## 1. Introduction

It has long been recognized that in the Southern Hemisphere the stratospheric polar vortex (SPV) edge acts as an effective barrier to air parcel crossings (e.g., Chen 1994; Beron-Vera et al. 2012). Despite this remarkable property, episodes of air crossing the edge develop sporadically (Bowman 1993; Öllers et al. 2002). The associated horizontal transport and mixing during those episodes may have nonnegligible effects on the intraseasonal evolution of the ozone hole (Sato et al. 2009). In a recent paper, de la Cámara et al. (2012, hereafter DLC12) examined the Lagrangian structure of the flow in the Antarctic lower stratosphere focusing on the mechanisms for transport across the SPV edge. The framework for DLC12 was the southern spring of 2005, during which the Vorcore field campaign released from Antarctica superpressure balloons able to drift for months at around the 50- and 70-hPa levels (Hertzog et al. 2007). The records of precisely monitored balloons' trajectories provide unique, quasi-Lagrangian information on the flow. Therefore, the principal results of DLC12 could be verified with in situ data. DLC12 found routes of transport across the entire width of the southern polar-night jet by application of a Lagrangian analysis. The suggested mechanism at work was given by occasional intersections between stable and unstable manifolds associated with hyperbolic trajectories outside and inside the SPV.

The characterization of hyperbolic trajectories, which trace out the movement of material hyperbolic points, inside the SPV provides important information on the structure of the complex nonstationary flow at this region. According to dynamical systems theory (e.g., Wiggins 1992), those Lagrangian structures play an organizing role in the dynamics: fluid parcels come nearer and move away from hyperbolic trajectories along stable and unstable manifolds, respectively, which act as material boundaries to transport because no parcels can cross them by advection processes. The intersections of the manifolds form lobes, and the mass transport across the boundaries formed by the manifolds can be explained in terms of the dynamics of these lobes (Wiggins 1992). Near hyperbolic trajectories, lobes become elongated in a filamentary structure. Thus, hyperbolic trajectories provide a relevant framework to study (Lagrangian) transport in general time-dependent flows and, in particular, in critical layers developed as planetary (Rossby) waves break (Ngan and Shepherd 1997, 1999).

The present paper builds on these concepts and applies them to examine the behavior of superpressure balloons released by Vorcore and by a follow-up project, Concordiasi, in the spring of 2010 (Rabier et al. 2010). Our focus is on the poleward (inner) side of the polar-night jet in the southern stratosphere, a region much less explored dynamically thus far compared with its counterpart outside the vortex. Our aim is the insights that Lagrangian structures of the flow provide of Rossby wave breaking (RWB) in the region. RWB processes are important contributors to quasi-horizontal air exchanges in the upper troposphere–lower stratosphere (e.g., Scott and Cammas 2002). Those processes also contribute to erode the wintertime SPV in both hemispheres through the formation of potential vorticity (PV) filaments (Polvani and Plumb 1992; Waugh et al. 1994; Abatzoglou and Magnusdottir 2007). The filamentation of PV contours leads to regeneration of the large PV gradients in the jet stream that define the vortex edge (e.g., Bowman 1993).

Linear, nondissipative wave theory provides a simple framework of RWB development. In this frame the equations describing Rossby wave propagation in a background flow that is a function of latitude only become singular at locations where the phase speed matches the background wind velocity (i.e., the critical latitude). This situation leads into the formation of a critical layer (e.g., Haynes 2003) whose mathematical description requires consideration of the nonlinear terms neglected in the wave propagation equation. If the flow is stationary in an appropriate comoving frame, the instantaneous stagnation points are at the boundaries of typical “cat's eye” structures comprising anticyclones that develop in the critical layer (Stewartson 1978; Warn and Warn 1978), where material contours are irreversibly deformed providing the RWB signature.

During winter and spring, critical layers for Rossby waves in the stratosphere develop at geographical locations of the region referred to as the “surf zone,” which covers the midlatitudes between the vortex edge and the subtropical jet (McIntyre and Palmer 1984). RWB also occurs poleward of the jet. Hitchman and Huesmann (2007) compiled an RWB climatology from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis and the Met Office (UKMO) analysis datasets and found that RWB occurs systematically on both sides of the Antarctic polar-night jet in winter and spring. Pierce et al. (1994) examined the mixing processes within the polar-night jet for the year 1992 using UKMO analysis data, and found less filamentation of PV contours (i.e., a transport barrier) in the jet region in both hemispheres at locations slightly poleward of the wind velocity maximum. The matching between phase speeds of Rossby waves and jet stream velocities is more likely in the northern than in the southern stratosphere during winter and spring because jet stream intensities cover a larger range of values in the former than in the latter hemisphere. RWB in the northern stratosphere can become vigorous enough to produce intrusions of midlatitude air into the vortex (Plumb et al. 1994), excursions of air out of the interior of the vortex (Waugh et al. 1994), and even vortex splits in major sudden warmings events (e.g., McIntyre 1982; Ayarzagüena et al. 2011). By contrast, the SPV in the southern stratosphere during winter and spring is generally much stronger and less variable, and phase speeds of Rossby waves are not as fast as to match velocities at the core of the polar-night jet (Bowman 1996). Irreversible deformation of PV contours at both sides of the jet core tends to occur separately, and associated transport across the vortex edge is very weak or even absent (Juckes and McIntyre 1987; Hartmann et al. 1989; Bowman 1993; Chen 1994; Öllers et al. 2002).

There have been several studies of RWB using numerical models. Juckes and McIntyre (1987) and Polvani and Plumb (1992), using barotropic models, found that filamentation of PV contours is more frequent outside the jet (outward wave breaking) than inside it. This special feature was attributed to the asymmetry in the rate of strain of the background flow. Using a shallow-water model, Nakamura and Plumb (1994) pointed out the existence of critical layers both outside and inside the vortex. They argued that outward breaking is more frequent because the critical latitude is generally closer to the jet in the outer than in the inner side. Mizuta and Yoden (2001) investigated chaotic mixing processes and transport barriers in the SPV using simulations of an unstable jet with an idealized barotropic model. Their results showed the existence of isolated regions of very weak mixing surrounding the core inside the vortex, which the authors explained in terms of a critical layer (stagnation points and cat's eyes) by decomposing the flow in zonal wave components.

This paper searches for evidence of RWB by examining both balloon trajectories and PV fields. We start by describing general characteristics of the flights of Vorcore and Concordiasi balloons inside the vortex. Case studies in the poleward (inner) side of the southern polar-night jet are selected for analysis, which is performed by following a two-pronged approach. First, we explore the evolution of PV filaments as given by the reverse domain-filling (RDF) technique (O'Neill et al. 1994; Sutton et al. 1994). The technique is applied to European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) products, which provide reliable data on the stratospheric temperature and velocity fields (and, hence, on PV). Second, we look for evidence of critical layers. This we do by estimating hyperbolic trajectories from the geometrical structure of the flow as described by the Lagrangian descriptor *M* (Mendoza and Mancho 2010, 2012). In both cases, we explore the consistency between the results of the analysis and the observed balloon trajectories.

The remainder of the paper is organized as follows. Section 2 provides a brief review of the basic concepts from dynamical systems theory employed in the present work. Section 3 describes the data and methodology. Section 4 describes general characteristics of the balloon trajectories during the Vorcore and Concordiasi field campaigns in 2005 and 2010. Section 5 presents evidence of RWB events inside the vortex and examines the Lagrangian structures of the flow during the events. Section 6 gives the summary and conclusions. The appendix provides a detailed description of the methodology employed in the explicit computation of invariant manifolds.

## 2. Basic concepts

*t*is time,

**x**(

*t*) represents the trajectory of a fluid parcel, and

**v**(

*t*,

**x**) is the velocity field that depends generally on the space–time coordinates. By using Eq. (1) it is implicitly assumed that no diffusion is present and that air parcels are passive scalars with no influence on the velocity field. Besides, diabatic heating/cooling in the extratropical stratosphere generally have longer time scales than those of horizontal advection. Hence, air parcels move on two-dimensional isentropic surfaces to a good approximation (they stay within ±15 K for 30 days; see section 3b) (e.g., Plumb 2007).

Solutions of Eq. (1) are represented in the phase space (the physical space in our case), where we look for geometrical objects that allow a qualitative description of the parcels' trajectories. To illustrate this we next analyze several examples increasing the complexity of the velocity fields.

### a. Transport in autonomous systems

The first example is the case of an autonomous system, in which the velocity field in Eq. (1) is time independent, **v** = **v**(**x**). Stagnation or fixed points of the flow—that is, points where **v**(**x**) = 0—are a very useful tool to understand parcels trajectories in autonomous systems. Of particular interest are the hyperbolic fixed points. A fixed point is hyperbolic if there are two sets of solutions linearly independent in its neighborhood: one comprising the trajectories that converge toward the hyperbolic point as *t* → ∞ (i.e., the stable manifold), and the other comprising the trajectories that converge toward the hyperbolic point as *t* → −∞ (i.e., the unstable manifold). This means that the behavior of fluid parcels near a hyperbolic point is as follows: parcels approach the hyperbolic point through a trajectory aligned with the stable manifold and move away by following a trajectory aligned with the unstable manifold. This perspective will help us interpret balloon trajectories near hyperbolic trajectories.

Figure 1a shows the geometry given by the hyperbolic fixed points (black circles) and their manifolds (thick lines), superimposed on the streamlines (thin lines), for the case of the cat's eye structure that appears as solution of the critical layer for Rossby waves (Stewartson 1978; Warn and Warn 1978). We can see in Fig. 1a that manifolds are at the boundary of the closed streamlines that characterize the anticyclonic eddies of the cat's eye. Since manifolds are material lines, no fluid parcel can cross them by advection processes; particularly in Fig. 1a, no parcel can be transported in and out of the eddies. Therefore, the geometric skeleton in Fig. 1a helps us understand at a glance the evolution of many fluid parcels.

This approach to Lagrangian transport is alternative and complementary to that traditionally followed by the atmospheric dynamics community, more commonly focused on the deformations of a tracer field. For example, techniques such as reverse domain filling (O'Neill et al. 1994; Sutton et al. 1994) or contour advection (Dritschel 1989; Waugh and Plumb 1994) do this job. Given an initial disposition of any tracer field (such as PV in the stratosphere), it will be deformed as dictated by the underlying structure of hyperbolic points and manifolds. Figure 1b illustrates the concept. A single tracer contour initially placed horizontally, dividing the panel in two equal parts, will be stretched and deformed as shown in Fig. 1b, stirring the air of different tracer values (different shadings in Fig. 1b) inside the eddies and providing the signature of RWB in the critical layer.

### b. Transport in periodically time-dependent systems

One way for fluid parcels to cross the edge of the cat's eye is to perturb the velocity field time periodically; that is, **v**(*t*, **x**) = **v**(*t* + *T*, **x**), with *T* being the period. In this case the hyperbolic periodic trajectories play the role equivalent to the hyperbolic fixed points in autonomous systems. In general they no longer coincide with the instantaneous stagnation or fixed points of the flow. Stable and unstable manifolds of these hyperbolic trajectories intersect and form lobes (Fig. 1c), and the dynamics of these lobes control transport in and out of the cat's eye. Consequently, although instantaneous stagnation points (in the appropriate comoving frame) have been commonly used to provide information about the critical layers for Rossby waves, they are of no relevance from the viewpoint of (Lagrangian) transport in realistic flows. Instead, hyperbolic trajectories are the appropriate feature. In this sense, Ngan and Shepherd (1999) argued that these Lagrangian tools are suitable for studying transport in the stratospheric surf zone.

### c. Transport in aperiodically time-dependent systems

The tools and concepts just described do not translate directly to aperiodically time-dependent systems, which is the relevant situation for applications to geophysical flows such as the atmosphere. In this framework, some studies have proposed definitions of distinguished hyperbolic trajectories (Ide et al. 2002; Madrid and Mancho 2009) and methods to compute their manifolds (Mancho et al. 2003, 2004). The explicit calculation of hyperbolic trajectories in aperiodic systems is a challenging task, however, and alternative methods have been developed such as finite versions of the Lyapunov exponents (Aurell et al. 1997; Haller 2000, 2001), Lagrangian descriptors (Mendoza and Mancho 2010, 2012), complexity methods (Rypina et al. 2011), and several others.

In the present paper we use the Lagrangian descriptor referred to as the function *M* (Mendoza and Mancho 2010, 2012; Mancho et al. 2013) to extract the geometric structure of the flow. The function *M* measures the arc length of trajectories; that is, it integrates a parcel property, the modulus of its velocity, along trajectories (more details in the next section). A heuristic argument on why *M* displays manifolds is as follows. The function *M* will present abrupt changes in space when the arc lengths of nearby trajectories are very different. This will occur at the boundary of regions bounded by manifolds in which trajectories have different dynamical fates. Figure 1d shows the geometric structure given by *M* for the perturbed cat's eye. A visual comparison with Fig. 1c evidences that the manifolds are well depicted by a discontinuity in the derivative of *M* (see also Mendoza and Mancho 2010, 2012). Mancho et al. (2013) prove rigorously in specific examples that lines of sharp gradients of *M* are aligned with the manifolds.

## 3. Data and methods

### a. Data

Three different databases are used in this study. The first two are in situ information from the superpressure balloons (SPBs) released by the Vorcore and Concordiasi field campaigns in the southern spring of 2005 and 2010, respectively. The 27 SPBs launched by Vorcore from McMurdo station (77.85°S, 166.67°E), Antarctica, drifted in the lower southern stratosphere on isopycnic surfaces near the 50- and 70-hPa pressure levels for an average period of 2 months. Instruments on board the SPBs recorded pressure, temperature, and position. Hertzog et al. (2007) provide a thorough report of the Vorcore campaign, and further information is given in the project's website www.lmd.ens.fr/STRATEOLE/. The 19 SPBs launched by Concordiasi also from McMurdo, Antarctica, were equipped with more sophisticated instruments than their predecessors. This allowed for increasing frequency of meteorological observations from 15- to 1-min intervals, among other improvements. Rabier et al. (2010) provide details on the Concordiasi project and the project website is www.cnrm.meteo.fr/concordiasi.

Hertzog et al. (2007) discussed the performance of SPBs as tracers of air parcels' motions. The design of such balloons minimizes the effect on flight level of the radiative heating/cooling owing to daily variations of solar radiation and to stay on a particular isopycnal. Hertzog et al. (2007) estimated that the difference between the balloons and wind horizontal velocities (averaged every 15 min) are not larger than 0.05 m s^{−1}, and SPBs can therefore be considered accurate tracers of horizontal motions. In contrast, the isopycnal restriction implies that SPBs cannot follow vertical motions of air parcels accurately. Hertzog et al. (2007) estimated that vertical displacements under the assumption of adiabatic motions are typically 4 times as large as those experienced by the balloons.

The third dataset we use is the four-times-daily isentropic fields at 475 K (lower stratosphere) from ERA-Interim (Dee et al. 2011). These fields are available on a horizontal grid with resolution 1.5° longitude × 1.5° latitude.

### b. Isentropic trajectories

As in many studies 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; DLC12) we will assume adiabatic flows; that is, that fluid parcels remain on the same isentropic surfaces during the interval of calculation. At this stage, we pause to address the limitations of the isentropic approximation and its validity to capture the isopycnal motions of superpressure balloons.

First, several studies have demonstrated the relevance of cross-isentropic motions on transport in the Antarctic polar vortex. For example, Mariotti et al. (2000) showed that in the periphery of the lower Antarctic vortex, ozone loss due to large-scale stirring on an isentropic surface is almost balanced by downward, diabatic transport of ozone in the region. For the late southern winter and spring of 1998, Öllers et al. (2002) found that three-dimensional trajectories provide higher transport rates across the vortex edge than isentropic trajectories. Typically, diabatic heating rates in the Antarctic lower stratosphere during spring are on the order of 0.5 K day^{−1}, although uncertainties in this magnitude remain large (e.g., Fueglistaler et al. 2009). During the time interval of our calculations of isentropic trajectories (~30 days), the material surface would experience an increase of potential temperature of around 15 K. Nevertheless, calculations of *M* (see below) using wind fields at 475 and 530 K have produced qualitatively similar results (not shown). This suggests that horizontal motions of the parcels will be affected by similar geometric structures at those isentropic levels and that the isentropic approach can be used in our problem.

Second, Hertzog et al. (2007) showed that isopycnic and isentropic surfaces remain fairly parallel during September and December 2005, but not quite so during October and November. In section 5, however, we present results that highlight the good agreement between both RDF and *M* calculations and the balloons' motions, which confirms the validity of the isentropic approximation to study balloon motions.

In our RDF and *M* computations the initial conditions for parcel trajectories are taken on a ⅓° longitude × ⅓° latitude reference grid south of 20°S on the isentropic surface 475 K, and a fourth-order Runge–Kutta scheme is applied to numerically solve Eq. (1) with a time step of 1 h. We work on three-dimensional Cartesian coordinates to avoid the singularity at the pole in spherical coordinates, but the parcel motions are restricted to the two-dimensional isentropic surface, which is a 2D sphere. Interpolation of the velocity field to parcel locations is done using a bicubic and cubic splines schemes in space and time, respectively (Mancho et al. 2006). The integration time *τ* needed to obtain useful information with RDF maps differs from the one needed with *M*. For the former we have used *τ* = 11 days, and for the latter *τ* = 15 days (see section 3d).

### c. The reverse domain filling technique

The RDF technique (O'Neill et al. 1994; Sutton et al. 1994) is a trajectory-based method used to analyze small-scale structures of a tracer field. The RDF calculation of the PV field in the present paper uses ERA-Interim in order to identify filamentation of the polar vortex associated to RWB (McIntyre and Palmer 1983). Under the assumption that PV is conserved along the trajectory of the air parcels, there are three steps to the RFD method. First, we compute two-dimensional backward trajectories from *t*_{0} to *t*_{0} − *τ*, starting from conditions distributed in a regular longitude–latitude grid. Second, we spatially interpolate the PV values to the final positions of the parcels at time *t*_{0} − *τ*. Last, we display those interpolated in the original regular grid at time *t*_{0}.

### d. The Lagrangian descriptor M

*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}), and

*z*

_{0}=

*z*(

*t*

_{0}) (Madrid and Mancho 2009). The descriptor measures the length of trajectories passing through (

*x*

_{0},

*y*

_{0},

*z*

_{0}) at

*t*=

*t*

_{0}over a range of times centered at

*t*=

*t*

_{0}: [

*t*

_{0}−

*τ*,

*t*

_{0}+

*τ*]. Calculation of

*M*, therefore, implies the integration of backward trajectories from

*t*

_{0}to

*t*

_{0}−

*τ*and of forward trajectories from

*t*

_{0}to

*t*

_{0}+

*τ*. The function

*M*as defined in Eq. (2) is able to provide a global dynamical picture of any arbitrary time-dependent flow. That is,

*M*can identify geometrical regions on the physical space (i.e., the isentropic surface) with different dynamical fates. As noted by Mendoza and Mancho (2010, 2012) and Mancho et al. (2013), convergence of the structure of

*M*toward the stable and unstable manifolds requires a large enough value of

*τ*. Following DLC12, we choose

*τ*= 15 days, so the total time interval for computing

*M*is 2

*τ*= 30 days.

Figure 2a displays the structure of *M* on the 475-K isentropic surface on 8 October 2005. As explained in section 2, manifolds align with lines of sharp color gradients in *M*. Crosshatched patterns are indicative of the manifold crossing and hence of the presence of hyperbolic trajectories (Mendoza and Mancho 2010) since these are at the intersection of the stable and unstable manifolds. In some special cases, these trajectories play key roles in organizing the geometry of the flow, and are referred to as “distinguished” (Ide et al. 2002; Madrid and Mancho 2009). In the present paper, we will simply refer to them as hyperbolic trajectories. Also, we will refer to the location of the hyperbolic trajectory at an instance as hyperbolic point.

In Fig. 2a three different hyperbolic points are labeled Q1, Q2, and Q3 at the intersections of lines of *M*. It is interesting to note the high values of *M* (red colors) signal the location of the polar-night jet where the trajectories lengths are larger. The smooth local structure of *M* means that manifolds do not usually cross the jet, thus highlighting its insulating role.

The interpretation of crossings of *M* lines as points belonging to hyperbolic trajectories is next demonstrated. Figure 2b shows portions of the stable and unstable manifolds explicitly computed using the same technique as Mancho et al. (2004; see also the appendix) in the neighborhood of Q3 in Fig. 2a, superimposed on *M*. The intersection of the manifolds clearly coincides with Q3, and the path described by the intersection of the manifolds—which is a trajectory by definition—matches well with the successive locations of Q3 (not shown). Higher color contrasts in Fig. 2b than in Fig. 2a around Q3 arise adjusting the color code to the values of *M* plotted in each panel. The difference in color code used in the panels of Fig. 2 is not relevant because the information given by *M* is encoded in sharp color changes.

It is worth mentioning that just analyzing a single map of *M* does not suffice to determine whether manifolds are stable or unstable. In the present paper we have determined the stability or instability of the manifolds by studying the time evolution of *M* maps.

### e. Identification of the stratospheric polar vortex

Since the seminal work by McIntyre and Palmer (1984), the SPV edge is generally identified in terms of PV isolines, across which the PV field has large horizontal gradients. Some works on stratospheric dynamics refer to the domain bounded by two such PV isolines as an SPV edge region (e.g., Fairlie and O'Neill 1988; Trounday et al. 1995; Öllers et al. 2002). Dahlberg and Bowman (1994) defined the vortex edge as the PV contour with the minimum flux of parcels across it. In the present paper we identify as the PV contour with the strongest horizontal PV gradient on the 475-K isentropic surface. As we will see, this straightforward definition is sufficiently precise for the objectives of our study.

## 4. Superpressure balloons inside the stratospheric polar vortex

Figure 3 shows the locations of the Vorcore and Concordiasi SPBs launched from McMurdo station superimposed on PV maps at 475 K on 2 days during the experiments. On 28 October 2005, the PV field depicts a vortex that elongates toward the South Pacific (Fig. 3a), while on 26 October 2010 the vortex is less disturbed and centered approximately over East Antarctica (Fig. 3b). These two situations are typical in October during the austral stratospheric final warming (e.g., Mechoso et al. 1988). Figure 4 shows the time series of the zonal-mean poleward eddy heat flux at 100 hPa averaged over 45°–75°S for the September–November periods (SON) of 2005 and 2010, which provides a good measure of the injection of tropospheric wave activity into the stratosphere (e.g., Hu and Tung 2003; Ayarzagüena et al. 2011). Negative values in Fig. 4 indicate poleward (southward) eddy heat flux. A higher number of episodes with strong and negative values will be associated with a more perturbed polar vortex. In 2005 there are three strong wave events at the beginning of October, mid-to-late October, and early-to-mid-November. In 2010, by contrast, there is only one event, at the beginning of September, with poleward eddy heat flux stronger than one standard deviation above the climatological mean, and another two weaker events in the first weeks of October and November. Therefore, it appears that the vortex experienced higher large-scale variability in spring 2005 than in 2010, which translates into more elongated shapes and more displacements off the pole in 2005 than in 2010, in the way shown in Fig. 3. De la Cámara et al. (2010) and DLC12 reported the transport of two Vorcore balloons out of the SPV during SON 2005, which occurred toward the end of the period. No Concordiasi balloon escaped from the SPV interior during SON 2010.

To characterize the behavior of an SPB in relation to the SPV, we calculate the distance between the vortex edge (as defined in section 3e) and the successive balloon locations from September to November (2005 in the case of Vorcore and 2010 in the case of Concordiasi). Figures 5a and 5b show the mean distance to the vortex edge in degrees of arc of each individual balloon, along with the standard deviation of the distance, for Vorcore (Fig. 5a) and Concordiasi (Fig. 5b) SPBs. In both campaigns, the mean distance lies between 5° and 15°, with a typical standard deviation of 5°. Figure 5c shows the distribution of the distance of Vorcore and Concordiasi balloons to the vortex edge in 2.5°-wide belts during SON of each campaign. A larger percentage of Vorcore balloons remains closer than 10° from the vortex edge. The statistical mode is also centered closer to the edge for Vorcore (on 7.5°) than for Concordiasi SPBs (on 10°). Although the metric used (i.e. distance to the edge) can be influenced by the shape of the vortex (and this in turn may explain the differences in Fig. 5c between Vorcore and Concordiasi balloons), we can see in Fig. 5 that the relatively small number of balloons launched by Vorcore and Concordiasi (27 and 19, respectively) achieved a large sampling coverage of the SPV interior as also reported by Hertzog et al. (2007) (see also www.cnrm.meteo.fr/concordiasi). This supports our use of the balloon data for both raising conjectures and testing consistency on the dynamics of the stratospheric polar vortex during its final warming, as described in the next section.

## 5. Case studies of Rossby wave breaking inside the vortex

Planetary (Rossby) wave breaking is very likely involved in the events during which balloons were transported between different regions within the stratospheric polar vortex. In this section we analyze RWB events evidenced by the balloons trajectories, PV field filamentation, and Lagrangian coherent structures of the flow.

In the linearized theory of waves superimposed on a background flow depending on latitude only, fluid parcels behave differently according to their locations in reference to the critical latitude (i.e., where the wave phase speed matches the background velocity). Away from the critical latitude they oscillate in reference to their unperturbed latitude. Near the critical latitude the wave breaks, and parcel trajectories become more complicated in a critical layer whose width depends on the amplitude of the breaking wave and the magnitude of the background shear. In the real stratosphere, drifters such as balloons do not necessarily experience large latitudinal displacements when involved in a RWB event in the interior of the SPV. Hence, in this subsection we search for RWB episodes analyzing both SPB trajectories (i.e., latitudinal displacements with respect to the vortex edge taking place on time scales of a few days) and RDF calculations of PV fields (i.e., development of filaments in the inner side of the jet stream).

The detailed examination of the Vorcore and Concordiasi SPB trajectories revealed several examples of balloons inside filaments of PV that elongate from the inner flank of the jet into the vortex interior. The results are summarized in Table 1. From this examination we selected several cases for further study.

Superpressure balloons involved in events of irreversible filamentation of the RDF-PV field, and time range of the events.

### a. Rossby wave breaking inside the vortex captured by RDF-PV

In the first two case studies, one from each campaign, the balloons do not experience large displacements with respect to the vortex edge, remaining close to it during the event (~10° of spherical distance or closer according to Figs. 6a and 6b). The first case (case 1) involves the Vorcore balloon labeled VOR4 and covers roughly the time period 16–19 October 2005. During these days, the wave activity entering the lower stratosphere is intensifying, as seen in Fig. 4. The second case study (case 2) implicates the Concordiasi balloon labeled CON19 covering the time period 10–14 November 2010, when the wave activity coming from the troposphere seems to be declining (Fig. 4). Filaments of PV contours due to RWB tend to become longer and thinner and to become undetectable in any Eulerian representation with limited horizontal resolution (McIntyre and Palmer 1984). Consequently, PV maps obtained directly from the reanalysis dataset may not be suitable to visualize filamentation due to wave breaking. To overcome this difficulty, we use the RDF method.

Figure 7 shows RDF calculations of PV (RDF-PV) on the 475-K surface together with the positions of the SPBs on 3 selected days during case 1 (Figs. 7a–c) and case 2 (Figs. 7d–f). Let us start with case 1, focusing on the evolution of the balloon VOR4 (white circle). The arrow in Fig. 7a points to a PV filament that elongates from around 70°E westward to 0° in the inner flank of the vortex edge, and that stays almost parallel to it. Balloon VOR4 is located near the vortex edge in the region from which the PV tongue stretches. During the following 2 days (Figs. 7b,c), both VOR4 and the structures depicted by RDF-PV rotate clockwise, and the PV tongue continues elongating and narrowing, while VOR4 drifts inside it.

Figures 7d–f shows selected days during case 2. In Fig. 7d, CON19 (white circle) is at 70°S, 60°E located inside a PV tongue that emerges from the edge into the interior of the vortex and that roughly extends from 120° to 50°E (pointed out by an arrow). During the following 3 days (Figs. 7e,f), the system rotates clockwise and the PV tongue is stretched in the inner periphery of the edge while CON19 remains in it. In neither of these two events do the balloons travel deep into the vortex core during the RWB events.

In all cases highlighted in Table 1, the balloons' motions match well with the evolution of PV filaments that appear in the inner flank of the jet. The two examples in Fig. 7 show that the small-scale structures of the PV field obtained from RDF calculations are consistent with the balloon trajectories in the lower stratosphere. Thus we are confident in concluding that RWB events—as characterized by irreversible deformation of PV contours as described by the RDF method—contribute to transport and stirring of air inside the vortex.

We next address the identification and role of hyperbolic trajectories in case 1 and case 2. Figure 8 shows the locations of VOR4 and CON19 (white circles in Figs. 8a–c and 8d–f, respectively) on *M* maps at 475 K on the same days as Fig. 7, along with all other SPBs flying at those times (magenta circles). There are large values of *M* (reddish colors) along the jet stream at the vortex edge. In the vortex interior (see Fig. 8) *M* displays structures that clearly resemble lines crossing at the points labeled H1 in case 1 (Figs. 8a–c) and H2 in case 2 (Figs. 8d–f). As explained in section 3d, we interpret H1 and H2 as hyperbolic points that describe paths belonging to hyperbolic trajectories and lines of sharp color change in the *M* field as their stable and unstable manifolds.

As expected, H1 and H2 are very helpful in the interpretation of air movements inside the vortex. In case 1 starting on 16 October 2005 (Fig. 8a), VOR4 comes near H1 (Figs. 8b,c) following a contour of *M* that passes through the hyperbolic point (i.e., a stable manifold). In case 2, CON19 is first located very near H2 (Fig. 8d), and moves away from it following inside a lobe that elongates from H2 (Figs. 8e,f). Taken together, Figs. 7 and 8 reveal that in both cases the presence of H1and H2 plays a role similar to that of the hyperbolic point in Fig. 1b with respect to the scalar field. They produce the elongation of the PV in filaments. In this respect, it is interesting to note that hyperbolic points seem to move faster than the balloons in these two cases (Fig. 8), exemplifying the different velocities of the air parcels near a hyperbolic trajectory that lead to the filamentation of the PV.

These results may be relevant to the slow recovery of Antarctic ozone content inside the SPV before the final breakdown, since the jet region is relatively rich in ozone owing to the diabatic descent of air in the area (e.g., Mariotti et al. 2000). Sato et al. (2009) estimated the contribution of lateral mixing in the recovery of ozone content at 20 km during early spring 2003 at 17% ± 4% of the contribution of the diabatic downwelling of the Brewer–Dobson circulation in the vortex core.

In most of the cases selected in Table 1, RWB seems to be “peripheral” in the sense that filaments of PV remain parallel and close to the vortex edge and the balloons are not transported very deep inside the vortex. We next analyze a case in which the transport from the edge reaches deep into the vortex core. This case (case 3) involves the Vorcore balloon VOR10, and roughly covers the period 4–12 October 2005. During these days, the southward heat flux at 100 hPa is declining from large values (Fig. 4). As shown in Fig. 6c, the minimum distance at 475 K between the VOR10 and the vortex edge is about 10° until 4 October 2005. Thereon, the distance starts increasing almost systematically up to more than 20° in 1 week. Such a behavior corresponds to the balloon drifting near the inner flank of the jet and then transported rapidly into the vortex core.

Let us examine the description given by RDF-PV (Figs. 9a–c). The arrow in Fig. 9a points to a PV filament that elongates from around 60°E westward to 45°W in the inner flank of the vortex edge, and that stays almost parallel to it. Balloon VOR10 (white circle) is located at 78°S, 45°W, that is, on the limit where the mentioned filament flips deeper into the vortex. Two days later (Fig. 9b), the system has rotated about 100° clockwise. The PV tongue has stretched farther and VOR10 is clearly inside the part of the filament that penetrates the vortex core. In Fig. 9c, the balloon is already well inside the vortex, but the PV tongue is hardly visible. This means that the air inside that filament comes from a region with weak PV gradients.

We now turn to the information given by *M*. H3 labels the intersection of *M* lines (Figs. 9d–f), which is interpreted as a hyperbolic point. VOR10 (white circle) locates very near H3 as shown in Fig. 9d. Starting on the first days of October 2005, VOR10 drifts into the vortex core (Figs. 9d–f) following a contour of *M* that passes through H3 (i.e., an unstable manifold, in light blue where VOR10 is placed). In this case 3, therefore, the balloon does travel into the vortex core while involved in a RWB process. A comparison between the panels in the two columns of Fig. 9 also reveals that the region where a PV filament flips into the core of the vortex is a hyperbolic region marked by the location of H3.

The hyperbolic points analyzed so far, therefore, are in special regions in reference to the filamentation of PV contours associated to RWB. In this sense hyperbolic points might be regarded as key organizing structures in the critical layer for Rossby waves, taking the role that stagnation fixed points usually have in the framework of a steady zonal background flow and a monochromatic zonal wave breaking (e.g., Haynes 2003). Scott and Cammas (2002) applied these concepts using reanalysis data to study RWB intensity in the subtropical tropopause. They did so by exploring the locations of stagnation points in the comoving velocity field (i.e., the velocity field relative to the phase speed of the breaking wave). Following such an approach requires previous knowledge of the breaking wave, such as its wavenumber and phase speed, which is not simple in our problem. The kinematic-based method for identifying hyperbolic trajectories, on the other hand, does not require any extra assumptions on the dynamics of the flow (i.e., mean flow plus zonal waves with relatively small amplitudes) or on the properties of the breaking waves, and thus its application is straightforward.

One may be tempted to infer a local phase speed from the velocity of the hyperbolic point. This would be accurate in the case of a simple flow, or even in the real stratosphere outside the vortex where RWB events are associated with specific larger-scale waves. For example, Joseph and Legras (2002) found instantaneous stagnation points—just by subtracting a solid body of rotation to the streamfunction field—in the region outside the vortex where they identified hyperbolic trajectories. In our case studies of RWB events inside the vortex, however, there is no appropriate comoving frame in which saddle structures in the streamfunction field clearly emerge. Our interpretation is that such events do not involve a single, monochromatic wave, but a complex interaction of waves with different phase speeds and wavenumbers (i.e., wave packets). The fact that most of the events identified in Table 1 (except case 1) occur in periods when the heat flux at 100 hPa (Fig. 4) is either decaying or fairly inactive suggests that declining wave activity coming from the troposphere may be relevant for the growth of such wave packets. Thus, on the basis of our case studies, we are not able to assign a phase speed to a particular wave from a hyperbolic point velocity.

### b. Rossby wave breaking inside the vortex not captured by RDF-PV

From the dynamical systems perspective adopted in the present study, we can identify transport events similarly as it was done in Table 1, but restricting consideration to the balloons' motions with respect to hyperbolic points and their invariant manifolds. Table 2 shows the results for the events not easily connected to filamentation of the RDF-PV field, and hence that are not in Table 1. There are two possible explanations for why these events are identified by *M* but not by the RDF-PV field: 1) RDF calculations of any tracer field (i.e., PV in our case) only capture filamentation in regions with relatively large tracer gradients, whereas Lagrangian structures do not depend on any tracer field, and/or 2) tracer tongues and filaments from RDF calculations tend to align with the unstable direction. Although we have analyzed a counterexample with case 1, the RDF may have difficulties in capturing transport along the stable direction (i.e., approaching a hyperbolic point). Let us further discuss this hypothesis. In such situations, a balloon would systematically approach the hyperbolic point along the stable direction, and subsequently move away from it along the unstable direction. Our analysis of Vorcore and Concordiasi flights suggests that this process lasts at least 1 week inside the vortex. In contrast, the dispersion of fluid parcels and Vorcore balloons away from hyperbolic points outside the vortex, as reported by Joseph and Legras (2002) and DLC12, is much faster. This is indicative of a higher stretching rate of material contours outside the vortex. The corresponding difference in time scales, along with the short existence of hyperbolic points, can explain the observations of balloons transport events either along the stable or unstable direction alone.

As in Table 1, but for events identified with *M*.

Figure 10 illustrates two of these events not captured by the RDF-PV approach, the first one involving Vorcore balloons VOR23 and VOR24 in mid-November 2005 (case 4), and the second one involving Concordiasi balloon CON18 in mid-to-late October 2010 (case 5). In each event, the hyperbolic points of interest are labeled H4 and H5, respectively. The vortex in mid-November 2005 during case 4 is very elongated and displaced off the pole (Figs. 10a–c). According to Fig. 10a, VOR23 and VOR24 approach H4 from opposite directions but along the same line of *M* (i.e., a stable manifold) that crosses H4. Notice that the line of *M* in Fig. 10a presents different values (colors) in the different branches where VOR23 and VOR24 are located. A few hours later (Fig. 10b), VOR24 has followed the unstable direction and is “trapped” in a region delimited by marked lines of *M* (reddish colors), while VOR23 is getting closer to H4. In Fig. 10c, VOR24 remains in the same region, and VOR23 is now located very close to H4. A couple of days later, VOR23 drifts along the unstable direction going away from H4 and approaching VOR24 (not shown). Overall, the motions of all balloons displayed are consistent with the dispositions of the lines of *M*, very vividly highlighted in this example.

The vortex is much less perturbed during case 5 in mid-to-late October 2010 (Figs. 10d–f). CON18 is initially located in the region of lower values of *M* (bluish colors) in the vortex core (Fig. 10d) and approaches the inner flank of the jet along a contour of *M* (i.e., a stable manifold, in relatively light blue) that passes through H5 (Figs. 10e,f). The whole sequence of Figs. 10d and 10f covers around 10 days, and although it is very difficult to follow the motion of CON18, our concern here is to make clear that the balloon is initially in the vortex core and 10 days later has reached the inner flank of the jet getting close to a hyperbolic point. The behavior of this balloon, thus, is the opposite of that described in case 3. While VOR10 was transported from the inner side of the jet to the vortex core in case 3, CON18 is transported from the core to the inner periphery of the edge in case 5. It is interesting to note again that, as for the events identified in Table 1, most of the events in Table 2 occur in periods when the southward heat flux at 100 hPa (Fig. 4) is either decaying or inactive.

The behavior of the observed balloons inside the SPV in the vicinity of hyperbolic trajectories and their manifolds is similar to that of two Vorcore balloons in the outer periphery of the vortex described by de la Cámara et al. (2010) and DLC12. In their case study, the balloon escapes from the outer periphery of the vortex through a tongue of PV that develops in a region where a hyperbolic trajectory is located during a wave breaking event. Our results in the present paper, supported by in situ balloon data, illustrate 1) that hyperbolic points are related to RWB inside the vortex and 2) the way in which they influence mass transport between the jet and the vortex core.

Plumb et al. (1994) found, in three case studies of intrusions of air inside the northern SPV, quasi-stationary anticyclones extending throughout the troposphere and the stratosphere up to 10 hPa. Those anticyclones caused strong and localized vortex ridging, which appeared to be important for air intrusions into the lower SPV. The perturbations in the Southern Hemisphere are much weaker than those in the Northern Hemisphere and exchange of air from both sides of the jet and, in particular, intrusions of midlatitude air into the SPV are very rare (e.g., Trounday et al. 1995). Instead, we have shown that air exchange does occur between the interior flank of the jet and the vortex interior.

## 6. Summary and conclusions

We have examined the evidence of Rossby wave breaking in the interior of the Antarctic polar vortex during the southern spring seasons (September–November) of 2005 and 2010. Our methodology used reverse domain-filling calculations of potential vorticity fields (RDF-PV) and Lagrangian coherent structures of the flow given by the descriptor *M*, based—in both instances—on isentropic trajectories obtained from the velocity fields in the ERA-Interim. The results were contrasted with observed isopycnic balloon trajectories from the Vorcore and Concordiasi field campaigns (2005 and 2010, respectively), which released respectively 27 and 19 isopycnal balloons from McMurdo, Antarctica, to drift for several months in the lower southern stratosphere. Even though the number of balloons released is relatively small, we found it sufficient to both raising conjectures and testing insights into the dynamics of the stratospheric polar vortex during its final warming. All balloons except for two drifted inside the polar vortex during the September–November period (Hertzog et al. 2007; De la Cámara et al. 2010; DLC12). We started by describing general characteristics of the balloons' locations with respect to the vortex edge. Overall, in Vorcore a larger percentage of SPBs remained nearer the vortex edge than in Concordiasi.

We looked for the signature of Rossby wave breaking in filaments of PV at the 475-K isentropic level that elongates from the vortex edge poleward and identified events in which balloons were located inside the filaments. Three cases were first selected for further analysis, all of which are captured by the RDF-PV. In both case 1 and case 2, either a Vorcore (VOR4) or a Concordiasi balloon (CON19), respectively, stayed relatively close to the jet stream while involved in the RWB events. By contrast, VOR10 in case 3 experienced an unusually large displacement from the poleward flank of the jet toward the vortex core in about 1 week. In all three cases, the evolution of the PV filament and the path followed by the balloons were entirely consistent with each other.

In these three cases, we explored the Lagrangian structures of the flow at the 475-K level. Our methodology relied on *M* (Mendoza and Mancho 2010, 2012; Mancho et al. 2013), which we introduced in the context of stratospheric circulation in a previous paper (DLC12). In the three events, we found in the interior flank of the jet outstanding structures in the *M* fields that carry the signature of hyperbolic trajectories. In dynamical systems theory hyperbolic trajectories are the generalization to aperiodic flows of the concept of stagnation points in stationary flows, and hence play an important organizing role in fluid parcel behavior. We showed that the selected balloons drifted either away from or toward the hyperbolic trajectories in the interior flank of the jet along the corresponding manifolds.

We further studied two cases not captured by the RDF-PV. In these cases, we showed that the dynamical systems approach (*M* function) can still provide information and argued that there are two possible reasons: 1) RDF calculations of any tracer field only capture filamentation in regions with relatively large tracer gradients, and/or 2) tracer tongues from RDF calculations tend to align with the unstable manifolds. If the transport event occurs in the stable direction, RDF will not capture it. Each one of the two cases of this type was selected from each campaign. Case 4 involved the balloons VOR23 and VOR24 and their motions in the very perturbed vortex in mid-November 2005. The other event (case 5) was the transport of a Concordiasi balloon (CON18) from the jet core toward the poleward flank of the jet in about 1 week in mid-to-late October 2010.

We found hyperbolic points in the region within the vortex from where PV filaments elongated. Therefore, we argued that hyperbolic points act as key organizing structures for Rossby wave breaking in the vortex interior. This close link between RWB and hyperbolic points could be elevated to a cause-and-effect relationship in large scales, since RWB is the dynamical process of nonlinear differential advection that leads to hyperbolic points. It is worth highlighting that such Lagrangian (geometrical) structures are based on the kinematics of the flow and allow studying transport in the SPV without making any extra assumptions on the velocity field (apart from assuming adiabatic and frictionless motions), which facilitates the interpretation of the dynamics involved. We further illustrated how the combination of the dynamical systems approach with reverse domain-filling calculations of PV from the ERA-Interim contribute to finding order in the apparent chaotic behavior of fluid parcels in the stratospheric polar vortex (and hence stratospheric balloons) and to identify the existence of RWB. Finally, the excellent agreement between the observed balloons' paths and both RDF-PV fields and the Lagrangian structures obtained from ERA-Interim confirms the quality of this dataset to perform trajectory-based studies of the stratosphere in the extratropics.

The authors thank an anonymous reviewer for insightful suggestions to a former version of the manuscript and two anonymous referees reviewing the present manuscript for their useful comments. This research was supported by the Spanish Ministry of Science under Grants CGL2008-06295, MTM2011-26696, ICMAT Severo Ochoa Project SEV-2011-0087, Spanish CSIC under Grant ILINK-0145, US NSF under Grant ATM-0732222, and US ONR N000140910418 and N000141010655. The computational part of this work has benefited from an ICTS-CESGA Project 109, which allowed priority access to supercomputer Finis Terrae. The first author was supported by the FPI-UCM fellowship program. Concordiasi was built by an international scientific group and is currently supported by the following agencies: Météo-France, CNES, IPEV, PNRA, CNRS/INSU, NSF, NCAR, Concordia consortium, University of Wyoming, and Purdue University. ECMWF also contributes to the project through computer resources and support and scientific expertise. The two operational polar agencies PNRA and IPEV are thanked for their support at Concordia station. Concordiasi is part of the THORPEX-IPY cluster within the International Polar Year effort.

# APPENDIX

## Computation of Unstable and Stable Manifolds of Hyperbolic Trajectories

The method we used to compute unstable and stable manifolds is similar to that presented by Mancho et al. (2004), except for some parts of the algorithm having been adapted to calculation on spherical surfaces (Dritschel 1989) to approximate atmospheric motions in the stratosphere. This appendix concentrates on unstable manifolds because the stable ones are obtained in the same way except for inverting the time direction.

*t*in a discrete set of time increments [

_{k}*t*,

_{k}*k*= 0, …,

*N*], the manifold is represented by a discrete set of points or nodes

**x**

_{j}determined by application of the following procedure. We start by considering a segment at

*t*

_{0}aligned along the unstable subspace of the hyperbolic trajectory, which in our case is determined from the features of

*M*(i.e., the point at the intersection of lines of

*M*forming an “X” structure that separate regions with markedly different colors). The segment at time

*t*

_{0}is represented by a discrete set of points. These points are evolved as trajectories of fluid parcels until time

*t*

_{1}. As they evolve they may grow apart, giving rise to unacceptable large gaps between adjacent points on the manifold. These gaps are considered to be unacceptable gaps if

*σ*=

_{j}*e*> 1, where

_{j}ρ_{j}*e*is the distance between nodes

_{j}**x**

_{j}and

**x**

_{j+1}, and

*ρ*is density of points along the computed manifold defined as the minimum value between

_{j}*L*is a typical length scale fixed to 1, and

*a*= ⅔. The parameter

*μ*controls the overall point density along the manifold and needs tuning for individual problems. Small values of

*μ*correspond to a high point density. In our computations

*μ*is fixed to 0.01. The quantity

*κ*in Eq. (A1) is defined in terms of

_{j}*ζ*serves in Eq. (A2) as a small-scale cutoff distance for resolving manifold details that we have fixed to 10

^{−6}.

*t*

_{1}is unacceptable according to the criteria just defined, a node is inserted between those in the previous time using an appropriate interpolation technique and advected until time

*t*

_{1}. For the interpolation, we use the scheme due to Dritschel (1989). This method represents the curve between points

**x**

_{j}and

**x**

_{j+1}aswhere 0 ≤

*p*≤ 1, and

*t*

_{1}and the procedure is iterated until there are no unacceptable gaps. Once this is achieved at

*t*

_{1}, we use the point redistribution algorithm described in Dritschel (1989) in an attempt to remove points from less computationally demanding parts of the manifold (Mancho et al. 2004). To describe this algorithm, let

*n*be the number of nodes at

*t*

_{1}:and define

*ñ*= [

*q*] + 2 (i.e., the nearest integer to

*q*plus 2). During redistribution, the end points of the manifold are held fixed. The

*n*− 2 “old” nodes between the end points will be replaced by

*ñ*− 1 entirely new nodes in such a way that the spacing of new nodes is approximately consistent with the desired average density, which is controlled by the parameter

*μ*. Let

*i*= 2, …,

*ñ*are found successively by seeking for each successive

*j*and

*p*such thatand placing each new node

*i*between the old nodes

*j*and

*j*+ 1 at the position

**x**(

*p*) given in Eq. (A3). The complete procedure to evolve the unstable manifold from

*t*

_{0}to

*t*

_{1}is repeated for successive times

*t*

_{k}_{−1},

*t*until the end time

_{k}*t*is reached.

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