 Because of the geostrophic approximation of the horizontal winds, we can expect that the eddy heat fluxes derived from Aura/MLS data are weaker than the total eddy fluxes. In particular, they do not include nongeostrophic perturbations due to gravity wave breaking processes, which become important at upper mesospheric altitudes. The effect of perturbations due to tides might also become significant above an altitude of about 80 km, but this is not discussed in this paper because the monthly mean zonal asymmetries in stratospheric ozone and in the middle atmospheric water vapor are only pronounced below altitudes of about 80 km.
4 Diagnosis of 3-D Residual Wind Components
 In this section, the zonal, meridional, and vertical residual wind components are examined for different cross sections of the Northern Hemisphere to illustrate the longitudinal variations in the residual circulation. In particular, the residual flow is considered for longitude-height cross sections at 60°N, latitude-height cross sections at specified longitudes over the Atlantic and Pacific regions, and horizontal cross sections at a specific height level of 30 km. In a first step, we additionally analyze the longitudinal variations in the time-mean eddy heat fluxes because they are the main factor in the transformation from Eulerian to residual circulation.
4.1 Zonal and Meridional Eddy Heat Fluxes
 Figure 1 shows the meridional eddy heat fluxes, , at northern midlatitudes (60°N) for all three data sets, including the quasi-geostrophic eddy fluxes of HAM(QG) 2001 to 2006 and ERA(QG) 2001 to 2006 and 2005 to 2010. In all data sets, exhibits a pronounced zonal asymmetry. In the upper stratosphere at around 40 km, of HAMMONIA (Figure 1a) shows local maxima over the northeast Pacific Ocean (180°E–300°E) but also strong values over the northern Atlantic/Europe (300°E–60°E). The extracted quasi-geostrophic eddy fluxes of HAM(QG) (Figure 1b) are mainly located over northern Atlantic/Europe. The pattern of the eddy fluxes of HAM(QG) agrees quite well with that of Aura/MLS (Figure 1c), whereas the maximum amplitudes are weaker by ≈30%, indicating that the quasi-geostrophic planetary waves are slightly underestimated in comparison to Aura/MLS. On the other hand, the structure of of HAMMONIA (Figure 1a) agrees quite well with that of ERA-Interim (Figure 1d), but the maximum amplitudes at 180°E to 300°E are stronger by ≈30%. This difference could be due to too strong transient wave perturbations in the HAMMONIA model. There might also be some deficiencies in the upper stratospheric winds of ERA-Interim because of the upper boundary of the assimilation model at 0.1 hPa, or because of the lack of observed upper stratospheric wind data in the assimilations. However, more research is needed to analyze these differences. The approximated eddy fluxes of ERA(QG) 2001 to 2006 (Figure 1e) and ERA(QG) 2005 to 2010 (Figure 1f) agree quite well, illustrating that the 6 year time-mean values are comparable and that the effects of interannual variability are of minor importance for the purpose of this paper. The pattern of ERA(QG) 2005 to 2010 (Figure 1f) also agrees quite well with that of Aura/MLS (Figure 1c) but the amplitudes are smaller by about 30%, indicating that quasi-geostrophic approximated planetary waves in the upper stratosphere might be slightly underestimated by ERA-Interim in comparison to Aura/MLS.
 The meridional eddy heat fluxes derived from HAMMONIA (Figure 1a) also show pronounced amplitudes in the mesosphere. The zonal asymmetries of the mesospheric fluxes are obviously coupled to those in the stratosphere due to vertical wave propagation. In the upper mesosphere, we find also a zonally symmetric component that can be referred to as the influence of zonal asymmetries in gravity wave activity. This later becomes more evident when analyzing the transport tendencies of water vapor (section 5). On the other hand, the meridional eddy heat fluxes derived from Aura/MLS (Figure 1c) show some more pronounced longitudinal variations in the lower mesosphere than those of HAM(QG), e.g., at altitudes of 50 to 70 km, the positive and negative values of at 150° to 240°E and 30° to 90°E using Aura/MLS are stronger than those of HAM(QG) by about 30%. The mesospheric eddy fluxes derived from Aura/MLS are weaker than those of HAMMONIA and are restricted to altitudes below 80 km because the quasi-geostrophic approach does not include gravity wave activity. Therefore, a comparison of HAMMONIA, HAM(QG), and Aura/MLS elucidates the role of zonal asymmetries in gravity wave breaking processes in configuring the zonal asymmetries in the 3-D residual circulation and transport in the upper mesosphere.
 A similar asymmetric behavior with local minima in the upper stratosphere and strong fluxes in the mesosphere can be seen in the zonal heat fluxes (Figure 2). The zonal eddy fluxes of HAMMONIA 2001 to 2006 (Figure 2a) are stronger by about 30% than those of ERA-Interim 2001 to 2006 (Figure 2d), and the quasi-geostrophic eddy fluxes of ERA(QG) 2005 to 2010 (Figure 2f) are smaller by about 30% than those of Aura/MLS (Figure 2c). In particular, the picture of derived from HAMMONIA and ERA-Interim (Figures 2a and 2d) reveals that the transient wave activity is much weaker in the area of the stratospheric polar vortex (≈60°E–180°E) than in other regions. Overall, we conclude that HAMMONIA overestimates transient wave activity in comparison to ERA-Interim but underestimates quasi-geostrophically balanced planetary wave activity in comparison to Aura/MLS. Note that this overestimation of transient waves at the cost of stationary waves is usually found in current general circulation models and chemistry-climate models [Boer and Lambert, 2008; SPARC-CCMVal, 2010].
4.2 Longitudinal Variation of Residual Winds
 Figure 3 shows the time-mean residual winds for a longitude-height cross section at 60°N as derived from the different data sets. Zonal and vertical winds are given as streamlines denoting the wind vector components in the plane, and meridional winds are given as contours and shaded areas. Here, we show streamlines of wind vectors but not a mass stream function that cannot be derived for a 3-D flow as easily as for a 2-D flow. Therefore, in some regions, the streamlines can lead to a somewhat perturbed picture; although the mass flow is very weak, particularly in the mesosphere. However, the streamlines document the spatial characteristics of the flow as well as a stream function, and a quantification of the meridional flow component (in this section) and vertical flow component (in section 4.4) are explicitly given.
 For all data sets shown in Figure 3, the picture of the meridional wind indicates a wave one structure with westward shift in phase with height, with maximum northward winds (red) in the upper stratosphere over northern Europe/Asia (0°–120°E), and southward winds (blue) over the northern Pacific/Aleutians (180°–270°E). This flow pattern is related to the shape and position of the stratospheric polar vortex, which is usually not zonally symmetric but with its center usually located over northern Siberia [Waugh and Randel, 1999; Karpetchko, 2005]. The other main feature of the pictures shown in Figure 3 is a pronounced downward and eastward flow over northern Europe/Asia toward the center of the polar vortex in the lower stratosphere. Here, we discuss the 3-D structure of the circulation derived from the different data sets in detail, revealing the effect of residual transformation and the role of balanced and unbalanced flow components.
 Figure 3a shows the time-mean residual winds derived from HAMMONIA representing the time-mean mass circulation. In the stratosphere and lower mesosphere, we find not only northward and downward residual streamlines over northern Europe/Asia (0°–120°E) but also southward and upward residual streamlines over the northern Pacific/Aleutians (180°–270°E). Accordingly, over northern Europe/Asia, the trace gas constituents are transported northward (red area) where the streamlines are mainly downward. This picture of poleward and downward flow is well-known from the zonal mean BDC. On the other side, over the north Pacific/Aleutians, the trace gas constituents are transported southward (blue area) where the streamlines indicate upwelling, which cannot be indicated by the zonal mean concept. Overall, we identify a wave one structure in the residual circulation that is averaged out when looking at the zonal means.
 For comparison, Figure 3b shows the untransformed Eulerian winds derived from HAMMONIA. The differences between the residual and Eulerian flow indicate the time-mean effect of the eddy heat fluxes introduced in equations (1–3), i.e., they result from a modulation and, partly, a cancellation of the 3-D Eulerian flow by the 3-D eddy time-mean flow analogously to the zonal mean flow components [Andrews et al., 1987]. For example, in the lower stratosphere over the north Pacific/Aleutians (≈120°E–270°E), the upwelling in the Eulerian flow is much more pronounced than the residual flow, but in the upper stratosphere, the residual flow is more pronounced than the Eulerian flow (highlighted by the thick arrows). Here, the streamlines only illustrate the change in the structure of the flow due to the eddy time-mean flow. In section 4.4, we show that the amplitude of the residual upwelling is weaker than the Eulerian upwelling by about 50%.
 In the stratosphere and lower mesosphere, the main wave one structures of the quasi-geostrophically approximated residual flows derived from HAM(QG) (Figure 3c) and Aura/MLS (Figure 3d) agree quite well, although the wind amplitudes of HAM(QG) are slightly smaller (≈30%) than those of Aura/MLS, indicating the underestimation of planetary wave activity in the model in comparison to the observations. On the one side, the structures of the streamlines of both HAM(QG) and Aura/MLS show more similarities with those of the Eulerian winds (Figure 3b) than with the residual winds (Figure 3a) of HAMMONIA. This leads to the conclusion that the effect of the eddy time-mean flow might be slightly underestimated in the pictures using the geostrophic approximations in comparison to the unapproximated pictures of HAMMONIA and ERA-Interim. On the other side, the residual streamlines of HAMMONIA (Figure 3a) indicate more perturbed structures in the flow compared to those of HAM(QG) and Aura/MLS because of synoptic-scale transient wave activity, but also a stronger cancellation of the upwelling over northern Europe/Asia (0°–120°E) due to the eddy time-mean flow compared to the residual streamlines of ERA-Interim (Figure 3e), whereas the quasi-geostrophic flow pattern of ERA(QG) (Figure 3f) agrees quite well with that of Aura/MLS. Conclusively, the effect of transient wave activity might be too strong in HAMMONIA.
 In the upper stratosphere, the streamlines of ERA-Interim show downward flow at nearly all longitudes, unlike ERA(QG), which is similar to Aura/MLS. This could be related to unrealistic effects of the upper boundary or to the general problem of reliable wind estimates from data assimilations in the upper stratosphere and mesosphere [Polavarapu et al., 2005]. More research is needed to clarify these differences. In the upper mesosphere (at ≈80 km), we find a pronounced downward flow in the residual wind streamlines in the HAMMONIA data that can be attributed to gravity wave breaking, whereas in the Aura/MLS data, the streamlines remain mainly horizontal because of the quasi-geostrophic approach, which do not include gravity waves. The different effects of the zonal asymmetries in the quasi-geostrophically balanced planetary wave activity and of the gravity wave breaking on the 3-D residual circulation in the mesosphere will become more evident in the next section when discussing the residual advection of H2O in the mesosphere. Overall, Figure 3 gives an insight into the structure of the 3-D residual circulation and the counteracting effects of the Eulerian and eddy time-mean flow and of the quasi-geostrophically balanced and the synoptic-scale transient wave activity.
4.3 Cross-Polar Transport
 In this section, the meridional structure of the time-mean residual winds derived from HAMMONIA is examined in relation to the stationary planetary waves in geopotential height. Figures 4 and 5 show that stationary zonal wave one structures are apparent in cross-polar latitude-height cross sections of both the residual wind streamlines and the zonal anomaly of geopotential height (Φ*) at the longitudes 180°E and 0° (Figure 4) and at the longitudes 90°W and 90°E (Figure 5). In particular, Figure 4 depicts the latitude-height cross section from the tropical Pacific Ocean via the North Pole to Atlantic Ocean/North Africa. In the tropical stratosphere at both 180°E over Pacific Ocean and at 0° longitude over Atlantic Ocean/North Africa, the air moves from the equator toward the polar latitudes, which is in accordance with the zonal mean residual circulation theory. On the other hand, starting from the lower midlatitudes, the movement of the air differs from the zonal mean in relation to the geopotential height anomalies. On the left-hand side of Figure 4, at 180°E, highly positive values of Φ* (red area) are found between 60° and 80°N at lower to higher stratosphere altitudes, indicating the Aleutian high anomaly. On the right-hand side of Figure 4, at 0°, low values of Φ* (blue area) are found between 60° and 80°N at lower to higher stratospheric altitudes, indicating the polar low anomaly. These high and low values of Φ* represent the stationary zonal wave one structure forced by planetary waves. In Figure 4, the center of both high and low geopotential height anomalies are located around 30 to 40 km.
 In relation to this pattern, at the North Atlantic side at 60° to 75°N, the streamlines proceed down toward the center of the polar low. On the other hand, at the Pacific side, the streamlines describe an anticlockwise circulation cell around the Aleutian high with upward movement of air masses in the polar lower stratosphere, southward movement in the midlatitude higher stratosphere, and then down toward the midlatitude and subtropical lower stratosphere. As discussed for Figure 3, this strong zonal asymmetry in the 3-D residual circulation is averaged out when looking at the 2-D zonal mean residual circulation. Note, again, that we show the streamlines of the wind vectors and not a residual mass stream function; therefore, the residual wind streamlines lead to a somewhat perturbed picture, particularly over the tropics and subtropics (as indicated in Figures 4 and 5), where the flow is perturbed by the effects of tropical cumulus convection and inertial instabilities.
 Figure 5 illustrates the cross-polar latitude-height cross section at the longitudes 90°W and 90°E. In comparison to Figure 4, the wave pattern is altered because of the westward shift in phase with height of the time-mean stationary wave one structure in temperature, geopotential height, and winds usually observed in the middle atmosphere (compare with the meridional residual wind component shown in Figure 3a). The stationary wave one structure in geopotential height anomaly Φ* shows that the center of the high-pressure and low-pressure systems is shifted upward and westward to altitudes of around 50 km. The streamlines indicate a cross-polar transport of air masses in the stratosphere and lower mesosphere compared to Figure 4. Particularly in the middle and upper stratosphere, the air is transported from the midlatitudes over Asia through regions over the North Pole toward midlatitudes over North America. Additionally, converging flow below the upper stratospheric low anomaly at 90°W and diverging flow below the upper stratospheric high anomaly at 90°E yields a wave one pattern. The described cross-polar flow pattern in the upper stratosphere is similar to that found by Callaghan and Salby  in a 3-D primitive equation model on isentropic coordinates with prescribed tropospheric wave forcing.
4.4 Vertical Time-Mean Mass Flux
 The time-mean transport of atmospheric constituents is closely related to the time-mean mass fluxes as indicated by the residual circulation. Hence, the mass flux components given in Figure 6 represent the direction of the movement of trace gases, where the vertical mass flux provides a quantitative measure of the local time-mean residual mass circulation. Here, the mass flux vector is defined by m⋅(ures, vres, wres), where m is the mass in the volume dV of a grid box (m = ρ dV = ρ r2 cosφ dφ dλ dz). Figure 6 shows the northern polar stereographic view of the 3-D residual mass fluxes derived from the three different data sets together with the Eulerian mass flux of HAMMONIA (Figure 6b) and the quasi-geostrophic residual mass fluxes of HAM(QG) and ERA(QG) (Figures 6c and 6f). In Figure 6, the streamlines display the horizontal mass flux component related to the residual zonal and meridional winds (ures, vres) and shades of blue and red display the downward and upward mass fluxes, respectively.
 In all three data sets, the most prominent downward movement takes place not directly over the pole but almost over the center of the polar low around 70°N (Figure 6, blue). On the other hand, Figure 6 (yellow and red) indicates an upward movement of air near the geographic North Pole extending from regions over the Aleutians to regions over Labrador/Greenland. This zonal asymmetry in the vertical mass fluxes is part of the wave one structure in the time-mean 3-D residual circulation in which the local mass fluxes are much stronger than the zonal mean vertical mass fluxes. Hence, the 3-D residual circulation is an important factor in configuring the local distribution of trace gas constituents but is not considered in a zonal mean residual approach. In particular, the mass flux might be reduced to a small upwelling over the North Pole when zonally averaged, which is usually not taken into account when analyzing the 2-D BDC. Analogously to Figure 3, a comparison of the different data sets can help to understand this structure in more detail.
 In particular, the belt of downward-moving air at around 60°N and upward-moving air near the pole as derived from HAMMONIA (Figure 6a) is much less intense than in the Eulerian mass flux (Figure 6b) because the eddy-induced time-mean flow is partly counteracting the Eulerian flow. As suggested in Figures 6a and 6b, at 30 km, the amplitudes of the Eulerian flow are cancelled by the eddy time-mean flow in the order of 50%. In Figure 6a, the streamlines indicate the cross-polar mass transport from Asia toward North America (highlighted by thick white arrows), which is related to the cross-polar flow pattern in the longitude-height cross section shown in Figure 5. The Aleutian high and polar low anomalies constitute the ridge and trough at 30 km, forming a pronounced zonal wave 1 pattern in the streamlines with anticlockwise rotation in the horizontal flow in the region of the Aleutian high anomaly and clockwise rotation in the horizontal flow in the region of the polar low anomaly.
 The quasi-geostrophically approximated vertical mass flux of HAM(QG) (Figure 6c) shows a very smoothed wave one structure with amplitudes much weaker (more than a factor of 2) than those of Aura/MLS (Figure 6d), which indicates that a large part of the quasi-geostrophic vertical mass flux in HAMMONIA is strongly damped by nonbalanced transient waves. On the other side, the vertical mass flux of ERA(QG) (Figure 6f) is stronger than that of Aura/MLS (Figure 6d) by about 40% to 50%. Furthermore, the large-scale wave one pattern of the vertical mass flux derived from ERA-Interim (Figure 6e) is very perturbed due to nonbalanced small-scale perturbations. In case we assume that ERA-Interim describes the large-scale wave one pattern depicted by ERA(QG) realistically, we have to conclude that the amplitude of the residual flow of Aura/MLS might be slightly underestimated in comparison to that of ERA(QG). On the other hand, in meteorological analysis data and assimilated data sets, the determination of vertical wind components is very problematic [Polavarapu et al., 2005; Wohltmann and Rex, 2008]. Gabriel et al. [2011b] assumed that the limitations due to the upper boundary of the ERA assimilation model might have an influence on the vertical structure of the stationary wave patterns in the upper stratosphere. This could also lead to the differences between ERA(QG) and Aura/MLS at 30 km shown in Figure 6, which become more evident when discussing the transport tendencies in section 5.2. However, further research is needed to clarify these discrepancies. Generally, the zonal asymmetry in the 3-D residual circulation is well captured by the HAMMONIA model, although large-scale planetary waves are obviously too weak whereas synoptic-scale transient wave perturbations are too strong in comparison to observations.
 As for the cross-polar flow component in the upper stratosphere, the zonal asymmetry of the vertical flow component shows a similar structure as derived by Callaghan and Salby  from a simplified model on isentropic coordinates. In combination with Figures 3, 4, and 5, a new aspect of the 3-D residual approach is the picture of a circulating 3-D conveyor belt including northward and southward transport of air masses in relation to the wave one pattern in the middle atmosphere, where both quasi-geostrophically balanced planetary waves and transient waves are important in configuring this circulation structure. Overall, we conclude that the identified zonal asymmetry in the 3-D residual circulation is a robust feature and is quite well captured by the HAMMONIA model.
5 Effect of the 3-D Residual Circulation on the Distribution of O3 and H2O
 In this section, we examine the effect of 3-D residual circulation on the regional distributions of stratospheric ozone (O3) and middle atmospheric water vapor (H2O). The residual winds derived through the formulae of Kinoshita et al.  are used to examine and to understand the longitudinal differences in O3 and H2O in relation to the residual circulation and wave-driven transport for the Northern Hemisphere during winter. For orientation, first we present the zonal asymmetries in O3 and H2O as derived from the three different data sets. Second, we analyze the advection of O3 and H2O by the residual wind components as given by the tracer transport equation.
5.1 Zonal Asymmetries in O3 and H2O
 The zonally asymmetric components O3* and H2O*, which are defined as the deviations from zonal mean (O3* = O3 - [O3]; H2O* = H2O - [H2O]), are examined for a longitude-height cross section at 60°N (Figures 7 and 8). For direct comparison with Aura/MLS, Figures 7 and 8 include the distributions when applying exactly the same procedure for deriving O3* and H2O* on the horizontal 10 × 10 low-grid (LG) resolution as used for the Aura/MLS data, here denoted by HAM(LG) and ERA(LG). In agreement with previous works [e.g., Gabriel et al., 2011a, 2011b], the spatial structures of O3* and H2O* are related to the stationary wave one pattern usually observed during northern winter, i.e., we find negative values in O3* over the North Atlantic/northern Europe and positive values over the northern Pacific/Aleutians (Figure 7), and a wave one pattern in H2O* with pronounced amplitude in the middle stratosphere and mesosphere including a jump in phase at upper stratospheric altitudes (Figure 8). However, the amplitudes of these wave patterns and the specific locations of maximum and minimum values differ considerably in the data sets.
 In the lower and middle stratosphere (20–30 km), the minimum of O3* of both HAMMONIA (Figure 7a) and HAM(LG) (Figure 7b) are weaker than in the Aura/MLS (Figure 7c) by more than 50%, whereas the amplitude of O3* derived from ERA Interim (Figure 7d) is twice as strong than in HAMMONIA and Aura/MLS. Because zonal mean ozone is captured quite well by the HAMMONIA simulations in comparison to observations [e.g., Schmidt et al., 2010], we can expect that the differences in the zonal asymmetries of the net mass transport might be primarily responsible for these differences. Similar to HAMMONIA and HAM(LG), the amplitudes of O3* derived from ERA(LG) 2001 to 2006 (Figure 7e) and ERA(LG) 2005 to 2010 (Figure 7f) are weaker by about 25% in comparison to ERA-Interim, quantifying the effect of the change to the 10 × 10 grid resolution, which is useful when considering the advection terms in the next sections. We also found that ERA(LG) 2001 to 2006 and ERA(LG) 2005 to 2010 are very similar in amplitude and structure, underlining that interannual variations due to the different periods are of minor importance for the comparison of the 6 year values of the different data sets.
 In the upper stratosphere, the minimum in O3* at around 0° and 40 km is stronger in the HAMMONIA simulations than in Aura/MLS and ERA-Interim by about 50%. As for the lower stratospheric O3*, the Aura/MLS data might be somewhat too small because of the limitations in spatial resolution. In the ERA-Interim data, the amplitude of the upper stratospheric O3* might be somewhat too small because of the effects at the upper boundary of the assimilation model (as discussed in section 4 for Figures 1, 3, and 6), or because of temperature-dependent photochemistry, which plays a significant role in producing stationary wave patterns at these altitudes but is not included in the ERA-Interim assimilation model.
 As shown in Figure 8, the stationary wave pattern in H2O* shows a similar structure in all data sets, but there are significant differences in the amplitude. In the lower stratosphere, the amplitude of H2O* is weaker in HAMMONIA (Figure 8a) than in Aura/MLS (Figure 8c) by a factor of about 3, and weaker than in ERA-Interim (Figure 8d) by a factor of about 4. Similar to the case of O3*, one reason might be a too strong transient wave activity in HAMMONIA as mentioned previously (section 4); hence, an underestimation of the transport by stationary flow components of the 3-D BDC in comparison to observations. In the stratosphere, parts of these differences might also be due to a dry bias in stratospheric H2O when small–space scale and short–time scale temperature fluctuations are not taken into account, as pointed out by Liu et al. , which is the case for the HAMMONIA simulations using a spectral T31 resolution.
 Comparison of observed lower stratospheric H2O* based on Aura/MLS (Figure 8c) and ERA-Interim (Figure 8d) data indicates agreement, but the amplitude of H2O* is slightly weaker (by about 25%) in Aura/MLS because of the lower grid resolution used for deriving the wave patterns. This is confirmed by H2O* of ERA(LG) 2001 to 2006 and 2005 to 2010 (Figures 8e and 8f), in which the amplitudes of H2O* are also weaker by about 25% in comparison to ERA-Interim and in agreement with Aura/MLS. In the lower mesosphere, the amplitude of H2O* is somewhat weaker in HAMMONIA than in Aura/MLS (by about 30%). Because both stratospheric and mesospheric H2O have long photochemical lifetimes, these differences might be primarily due to an underestimation of the zonal asymmetries in the wave-driven 3-D BDC and associated net mass transport in the models.
 In the next section, we demonstrate that the diagnosis of the 3-D residual circulation is a useful tool to understand these wave patterns as well as their differences. In addition, by analyzing the advection of the trace gas constituents by the residual winds, it will become more evident where the models have to be improved to capture the observed structures in the zonally asymmetric components O3* and H2O*.
5.2 Zonally Asymmetric Transport of O3 and H2O by the Residual Winds
 Here, we analyze the zonally asymmetric distribution of zonal, meridional, and vertical advection of O3 and H2O by the residual winds to examine their contributions to the observed zonal asymmetries O3* and H2O* shown in Figures 7 and 8. Again, we focus our analysis on higher midlatitudes, 60°N, in northern hemisphere winter.
 Following Kinoshita et al. , the time-mean transport equation including the residual wind components is given by:
where μ is the tracer, is the remaining eddy flux divergence due to wave dissipation processes after introducing the transformation from Eulerian to residual winds, S represents the chemical sources, and the subscripts t, x, y, and z denote the derivatives with time, in eastward and northward directions, and with height. For stationary steady-state conditions (∂μ/∂t = 0) the advection terms are balanced by the source terms and the eddy flux divergence term. On the one hand, it is well known that, in the residual framework, the eddy flux divergence term is counteracting the advection term but is smaller in amplitude [e.g., Holton, 1986; Randel et al., 1998; Gabriel and Schmitz, 2003; Kinoshita et al., 2010]. On the other hand, because of the long photochemical lifetimes of O3 and H2O, zonal asymmetries in temperature-dependent chemistry do not play a significant role in producing the zonal asymmetries in O3 and H2O, except for upper stratospheric ozone [e.g., Gabriel et al., 2011a]. Therefore, in this paper, we focus on the analysis of the advection terms as the primary process configuring the zonal asymmetries of a long-lived tracer.
5.2.1 Examination of Ozone Advection
 Figures 9, 10, and 11 show the zonal asymmetries in the zonal, meridional, and vertical advection of O3, respectively. For all data sets, the time-mean patterns of the zonal ozone advection terms (Figures 9a–9c) and the meridional ozone advection terms (Figures 10a–10d) are largely balanced, indicating the balanced rotational flow component in connection with the zonal asymmetry of the northern winter polar vortex. In particular, for the meridional ozone advection term (Figure 10), we find positive tendencies (dark red shaded region) over northern Europe/west Siberia in the middle and upper stratosphere, where a northward flow is observed as shown in Figure 3 (note that ∂O3/∂y < 0 at northern midlatitudes in the middle and upper stratosphere). Over northern North America, negative tendencies (dark blue shaded area) occur following the southward flow shown in Figure 3.
 Comparing the tendencies of the meridional ozone advection (Figure 10) and the stationary wave patterns in O3* (Figure 7), a zonal shift in phase between the maxima in the tendencies and the mean values can be identified. This phase shift is related to the eastward advection of ozone within the zonal westerlies (Figure 9) balancing to a first-order meridional advection (Figure 10), as evident from equation (5), when neglecting the chemical source term and the eddy diffusion. For illustration, assuming a simplified first-order approximation of equation (5) for the zonally asymmetric component with horizontal advection only by geostrophic winds, ∂μg*/∂t + [ug]∂μ*/∂x + vg*∂[μ]/∂y ≈ 0 (where μ* = μ−[μ], [ ]: zonal mean; ug and vg: geostrophic winds; note that [vg] = 0 per definition), a zonal wave one perturbation in the meridional wind vg* = v0⋅cos(x) (where v0 is the amplitude), leads to a steady-state solution μ* = μ0⋅sin(x) via the balance ∂μ*/∂x ≈ −[ug]−1⋅vg*∂[μ]/∂y, i.e., to a zonal shift in phase of 90° in μ* compared to v*.
 For a more detailed comparison and validation purposes, Figure 10 also shows the quasi-geostrophically balanced advection terms of HAM(QG) (Figure 10b) and ERA(QG) 2001 to 2006 and 2005 to 2010 (Figures 10e and 10f). We found that the amplitude of the meridional ozone advection is stronger (about 20%) in HAMMONIA (Figure 10a) but weaker (about 20%) in HAM(QG) (Figure 10b) in comparison to Aura/MLS (Figure 10c). Here, we might conclude that, if considering the reducing effect due to the transformation to the 10 × 10 grid, the meridional ozone advection of HAM(QG) is weaker than HAMMONIA (~25%) due to the quasi-geostrophic approximation. On the other hand, the effect of zonal asymmetries in the synoptic-scale transient waves is strong enough to overcompensate for the deficiencies in the quasi-geostrophic planetary waves, which at least results in a stronger meridional ozone advection in HAMMONIA in comparison to Aura/MLS. The meridional ozone advection of ERA-Interim (Figure 10d) is also reduced when using the quasi-geostrophic approach of ERA(QG) (Figures 10e and 10f), but the terms of ERA(QG) 2005 to 2010 are in agreement with those of Aura/MLS. Here, we have to consider that the amplitude of O3* (Figure 7) is stronger in ERA-Interim than in Aura/MLS, which compensates for the effects of the quasi-geostrophic approximation so that Figures 10c and 10f become quite similar.
 Figures 9 and 10 primarily illustrate the effects of the balanced rotational component of the residual flow. However, there are strong deviations from this balance owing to the divergent flow and the eddy-induced time-mean flow components, which is illustrated in Figure 11.
 For all three data sets, the vertical ozone advection terms shown in Figure 11 indicate negative tendencies in the upper stratosphere over northern Europe/western Siberia, and positive tendencies in the lower stratosphere over eastern Siberia (note here that, at northern midlatitudes, ∂O3/∂z changes its sign at altitudes of around 35 km). This structure mirrors the effect of the downward branch of the BDC, which is located approximately over the center of the polar low anomaly (compare with Figure 5), i.e., ozone-enriched air masses are transported poleward from the tropics to the midlatitudes (positive tendencies in the meridional advection term) but also downward from upper stratospheric altitudes (negative tendencies in the vertical advection term) to lower stratospheric altitudes (positive vertical advection terms). This picture corresponds with the time-mean meridional transport of O3 in a zonal mean framework, but it shows a pronounced westward shift in phase in the vertical. Analogously, but with an opposite sign, an imprint of the upward branch of the 3-D BDC over the northern north Pacific/Aleutians is indicated, which is hidden when looking at zonal mean ozone transports. For all three data sets, the vertical ozone advection has a strong connection to the wave one pattern in the lower and middle stratosphere, with prominent positive O3 tendencies in the center of the polar low (at around 60°E–120°E). Therefore, the spatial structure of downward ozone advection due to the zonal asymmetries in the BDC can be identified as a robust feature. Here, we only note that the advection terms of the Eulerian and the eddy-induced time-mean flow are partly counteracting as described for the wind components in sections 'Zonal and Meridional Eddy Heat Fluxes' and 'Cross-Polar Transport'.
 The satellite observations, model simulations, and reanalysis data shown in Figure 11 are mainly consistent with each other, but there are also pronounced differences. For example, the amplitude of the vertical ozone advection at ≈20 km is twice as strong in HAMMONIA (Figure 11a) than in HAM(QG) (Figure 11b), demonstrating the important role of the nonbalanced vertical and eddy time-mean flow components contributing to the stationary wave patterns. On the other hand, in the upper stratosphere, the amplitude of the vertical ozone advection is weaker in HAM(QG) than in Aura/MLS (Figure 11c) by about a factor of 2, except a perturbation at around 120°W and 50 km occurring in the upper stratosphere, which is compensated by transient wave activity and therefore not occurring in Figure 11a.
 The picture of ERA-Interim (Figure 11d) shows more perturbed vertical ozone advection terms in the upper stratosphere than the other data sets, corresponding to the perturbed picture of the vertical mass flux shown in Figure 6. The pictures of the vertical ozone advection derived from ERA(QG) 2001 to 2006 and ERA(QG) 2005 to 2010 are very similar, underlining the comparability of the data sets, although the periods do not match exactly. The amplitudes are smaller than in Aura/MLS (Figure 11c) by about 50%, particularly in the middle stratosphere, although the amplitudes of both vertical residual wind (Figure 6) and O3* (Figure 7) are slightly stronger in ERA(QG) than in Aura/MLS. Here, we have to consider that the assimilation model does not include a feedback of the zonally asymmetric component O3* to the transport circulation via radiative heating, which can lead to a significant shift in the phase of O3* (Gabriel et al., 2011b). Therefore, the phase relation between the vertical residual wind and O3* leads to stronger vertical ozone advection terms in the data sets of HAM(QG) and Aura/MLS, where the ozone-dynamic feedbacks are operating more consistently. In the upper stratosphere, ERA-Interim might also underestimate the vertical ozone advection terms in comparison to Aura/MLS because the feedback of temperature-dependent photochemistry on the vertical distribution of O3* is not included in the ERA assimilations. Overall, we conclude that the transport due to the rotational components of the flow is better captured by the models and assimilations compared with the divergent component of the flow.
 In the lower stratosphere, both the horizontal and vertical ozone advection terms of HAMMONIA are stronger than those of Aura/MLS and ERA-Interim, although the amplitude of O3* derived from HAMMONIA is particularly weak (compare with Figure 7). Here, we have to consider that the lower stratospheric eddy heat fluxes of HAMMONIA shown in Figures 1 and 2 are much stronger than those of the other two data sets, i.e., the effect of ozone advection by the stationary flow components is reduced by eddy diffusion due to transient baroclinic wave activity influencing the lower stratosphere from the lower levels. As a subsequent consequence of the overestimation of transient waves at the cost of stationary waves in the model, the vertical propagation of planetary waves from the lower stratosphere to the upper stratosphere and mesosphere is reduced, which leads to the underestimation of the zonal asymmetries in the vertical advection terms and in the feedback of temperature-dependent chemistry in the upper stratosphere. Thus, an improvement of the stationary components in tropospheric wave activity in the model might lead to an improvement of the wave one patterns in upper stratospheric O3*.
5.2.2 Examination of Water Vapor Advection
 Figures 12 and 13 show, respectively, the meridional and vertical advection of H2O by the residual winds up to mesosphere altitudes. In meridional H2O advection (Figure 12), we find negative tendencies in the lower and middle stratosphere over eastern Siberia and positive tendencies in the lower mesosphere over northern Europe/west Siberia, in which northward flow is found (see Figure 3). On the other hand, in relation to the stratospheric/mesospheric wave one pattern, the meridional advection term shows positive tendencies in the lower and middle stratosphere over western Canada/Labrador but negative tendencies in the lower mesosphere over the Aleutians/eastern Canada. For understanding this pattern, we have to consider that the time-mean concentrations of H2O increase with height in the lower and middle stratosphere because of the oxidation of methane (CH4), which is transported from the troposphere into the stratosphere, but that it decreases slowly in the mesosphere because of photochemistry [Brasseur and Solomon, 2005]. Therefore, at 60°N, the time-mean values of H2O reach a maximum at upper stratospheric altitudes and, subsequently, both the midlatitudinal meridional and vertical gradients of H2O change their sign at these altitudes. Thus, the jump in phase of the wave one pattern at upper stratospheric altitudes is primarily related to the change in sign of the related tendency terms −vres*(H2O)y and −wres*(H2O)z of the transport equation [equation (5)]. Here, we only note that, as in case of the O3 advection, large fractions of the meridional H2O advection are balanced by the zonal H2O advection. Similar to the horizontal ozone advection terms, the horizontal H2O advection terms mirrors the effect of the rotational flow component, but up to an altitude of about 80 km in relation to the time-mean zonal wave one pattern in the northern winter stratosphere and mesosphere.
 Overall, the spatial patterns of the wave one structures shown in Figures 12a and 12d resemble each other, but there are also considerable differences between the data sets, which can help to understand the differences of the zonally asymmetric components H2O* shown in Figure 8. For example, in the lower mesosphere, the amplitude of the meridional H2O advection is smaller in HAMMONIA (Figure 12a) than in Aura/MLS (Figure 12b) by about 30%. A part of these differences can be explained by the underestimation of the large-scale planetary wave one pattern and the associated meridional wind in the upper stratosphere and mesosphere in the model. This becomes more evident when comparing the meridional advection terms of HAMMONIA(QG) and Aura/MLS, which shows differences in the amplitude of a factor of about 4 in the area of northward flow (0°–120°E). Conclusively, in the model, a large fraction of lower mesospheric H2O* is produced by zonal asymmetries in synoptic-scale transient wave activity, whereas the effect of large-scale planetary waves is strongly underestimated in comparison to Aura/MLS.
 In the lower stratosphere, the minimum in H2O* at around 120°E is smaller in HAMMONIA (Figure 12a) than in both Aura/MLS (Figure 12b) and ERA-Interim (Figure 12d) by a factor of about 2, which might be due to too strong transient baroclinic wave perturbations in the model propagating from troposphere into lower stratosphere because the effect of eddy mixing is particularly strong in the upper troposphere/lower stratosphere region. Here, we only note that the H2O advection terms of ERA(QG) 2001 to 2006 and 2005 to 2010 are very similar and that they are reduced in comparison to ERA-Interim by about 25% as in case of the ozone advection terms. Given the similarity between the H2O advection of Aura/MLS and ERA-Interim shown in Figures 12b and 12d, the quasi-geostrophic advection in ERA-Interim is underestimated by about 25%. Because both H2O* (Figure 8) and the meridional wind vres (Figure 3) of ERA(QG) and Aura/MLS agree quite well, a shift in phase of the related wave one patterns in H2O* and vres might be responsible for this difference, e.g., a feedback of zonal asymmetries in the production of H2O by temperature-dependent oxidation of methane to the development of the wave one pattern.
 The vertical H2O advection of both HAMMONIA and Aura/MLS (Figures 13a and 13b) indicates the effect of the divergent flow component up to mesospheric altitudes. We find tendencies in the order of 30% of the horizontal H2O advection terms, and, in relation to the wave one pattern, positive tendencies in the lower stratosphere over northern Europe/west Siberia and negative tendencies in the upper mesosphere between northern North America/northern North Atlantic. This structure again mirrors the poleward and downward transport of H2O by the downward branch of the BDC (compare with Figure 3) analogously to the zonal mean framework, but with a westward shift in phase with height from the lower stratosphere to the upper mesosphere. In Aura/MLS (Figure 13b), we also find an imprint of the upward branch of the 3-D residual circulation producing negative tendencies in the lower stratosphere over northern north Pacific/northern North America and positive tendencies in the mesosphere over eastern Europe/Siberia, in relation to the westward shift in phase with height of the wave one structure and to the change in sign of the vertical gradient in time-mean H2O mixing ratio at upper stratosphere altitudes. On the other hand, in the HAMMONIA data (Figure 13a), we find negative vertical H2O advection terms in the upper mesosphere at nearly all longitudes in relation to the pronounced zonally symmetric downward flow component in the upper mesosphere (Figure 3a). This effect results from the parameterized gravity wave drag in these altitudes.
 In order to reveal the zonal asymmetry in the vertical H2O advection of HAMMONIA more clearly, Figure 13c depicts the deviation from the zonal mean vertical H2O advection. This zonally asymmetric component shows negative tendencies in the mesosphere between 90°W and 0° and positive tendencies between eastern Asia (120°E) and northwestern North America (120°W). Overall, the spatial pattern of the zonally asymmetric vertical advection of HAMMONIA shows reasonable agreement with the vertical advection derived from Aura/MLS, but the amplitude is stronger by about 25% and the phase of the wave one pattern is shifted eastward. These differences can be attributed to zonal asymmetries in gravity wave breaking processes as identified by Smith , which are dependent on the modulation of the parameterized vertical gravity wave propagation due to the planetary wave patterns in the stratosphere and lower mesosphere. Similar to meridional H2O advection, the vertical H2O advection is much smaller in HAM(QG) (Figure 13d) than in Aura/MLS (smaller by a factor of about 3), i.e., a large fraction of mesospheric H2O* is produced by zonal asymmetries in nonbalanced transient wave perturbations. Note here that in the upper mesosphere, the vertical H2O advection is more pronounced than in the meridional H2O advection because of the spatial structure of the horizontal and vertical gradients of the eddy flux terms introduced in the residual transformation described by equations (1–4). Above 85 to 90 km, the time-mean zonal asymmetries disappear because of a strong increase in wave turbulence at these altitudes [Offermann et al., 2009]. Here, we conclude that the model captures a large fraction of the observed wave one pattern in the mesosphere primarily due to the zonal asymmetries in the downward branch of the 3-D residual circulation driven by gravity wave breaking processes.
6 Discussion and Outlook
 The zonal asymmetries of the BDC are investigated based on the concept of the 3-D residual circulation of Kinoshita et al. , which provides a new perspective in addition to the zonal mean BDC usually investigated. The 3-D residual circulation includes the Eulerian and the counteracting eddy-induced time-mean flow leading to the picture of the 3-D residual flow as a reliable proxy for the 3-D BDC. The effects of the 3-D residual circulation on the local distributions of stratospheric ozone (O3) and middle atmospheric water vapor (H2O) via the related tracer transport processes are examined based on three different data sets for the 2000s: model simulations of the general circulation and chemistry model HAMMONIA, satellite data provided by the Aura/MLS observing platform of the NASA, and assimilated data of ECMWF reanalysis (ERA-Interim). In the northern winter hemisphere, we find that the spatial structure of both the 3-D residual circulation and the advection of O3 and H2O by residual winds are related to the zonal wave one pattern observed in the stratosphere and mesosphere. The differences between the data sets, including either total time-mean winds and eddy fluxes or quasi-geostrophically balanced winds and eddy fluxes on a low-resolution 10 × 10 grid, elucidate the different roles of planetary and transient synoptic-scale wave activity in producing the zonal asymmetries in O3 and H2O. Overall, it is demonstrated that the 3-D residual mean circulation can be understood as a circulating 3-D conveyor belt in relation to the wave one pattern, providing an extended picture beyond the 2-D residual approach, and that it is a beneficial tool to analyze the zonal asymmetries in the atmospheric transport, contributing to the understanding of the regional distribution of middle atmospheric trace gas constituents.
 Similar to the zonal mean framework, we find upward and northward transport of air masses from the equator toward midlatitudes in the tropical and subtropical stratosphere and mesosphere, and downward and northward flow in regions over Europe/Asia from the upper levels toward the lower stratosphere and upper troposphere at higher midlatitudes. On the other side, over the Pacific Ocean, a circulation cell indicates that air masses are driven southward and upward from lower levels at polar and high midlatitudes into the upper stratosphere, and then southward and down toward the lower midlatitudes and subtropics. This circulation cell counteracts the usual picture of the 2-D BDC because it is averaged out when deriving a zonal mean circulation including the circulation cell structure over the North Atlantic/Europe, which is similar to but much stronger than the zonal mean circulation. In summary, we identify a stationary wave one pattern in the structure of the 3-D residual circulation, which is an essential factor producing the stationary wave patterns in O3 and H2O.
 Cross-polar latitude-height cross sections for specific longitudes reveal the downward branch of the BDC toward the center of the northern winter polar low anomaly over Siberia. Consistently, the residual mass transport at 30 km indicates downward motion over northern Asia but also upward motion over northern North America. These branches of the circulation are closed by a cross-polar movement of middle stratospheric air masses from eastern Asia via the North Pole toward North America, as reported by Callaghan and Salby  in a 3-D primitive equation model on isentropic coordinates with prescribed tropospheric wave forcing. The spatial structure in the 3-D residual circulation is clearly related to the zonal wave one pattern in the northern winter hemisphere, as indicated by the deviation of geopotential height from zonal mean, i.e., by the Aleutian high and polar low anomalies forming a ridge and trough in the middle stratosphere. In this context, we can attribute a large part of the horizontal components of the described circulation to the nearly geostrophically balanced rotational flow component of the Eulerian flow. However, a comparison of the residual and the Eulerian flow demonstrates the important role of the divergent part of the flow and of the eddy-induced time-mean flow in configuring the 3-D net mass transport. In particular, the latter one compensates the large amplitudes of the meridional and vertical Eulerian wind components in the middle stratosphere by about 50%, where the strength of this effect largely depends on the spatial distribution of transient wave activity.
 Based on a detailed investigation of the zonally asymmetric structures at 60°N during January, we conclude that ozone and water vapor tendencies due to horizontal and vertical advection by the 3-D residual circulation produce the observed stationary wave one patterns in these trace gases. In particular, we demonstrate that these wave one structures in O3 and H2O are the result of both zonal asymmetries in the Eulerian time-mean winds and in the eddy-induced time-mean flow, driving the net mass transport. Here, again the tendencies due to advection by the eddy-induced time-mean flow compensates large fractions of the tendencies due to the Eulerian time-mean flow.
 The analyzed distributions of the time-mean zonal asymmetries in ozone and water vapor, denoted as O3* and H2O*, highlight substantial differences between the model simulations, satellite observations, and reanalysis data. The amplitudes of the stationary waves in stratospheric O3* and mesospheric H2O* are underestimated in the HAMMONIA data in comparison to observations. Our examinations show that these differences are related to differences in the 3-D residual circulation and associated tracer transport by the residual winds. In particular, the effect of quasi-geostrophically balanced planetary waves is strongly underestimated in the residual wind components of the HAMMONIA model in comparison to both Aura/MLS and ERA-Interim. On the other hand, the HAMMONIA model overestimates transient wave activity in comparison to ERA-Interim, reducing the amplitudes of the quasi-geostrophically balanced planetary waves, but contributing to the configuration of the stationary wave one patterns in O3* and H2O* because of significant zonal asymmetries in the transient wave activity. We conclude that an improvement of the stationary components in tropospheric wave activity in the model would lead to an improvement of the wave one patterns in upper stratospheric O3* and middle atmospheric H2O*.
 In the ERA-Interim data, drier atmospheric conditions in the lower stratosphere are found compared with observations [e.g., Liu et al., 2010], which might lead to deficiencies in describing the advection of H2O at these altitudes. Also, deficiencies in the description of vertical wind components in reanalysis data due to upper boundary restrictions [e.g., Polavarapu et al., 2005] might lead to an underestimation of the zonal asymmetries in the advection by the residual winds and, hence, to an underestimation of the zonal asymmetries in O3 and H2O at upper stratospheric altitudes in the ERA-Interim data in comparison to the satellite data.
 The results presented in this study contribute to the understanding of the time-mean 3-D transport of middle atmospheric constituents such as O3 and H2O. The quality of the modeled zonally asymmetric distributions of these constituents is strongly related to the quality in describing the 3-D residual winds, as demonstrated by introducing the residual wind components into the transport equation. It is important to examine these relationships in order to understand the observed local behavior and regional differences in middle atmospheric tracers such as O3 and H2O. In particular, the 3-D residual approach might be helpful to understand the processes driving the stratospheric polar vortex out of zonal symmetry, which is important for assessing polar ozone changes. Further examinations might help to identify the influence of 3-D transport processes and of chemical reactions introduced by anthropogenic emissions on observed local trends in O3 and H2O separately.
 Recently, several studies have shown that climate change might have an effect on the zonal mean BDC and related tracer transport [Butchart et al., 2006, 2010; Garcia and Randel, 2008; Li et al, 2008; Haklander et al., 2008; McLandress and Shepherd, 2009; Li et al., 2010; Okamoto et al., 2011; Monier and Weare, 2011; Weber et al., 2011; Seviour et al., 2011]. In this context, a 3-D viewpoint might be of interest to extend these findings. The time-mean longitudinal variability in atmospheric wave fluxes are essential to improve the understanding of the wave-driven transports and, furthermore, they might provide an important validation tool for predictions with general circulation and chemistry models in the framework of stratospheric ozone depletion and climate change induced by greenhouse gases. For example, the model simulations mentioned above suggest a long-term increase in the zonal mean BDC and an accompanying decrease in the mean age of stratospheric air resulting from the increase of greenhouse gases, whereas balloon observations for the period 1975 to 2005, and satellite observations for the period 2002 to 2010, suggest an increase in the local and zonal mean age [Engel et al., 2009; Stiller et al., 2012]. Based on our examinations, we might speculate that a too strong transient wave activity at the cost of stationary wave activity prefers a decrease of the mean age due to increased greenhouse gases leading to the contrast in observations. On the other hand, the picture of the 3-D residual circulation suggests strong zonal asymmetries in the mean age of air. Subsequently, the local trend of the mean age can be much stronger than or counteracting those of zonal mean values. In this context, the 3-D residual approach could help to improve the understanding of the discrepancies between local trend profiles derived from observations and from model calculations.
 It has been shown that the wave-driven zonal asymmetries in stratospheric ozone might lead to an important feedback in tropospheric and stratospheric wave developments because ozone has an effect on radiative heating rates [e.g., Kirchner and Peters, 2003; Nathan and Cordero, 2007; Gabriel et al. 2007; Crook et al. 2008; McCormack et al., 2011, Gabriel et al., 2012]. In particular, the related zonally asymmetric radiation perturbation might play an important role in troposphere-stratosphere coupling processes and associated climate-relevant changes in the atmosphere [Sassi et al., 2005; Gillett et al., 2009; Waugh et al., 2009]. However, currently, general circulation models and chemistry-climate models do not sufficiently reproduce the observed zonal asymmetries in stratospheric ozone [SPARC, 2010]. The use of the 3-D residual circulation and its effects on the spatial distribution of the stationary wave patterns of important absorbers like O3 and H2O might help us to understand these deficiencies and to improve complex coupling processes in model simulations.
 We thank ECMWF, which provided the ERA-Interim data, and NASA, which provided Aura/MLS data. In particular, we thank Hauke Schmidt from the Max-Planck-Institute for Meteorology for providing the HAMMONIA data, and three anonymous reviewers for helpful comments and suggestions significantly improving the paper. The work in this paper was supported by the International Meteorological Institute in Stockholm, and by the Deutsche Forschungsgemeinschaft. Thanks also to the Deutsches Klimarechenzentrum (Hamburg) for providing computer resources.