Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2014-02-01
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| Series: | Nonlinear Processes in Geophysics |
| Online Access: | http://www.nonlin-processes-geophys.net/21/165/2014/npg-21-165-2014.pdf |
| _version_ | 1811325416714010624 |
|---|---|
| author | S. Wiggins A. M. Mancho |
| author_facet | S. Wiggins A. M. Mancho |
| author_sort | S. Wiggins |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T14:32:36Z |
| format | Article |
| id | doaj.art-3f9352ee9370488588ca118fb3d2a8cf |
| institution | Directory Open Access Journal |
| issn | 1023-5809 1607-7946 |
| language | English |
| last_indexed | 2024-04-13T14:32:36Z |
| publishDate | 2014-02-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Nonlinear Processes in Geophysics |
| spelling | doaj.art-3f9352ee9370488588ca118fb3d2a8cf2022-12-22T02:43:09ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462014-02-0121116518510.5194/npg-21-165-2014Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" toriS. Wiggins0A. M. Mancho1School of Mathematics, University of Bristol, Bristol BS8 1TW, UKInstituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/Nicolás Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, SpainIn 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.http://www.nonlin-processes-geophys.net/21/165/2014/npg-21-165-2014.pdf |
| spellingShingle | S. Wiggins A. M. Mancho Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori Nonlinear Processes in Geophysics |
| title | Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori |
| title_full | Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori |
| title_fullStr | Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori |
| title_full_unstemmed | Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori |
| title_short | Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori |
| title_sort | barriers to transport in aperiodically time dependent two dimensional velocity fields nekhoroshev s theorem and nearly invariant tori |
| url | http://www.nonlin-processes-geophys.net/21/165/2014/npg-21-165-2014.pdf |
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