Over the past several decades the mathematical theory of dynamical systems has provided a fruitful framework to describe the transport and mixing processes that take place in fluids and to understand the underlying flow structures associated with these phenomena. The seminal paper by on chaotic advection sparked interest in this perspective, which is inspired by the work of Poincaré. For the understanding of particle dynamics, Poincaré sought a geometrical approach that was based on geometrical structures and their role in organizing all trajectories into regions corresponding to qualitatively different dynamical fates. These structures have been referred to as Lagrangian coherent structures (LCS) in the fluid mechanics community .

Many studies of LCS in the atmosphere and in the ocean have been performed in a two-dimensional (2-D) scenario. This is because in an appropriate range of space scales and timescales a Lagrangian property of the particles is approximately unchanged in time. Hence, the flow can be assumed to occur on surfaces on which that property is constant. For instance, stratospheric flows on the timescale of stratospheric sudden warmings (∼ 10 days) are, to a first approximation, adiabatic and frictionless, and thus fluid particles and their trajectories are constrained to remain on surfaces of constant specific potential temperature (isentropic surfaces). and have examined transport processes across the Antarctic stratospheric polar vortex (SPV) on isentropic surfaces, which are quasi-horizontal in the atmosphere. Also, for oceanic flows it is often assumed that fluid parcels remain on surfaces of constant density (isopycnals), which are quasi-horizontal. , and followed the isopycnal approach for oceanic applications in the Mediterranean Sea, for the Gulf of Mexico and for other ocean areas.

Geophysical flows, however, are not 2-D. The study of transport processes in 3-D flows brings into the discussion issues about the three-dimensional (3-D) visualization of Lagrangian structures (see e.g. ). In idealized 3-D time-dependent flows Poincaré sections have been used to recognize significant Lagrangian structures . Invariant manifolds acting as transport barriers in 3-D flows may have the structure of convoluted 2-D surfaces embedded in a volume . In oceanic contexts these surfaces have been identified by stitching together 2-D Lagrangian structures in different layers or connecting ridges computed from finite-size Lyapunov exponent fields . In the field of atmospheric sciences, these structures have similarly been obtained by connecting ridges computed from finite-time Lyapunov exponents (FTLE) . More recently, also in atmospheric contexts, 3-D Lagrangian information has been extracted by means of 2-D slices of the full 3-D FTLE field computed from 3-D trajectories .

The methodology used in this paper for visualizing 3-D Lagrangian structures in the stratosphere is based on the Lagrangian descriptor (LD) known as the M function . So far, in the stratosphere context, the M function has been used to gain insight into key dynamical and transport processes in 2-D settings . More recently, and have applied the M function to the visualization of structures in idealized 3-D flows. have applied the M function to visualize coherent structures in full 3-D direct numerical simulations of the compressible magnetohydrodynamic equations. The M function has the advantage of highlighting simultaneously invariant manifolds by means of singular features and also tori-like coherent structures (see ). Here we apply M to produce a full 3-D description from 3-D flows above Antarctica during a period in the spring of 1979 in which the stratosphere was both rather stable (August) and subjected to rapid changes (October) . In this region, the later period selected for analysis comprises an interval when the winter circulation, which is characterized by a strong circumpolar westerly (cyclonic) flow known as the SPV, breaks down as the final warming to summer conditions develops. Although final warmings in the Southern Hemisphere are broadly similar each year , they can be punctuated by periods of rapid changes. During the final warming of 1979, the transition from winter to summer circulation accelerated during mid-October, a period when perturbing waves were very active and the vertical energy flux from the troposphere intensified . Our aim is to describe and visualize these phenomena from a full 3-D perspective using reanalysis data. To our knowledge, this is the first time that the potential of M to achieve this goal is explored.

The article is organized as follows. Section 2 describes the dynamical systems approach to the analysis of 3-D Lagrangian structures. Section 3 describes the dataset used in this study, the calculation of M in 3-D, the data post-processing needed to this end, the computational procedures and other issues involved in this task. Section 4 discusses some benchmark calculations and their interpretation. Section 5 provides the results and findings of M on the 3-D Lagrangian structure of the polar vortex on August and in mid-October 1979. Finally, Sect. 6 includes a discussion and presents our conclusions.

The theory of dynamical systems provides an ideal framework for studying nonlinear transport and mixing processes in the atmosphere. The geometrical structures that vertebrate the Lagrangian skeleton act as material barriers that fluid particles cannot cross. A key element in the dynamical description is the presence of hyperbolic regions defined by rapid fluid contracting and expanding rates along directions that are respectively associated with the stable and unstable manifolds. In 2-D flows, these manifolds are curves, while in 3-D settings, other possibilities arise. We discuss some particularities for the system under study next.

If we assume that air parcels are passively advected by the flow, the dynamical system that governs the atmospheric flow is given by x˙=vxt,t,xt0=x0, where x(t;x0) represents the trajectory of an air parcel that at time t0 is at position x0, and v is the velocity field. For the geophysical context we are focusing on, the velocity components will be supplied by the ERA-Interim Reanalysis dataset produced at the European Centre for Medium-Range Weather Forecasts, as explained in detail in the next section. This dataset confirms that the magnitude of the vertical velocity component in the stratosphere is very small, so that vertical displacements of fluid parcels compared to the horizontal displacements are also small for the timescales of interest in this study (∼ days). These considerations motivate the discussion of a system with the particular structure given in Eq. () as it will support the interpretation of the findings described in Sect. 5 in this region of the atmosphere.

For two-dimensional flows hyperbolic trajectories and their stable and unstable manifolds are the key kinematical features responsible for the geometrical template governing transport. However, in three dimensions there are new types of three-dimensional structures that may form a geometrical template that governs transport. The “weak three-dimensionality” of many geophysical flows, like the one considered in this paper, gives rise to a (normally) hyperbolic invariant curve (i.e. not a single trajectory) that has two-dimensional stable and unstable manifolds embedded in the 3-D space. In this case the stable and unstable manifolds of this invariant curve are codimension one in the flow and therefore provide barriers to transport. Moreover, since in this case the stable and unstable manifolds are both codimension one, they can intersect to form lobes, resulting in a three-dimensional version of lobe dynamics. We now describe the special form of the flow giving rise to this structure. The form of the flow follows from and was described in the context of fluid mechanics in .

Here we examine a simple model three-dimensional velocity field that captures the form of the velocity field given by the dataset that we study and, therefore, allows us to describe this less familiar notion of a normally hyperbolic invariant curve in a simple setting. The velocity field has the following form: dxdt=∂H(x,y,z,t)∂y=vx(x,y,z,t)dydt=-∂H(x,y,z,t)∂x=vy(x,y,z,t)dzdt=0, where H(x,y,x,t)=A(z)sin⁡(πy)sin⁡(πx)/π and A(z)=1+sin⁡(πz/2). The system is defined in the domain (x,y,z)∈[0,1]×[-1,1]×[-1,1]. We refer to x and y as the horizontal coordinates and to z as the vertical coordinate. This velocity field is certainly “weakly” three-dimensional as there is no motion in the vertical direction, yet the horizontal motion does depend on the height. This model contains the essence of the geometrical structures governing transport in the dataset that we analyse.

More specifically, note that for each z the system () has a hyperbolic fixed point at (x=0,y=0), for which the linearized system is dxdt=A(z)πx,dydt=-A(z)πy. The curve (0,0,z) is clearly an invariant curve (in particular, it is a curve of fixed points). The linearized stability described by Eqs. (3) and () quantifies linearized stability normal to the invariant curve (0,0,z) and, hence, is the origin of the phrase normal hyperbolicity. For each z, the fixed point has a one-dimensional stable manifold and a one-dimensional unstable manifold. Therefore as a function of z the curve (0,0,z) has two-dimensional stable and two-dimensional unstable manifolds in three dimensions. Hence, these two-dimensional invariant surfaces provide barriers to transport in the three-dimensional flow . The geometrical representation of the manifolds is shown in Fig. a.

The 3-D Lagrangian structure of Eq. () visualized in Fig. b is achieved by means of the M function, which is represented on slices intersecting the manifolds. Other slice choices are possible, but here we just show two simple possibilities on perpendicular planes which help to capture the essential features of the full 3-D motion. The M function is defined as follows: Mx0,t0,τ=∫t0-τt0+τ‖v(x(t;x0),t)‖dt, where v(x,t) is the velocity field and ‖⋅‖ denotes the Euclidean norm. At a given time t0, M corresponds to the length of the trajectory traced by a fluid parcel starting at x0=x(t0) as it evolves forwards and backwards in time for a time interval τ. For sufficiently large τ values the sharp changes that occur in narrow gaps in the scalar field provided by M, which we will refer to as singular features, highlight the stable and unstable manifolds and, at their crossings, hyperbolic trajectories as confirmed by Fig. b. Recently, have established a rigorous mathematical foundation for specific LDs for a class of examples in continuous time-dynamical systems.

Figure b shows that the M function also has the capability of revealing vortices present in the fluid. In particular for this example two counterrotating vortices which are vertically extended are visible. The yellowish colours highlight the parts of the vortices with the highest speeds. Typically vortex- or jet-like structures (for periodic domains) are related to 2-tori in three-dimensional flows. This is discussed in and . This notion is related to fluid regions trapping fluid parcels in their interior and isolating them from the surrounding fluid, as for instance is the case for the circulating strong jet forming the SPV. There exist formal results linking contour lines of the time average of M with tori-like invariant sets. In this manner singular lines in Fig. b highlight invariant manifolds and contour lines of converged averages of M highlight invariant tori (see ).

To benchmark the procedure described in the previous section, we compare the Lagrangian outputs obtained from the full 3-D scenario at constant heights (spherical shells) with those obtained from the 2-D scenario for potential temperature surfaces that are approximately at the same height. Figure  compares, for different τ values, the evaluation of the M function for a 2-D integration in the 850 K isentropic surface with that obtained on a spherical shell at 31.3 km height, which approximately corresponds to the 850 K surface. For τ=5, 10 days, the figure highlights well-defined structures that are very similar in both cases. These consist of a large circulating coherent vortex in yellow, representing high values of M that are related to particles exhibiting large displacements and blueish zones corresponding to calmer regions. Lobes eroding the outer part of the vortex are clearly identified by coloured filaments. Also, crossings of contours of M highlighting hyperbolic trajectories are noticed along longitudes 15∘ W and 165∘ E. These results confirm that in the upper stratosphere the potential temperature surface and the spherical shell are almost identical.

Increasing the τ values to 20 or 30 days adds more Lagrangian detail to the figures, which thus contain long-term transport issues reflected in very thin and long filamentous structures. However, the figures at constant potential temperature and constant height are still very similar and contain the gross structure already observed at smaller τ. Differences are restricted to those filamentous structures which are difficult to follow and to compare. Given that long-term transport issues are difficult to interpret and that we do not require them to describe the phenomena of our interest which occur in time intervals varying from hours to 10 days, we fix in this range the selected τ values for this article.

Comparisons of transport between both approaches are rather different, as Fig.  confirms. The first row in this figure shows the 2-D calculations for a 2-D integration in the 330 K isentropic surface for a period of τ=5, 10 days, and the second row a 3-D output at the corresponding height (10 km) for the same integration intervals. The panels on the top row of Fig.  (at a constant potential temperature) are quite similar to those in the bottom row (at constant height), although differences are more evident than those in Fig.  for the upper stratosphere. At this level constant potential temperature surfaces are not so close to a sphere, but flow structures in the troposphere are deep, i.e. they have a vertical scale of about 10 km, which is larger than the vertical separations between points on the 10 km high sphere and 330 K surfaces. Results discussed in the next section will show further evidence of the 2-D–3-D motion transitions.

Figure  contrasts the more classical description of events in the middle–upper stratosphere with the one provided by our Lagrangian descriptor. The left-hand column of Fig.  shows the geopotential height at 10 mb. This field is particularly useful because it is equivalent to the velocity streamfunction at the corresponding pressure level in the extratropics (see ). This field provides a purely Eulerian description of the flow; however, in the time-dependent case this point of view is limited as it does not address issues regarding the fate of particle trajectories. The middle column shows the picture provided by the potential vorticity at the 850 K equivalent temperature (isentropic) surface. This field is conserved along particle trajectories; thus, the image is Lagrangian in the sense that we observe the field of a purely advected quantity. The right-hand column shows the M function at the surface z=31.3 km obtained from particle trajectories in full 3-D calculations. Similarly to the second column, the displayed information is Lagrangian, but here we obtain more fundamental information in this regard. This figure provides feedback for characterizing the time evolution of any purely advected scalar field, while the previous one displays just the realization of one particular initial datum. More specifically, the third column highlights the position and evolution of two hyperbolic points in the outer part of the vortex, as well as the vortex itself. As discussed by , hyperbolic points are responsible for filamentation processes. Whether or not these filaments are eventually observed depends on the distribution of the scalar field. For instance, if the scalar field is completely uniform in the whole domain, then its time evolution will show nothing about the features highlighted by M. How the features of the M field are visible in a scalar field depends on how the initial distribution of the advected field is with respect to the features of M. Figure  illustrates these facts in a very simple example. Figure a highlights the Lagrangian skeleton as obtained for the stationary cat eyes, i.e. x˙=y, y˙=0.5sin⁡x. The hyperbolic fixed point at the origin and its stable and unstable manifolds are clearly visible. Also, elliptic fixed points at positions (-π,0) and (π,0) are visible. Figure b shows a set of initial scalar fields and c shows the evolution of the three patches: the one in cyan shows filamentation, the other staying coherent (in blue) and the third done showing a tongue formation (in red) similar to the one observed for the potential vorticity in Fig. . Colours are chosen to highlight the analogous features visible in Fig. . Clearly Fig. b and c show time-dependent patterns; however, the Lagrangian skeleton shown in Fig. a is stationary.

The strong and cyclonic SPV characteristic of the winter circulation above Antarctica has been typically represented in the literature by cross sections such as those in Figs.  and . In this section we will improve this representation with figures that are more revealing of the full 3-D description of the circulation. Figure  shows for a day in late winter 1979 (15 August) the representation of M obtained for τ=5 for the vertical slice passing through longitudes 90∘ W and 90∘ E. An outstanding feature in this representation is the bright yellow colour highlighting a coherent structure. This feature captures the SPV as a tubular structure, similar to those described in Fig. b, with walls around 60∘ S and an approximately vertical axis coinciding with that of the Earth and extending from the uppermost level of data down to between 15 and 20 km in height, i.e. the transition between the troposphere and stratosphere (tropopause). In this case however the average of M does not converge as the flow is aperiodic (see ), and thus contours of M do not strictly represent invariant sets. Despite this, the setting is analogous to that described for the vortices in the example of Sect. 2. Additionally, the greenish colours that extend equatorward between 15 and 20 km, both in the west and the east, capture the upper tropospheric subtropical westerly jets. Specifically in the west, the greenish colours extend downwards up to the Equator (90∘ W), suggesting that the structures involve the entire atmospheric layer. We can also clearly see evidence of the very different dynamical characters of the troposphere and the stratosphere. Whilst the former region is practically dominated by the SPV, the latter shows much finer detail reflected by an intricate line pattern. This tangled pattern is the manifestation of crossings of stable and unstable manifolds, which are associated with strong and fast mixing processes at the lowest atmospheric levels. This image is consistent with and complementary to the projection at 10 km presented in Fig. c and d and also to the one described next.

The description of the vortex on 15 August 1979 is supplemented by Fig. , which shows the M function computed for τ=10 along a vertical slice through latitude 60∘ S, where most of the SPV walls are. The intricate structures in the troposphere are also apparent in this figure. In addition, a wavy structure is clearly visible at the boundary between troposphere and stratosphere. In terms of a Fourier decomposition of M at constant height in Fig. , we can see the classical pattern of longer wavelengths dominating the field as height increases. At tropopause level, a wavenumber 4 component is clearly visible, while in the upper part of the vortex, a wavenumber 2 is evident . The vertical propagation of these features across the stratosphere from 8 to 21 August of 1979 is clearly visible in the attached Movie S1 in the Supplement.

We next focus on the description of the 3-D Lagrangian structures for the period 6–18 October 1979. Figure  shows the rapid changes in the SPV that took place in October 1979 as the lower polar stratosphere warmed up strongly during the spring season . The figure shows the M function computed for τ=5 along a vertical slice passing through latitude 60∘S for several October dates. Figure a, b, c and d confirm that Lagrangian structures in the stratosphere become more complex in the warming period. On 18 October no yellow coherent features are visible in the upper stratosphere.

To help in the interpretation of Fig. , Fig.  displays horizontal sections of M on 6 October at different heights z=10, 21.2, 31.3 and 40 km as well as a vertical section along meridians 90∘ E and 90∘ W. Figures  and show important differences with the winter conditions 2 months earlier, visible in Figs.  and . The 60∘ S section in Fig. a intersects the cyclonic vortex four times, mostly in the Western Hemisphere. Another deep, anticyclonic vortex appears in the Eastern Hemisphere above 25 km. These cyclonic and anticyclonic vortices are also evident in Fig. a at z=31.3 km and z=40 km, and in Fig. b. In particular in the projection of Fig. a at z=31.3 km, along the longitude 0∘, between latitudes 30 and 45∘ S, crossing lines marked with a white arrow show the presence of a hyperbolic point, its unstable manifold being the separatrix between the two vortices. The vertical extension of this hyperbolic trajectory is depicted with the red line in Fig. c. This figure shows the analogue to the normally hyperbolic invariant curve explained in Sect. 2 with the additional feature that here the flow is time-dependent, while, for simplicity, the example in Sect. 2 was stationary. Figure b sketches the corresponding typical configuration for two counterrotating vortex tubes illustrating this description.

The vertically extended unstable manifold of the normally hyperbolic invariant curve that separates the two vortex tubes is captured by M and is visible in Figs. a and b as a narrow dark blue line in an analogous way to that presented in Fig. . The first of these two figures shows it near the edge of the polar vortex, around 45∘ E marked with a white arrow, and for the latter a white arrow also points to the described feature. This manifold structure separates the two counterrotating vortex tubes just described. It acts as a vertical barrier, which is several kilometres deep, as is the case for the unstable manifolds associated with a normally hyperbolic invariant curve formed from hyperbolic trajectories at each h level. Further information on the singular features is given in Fig. . The black dots in Fig. a and b represent, for 6 October 1979, 00:00:00 UTC, the horizontal and vertical positions of a particle located on the feature indicated by the dark blue line at a height of 31.3 km, longitude 90∘ E and latitude 77.74∘ S. The black dots in Fig. c and d show the corresponding locations of the same particle 6 h later. The invariant character of the singular structure is confirmed as the particle remains on it during its evolution, and its unstable character is confirmed by the fact that the particle moves away from the hyperbolic point.

have suggested that the fact that the preferred geographical location (ridge south of Australia) for the development of this anticyclonic vortex in this particular event indicates that the stratospheric circulation is governed to a significant extent from below (see also ). This anticyclonic vortex will strengthen and eventually dominate at high levels. In terms of Fourier components, a quasi-stationary wave 1 amplifies on this date, in conjunction with the displacement of the cyclonic vortex from the polar position.

In the present paper we discuss the visualization of three-dimensional Lagrangian structures in atmospheric flows. Specifically, we have explained mathematical aspects about the Lagrangian geometrical structures to be expected in the atmospheric setting in 3-D and have introduced the concept of normally hyperbolic invariant curves in a specific example which recovers features of those observed in the stratosphere. The algorithm used to represent the 3-D Lagrangian structures is based on the methodology of Lagrangian descriptors (LDs). We have explored the application of the full power of the M function computed with 3-D trajectories, which hitherto had been used in 2-D settings. The consistency of our development has been verified by comparing the 3-D scenario results at a constant height with those obtained from the 2-D scenario in potential temperature surfaces at equivalent heights.

To demonstrate the methodology we have applied it to a numerical dataset describing the flow above Antarctica during the southern mid–late winter and spring. The dataset was obtained from ERA-Interim Reanalysis data provided by the ECMWF. Our findings show the vertical extension and structure of the stratospheric polar vortex and its evolution. We also characterize, from the Lagrangian point of view, the boundary between the troposphere and the stratosphere. Very complex Lagrangian patterns are identified in the troposphere, which support the presence of strong mixing processes. The “final stratospheric warming” is characterized by the breakdown of the westerly SPV during the transition from winter to summer circulation. Our results confirm that the onset of this process is characterized by an initial decay of the vortex in the upper stratosphere where the circulation weakens, albeit remaining strong at lower heights. We have also captured the anticyclonic circulation that develops during October preferentially above the southern part of Australia. We illustrate the vertical structure of these two counterrotating vortices, and the invariant separatrix that divides them. The particular feature found is several kilometres deep and we demonstrated that fluid parcels remain in this feature during intervals of the order of days. Such features highlight the complexities in the transport of chemical tracers in the stratosphere.