Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-28T22:35:11.976Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ablowitz, M. J., Ramani, A. & Segur, H. 1978 Lett. Nuovo Cim. 23, 333.
Ablowitz, M. J., Ramani, A. & Segur, H. 1980 J. Math. Phys. 21, 715 and 21, 1006.
Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. Applied Math. Series 55, National Bureau of Standards.
Aref, H. 1984 J. Fluid Mech. 143, 1.
Arnold, V. I. 1965 C.R. Acad. Sci. Paris 261, 17.
Arnold, V. I. 1974 Les Méthodes Mathématiques de la Mécanique Classique. Moscow: Mir. English translation Mathematical Methods of Classical Mechanics. Springer (1980).
Arnold, V. I. 1978 Chapitres Supplémentaires de la Théorie des Equations Différentielles Ordinaires. (Translated from Russian original.) Moscow: Mir.
Arnold, V. I. 1984 In Nekotore Voprocy Sovremennovo Analisa, p. 8. University of Moscow.
Arnold, V. I. & Korkina, E. I. 1983 Vest. Mosk. Un. Ta. Ser. 1. Matematika Mecanika 3, 43.
Arnold, V. I., Zel'dovich, Ya.B., Ruzmaikin, A. A. & Sokolov, D. D. 1981 Zh. Eksp. Teor. Fiz. 81, 2052; Sov. Phys. JETP 54, 1083.
Arter, W. 1983 Phys. Lett. 97A, 171.
Batchelor, G. K. & Townsend, A. A. 1949 Proc. Roy. Soc. Lond. A199, 238.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw Hill.
Birkhoff, G. D. 1927 Dynamical systems. American Mathematical Society.
Cary, J. R. & Littlejohn, R. G. 1983 Ann. Phys. 151, 1.
Chang, Y. F., Greene, J. M., Tabor, M. & Weiss, J. 1983 Phyaica 8D, 183.
Chang, Y. F., Tabor, M., Weiss, J. & Corliss, G. 1981 Phys. Lett. A85, 211.
Chang, Y. F., Tabor, M. & Weiss, J. 1982 J. Math. Phys. 23, 531.
Childress, S. 1967 Rep. AFOSR-67-0124. Courant Institute, New York.
Childress, S. 1970 J. Math. Phys. 11, 3063.
Collet, P., Eckmann, J.-P. & Koch, H. 1983 Physica 3D, 457.
Cowley, S. W. H. 1973 Radio Sci. 8, 903.
Frisch, U. & Morf, R. 1981 Phys. Rev. A23, 2673.
Gagne, Y. 1980 Intermittence en turbulence developpeé. thesis, Institut de Mécanique de Grenoble, unpublished.
Galloway, D. J. & Frisch, U. 1984 Geophys. Astrophys. Fluid Dyn. 29, 13.
Galloway, D. J. & Frisch, U. 1985 The Hydrodynamic Stability of the ABC Flows. Preprint.
Galloway, D. J. & Frisch, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. (To appear).Google Scholar
Gautero, J.-L. 1985 C. R. Acad. Sci. Paris 301, 1095.
Greene, J. M. 1982 In Nonlinear Problems: Present and Future (ed. A. Bishop, D. Campbell & B. Nicolaenko), p. 423. North-Holland.
Greene, J. M., Mackay, R. S., Vivaldi, F. & Feigenbaum, M. J. 1983 Physica 3D, 468.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Helleman, R. H. G. 1980 In Fundamental Problems in Statistical Mechanics (ed. E. G. D. Cohen), vol. 5, p. 165. North-Holland.
Hénon, M. 1966 C.R. Acad. Sci. Paris 262, 312.
Hénon, M. 1983 In Chaotic Behaviour of Deterministic Systems, Les Houches Summer School, Session XXXVI (ed. G. Iooss, R. H. G. Helleman & R. Stora), p. 55. North-Holland.
Hille, E. 1976 Ordinary Differential Equations in the Complex Domain. Wiley.
Iooss, G., Helleman, R. H. G. & Stora, R. (ed.) 1983 Chaotic Behaviour of Deterministic Systems, Les Houches Summer School, Session XXXVI. North-Holland.
Jimbo, M., Kruskal, M. D. & Miwa, T. 1982 Phys. Lett. A92, 59.
Kowalevskaya, S. 1889 Acta Math. 12, 177.
Kowalevskaya, S. 1890 Acta Math. 14, 81.
Kuo, A. Y. & Corrsin, S. 1971 J. Fluid Mech. 50, 285.
Landau, E. 1927 Elememtare Zahlentheorie. Leipzig: Hirzel. Translated as Elementary Number Theory, (1958). Chelsea.
Lochak, P. 1985 Phys. Lett. A108, 188.
Melnikov, V. K. 1963 Trans. Moscow Math. Soc. 12, 1.
Moffatt, H. K. 1985 J. Fluid Mech. 159, 359.
Moffatt, H. K. & Proctor, M. R. E. 1985 J. Fluid Mech., 154, 493.
Moser, J. 1973 Stable and Random Motions in Dynamical Systems. Princeton University Press.
Nakach, R. 1977 In Plasma Physics Nonlinear Theory and Experiments (ed. H. Wilhelmson), p. 456, Plenum.
Painlevé, P. 1897 LecLons sur la Théorie des Equations Différentielles. Hermann.
Parker, E. N. 1985 Dynamical Nonequilibrium of Magnetic Fields With Arbitrary Interweaving of the Lines of Force. Preprint, University of Chicago.
Poincaré, H. 1892 Les Méthodes Nouvelles de la Mécanique Celeste. Gauthier-Villars, Paris.
Ramani, A., Dorizzi, B. & Grammaticos, B. 1892 Phys. Rev. Lett. 49, 1539.
šil'nikov, L. P. 1969 Dokl. Akad. Nauk SSSR 189; translation Soviet Math. 10, 1368.
Tabor, M. & Weiss, J. 1981 Phys. Rev. A24, 2157.
Thual, O. & Frisch, U. 1984 Natural Boundary in the Kuramoto Model. In Combustion and Nonlinear Phenomena, (Les Houches lectures) (ed. P. Clavin, B. Larrouturou and P. Pelcé), Editions de Physique (Paris).
Weiss, J., Tabor, M. & Carnevale, G. 1983 J. Math. Phys. 24, 522.
Yoshida, H. 1984 In Chaos and Statistical Mechanics, Proc. Sixth Kyoto Summer Institute (ed. Y. Kuramoto), p. 42. Springer.
Zel'dovich, Ya. B., Ruzmaikin, A. A. & Sokolov, D. D. 1983 Magnetic Fields in Astrophysics. In The Fluid Mechanics of Astrophysics and Geophysics; vol. 3 (ed. P. H. Roberts). Gordon and Breach.
Ziglin, S. L. 1981 Dokl. Akad. Nauk SSSR 257, no. 2; translation Soviet Math. 23, 220.