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Chaotic streamlines in the ABC flows

Published online by Cambridge University Press:  21 April 2006

T. Dombre ,
U. Frisch ,
J. M. Greene ,
M. Hénon ,
A. Mehr  and
A. M. Soward
T. Dombre
Affiliation:
CNRS, Groupe de Physique des Solides, école Normale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05, France
U. Frisch
Affiliation:
CNRS, Observatoire de Nice, BP 139, 06003 Nice Cedex, France
J. M. Greene
Affiliation:
GA Technologies Inc. PO Box 81608, San Diego, California 92138, USA
M. Hénon
Affiliation:
CNRS, Observatoire de Nice, BP 139, 06003 Nice Cedex, France
A. Mehr
Affiliation:
Observatoire de Nice, BP 139, 06003 Nice Cedex, France
A. M. Soward
Affiliation:
School of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, UK
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Abstract

The particle paths of the Arnold-Beltrami-Childress (ABC) flows \[ u = (A \sin z+ C \cos y, B \sin x + A \cos z, C \sin y + B \cos x). \] are investigated both analytically and numerically. This three-parameter family of spatially periodic flows provides a simple steady-state solution of Euler's equations. Nevertheless, the streamlines have a complicated Lagrangian structure which is studied here with dynamical systems tools. In general, there is a set of closed (on the torus, T3) helical streamlines, each of which is surrounded by a finite region of KAM invariant surfaces. For certain values of the parameters strong resonances occur which disrupt the surfaces. The remaining space is occupied by chaotic particle paths: here stagnation points may occur and, when they do, they are connected by a web of heteroclinic streamlines.

When one of the parameters A, B or C vanishes the flow is integrable. In the neighbourhood, perturbation techniques can be used to predict strong resonances. A systematic search for integrable cases is done using Painlevé tests, i.e. studying complex-time singularities of fluid-particle trajectories. When ABC ≠ 0 recursive clustering of complex time singularities occurs that seems characteristic of non-integrable behaviour.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 167 , June 1986 , pp. 353 - 391
Copyright
© 1986 Cambridge University Press

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