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Hamiltonian Chaos and Anomalous Transport in Two Dimensional Flows

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Hamiltonian Chaos Beyond the KAM Theory

Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

In this chapter we discuss the dynamics of particles advected in regular and chaotic flows. We first address the dynamics of point vortices and show the great variety of the dynamics of three point vortices near the singularity giving rise to vortex collapse. We discuss the strong influence of the existence of a finite time singularity on the dynamics, especially on how the period of the motion evolves as we get closer to the singular conditions. We then analyze transport properties of passive tracers in various flows. We start with integrable flows governed by three vortices, then switch to chaotic flows generated by four and sixteen vortices, and end up with a turbulent flow governed by the Charney-Hasegawa-Mima. For all cases, anomalous superdiffusive transport with a characteristic exponent μ ∼ 1.5 – 1.8 is observed. The origin of the anomaly is explained by the phenomenon of stickiness around coherent structures in regular flows, and by the presence of regular chaotic jets for the chaotic and turbulent ones. Finally we illustrate how the Hamiltonian nature of chaos can be used to localize 3-dimensional coherent structures or how to improve mixing properties in cellular flows while keeping the cellular structure of the flow.

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Author information

Authors and Affiliations

  1. Centre de Physique Théorique, Luminy Case 907, 13288, Marseille Cedex 9, France

    Xavier Leoncini

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  1. Xavier Leoncini
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Editors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1805, USA

    Albert C. J. Luo

  2. IICO-UASLP, Av. Karakorum 1470 Lomas 4a Seccion, San Luis Potosi SLP, 78210, Mexico

    Valentin Afraimovich

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© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Leoncini, X. (2010). Hamiltonian Chaos and Anomalous Transport in Two Dimensional Flows. In: Luo, A.C.J., Afraimovich, V. (eds) Hamiltonian Chaos Beyond the KAM Theory. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12718-2_3

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