In this paper we study the three-dimensional (3-D) Lagrangian structures in the
stratospheric polar vortex (SPV) above Antarctica. We analyse and visualize
these structures using Lagrangian descriptor function

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

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 (

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.

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

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

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

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

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:

More specifically, note that for each

The 3-D Lagrangian structure of Eq. (

Figure

We use the ERA-Interim Reanalysis dataset produced by European Centre for
Medium-Range Weather Forecasts (ECMWF;

From the ERA-Interim dataset we extract the horizontal wind velocity components

The procedure to obtain the

Step 1. The data are downloaded from ERA-Interim in

Step 2. The data files are converted from

Step 3. The vertical velocity

Step 4. Then, the 3-D velocity

Step 5. The data are converted from sigma levels to the height levels
specified for the analysis, and required data are produced by interpolation,
with all velocity components expressed in metres per second (m s

Step 6. To avoid issues at the pole in the calculation of trajectories from
equations expressed in spherical coordinates, the velocities are written in
Cartesian coordinates from the velocity data available in spherical
coordinates (see

Step 7. The trajectories are calculated in a Cartesian coordinate system,
which are obtained by solving

To integrate the system (

Step 8. The

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

Increasing the

Comparison of geopotential at constant pressure 10 mb, potential
vorticity at 850 K and

Comparisons of transport between both approaches are rather different, as
Fig.

Figure

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.

The description of the vortex on 15 August 1979 is supplemented by
Fig.

15 August 1979 00:00:00 UTC.

We next focus on the description of the 3-D Lagrangian structures for the
period 6–18 October 1979. Figure

15 August 1979 00:00:00 UTC.

Vertical slices showing

6 October 1979 00:00:00 UTC.

To help in the interpretation of Fig.

Evaluation of the

The vertically extended unstable manifold of the normally hyperbolic
invariant curve that separates the two vortex tubes is captured by

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

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.

The datasets used in this work are described in Sect. 3, where links are provided to the official websites from which they have been downloaded.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Current perspectives in modelling, monitoring, and predicting geophysical fluid dynamics”. It is not part of a conference.

Jezabel Curbelo, Víctor José García-Garrido and Ana Maria Mancho are supported by MINECO grant MTM2014-56392-R. Coumba Niang acknowledges Fundacion Mujeres por Africa and ICMAT Severo Ochoa project SEV-2011-0087 for financial support. Ana Maria Mancho and Coumba Niang are supported by CSIC grant COOPB20265. The research of Stephen Wiggins is supported by ONR grant no. N00014-01-1-0769. Carlos Roberto Mechoso was supported by U.S. NSF grant AGS-1245069. We also acknowledge support from ONR grant no. N00014-16-1-2492. Thanks are owed to CESGA and ICMAT for computing facilities. Edited by: Emilio Hernández-García Reviewed by: two anonymous referees