The present two-part paper provides a Lagrangian perspective of the final southern warming in 2002, during which the stratospheric polar vortex (SPV) experienced a unique splitting. Part I focuses on the understanding of fundamental processes for filamentation and ultimately for vortex splitting on a selected isentropic surface in the middle stratosphere. Part II discusses the three-dimensional evolution of the selected sudden warming event. We approach the subject from a dynamical systems viewpoint and search for Lagrangian coherent structures using a Lagrangian descriptor as a tool. In this Part I we work in the idealized framework of a kinematic model that allows for an understanding of the contributing elements of the flow in late September during the splitting. We introduce a definition of kinematic SPV boundary based on a criterion for binning parcels inside and outside the vortex according to their values of the Lagrangian descriptor and associated PDF. This definition is justified by using arguments that go beyond heuristic considerations based on the potential vorticity. Next, we turn to the filamentation processes along this kinematic boundary, and the role of Lagrangian structures in the SPV splitting. We determine a criterion for splitting based on the structure of the evolving unstable and unstable manifolds.
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J. Curbelo and A. M. Mancho were supported by MINECO Grant MTM2014-56392-R. A. M. Mancho and J. Curbelo are supported by ONR. Grant No. N00014-17-1-3003. C. R. Mechoso was supported by the U.S. NSF Grant AGS-1245069. The research of S. Wiggins is supported by ONR Grant No. N00014-01-1-0769. This paper has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 777822.
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There exist different dynamical objects that support the qualitative description of particle time evolution. This qualitative description is based on Poincaré’s ideas, and consist of determining geometrical structures that define regions where particle trajectories have qualitatively different behaviours. The boundaries between these regions are dynamical barriers. These geometrical structures provide a template for a specific velocity field and emphasize the essential transport features associated with it.
Hyperbolic trajectories are one type of recognisable dynamical features in flows. These are special trajectories, in the neighbourhood of which air masses are elongated along the unstable direction and compressed along the stable direction. An important feature here is that two parcels placed at nearby locations close to a hyperbolic trajectory may evolve in time quite differently, separating at exponential rates.
Trajectories of particles initially placed in the neighbourhood of a hyperbolic trajectory become aligned as time evolves forward with a curve called the unstable manifold; similarly, trajectories of particles in the neighbourhood of a hyperbolic trajectory as time evolves backwards become aligned with a curve called the stable manifold. Figure 11a illustrate this forwards and backwards alignment of the green blobs with the orange and cyan blobs. For time dependent flows, hyperbolic trajectories do not correspond to hyperbolic instantaneous stagnation points of the velocity field. Their positions can be rather different, thus being the behavior of particles in a time interval quite counter-intuitive from what is observed for a velocity field that is frozen in time.
Stable and unstable manifolds are aligned with singular features of M. Figure 11b shows two lines crossing in the middle of the yellowish region. These lines are aligned with the stable and unstable manifolds appearing in panel (a) and correspond, respectively, to the stable and unstable manifolds of the hyperbolic trajectory at the crossing point.
Unstable and stable manifolds are invariant curves. This means that as time evolves either forwards or backwards, respectively, particles stay on those curves. Particles do not cross these curves, therefore they are material curves, and are barriers to transport. Stable and unstable manifolds act as repelling and attracting material lines, respectively (Haller 2002).
Apart from hyperbolic trajectories and their stable and unstable manifolds, other types of dynamical flow structures exist, in which particles tend to stay together, coherently, without dispersing, such as vortices that keep fluid parcels inside them, or jets. Invariant tori are the dynamical objects related to that behaviour. These objects are invariant in the sense defined above, i.e. particles evolve in time staying on them. Further details on tori are given in Sect. 4.
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Curbelo, J., Mechoso, C.R., Mancho, A.M. et al. Lagrangian study of the final warming in the southern stratosphere during 2002: Part I. The vortex splitting at upper levels. Clim Dyn 53, 2779–2792 (2019). https://doi.org/10.1007/s00382-019-04832-y
- Stratospheric sudden warming
- Vortex boundary
- Kinematic models
- Vortex split