In this work, we study the Lagrangian footprint of the planetary waves present in the Southern Hemisphere stratosphere during the exceptional sudden Stratospheric warming event that took place during September 2002. Our focus is on constructing a simple kinematic model that retains the fundamental mechanisms responsible for complex fluid parcel evolution, during the polar vortex breakdown and its previous stages. The construction of the kinematic model is guided by the Fourier decomposition of the geopotential field. The study of Lagrangian transport phenomena in the ERA-Interim reanalysis data highlights hyperbolic trajectories, and these trajectories are Lagrangian objects that are the kinematic mechanism for the observed filamentation phenomena. Our analysis shows that the breaking and splitting of the polar vortex is justified in our model by the sudden growth of a planetary wave and the decay of the axisymmetric flow.

The availability of high-resolution and high-quality reanalysis data sets
provides us with a powerful tool for obtaining a detailed view of the
space–time evolution of the stratospheric polar night vortex (SPV), which has
implications for the geophysical fluid dynamics of the entire Earth. The
complexity of such a detailed view, however, makes it difficult to extract
the physical mechanisms underlying notable transport features in the observed
behaviour. The goal of this work is to gain new insights into the fundamental
mechanisms responsible for complex fluid parcel evolution, since these lie at
the heart of our understanding of the dynamics and chemistry of the
stratosphere. To this end, we extract, directly from the data, a simple model
with stripped-down dynamics in order to directly probe, in a controlled and
systematic manner, the physical mechanisms responsible for the key observed
transport features of the SPV. Models of this kind, termed “kinematic
models”, have provided a simple approach for studying Lagrangian transport
and exchange associated with flow structures such as meandering jets and
travelling waves

The importance of an increased understanding of the SPV was dramatically
demonstrated by the intense research effort that followed the discovery of
the “Antarctic ozone hole” phenomenon in the 1970s

Why does this occur over Antarctica and not over the Arctic (since pollution sources are stronger in the Northern than in the Southern Hemisphere)?

Why does this occur in the spring season?

Will ozone depletion extend worldwide?

Dynamical systems theory provides valuable insights into the transport
processes described in the previous paragraph. Tools of the theory include
the geometrical structures, referred to as hyperbolic trajectories (HTs),
their stable and unstable manifolds, and their intersection in homoclinic and
heteroclinic trajectories that provide the theoretical and computational
basis for describing the filamentation process. A challenge in the
application of these concepts to realistic geophysical flows is that while
the structures mentioned are defined for infinite-time autonomous or periodic
systems, geophysical flows are typically defined as finite-time data sets and
are not periodic.

We focus on the SPV behaviour during the major stratospheric sudden warming
that occurred in the southern stratosphere during September 2002. In this
unusual event, the SPV broke down in the middle stratosphere

The structure of the paper is as follows. Section 2 describes the data and methods we used. Section 3 describes the planetary waves in the reanalysis data in the year 2002 in the stratosphere at selected pressure levels (10 hPa). We relate these to filamentation phenomena and the polar vortex breakdown that occurred in that year. Section 4 reproduces the findings obtained with our analytical kinematical model, confirming the role played by the HTs in the 2002 vortex filamentation and breakdown. Section 5 discusses the consistency of the kinematic model as representative of atmospheric flows that conserve potential vorticity. Finally, in Sect. 6, we present the conclusions.

To achieve a realistic representation of the atmospheric transport processes,
it is crucial to use a reliable and high-quality data set. We use in this work
the ERA-Interim reanalysis data set produced by a weather forecast
assimilation system developed by the European Centre for Medium-Range Weather
Forecasts (ECMWF;

The ERA-Interim data set that we selected for this study is available four
times daily (00:00, 06:00, 12:00 and 18:00 UTC), with a horizontal resolution of

The geopotential height

For the analysis of planetary waves, we apply a zonal Fourier decomposition
to the geopotential height field on the 10 hPa pressure level (approximately
850 K potential temperature). The zonal wave decomposition yields

Dynamical systems theory provides a qualitative description of the evolution
of particle trajectories by means of geometrical objects that partition the
phase space (the atmosphere in our case) into regions in which the system
shows distinct dynamical behaviours. These geometrical structures act as
material barriers to fluid parcels and are closely related to flow regions
known as hyperbolic, where rapid contraction and expansion takes place.
Several Lagrangian techniques have been developed in order to detect such
structures in geophysical fluids. This is challenging because, while
classical dynamical systems theory is defined for infinite-time autonomous or
periodic systems, in geophysical contexts, the velocity fields are generally
time dependent, aperiodic in time and defined over a finite discrete
space–time domain. Among others, techniques developed are finite-size
Lyapunov exponents (FSLEs)

The dynamical system that governs the atmospheric flow is given by

To compute fluid parcels trajectories, it is necessary to integrate
Eq. (

The consistency between the output field of Eq. (

As an example relevant to the case that motivates the present study, we show
in Fig.

Stereographic projection of Lagrangian descriptors evaluated using

As we indicated in the previous section, in order to characterize the
planetary waves that propagate in the stratosphere, we carry out a Fourier
decomposition of the geopotential height. In Fig.

On the 10 hPa pressure level, the winter SPV in the Southern Hemisphere can
be broadly defined as a circumpolar westerly jet. Figure

Stereographic projection of the geopotential height field and its
Fourier decomposition for the 10 hPa pressure level on 22 September 2002
at 00:00:00 UTC:

On the 10 hPa pressure level:

The contribution of these different waves to the evolution of the SPV and
their transport implications are clearly observed in movie S5. A regime giving
rise to the stretching of material lines and the appearance of hyperbolic
regions and the associated filamentation processes is observed. These
filamentous structures and HTs are clearly highlighted by the application of
LDs to the wind fields, as shown in Figs.

Kinematic models have a long history in the geophysical fluid dynamics
community. They allow for a detailed parametric study of the influence of
identified flow structures on transport and exchange of fluid parcels. All
early studies utilizing the dynamical systems approach for understanding
Lagrangian transport and exchange associated with flow structures, such as
meandering jets and travelling waves, have employed kinematic models (see

Continuing in this spirit, we propose a kinematic model that allows us to
identify, in a controlled fashion, the characteristics of the distinct
propagating waves that are responsible for the different Lagrangian features
observed in the SPV. Our kinematic model is inspired by the Fourier component
decomposition of the geopotential extracted from the ERA-Interim data as
discussed in the previous section. The analysis of data from August and
September 2002 shows a mean axisymmetric flow, disturbed mainly by waves with
planetary wave numbers

We will work in a plane

The particular forms of

The other streamfunction components are

Stereographic projection of the

Representation of the three components of the streamfunction.
Panel

Some illustrative parameter choices for the kinematic model.

Lagrangian patterns obtained at

Lagrangian patterns obtained at

The velocity of fluid parcels in the Cartesian coordinates

We begin by considering the case of a mean flow with

Next, we consider the case with the same parameters except for

Evolution of the Lagrangian template for the case in which the mean
flow decreases and wave 2 increases. The sequence reproduces many of the
Lagrangian features observed in the splitting event that occurred at the end
of September 2002 (see movie S5).

Evolution of a vorticity patch.

Evolution of the scaled lower boundary

Figure

Figure

Figure

In this section, we discuss the connection between the kinematic model
introduced in the previous section and a fundamental dynamical principle of
geophysical fluids. Geophysical flows that are adiabatic and frictionless
conserve the potential vorticity

We assume that at the initial time,

In order to preserve Eq. (

In this work, we propose a simple kinematic model for studying transport phenomena in the Antarctic polar vortex. We are interested in gaining insights into the processes which carry material outward from the vortex structure and inward to the vortex structure.

The construction of the kinematic model is realized by analysing geopotential height data produced by the ECMWF. In particular, our focus is on the stratospheric sudden warming event that took place in 2002, producing the pinching and then breaking of the stratospheric polar vortex. We identify the prevalent Fourier components during this period, which consist of a mean axisymmetric flow and waves with wave numbers 1 and 2. The kinematic model is based on analytical expressions which recover the spatial structures of these representative Fourier components. The model can be controlled so that waves with wave numbers 1 and 2 can be switched on and off independently. We are also able to adjust the relative position of the waves with respect to the mean axisymmetric flow.

The study of Lagrangian transport phenomena in the ERA-Interim reanalysis
data by means of Lagrangian descriptors highlights hyperbolic trajectories.
These trajectories are Lagrangian objects “seeding” the observed
filamentation phenomena. The Lagrangian study of the kinematic model sheds
light on the role played by waves in this regard. The model is adjusted to a
stationary case which considers a mean flow and a stationary wave 2 that
perturbs the mean flow in its outer part, producing hyperbolic trajectories.
For the stationary case, hyperbolic trajectories are easily identified. This
framework is modified by transforming it to a time-dependent problem by
making the wave phase speed different from zero or by introducing time-dependent
amplitudes. This allows to relate the time-dependent structures
with those easily identified in the stationary case. The setting is repeated
with wave 1 and both wave 1 and wave 2 together. The joint presence of
these waves produces complex Lagrangian patterns remarkably similar to those
observed from the analysis of the complex reanalysis data and confirms the
findings discussed by

The data used in this work are described in Sect. 2.1, where links are also provided to the official websites from which they have been downloaded.

The authors declare that they have no conflict of interest.

V. J. García-Garrido, J. Curbelo and A. M. Mancho are supported by MINECO grant MTM2014-56392-R. The research of C. R. Mechoso is supported by the U.S. NSF grant AGS-1245069. The research of S. Wiggins is supported by ONR grant no. N00014- 01-1-0769. We also acknowledge support from ONR grant no. N00014-16-1-2492.Edited by: C. López Reviewed by: two anonymous referees