Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-25T10:13:49.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaos in models of double convection

Published online by Cambridge University Press:  26 April 2006

A. M. Rucklidge
A. M. Rucklidge
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 237 , April 1992 , pp. 209 - 229
Copyright
© 1992 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

ArneAodo, A., Coullet, P. H. & Spiegel, E. A. 1982 Chaos in a finite macroscopic system. Phys. Lett. 92A, 369–373.Google Scholar
ArneAodo, A., Coullet, P. H. & Spiegel, E. A. 1985 The dynamics of triple convection. Geophys. Astrophys. Fluid Dyn. 31, 1–48.Google Scholar
ArneAodo, A. & Thual, O. 1985 Direct numerical simulations of a triple convection problem versus normal form predictions. Phys. Lett. 109A, 367–373.Google Scholar
Arnol'd, V. I.1977 Loss of stability of self-oscillations close to resonance and versa1 deformations of equivariant vector fields. Funct. Anal. Applic. 11, 85–92.Google Scholar
Arter, W. 1983 Nonlinear convection in an imposed horizontal magnetic field. Geophys. Astrophys. Fluid Dyn. 25, 259–292.Google Scholar
Coullet, P. H., Gambaudo, J.-M. & Tresser, C. 1984 Une nouvelle bifurcation de codimension 2: le collage de cycles. C.R. Acad. Sci. Paris 299I, 253–256.Google Scholar
Coullet, P. H. & Spiegel, E. A. 1973 Amplitude equations for systems with competing instabilities. SIAM J. Appl. Maths 43, 776–821.Google Scholar
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Benard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Da Costa, L. N., Knobloch, E. & Weiss, N. O. 1981 Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25–43.Google Scholar
Dangelmayr, G., Armbruster, D. & Neveling, M. 1985 A codimension three bifurcation for the laser with saturable absorber. Z. Phys. B 59, 365–370.Google Scholar
Dangelmayr, G. & Knobloch, E. 1986 Interaction between standing and travelling waves and steady states in magnetoconvection. Phys. Lett. 117A, 394–398.Google Scholar
Deane, A. E. & Sirovich, L. 1991 A computational study of Rayleigh-BeAnard convection. Part 1. Rayleigh-number scaling. J. Fluid Mech. 222, 231–250.Google Scholar
Doedel, E. 1986 AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Pasadena: CTT Press.
Glendinning, P. & Sparrow, C. 1984 Local and global behaviour near homoclinic orbits. J. Statist. Phys. 35, 645–696.Google Scholar
Glendinning, P. & Sparrow, C. 1986 T-points: a codimension two heteroclinic bifurcation. J. Statist. Phys. 43, 479–488.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Guckenheimer, J. & Knobloch, E. 1983 Nonlinear convection in a rotating layer: amplitude expansions and normal forms. Geophys. Astrophys. Fluid Dyn. 23, 247–272.Google Scholar
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Knobloch, E. 1986a On convection in a horizontal magnetic field with periodic boundary conditions. Geophys. Astrophys. Fluid Dyn. 36, 161–177.Google Scholar
Knobloch, E. 1986b On the degenerate Hopf bifurcation with O(2) symmetry. Contemp. Math 56, 56–193.Google Scholar
Knobloch, E., Moore, D. R., Toomre, J. & Weiss, N. O. 1986 Transitions to chaos in two-dimensional double-diffusive convection. J. Fluid Mech. 166, 409–448.Google Scholar
Knobloch, E. & Proctor, M. R. E. 1981 Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech. 108, 291–316.Google Scholar
Knobloch, E., Proctor, M. R. E. & Weiss, N. O. 1992 Heteroclinic bifurcations in a simple model of double-diffusive convection. J. Fluid Mech. (in press).Google Scholar
Knobloch, E. & Silber, M. 1990 Travelling wave convection in a rotating layer. Geophys. Astrophys. Fluid Dyn. 51, 195–209.Google Scholar
Knobloch, E., Weiss, N. O. & Da Costa, L. N. 1981 Oscillatory and steady convection in a magnetic field. J. Fluid Mech. 113, 153–186.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Lyubimov, D. V. & Zaks, M. A. 1983 Two mechanisms of the transition to chaos in finite-dimensional models of convection. Physica 9D, 52–64.Google Scholar
May, R. M. 1976 Simple mathematical models with very complicated dynamics. Nature 261, 459–467.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh-BeAnard convection. J. Fluid Mech. 58, 289–312.Google Scholar
Proctor, M. R. E. & Weiss, N. O. 1990 Normal forms and chaos in thermosolutal convection. Nonlinearity 3, 619–637.Google Scholar
Rucklidge, A. M. 1992 Chaos in a low-order model of magnetoconvection. Physica D (to appear).Google Scholar
Rucklidge, A. M., Weiss, N. O., Brownjohn, D. P. & Proctor, M. R. E. 1992 Oscillations and secondary bifurcations in nonlinear magnetoconvection. Geophys. Astrophys. Fluid Dyn. (submitted).Google Scholar
Shil'nikov, A. L.1986 Bifurcations and chaos in the Marioka-Shimizu system (in Russian). In Methods of Qualitative Theory of Differential Equations, pp. 180–193. Gorky.
Shil'nikov, A. L.1989 Bifurcations and chaos in the Marioka-Shimizu model: II (in Russian). In Methods of Qualitative Theory of Differential Equations and Theory of Bifurcations, pp. 130–138. Gorky.
Shil'nikov, L. P.1965 A case of the existence of a countable number of periodic motions. Sov. Maths Dokl. 6, 163–167.Google Scholar
Shimizu, T. & Morioka, N. 1980 On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. 76A, 201–204.Google Scholar
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer.
Veronis, G. 1965 On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 1–17.Google Scholar
Veronis, G. 1966 Motions at subcritical values of the Rayleigh number in a rotating fluid. J. Fluid Mech. 24, 545–554.Google Scholar
Weiss, N. O. 1981a Convection in an imposed magnetic field. Part 1. The development of nonlinear convection. J. Fluid Mech. 108, 247–272.Google Scholar
Weiss, N. O. 1981b Convection in an imposed magnetic field. Part 2. The dynamical regime. J. Fluid Mech. 108, 273–289.Google Scholar