A three-dimensional model of dynamics and photochemistry is used to investigate the influence of planetary wave activity on the seasonal evolution of the wintertime stratosphere, which dictates springtime conditions. The final warming and springtime ozone are each found to depend strongly upon planetary wave activity during the disturbed season. The integrations reproduce their observed dependence, which enters through anomalous upward Eliassen-Palm (EP) flux from the troposphere and equatorial wind associated with the Quasi-Biennial Oscillation (QBO). Of those major influences, changes of upward EP flux are predominant. Changes representative of those in the observed record alter the timing of the final warming by as much as 1–2 months. Much the same lag distinguishes warm and cold winters in the observed record. Accompanying the shift in the final warming is a change of ozone at spring equinox. Magnified over the Arctic, anomalous springtime ozone develops largely through anomalous isentropic mixing by planetary waves. Such mixing, which precedes the final warming, incorporates ozone-rich air from lower latitude, leading to enriched polar ozone during spring. Relative to disturbed conditions, springtime polar ozone under undisturbed conditions appears depleted by some 60 DU. Derived through anomalous transport, the same difference characterizes observed changes between warm and cold winters. Much of the apparent depletion is eventually eliminated with the onset of isentropic mixing, as it is in the observed record. Together with anomalous dynamical structure, such behavior has implications important to the interpretation of interannual changes.
 The wintertime evolution of the Northern Hemisphere stratosphere involves a spinup of the polar-night vortex and a monotonic increase of totaltotal ozone. Both culminate in the final warming, when circumpolar westerlies are replaced by circumpolar easterlies and, simultaneously, totaltotal ozone achieves its spring maximum.
 Intrinsic to this evolution is the residual mean circulation, which shapes thermal and chemical structure. Through adiabatic warming and poleward transport, the Brewer-Dobson circulation controls how cold temperature becomes during winter, as well as how much totaltotal ozone increases [Brewer, 1949; Dobson, 1956; Murgatroyd and Singleton, 1961].
 The residual circulation is driven by planetary waves that transmit momentum upward from the troposphere. When absorbed, that momentum drives a poleward drift, which converges at high latitude to force mean downwelling. The accompanying adiabatic warming maintains polar temperature warmer and the vortex weaker than each would be under conditions of radiative equilibrium. Simultaneously, the poleward drift transfers ozone-rich air from its chemical source at low latitude into the winter hemisphere.
 Each process comes to a halt at the final warming, when the residual circulation collapses. Ozone-rich air that has been sequestered outside the vortex then invades the Arctic, where it mixes with ozone-lean air that has been sequestered inside the vortex. This process occurs through a complex rearrangement of air: Contours of ozone mixing ratio, previously aligned with latitude circles, fold, are drawn poleward and then wrap around the Aleutian high, which eventually replaces the polar low and ozone-lean air. (A signature in total ozone is presented by Salby ). Isentropic mixing of these distinct air masses subsequently drives ozone mixing ratio surfaces into coincidence with θ surfaces, with a commensurate increase of Arctic ozone [Salby and Callaghan, 2007].
 The timing of the final warming, along with isentropic mixing that precedes it, figures in observed changes of Arctic ozone. Each is closely related to the strength of the polar-night vortex (ibid; Waugh et al. ). During years when the vortex is anomalously strong, the final warming is delayed. Springtime ozone over the Arctic is then anomalously low, achieving its maximum later. During years when the vortex is anomalously weak, the final warming is advanced. Springtime ozone over the Arctic is then anomalously high, achieving its maximum earlier.
 The strength of the vortex, in turn, is closely related to planetary wave activity. It depends upon the net momentum that is transmitted to the middle atmosphere and where that momentum is absorbed. These features of planetary wave activity represent anomalous forcing of the residual circulation. They bear a close relationship to observed changes of the polar-night vortex and ozone, as well as to the final warming [Fusco and Salby, 1999; Waugh et al., 1999; Newman et al., 2001]. The latter track changes of momentum transmitted to the middle atmosphere, as measured by the net upward Eliassen-Palm (EP) flux at the tropopause (). Along with changes of equatorial wind associated with the Quasi-Biennial Oscillation (QBO) (uEQ), changes of account for much of the interannual variance of wintertime temperature and ozone [Hadjinicolaou et al., 1997, 2002, 2005; Salby and Callaghan, 2002; Hu and Tung, 2002; Salby et al., 2002].
 Here, we use a three-dimensional model of dynamics and photochemistry to investigate the influence of planetary wave activity on the final warming and spring ozone. Following a description of the model, section 3 presents dynamical and chemical structure under climatological mean conditions. Their wintertime evolution is then considered in the structure achieved during spring. Relative to initial conditions during autumn, springtime structure reflects the wintertime mean tendency, which figures importantly in observed interannual changes [ibid]. How springtime dynamical and chemical structure depend upon wave activity during the disturbed season is explored in section 4, through the collective influence of anomalous and the QBO, and in section 5, through the influence of the QBO alone. The integrations reproduce the observed dependence of the final warming and spring ozone, dependence that has recently been composited from a large population of winters in the observed records of National Centers for Environmental Prediction (NCEP) and SBUV-V8. Also Also reproduced is the apparent depletion of Arctic ozone following cold winters. The integrations indicate that much of the discrepancy in springtime ozone is introduced by anomalous isentropic mixing of planetary waves, which develops before spring equinox during disturbed years but afterward during undisturbed years. Implications of the results to Arctic ozone are then drawn in section 6, followed by conclusions.
2. Numerical Framework
 The three-dimensional model integrates the nonlinear primitive equations through spectral technique, along with a family treatment of photochemistry [Callaghan et al., 1999; Fusco and Salby, 1999]. It has horizontal and vertical resolution comparable to T42 and 3.5 km, respectively. Formulated in isentropic coordinates, the model extends from an isentropic surface in the upper troposphere (θ = 360 K) through the mesosphere. It is capped overhead by a deep thermal sponge layer (extending to 120 km), which achieves the radiation condition by absorbing wave activity above 85 km.
 Coupled to the lower boundary is a reservoir layer, which serves as an amorphous troposphere by enforcing conservation of mass. Air rejected from the stratosphere at high latitude is compensated at lower latitude by air that is absorbed from the troposphere at the same rate. Inside the reservoir layer is collected, among other species, stratospheric ozone that has been transferred across the lower boundary through mean downwelling. There, ozone is destroyed on a timescale of 3 months, consistent with the observed seasonality of tropospheric ozone and global estimates of its lifetime [London, 1985; Muller and Brasseur, 1995].
 The family treatment of photochemistry calculates the distributions of some 50 species that influence ozone [Fusco, 1997; Fusco and Salby, 1999]. It has been reformulated through an asymptotic partitioning of chemical species. Within an individual family is determined the limiting reaction rates, those which operate fastest and dictate the prevailing balance among members. The governing reactions are then considered recursively in the limit of those rates passing to infinity, first for the fastest rate, then for the next fastest, and so forth. Neglecting terms with higher order in inverse reaction rate then yields a system of equations that describes the prevailing balance among family members. When solved at each site, the resulting system determines the distributions of those chemical species.
 This formal treatment of chemical families achieves a robust description of species whose lifetimes vary widely. While reproducing observed zonal mean structure, the asymptotic partitioning of species remains well behaved in three-dimensional integrations, wherein diurnally varying photolysis causes short-lived species to vary rapidly. Yet it retains the computational advantages of the family representation.
 Forced at its lower boundary by observed tropospheric behavior, the three-dimensional model reproduces major dynamical and chemical structure of the wintertime stratosphere. Included are the residual mean circulation and ozone, as well as interannual changes associated with each [Francis and Salby, 2001; Callaghan and Salby, 2002; Salby et al., 2002].
 In the present study, parallel integrations are performed to isolate the dependence of the final warming and spring ozone on anomalous wave activity during the winter season. An integration in which tropospheric planetary waves are amplified yields the seasonal evolution under conditions of intensified EP flux. A counterpart integration in which tropospheric planetary waves are diminished yields the seasonal evolution under conditions of weakened EP flux. Analogous integrations are performed for extremal phases of the QBO, which, through interaction with planetary waves, likewise modulates the residual circulation [Tung and Yang, 1994]. Comparing evolutions in these parallel integrations then characterizes how the final warming and spring ozone depend upon anomalous forcing of the residual circulation.
3. Climatological Mean Conditions
 Forming the control is a seasonal integration, from autumnal equinox to spring equinox. In it, tropospheric structure at the lower boundary is prescribed from the NCEP record as the climatological mean seasonal variation (composited from a population of 20 winters). The QBO is imposed through a momentum source that drives equatorial zonal wind toward a particular phase of the QBO. Parallel integrations are performed, one with the QBO in its easterly phase and another with the QBO in its westerly phase. Averaging those integrations then recovers the seasonal evolution under climatological mean conditions.
Figure 1 plots, for the control, DJF zonal wind and temperature. Zonal wind (Figure 1a) is characterized by westerlies of the polar-night jet that intensify upward to ∼65 m/s. This is broadly consistent with observed structure, wherein the jet exceeds 60 m/s [e.g., Fleming et al., 1988]. In the opposite hemisphere are easterlies that likewise intensify upward, achieving a similar maximum in the summer mesosphere.
 Temperature (Figure 1b) assumes a minimum at the tropical tropopause, where T ∼ 200 K, and another in the Arctic stratosphere, where T decreases to ∼205 K. Like zonal wind, computed thermal structure is broadly consistent with that observed during northern winter, wherein T decreases at those sites to 200–210 K [ibid; Randel, 1987].
 Plotted in Figure 2 is DJF Montgomery stream function at θ = 1018 K (near 10 mb). The polar-night vortex is displaced over the Greenwich meridian by the Aleutian high, which is centered over the dateline near 50 N. Both features are consistent with observed structure [ibid].
Figure 3 plots zonal mean ozone mixing ratio for DJF. approaches 9.5 ppmv near 30 km and just south of the equator. Mixing ratio surfaces, which are plotted relative to θ surfaces, are deflected upward over the tropical tropopause and downward in the Arctic stratosphere. The upward deflection of surfaces in the lower stratosphere reflects mean upwelling in the tropics. It is apparent in other long-lived species, like N2O5, HNO3, and ClONO2 (not shown). The downward deflection over the Arctic reflects mean downwelling. Like dynamical structure, the computed structure of reproduces the major features of observed chemical structure during northern winter (see, e.g., Brasseur and Solomon ).
 Plotted in Figure 4 is the computed distribution of total ozone during January. Marked by crescent structure, has a maximum over Siberia, where total ozone approaches 500 DU. A secondary maximum appears over Eurasia and another over North America. These features reproduce the salient structure of the observed (climatological mean) distribution from Transactions on Mathematical Software (TOMS) during January (see, e.g., Fusco and Salby ). In the zonal mean, the computed structure in Figure 4 leads to a maximum near 60 N of ∼400 DU (not shown). Jointly with a minimum over the Arctic, where ∼320 DU, the computed zonal mean distribution corresponds well to the climatological mean distribution observed by TOMS.
 In the control, the final warming develops in early April. Zonal mean easterlies first appear in the mesosphere, descending into the stratosphere over the course of a week. The computed timing of the final warming is broadly consistent with that observed in the climatological mean [Salby and Callaghan, 2007].
 Plotted in Figure 5 is the latitudinal distribution of total ozone at spring equinox (solid). Springtime increases from ∼270 DU in the tropics to a maximum of ∼440 DU near 60 N, just outside the vortex. Further poleward, decreases. The maximum near 60 N reflects an accumulation of ozone outside the vortex, where, in the lower stratosphere, surfaces are deflected downward (cf., Figure 3). This structure is consistent with the climatological mean distribution of during spring, as observed by TOMS (see, e.g., WMO ).
 Superimposed in Figure 5 is total ozone following the final warming (dashed). Reflecting structure 2–3 weeks after equinox, then increases with latitude monotonically. Its gradient becomes flat over the Arctic, where total ozone approaches 480 DU. The uniform distribution of at polar latitudes reflects surfaces that are nearly level, having been driven into coincidence with θ surfaces by isentropic mixing. These features characterize the computed evolution of chemical structure surrounding the final warming. Very similar behavior is observed by SBUV-V8 [Salby and Callaghan, 2007].
4. Dependence on Anomalous Wave Activity
 Considered now is the influence on springtime conditions of anomalous planetary wave activity. Forcing anomalous residual mean motion, anomalous wave activity is dictated chiefly by anomalous and planetary wave structure in the troposphere. In the observed record, interannual variance of increases during winter, from a standard deviation of ∼15% during autumn to ∼30% near winter solstice [Salby and Callaghan, 2002]. Its impact upon springtime conditions is reflected in the cumulative momentum that has been transmitted to the middle atmosphere during the winter season and, hence, in wintertime mean .
 Perturbed about the control are two parallel integrations: In one, the seasonal variation of tropospheric wave structure is amplified uniformly to increase wintertime mean by 20%. (This may be regarded as the ensemble mean of many transient seasonal amplificatons, each one punctuated by sporadic episodes of different timing and wavenumber composition, but all with the sane cumulative momentum transfer). Simultaneously, the QBO is maintained in its easterly phase. In the other integration, tropospheric wave structure is weakened to decrease wintertime mean by 20%, while the QBO is maintained in its westerly phase. These combinations of anomalous and the QBO correspond to ±1 standard deviation of anomalous temperature and ozone in the observed record [Salby and Callaghan, 2002; Salby and Callaghan, 2007]. The parallel integrations thus reflect a 1-standard deviation increase and decrease of wintertime mean residual motion. The anomalous residual circulation between the integrations (not shown), in fact, is comprisecomprised of poleward motion that converges over the Arctic to form anomalous downwelling. It will be seen in section 5 that most of the induced anomaly follows from anomalous , as it does in the observed record (ibid; Newman et al. ; Hu and Tung ).
 Compared in Figure 6 is the DJF Montgomery stream function near 10 mb. Under conditions of amplified tropospheric wave activity and QBO easterlies (Figure 6a), the Aleutian high is advanced poleward of its location under climatological mean conditions (Figure 2). It now maximizes near the Arctic circle, invading the polar cap. Amplified over those which are under climatological mean conditions, the Aleutian high has further displaced the polar-night vortex, which has contracted and retreated equatorward along the Greenwich meridian. These changes reflect a weakened polar-night jet (not shown), the DJF mean reaching only 56 m/s (cf., Figure 1a). The poleward advance of the Aleutian high and equatorward retreat of the vortex represent an expansion of the planetary wave critical region, wherein nonlinearity leads to isentropic mixing [Juckes and McIntyre, 1987; Salby et al., 1990].
 Under conditions of weakened tropospheric wave activity and QBO westerlies (Figure 6b), the Aleutian high is weaker and removed equatorward. Now maximizing near 45 N, it leaves the polar-night vortex deeper, more extensive, and centered closer to the pole. The polar-night jet is therefore stronger, the DJF mean exceeding 82 m/s. The change of wave structure represents a contraction of the critical region, along with isentropic mixing. The difference in computed structure in Figures 6a and 6b reproduces that observed between ±1 standard deviation in anomalous forcing of the residual circulation [Salby and Callaghan, 2002].
 The more disturbed structure of the vortex is readily apparent in the isentropic distribution of near 10 mb during January (Figure 7). Under conditions of amplified tropospheric wave activity and QBO easterlies (Figure 7a), ozone mixing ratio is punctuated by two distinct anomalies ( 5.6 ppmv shaded). An absolute minimum appears inside the polar-night vortex, where decreases to ∼1.3 ppmv. Opposite the pole, neighboring the Aleutian high, is a local minimum, where decreases to ∼4.5 ppmv. Low values extend eastward and poleward, joining the absolute minimum. They reflect ozone-lean air that is detrained from the vortex and, together with ozone-rich air drawn poleward, is entrained into the Aleutian high in the region of isentropic mixing. Low ozone inside the secondary anomaly is in fact strongly correlated with high potential vorticity (PV) (not shown), both characteristic of vortex air. Introduced through advection, the secondary minimum then adjusts gradually to local photochemical equilibrium (Harvey et al. ; see also Manney et al. ), while being reinforced through sporadic episodes of detrainment and mixing. (The photochemical lifetime of O3 at this location and time (calculated in the integration) is a couple of weeks, still long compared to the timescaletime scale of advection). The secondary ozone minimum disappears above 40 km, where the photochemical lifetime becomes short enough for the diurnal variation of O3 to prevail. Between the two anomalies of ozone-lean air, contours fold, reflecting a tongue of ozone-rich air that is drawn sporadically from lower latitude. Entrained into the region of isentropic mixing, the tongue of high acts to enrich ozone in both anomalies.
 Under conditions of weakened tropospheric wave activity and QBO westerlies (Figure 7b), the absolute minimum of is about the same as under conditions of amplified wave activity and QBO easterlies. The resemblance reflects the efficient isolation of air at the core of the vortex under both conditions. However, the region of 5.6 ppmv that demarcates the vortex (shaded) is noticeably expanded over those which are under conditions of amplified wave activity and QBO easterlies (Figure 7a). Likewise, the local minimum found opposite the pole is shallower. Characterized by higher it reflects weaker detrainment of ozone-lean air from the vortex. The local minimum has receded equatorward, along with the Aleutian high (Figure 6b). The region of isentropic mixing, marked by the folding and wrap up of contours, now barely reaches the Arctic circle. Ozone-lean air associated with the vortex is then left more extensive, occupying most of the Arctic, and centered nearly over the pole.
 Plotted in Figure 8, as a function of month, is the evolution of minimum zonal wind over the polar cap (60 N–80 N) near 10 mb. Under conditions of amplified tropospheric wave activity and QBO easterlies (solid), the vortex reaches springtime anomalously weak. By March, polar westerlies have been weakened to less than 15 m/s. The vortex is, therefore, more vulnerable to subsequent disturbances, which encourage the final warming. Defined by the prevalence of zonal mean easterlies, the final warming then develops in mid March, about 2 weeks earlier than under climatological mean conditions. Advanced likewise is isentropic mixing of ozone-rich air into the Arctic, which precedes the final warming. Such behavior develops when the mean flow has weakened sufficiently for eddy motion to overturn the distributions of PV and chemical tracers (Figure 7). The critical region of planetary waves then expands poleward, along with eddy mixing of ozone-rich air from lower latitude. Springtime ozone over the Arctic is therefore enriched.
 Under conditions of weakened tropospheric wave activity and QBO westerlies (dashed), the vortex reaches springtime anomalously strong. It is, therefore, more resilient against subsequent disturbances. The final warming then does not develop until mid April, about 2 weeks later than under climatological mean conditions. This places it about a month later than under conditions of amplified tropospheric wave activity and QBO easterlies (solid). At spring equinox, polar westerlies remain stronger than 25 m/s, almost as strong as earlier in the winter season. The Arctic then remains confined largely within the polar-night vortex. It is, consequently, sequestered from ozone-rich air at lower latitude. The difference of computed seasonality in Figure 8 is very similar to that observed between warm and cold winters, recently composited from a large population of years in the NCEP record [Salby and Callaghan, 2007]. As in the computed evolution, the observed final warming is delayed by about a month during cold winters relative to that during warm winters.
 Plotted in Figure 9 is total ozone at spring equinox (solid). Under conditions of amplified tropospheric wave activity and QBO easterlies (Figure 9a), the structure at subpolar latitudes is quite similar to that under climatological mean conditions (Figure 5). increases from ∼270 DU in the tropics to ∼450 DU near 60 N. The strong resemblance with Figure 5 is consistent with the wintertime increase at those latitudes having a major contribution from seasonal transience of diabatic cooling, which is invariant between the integrations [Salby and Callaghan, 2006]. Differing more noticeably is over the Arctic. Under conditions of amplified tropospheric wave activity and QBO easterlies (Figure 9a), springtime ozone still decreases poleward of 60 N. However, its gradient has been flattened. now decreases to only 415 DU, whereas, under climatological mean conditions (Figure 5), it decreases to 380 DU. The enrichment of polar ozone under conditions of amplified tropospheric wave activity and QBO easterlies reflects the poleward advance of ozone-rich air neighboring the Aleutian high and, simultaneously, the equatorward retreat of ozone-lean air inside the vortex (Figure 7a).
 Superimposed in Figure 9a is total ozone following the final warming (dashed), when zonal mean easterlies have just invaded the lower stratosphere. now increases monotonically to the pole, where it approaches 540 DU. Two weeks later (not shown), the gradient flattens over the Arctic, leaving uniform values of ∼520 DU. Thereafter, decreases steadily, reaching 480 DU by May.
 Under conditions of weakened tropospheric wave activity and QBO westerlies (Figure 9b), at spring equinox (solid) is, at subpolar latitudes, again quite similar to its distribution under climatological mean conditions (Figure 5). The maximum is found about 10° equatorward of its position in Figure 9a. At higher latitudes, however, the distributions in Figures 9a and 9b differ conspicuously. Total ozone now decreases even more sharply inside the vortex. Left comparatively undisturbed, it occupies much of the Arctic (Figure 7b). Accordingly, springtime (solid) decreases over the pole to values of only ∼350 DU. Under conditions of amplified tropospheric wave activity, and QBO easterlies (Figure 9a), polar ozone is some 65 DU greater. This happens to be close to the observed difference between warm and cold winters, which approaches 60 DU (Salby and Callaghan, 2007; see also Muller et al. ; Tilmes et al. ; Rex et al. ). Comparison with thermal structure [ibid] indicates that the observed difference in springtime total ozone over the Arctic follows largely from surfaces during warm winters being driven into coincidence with θ surfaces, a signature of isentropic mixing. The same is true in the integrations.
 Following the final warming (dashed), under conditions of weakened tropospheric wave activity and QBO westerlies assumes a form similar to that under conditions of amplified tropospheric wave activity and QBO easterlies. The gradient has then flattened over the Arctic, with nearly uniform values of ∼440 DU. This limiting distribution is achieved about a month later than under conditions of amplified tropospheric wave activity and QBO easterlies, about the same lag as the final warming. Much the same behavior distinguishes warm and cold winters in the observed record [ibid]. In the integrations, Arctic ozone is substantially enriched over at spring equinox (solid). Nevertheless, it remains distinctly leaner than under conditions of amplified tropospheric wave activity and QBO easterlies (Figure 9a).
 The computed difference between Figures 9a and 9b reflects enhanced downwelling and isentropic mixing of ozone-rich air into the Arctic under disturbed conditions. Relative to those conditions, springtime ozone over the pole under undisturbed conditions appears depleted by some 60 DU. The apparent depletion reproduces the observed difference between warm and cold winters [Salby and Callaghan, 2007]. It arises chiefly from anomalous transport, which, under undisturbed conditions, leaves ozone-lean air over the pole more isolated.
 The gap in polar ozone between disturbed and undisturbed conditions narrows when ozone-lean air inside the vortex is eventually homogenized with ozone-rich air from lower latitude. Reflecting the onset of isentropic mixing (delayed under undisturbed conditions), this too mirrors ozone behavior in the observed record. The observed difference remaining at the end of April, however, is actually smaller than the computed difference, at least relative to monthly mean observations. Nonetheless, by May, over half of the apparent depletion during cold winters has been eliminated. Thereafter, the computed and observed differences converge in magnitude as well.
5. Dependence on the QBO
 Considered next is the influence on springtime conditions from the QBO alone. Perturbed about the control are two parallel integrations. In one, the QBO is maintained in its easterly phase. In the other, its is maintained in its westerly phase.
 Compared in Figure 10 is the DJF Montgomery stream function near 10 mb. Extremal phases of the QBO lead to differences in wintertime mean structure that are comparatively minor. In the easterly phase (Figure 10a), the Aleutian high is amplified and expanded over that of the westerly phase (Figure 10b). It therefore penetrates slightly deeper into the Arctic, just grazing the Arctic circle. The polar vortex is centered in nearly the same location in the two integrations. However, it is visibly smaller in the easterly phase than in the westerly phase of the QBO. The distinction reflects the poleward advance of the planetary wave critical region, wherein isentropic mixing detrains high-PV air from the vortex more vigorously in the easterly phase of the QBO.
 These differences are mirrored in the isentropic distribution of near 10 mb during January (Figure 11). In the presence of QBO easterlies (Figure 11a), ozone mixing ratio attains an absolute minimum inside the vortex of ∼1.3 ppmv. Opposite the pole, the local minimum associated with the Aleutian high is centered near 50 N. The region of mixing, marked by the folding and wrap up of contours, invades the Arctic, but just barely. In the presence of QBO westerlies (Figure 11b), ozone mixing ratio attains nearly the same absolute minimum inside the vortex. The resemblance reflects the isolation of air at the core of the vortex in both integrations. However, the region of 5.2 ppmv that demarcates the vortex (shaded) is slightly larger in the presence of QBO westerlies, then occupying more of the Arctic. The difference reflects a better isolation of ozone-lean air just inside the vortex from ozone-rich air outside it. This difference is consistent with the equatorward retreat of isentropic mixing inside the critical region, which attends the swing from QBO easterlies to QBO westerlies. Similarly, the local minimum of associated with the Aleutian high is, under QBO westerlies, displaced slightly equatorward. Now near 45 N, it is characterized by higher that reflects weaker detrainment of ozone-lean air from the vortex. The folding of contours, wherein ozone-rich air is drawn poleward, now just grazes the Arctic circle.
 In the presence of QBO easterlies, the final warming develops around the first of April, only slightly earlier than in the climatological mean (section 3). In the presence of QBO westerlies, it is postponed by about a week. Differences in the lower stratosphere, where the QBO is pronounced and ozone is concentrated, are somewhat greater. There, the reversal of zonal wind occurs about 2 weeks earlier under QBO easterlies than under QBO westerlies. Ozone-rich air therefore invades the Arctic that much sooner, with a commensurate advancement of the spring maximum.
 Plotted in Figure 12 is total ozone at spring equinox (solid). As for stream function and extremal phases of the QBO lead to differences in that are comparatively minor. In each phase of the QBO, springtime ozone maximizes near 60 N, similar to the distribution under climatological mean conditions (Figure 5). exceeds 450 DU under QBO easterlies (Figure 12a) and only slightly less under QBO westerlies (Figure 12b). Analogous differences appear in a neighborhood of the equator, where the QBO modulates mean upwelling. There, achieves minimum values of ∼270 DU under QBO easterlies but ∼280 DU under QBO westerlies. In contrast, air over the pole is efficiently isolated from air at lower latitudes. Consequently, polar at spring equinox remains very close in the two integrations.
 Superimposed in Figure 12 is total ozone following the final warming (dashed). Differences are, likewise, comparatively minor. In the presence of QBO easterlies (Figure 12a), increases poleward, exceeding 490 DU. In the presence of QBO westerlies (Figure 12b), has much the same form but with maximum values over the pole that are smaller by ∼10 DU. This difference, which reflects conditions after spring equinox, characterizes polar ozone that has been slightly enriched under QBO easterlies. It follows from ozone-enriched air at the edge of the vortex, which is eventually mixed into the Arctic during the final warming.
6. Implications for Arctic Ozone
 Weeks neighboring the final warming witnessed a sharp increase of total ozone over the Arctic (Figure 5). The increase of ozone there is achieved through a transformation of its latitudinal profile: Before the final warming, maximizes at the edge of the vortex, decreasing poleward of 60 N. Afterward, increases monotonically to the pole, eventually approaching uniform values over the Arctic. Equatorward of 60 N, seasonal changes of are comparatively minor.
 The sharp enrichment of Arctic ozone follows from isentropic mixing by planetary waves, which prevails over the Arctic a couple of weeks before the final warming. Figure 13 compares zonal mean before and after the final warming, each under conditions of weakened tropospheric wave activity and QBO westerlies. Before the final warming (Figure 13a), mixing ratio surfaces are deflected sharply downward over the Arctic but only in the upper stratosphere. That structure reflects ozone-lean air that has been transported downward into the Arctic stratosphere from the mesosphere, where mean downwelling is strong [Fischer and O'Neill, 1993]. Although it produces ozone-lean conditions in the upper stratosphere, downward transport there has only a modest impact upon the column abundance of ozone, which is concentrated at lower levels.
 In the lower stratosphere, the downward deflection of mixing ratio surfaces shifts to 60 N, just outside the vortex. (Ozone structure in Figure 13 is plotted relative to isentropic surfaces, which, in the lower stratosphere, slope downward towardtowards the pole. Relative to pressure surfaces, the downward deflection of mixing ratio surfaces at those levels is even greater). There, ozone has been enriched by downwelling that, under undisturbed conditions, is concentrated at the edge of the vortex. It is that enrichment of which leads to the maximum of total ozone at 60 N, decreasing further poleward (Figure 9b). Very similar structure is observed by SBUV-V8 during anomalously cold winters [Salby and Callaghan, 2007].
 Following the final warming (Figure 13b), the same mixing ratio surfaces have been flattened to the pole. There, they have been driven into coincidence with θ surfaces (which are level in this representation). Note: Although most conspicuous in the upper stratosphere, the flattening of mixing ratio surfaces is evident at all levels. The attendant increase of fills the ozone void in the upper stratosphere that developed during winter. Compensating the increase of ozone over the Arctic is a poleward displacement of the maximum, from just south of the equator before the final warming to just north of the equator afterward. Much the same displacement is, in fact, evident for the entire ozone distribution.
 In the zonal mean representation in Figure 13, ozone-rich air has advanced poleward along isentropic surfaces. Together with the downward slope of isentropic surfaces in the lowermost stratosphere, this results in a sharp increase of ozone number density and hence of total ozone over the Arctic. then increases monotonically toward the pole (Figure 9b). The same evolution of ozone structure appears in the observed record from SBUV-V8 [ibid].
 An analogous transformation occurs under conditions of amplified tropospheric wave activity, albeit sooner. However, the expanded critical region that prevails under those conditions leads to increased mixing of ozone-rich air into the Arctic throughout the winter season. Figure 14 plots at the same time as in Figure 13a but under conditions of amplified tropospheric wave activity and QBO easterlies. Although still 2 weeks before the final warming, isentropic mixing by planetary waves has mediated the downward deflection of surfaces at the edge of the vortex. Consequently, mixing ratio surfaces assume a shallower slope into the pole, achieving minimum values that are some 50% greater than in Figure 13a. Total ozone over the Arctic then arrives at the final warming already enriched, with closer to its limiting value after the final warming.
 In three-dimensional form, the transformation of chemical structure over the Arctic occurs through a complex rearrangement of ozone-lean air inside the vortex and ozone-rich air outside it. Figure 15 plots, under the conditions of Figure 14, the isentropic distribution of near 10 mb during the final warming, shortly after zonal mean easterlies have invaded the middle stratosphere. The absolute minimum of associated with the polar-night vortex is substantially enriched over earlier values (Figure 7). It has been displaced equatorward, almost to the Arctic circle. Replacing it over the Arctic is the tongue of high that has been drawn poleward. As the ozone minimum recedes equatorward along the Greenwich meridian, higher ozone from the opposite hemisphere is swept poleward about the secondary minimum that is associated with the Aleutian high. That ozone-rich air wraps up anticyclonically, enriching the Aleutian high, which eventually moves over the pole to form circumpolar easterlies with increased ozone.
 Integrations forced by anomalous wave driving representative of that observed reproduce observed changes of the final warming and of spring ozone. Both depend strongly upon planetary wave activity, through its forcing of residual mean motion, in concert with isentropic mixing. Of the major influences considered, changes of EP flux from the troposphere carry the lion's share of the impact. Changes of corresponding to ±1 standard deviation in the observed record alter the timing of the final warming by as much as a month. Much the same lag distinguishes the observed record of warm and cold winters, which are associated with similar changes of [Salby and Callaghan, 2002; Salby and Callaghan, 2007]. Larger changes of (e.g., ±2 standard deviations) can advance the final warming to as early as February, reflecting a highly disturbed Arctic vortex, or delay it to as late as May, reflecting a comparatively undisturbed Antarctic vortex.
 Accompanying the shift in the final warming is anomalous springtime ozone. Magnified over the Arctic, it develops largely through anomalous isentropic mixing by planetary waves. Such mixing incorporates ozone-rich air from lower latitude, leading to enriched polar ozone during spring. Surfaces of ozone mixing ratio are then driven toward isentropic surfaces, reproducing the signature of mixing in the observed record [ibid].
 Relative to disturbed conditions, springtime polar ozone under undisturbed conditions appears depleted by some 60 DU. Derived through anomalous transport, much the same difference characterizes observed changes of ozone between warm and cold winters. The latter, in turn, accompany observed changes of EP flux that are similar to those which distinguish the integrations.
 The final warming is also sensitive to the QBO, which controls where planetary wave activity is absorbed. By advancing isentropic mixing poleward or removing it equatorward, the QBO exerts a similar influence on the wintertime vortex and, hence, on the state it achieves at spring equinox. However, relative to observed changes of , the influence of uEQ is comparatively minor. This finding is consistent with two features in the observed record of interannual variability: (1) The final warming is strongly correlated to observed changes of . (2) The variance of springtime temperature and ozone are each mostly accounted for by observed changes of [Waugh et al., 1999; Newman et al., 2001; Salby and Callaghan, 2002].
 The advance or delay of the final warming has important implications for the interpretation of interannual changes. Conditions at spring equinox, for example, can differ dramatically, simply through the timing of the final warming. If sufficiently advanced, the final warming will result in equinoctial conditions over the Arctic that are anomalously warm and ozone rich. Conversely, if sufficiently delayed, the final warming will result in equinoctial conditions over the Arctic that are anomalously cold and ozone lean. Relative to warm winters, cold winters therefore yield a springtime Arctic in which ozone appears substantially depleted, as is observed [Rex et al., 2004]. Much of that anomaly is eventually erased with the development of isentropic mixing, which subsequently transfers ozone-rich air into the Arctic [Salby and Callaghan, 2007].
 The integrations reproduce the salient features that distinguish warm and cold winters. They illustrate that much of the observed depletion in springtime Arctic ozone following undisturbed winters can be understood in terms of the delayed final warming, when ozone-rich air eventually invades the Arctic. Comparing ozone, not at a fixed time of year, but relative to the final warming, eliminates much, but not all, of the apparent depletion, as it does in the observed record.
 The authors are grateful for the constructive remarks provided during review. This work was supported by NSF grant ATM-0121853.