## 1. Introduction

During the last two decades, much attention has been paid to the study of large-scale horizontal transport and stirring in the stratospheric polar night vortex (SPV) in view of its importance for the dissipation of the “Antarctic ozone hole” (e.g., Shepherd 2007, and references therein). In this regard, both observational and numerical model studies demonstrate that horizontal air exchange is inhibited across the vortex edge where the potential vorticity (PV) field on isentropic surfaces has sharp gradients (e.g., Manney et al. 1994). Large-scale Rossby wave breaking in the SPV is responsible for the formation of what in the Lagrangian perspective is referred to as hyperbolic trajectories, and for the continuous erosion and regeneration of the PV gradients by filamentation (Bowman 1993). On the basis of results using a three-dimensional transport model, Li et al. (2002) suggest that horizontal air transport out of the vortex into stratospheric midlatitudes is much smaller than that descending to the troposphere. Chen et al. (1994) measure the stretching of material lines in the middle stratosphere (i.e., the 600-K isentropic surface) during the southern winter and spring of 1992 using the contour advection technique (Dritschel 1989). Their findings portray stronger stirring either in the surf zone outside the SPV (McIntyre and Palmer 1984) or inside the vortex than on the vortex edge. The local differences in stirring have impacts on the Antarctic ozone hole development. Lee et al. (2001), working with a three-dimensional chemical transport model, show that greater ozone losses take place in the vortex edge until mid-September because of weak mixing in this region. Chen (1994) investigates the permeability of the SPV in the southern late winter of 1993. According to his results, in the period considered the vortex becomes more isolated as altitude increases, and there is virtually no mass transport across the vortex edge above 425 K.

A fundamental approach to the study of horizontal transport in the SPV has been based on concepts from dynamical systems theory (Wiggins 1992; Samelson and Wiggins 2006). The key tools from this perspective are hyperbolic trajectories and their stable and unstable manifolds (Wiggins 2003; Mancho et al. 2006a; Mendoza and Mancho 2010b, manuscript submitted to *Phys. D*.) Hyperbolic trajectories are the extension to time-dependent flows of the concept of hyperbolic points, which can be visualized as stagnation points of the streamfunction field in two-dimensional stationary flows. In this context, convergence and divergence toward the hyperbolic trajectory occur along the stable and unstable manifolds, respectively. These are invariant objects that act as material boundaries to transport since no particles can cross them by advection processes. Stable and unstable manifolds of hyperbolic trajectories intersect forming lobes, and the dynamics of these lobes are responsible for mass transport across the boundaries formed by the manifolds (Wiggins 1992; Mancho et al. 2006a). In the SPV context, the direct computation of manifolds has not been the norm, and approximate methods—such as finite-time or finite-size Lyapunov exponents (FTLEs and FSLEs, respectively)—have been used. The basic ideas for what are now referred to as FTLEs were originally proposed by Bowman (2000), made more rigorous in a series of papers by Haller (2000, 2001), and applied to the Antarctic stratospheric flow in several studies (e.g., Lekien and Ross 2010; de la Cámara et al. 2010, hereafter DLC10). DLC10 calculate FTLEs to visualize the Lagrangian coherent structures of the flow in the lower southern stratosphere during spring 2005. This particular period corresponds to the Vorcore project, which released isopycnal, superpressure balloons in the polar stratosphere over Antarctica (Hertzog et al. 2007). Since such balloons can be taken as quasi-Lagrangian tracers, DLC10 support their Lagrangian analysis with Vorcore data by showing that outstanding features in some balloon trajectories are consistent with the flow structure described by the FTLEs. For example, the sudden split in the trajectories of balloons previously drifting as a group is interpreted as a consequence of the balloons coming to locations that are very close geographically but on different sides of a ridge in the FTLE field. Aurell et al. (1997) introduce FSLE and Joseph and Legras (2002) and Koh and Legras (2002) apply them in studies of the Antarctic stratospheric flow. All these studies have confirmed that methods based on Lyapunov exponents can succeed in capturing the essentials of the Lagrangian structure, but they only provide a blurry view of the manifolds.

The present paper documents the first application to the study of transport in the SPV of a recently proposed Lagrangian descriptor: the function *M* (Madrid and Mancho 2009; Mendoza and Mancho 2010a, hereafter MM10). MM10 demonstrate the usefulness of *M* as a global Lagrangian descriptor in dynamical systems with arbitrary time dependence in applications of the technique to studies of ocean flows estimated from altimeter data (Río and Hernández 2004). In particular, MM10 show that the function “*M* displays all stable and unstable manifolds from all possible hyperbolic trajectories in the neighborhood of a region, without need to identify hyperbolic trajectories a priori, as required by the manifold algorithm (see Mancho et al. 2004).” We show that a methodology based on *M* results in a significantly improved visualization of Lagrangian coherent structures in the SPV context. We also show that the methodology facilitates the identification of transport routes across the vortex edge, even without a direct computation of manifolds from selected hyperbolic trajectories.

Our focus is on the southern spring of 2005, which was examined by DLC10 using Lyapunov exponents. We start by characterizing accurately the Lagrangian geometry of the SPV and by showing Lagrangian structures within the vortex core. Next, we examine routes of large-scale horizontal transport across the vortex edge and highlight the importance of lobe dynamics as transport mechanism. The remainder of the paper is organized in the following way: section 2 describes the data and methods, section 3 presents the results, and section 4 contains the conclusions.

## 2. Data and methods

*M*is defined as follows:where [

*x*(

*t*),

*y*(

*t*),

*z*(

*t*)] define the trajectory of a fluid parcel such that

*x*

_{0}=

*x*(

*t*

_{0}),

*y*

_{0}=

*y*(

*t*

_{0}),

*z*

_{0}=

*z*(

*t*

_{0}) (Madrid and Mancho 2009). Accordingly,

*M*measures the length of trajectories in the phase space passing through (

*x*

_{0},

*y*

_{0},

*z*

_{0}) at

*t*=

*t*

_{0}over a range of times [

*t*

_{0}−

*τ*,

*t*

_{0}+

*τ*]. Calculation of

*M*implies the integration of “backward” trajectories from

*t*

_{0}to

*t*

_{0}−

*τ*, and of “forward” trajectories from

*t*

_{0}to

*t*

_{0}+

*τ*.

In the present study we assume adiabatic flows (i.e., that fluid parcels remain on the same isentropic surfaces during the interval of calculation). Many studies have demonstrated the usefulness of such approximation for a better understanding of transport and mixing properties in the lower stratosphere (e.g., Bowman 1993; Chen 1994; Chen et al. 1994; Lee et al. 2001; Joseph and Legras 2002). We compute trajectories on the 475-K isentropic surface (~20 km) by using the 4-times-daily horizontal wind fields from the interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim; Simmons et al. 2006), which is available on a 1.5° longitude × 1.5° latitude grid. To bypass the singularity at the pole in spherical coordinates we work on a three-dimensional system of Cartesian coordinates. Trajectories are computed from points on a ⅓° longitude × ⅓° latitude grid on the 475-K isentropic surface south of 20°S by application of a fourth-order Runge–Kutta scheme with a time step of 1 h. We have checked the sensitivity of the results to degrading the horizontal resolution of the ERA-Interim fields to 3° longitude × 3° latitude and 6° longitude × 6° latitude, and to changing the resolution of the grid for the calculation of *M*. The results are marginally sensitive to these changes (not shown). The velocity field at fluid parcel locations is obtained with a combination of a bicubic interpolation in space and a third-order Lagrange polynomial interpolation in time (Mancho et al. 2006b). The total length of the trajectory—that is, the value of *M*—is estimated assuming that consecutive points in time are joined by straight lines.

Figure 1 displays the structure of the function *M* on the 475-K isentropic surface (with *τ* = 15 days) on 1 November 2005. The large-scale pattern corresponds to the typical structure of the flow during the southern final warming (Mechoso et al. 1988). In inspecting maps of *M*, one looks for sharp color gradients, which are aligned with stable and unstable manifolds (MM10). One also looks for crosshatched patterns, which are indicative of the manifold crossing and hence of the presence of hyperbolic trajectories (MM10). In some special cases, these trajectories play key roles in organizing the geometry of the flow and are referred to as “distinguished” (Madrid and Mancho 2009). In the present paper, we perform a visual inspection of *M* maps looking for geometric structures characterized by crosshatched patterns and interpret them as evidence of hyperbolic trajectories. Note that these will not be rigorously characterized in our flow, but we are applying a criterion derived from MM10, in which they were computed in detail. For example, according to our method, we interpret the outstanding crosshatched pattern of *M* in the neighborhood of the point labeled “H1” in Fig. 1 as indicative of a hyperbolic trajectory. By contrast, Fig. 1 shows high values of *M* (red colors) with a smooth structure in locations of the jet stream. These correspond to long trajectories in the time period 2*τ* under consideration and indicate how difficult it is for manifolds to cross the jet, thus highlighting its insulating role.

## 3. Discussion of lobe transport in the interior and exterior of the stratospheric polar vortex

### a. Lobe dynamics in the collar

The region surrounding the SPV is traditionally interpreted as one in which wave breaking and stretching and folding of material lines lead to transport processes (Shepherd 2007). Bowman (2000) notes the tangle of manifolds in the surf zone and boundary of the vortex edge. Joseph and Legras (2002) find lobe dynamics to occur in the periphery of the vortex and conclude that transport by lobes is responsible for mass exchange between the surf zone and a “stochastic layer” that surrounds the vortex. The latter region is bounded by the tangle of manifolds and lobes in the equatorward side, and by the vortex edge in the poleward side. We will refer to this region as the “collar” as in the paper by Mariotti et al. (2000).

In the collar, the structure of *M* displays very sharp features and one can clearly identify that lobe transport is at work. Figure 2 shows the structures of *M* on 2, 4, 5, and 6 November 2005, in which the same hyperbolic point H1 as in Fig. 1 is highlighted. Figures 2a–c show a lobe “L” emanating from H1, which shortens and widens in time. From 5 to 6 November 2005, L stretches while the air inside penetrates into the outer periphery of the jet (Figs. 2c,d). The dashed lines in all panels of Fig. 2 approximately represent the vortex edge defined as a single PV contour where this field has the strongest gradient. The vortex edge is located approximately at the jet center determined by the highest values of *M*, and the transport process described in this paragraph occurs outside the vortex edge (i.e., in the collar).

### b. Lagrangian structure inside the SPV

There is no a priori estimate of the optimum value of *τ* to visualize the geometry of the flow, as it depends on the specific case considered (Madrid and Mancho 2009; MM10). In this particular analysis, we have noticed that a lower *τ* suffices for convergence to Lagrangian structures in the perturbed SPV configuration characteristic of later times in the spring season.

Figures 3a and 3b show the dependence on *τ* of the Lagrangian structure portrayed by *M*. The structure of *M* in the jet region (red values) is largely insensitive to the value of *τ*, remaining smooth as *τ* is increased from 3 to 15 days (Figs. 3a and 3b, respectively). This confirms the robustness of the jet, which makes it difficult for the stable and unstable manifolds of hyperbolic trajectories located outside the SPV to influence the inner core of the vortex (i.e., lines of *M* hardly cross the vortex edge from the exterior for the integrations performed in this study). Such an efficiency of the lower stratospheric vortex edge in obstructing transport during periods longer than 1 month during the austral spring is in agreement with a number of previous studies (e.g., Chen 1994; Paparella et al. 1997). The homogeneity of the *M* values at the SPV edge—which in our context means that the parcels follow similar trajectories—suggests that the transport barrier in that region may be associated with a Kolmogorov–Arnold–Moser invariant torus (Rypina et al. 2007a,b; Beron-Vera et al. 2010). In the following section, however, we discuss results consistent with the existence of breaks in this barrier.

The structure of *M* in the vortex interior, on the other hand, is sensitive to *τ*. In studies of ocean eddies (MM10), the function *M* for small values of *τ* has a smooth structure in the eddy interior, indicating isolation from the neighborhood. For increasing values of *τ*, however, the interior becomes foliated by stable and unstable manifolds of nearby hyperbolic trajectories, indicating loss of isolation. Similarly in the SPV, the function *M* computed using *τ* = 3 days does not capture Lagrangian structures in the vortex interior (Fig. 3a), and it appears smooth without regions of crosshatched patterns. If *τ* is increased to 15 days (Fig. 3b), however, Lagrangian features emerge in a pattern that becomes more complex, and hyperbolic trajectories are identifiable.

One example of hyperbolic trajectory in the inner part of the jet is the feature labeled “C” in Fig. 3b. This trajectory perturbs the inner part of the jet, causing penetration of manifolds. Such trajectories play an important role in controlling transport between the jet and the vortex interior. The distribution of ozone mixing ratio at 475 K from the ERA-Interim dataset for the same day as Fig. 3b illustrates that role (see Fig. 4). The lower content of ozone is indicated by warmer colors, which clearly signal the Antarctic ozone hole. The selected contour of ozone mass mixing ratio highlighted in Fig. 4 surrounds the ozone hole and shows a region of relatively high ozone concentration near the pole. A tongue of higher ozone mixing ratios that extends toward the vortex interior is also visible in the vicinity of the hyperbolic trajectory C, evidencing enhanced transport in that region.

The presence of organizing structures around the outer periphery of the vortex discussed in subsection 3a, such as hyperbolic trajectories and invariant manifolds, is interpreted as a signature of wave breaking and stretching and folding of material lines leading to transport processes (Shepherd 2007). An extension of this argument on the poleward side of the jet suggests that active mixing and wave breaking also occurs in this zone. This argument is consistent with the climatology of wave breaking obtained by Hitchman and Huesmann (2007), who find persistent Rossby wave breaking on the poleward side of the Antarctic polar night jet during the austral spring. Hence, large-scale stirring occurs in the inner core, but it is expected to be weaker than in the outside. Consistently, color contrasts in Fig. 3b are fainter in the vortex core than in the collar as gradients on trajectory lengths are weaker in the former region than in the latter one.

### c. Transport routes across the vortex edge

We start this subsection by revisiting the transport cases examined by DLC10. These authors visualize coherent structures of the flow in the lower southern stratosphere during spring 2005 by plotting extreme values of FTLE computed on isopycnal surfaces with GEOS5 reanalysis products. The FTLE technique quantifies the dispersion rates of fluid parcels by measuring the separation between trajectories after a predetermined time. Therefore, the largest FTLE will align with stable and unstable manifolds when computing time-forward and time-backward trajectories, respectively. DLC10 combine FTLE and PV distributions to suggest that air is stripped away from the jet as stable manifolds eventually cross the vortex edge. This suggestion is supported by the approximations to Lagrangian trajectories given by the tracks of superpressure balloons (SPBs) released by the Vorcore project (Hertzog et al. 2007). The emphasis is on the behavior of two SPBs that escape from the vortex within high-PV tongues that develop in association with wave breaking at locations along the vortex edge where forward and backward FTLE maxima approximately intersect. The PV analysis by itself was insufficient to explain the differential behavior of those SPBs relative to others in their vicinity.

Figures 5a–c show the function *M* on the 475-K isentropic surface corresponding to the SPB escape examined by DLC10. On 17 November 2005 (Fig. 5a) it is easy to identify a distinct region in the *M* structure (light yellow) in which the SPB (green circle) is embedded. We interpret that structure as a lobe that penetrates the jet from the vortex boundary, stripping air from the jet region. By 21 November (Fig. 5b) the SPB, which has remained inside the lobe while this stretches, is approaching the hyperbolic trajectory labeled H2. On 25 November (Fig. 5c), the SPB has escaped the vortex periphery and remains within a lobe that emanates from H2 and stretches into midlatitudes.

Figures 5d–f show the corresponding plots using FTLE, with blue and red colors representing stable and unstable manifolds, respectively. The methodology for computing FTLE is the same as in DLC10 [see Eq. (2) of DLC10 and Eq. (12) of Shadden et al. (2005) for further details], except that here we use a fourth-order Runge–Kutta scheme with a 1-h time step, a horizontal grid of ⅓° longitude × ⅓° latitude, an initial parcel separation of 1 km, and a time integration of 9 days. The Lagrangian features in Figs. 5d–f are broadly consistent with those in Figs. 5a–c. Yet, the maps of *M* systematically display sharper plots that facilitate the identification of Lagrangian coherent structures and lobe dynamics. The reader is referred to MM10 and Mancho and Mendoza (2011, manuscript submitted to *Chaos*) for a detailed comparison of the two methodologies.

The situation described in the previous paragraphs of this subsection corresponds to the latest stages of the escape of an SPB released in the vortex edge. DLC10 also analyze another case in which the SPB was released inside the vortex. In this case, however, lobes could not be clearly identified. The relatively small number of Vorcore balloons does not allow for more examples of crossing. An inspection of the daily maps of *M* paying particular attention to the trajectories of parcels in the vortex region provides an example.

Figure 6a shows the structure of *M* on 27 September 2005, on the 475-K isentropic surface. One can clearly identify the signatures of wave breaking in the vortex periphery as represented by tongues of *M* expanding from the vortex edge to midlatitudes. In the outskirts of the vortex there are three good candidates for hyperbolic trajectories with potential to play an important organizing role on the flow. These are over South America, south of Africa, and south of Australia, from where lobes emanate and stretch into midlatitudes. We next focus on the region marked by the white circle in Fig. 6a, a closer view of which is illustrated in Fig. 6b where *M* is calculated with a higher *τ* (i.e., 20 days) for enhanced clarity. Within the jet domain of the selected region, one lobe (labeled L1) penetrating from the vortex interior intersects another lobe (labeled L2) stretching along the equatorward side of the jet. To examine whether this feature is associated with an air exchange across the jet, we calculate a set of forward and backward trajectories from initial conditions at the intersection of the two lobes (see Fig. 6b). The panels in Fig. 7 show the results of the forward and backward integrations from 27 September 2005, as displayed by the position of the parcels on selected dates superimposed on *M* maps (see also Fig. 8 to get details on the geographical locations of panels in Fig. 7). Since lobes L1 and L2 intersect on 27 September, they intersect at all times. However, they may become so distorted and elongated that they both are not simultaneously distinguished at all dates. Therefore, lobes L1 and L2 are only labeled in Fig. 7 when visible. It is also important to point out that parcels must remain within the region bounded by the lobes according to lobe dynamics theory (Wiggins 1992; Mancho et al. 2006a). In the backward integration (Figs. 7a–c), the parcels drift to the interior side of the jet within lobe L1. In the forward integration (Figs. 7d–f), parcels remain within lobe L2 drifting in time to the collar. Figure 9 displays the final positions of the parcels in the backward and forward integration (Figs. 9a and 9b, respectively), superimposed on *M* maps and on the position of the SPV edge as defined by the single PV contour where this field has the strongest gradient. The parcels are initially located in the interior side of the jet on 17 September (Fig. 9a), and they are finally expelled out into midlatitudes from the collar on 15 October (Fig. 9b). This is further confirmed by the corresponding distribution of the PV field (Fig. 10).

Our results provide the kinematic structure of the isentropic excursions of air across the vortex edge. For the late southern winter and spring of 1998, Öllers et al. (2002) compare quasi-horizontal transport across the vortex edge given by isentropic and three-dimensional (3D) trajectories. The latter are found to provide higher transport rates than the former ones. We conjecture that routes of transport such as those found in the present study will appear more frequently in a 3D analysis.

## 4. Conclusions

We have examined the Antarctic lower stratospheric flow during the southern spring of 2005 by means of a new Lagrangian descriptor, which is referred to as the function *M* (Madrid and Mancho 2009; MM10). The selected methodology provides key Lagrangian information on hyperbolic trajectories and their stable and unstable manifolds, which are responsible for transport and large-scale stirring in nonstationary flows (MM10).

We find that lobe dynamics is a persistent mechanism for transport in the collar throughout the southern spring (not shown), which is consistent with results in previous works (Bowman 2000; Joseph and Legras 2002), and illustrate this property by showing a selected event. Moreover, we identify a rich Lagrangian structure in the SPV interior and find that the vortex core is also populated by hyperbolic trajectories with a complex structure of invariant manifolds. Such a structure allows for intricate intersections of manifolds that eventually lead to effective transport of air mass across the vortex edge. We reexamine the escape of a Vorcore superpressure balloon from the jet region during the spring of 2005 already studied by DLC10. We find that lobe dynamics effectively explain such an event, and that is easily captured by the Lagrangian descriptor *M*. The successful comparison between Lagrangian analysis and observational data (Vorcore data) underlines the accuracy of the ERA-Interim velocity field. Furthermore, we present an example of full jet crossing by a set of fluid parcels. In this case, one lobe penetrating the jet from its poleward side intersects another lobe entering from the equatorward side in such a way that parcels in the common domain are eventually ejected from the interior part of the vortex edge out to midlatitudes. Although such instances exist, the associated transport across the vortex edge is confirmed to be small. In this sense, our results are in agreement with the SPV as a “containment vessel” for horizontal transport (Juckes and McIntyre 1987; Hartmann et al. 1989; Bowman 1993; Chen 1994; Öllers et al. 2002).

Koh and Plumb (2000), based on the results of a numerical quasigeostrophic, shallow water model on an *f* plane, argue that lobes do not provide a good framework for measuring transport across the vortex edge. This work, based on reanalysis data, shows that lobe dynamics can be invoked to describe transport across the jet.

## Acknowledgments

The authors are grateful to two anonymous reviewers for their constructive and useful comments that helped improve the manuscript. This research was supported by the Spanish Ministry of Science under Grants CGL2008-06295 and MTM2008-03754, Spanish CSIC under Grant ILINK-0145, U.S. NSF under Grant ATM-0732222, and U.S. ONR N000140910418. The computational part of this work has benefited from an ICTS-CESGA Project 109, which allowed priority access to supercomputer Finis Terrae.

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