Journal of Fluid Mechanics

Chaotic streamlines in the ABC flows

T.  Dombre a1, U.  Frisch a2, J. M.  Greene a3, M.  Hénon a4, A.  Mehr a5 and A. M.  Soward a6
a1 CNRS, Groupe de Physique des Solides, école Normale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05, France
a2 CNRS, Observatoire de Nice, BP 139, 06003 Nice Cedex, France
a3 GA Technologies Inc. PO Box 81608, San Diego, California 92138, USA
a4 CNRS, Observatoire de Nice, BP 139, 06003 Nice Cedex, France
a5 Observatoire de Nice, BP 139, 06003 Nice Cedex, France
a6 School of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, UK

Article author query
dombre t   [Google Scholar] 
frisch u   [Google Scholar] 
greene jm   [Google Scholar] 
hénon m   [Google Scholar] 
mehr a   [Google Scholar] 
soward am   [Google Scholar] 


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 [not equal] 0 recursive clustering of complex time singularities occurs that seems characteristic of non-integrable behaviour.

(Published Online April 21 2006)
(Received May 15 1985)
(Revised November 20 1985)