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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 21, issue 1
Nonlin. Processes Geophys., 21, 165–185, 2014
https://doi.org/10.5194/npg-21-165-2014
© Author(s) 2014. This work is distributed under
the Creative Commons Attribution 3.0 License.

Special issue: Nonlinear dynamics in oceanic and atmospheric flows: theory...

Nonlin. Processes Geophys., 21, 165–185, 2014
https://doi.org/10.5194/npg-21-165-2014
© Author(s) 2014. This work is distributed under
the Creative Commons Attribution 3.0 License.

Review article 04 Feb 2014

Review article | 04 Feb 2014

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

S. Wiggins1 and A. M. Mancho2 S. Wiggins and A. M. Mancho
  • 1School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
  • 2Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/Nicolás Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain

Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

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