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Asymmetric oscillations in thermosolutal convection

Published online by Cambridge University Press:  26 April 2006

D. R. Moore ,
N. O. Weiss  and
J. M. Wilkins
D. R. Moore
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, UK
N. O. Weiss
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
J. M. Wilkins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

Thermosolutal convection provides a testbed for applications of nonlinear dynamics to fluid motion. If the ratio of solutal to thermal diffusivity is small and the solutal Rayleigh number RS is large, instability sets in at a Hopf bifurcation as the thermal Rayleigh number RT is increased. For two-dimensional convection in a rectangular box the fundamental mode is a single roll with point symmetry about its axis. The symmetries of periodic and steady solutions form an eighth-order group with invariant subgroups that describe pure single-roll and multiroll solutions. A systematic numerical investigation reveals a rich variety of spatiotemporal behaviour in the regime where RS [Gt ] RT−RS > 0. Point symmetry is broken and there is a branch of spatially asymmetric periodic solutions. These mixed-mode oscillations lose their temporal symmetry in a subsequent bifurcation, followed eventually by a transition to chaos. The numerical experiments can be interpreted by relating the physical form of the solutions to an appropriate bifurcation structure.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 233 , December 1991 , pp. 561 - 585
Copyright
© 1991 Cambridge University Press

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References

Arnol'D, V. I. 1983 Geometrical Methods in the Theory of Ordinary Differential Equations. Springer.
Armbeuster, D., Guckenheimer, J. & Kim, S. 1989 Chaotic dynamics in systems with square symmetry.. Phys. Lett A 140, 416–420.Google Scholar
Bretherton, C. & Spiegel, E. A. 1983 Intermittency through modulational instability.. Phys. Lett. A 96, 152–156.Google Scholar
Cash, J. R. & Moore, D. R. 1980 A high order method for the numerical solution of two-point boundary value problems. BIT 20, 44–53.Google Scholar
Coullet, P. H. & Spiegel, E. A. 1983 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 Bénard 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
Deane, A. E., Knobloch, E. & Toomre, J. 1988 Travelling waves in large-aspect ratio thermosolutal convection.. Phys. Rev. A 37, 1817–1820.Google Scholar
Goldhirsch, I., Pelz, R. B. & Orszag, S. A. 1989 Numerical simulation of thermal convection in a two-dimensional finite box. J. Fluid Mech. 199, 1–28.Google Scholar
Golubitsky, M. & Schaeffer, D. 1985 Singularities and Groups in Bifurcation Theory. Springer.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Jennings, R. L. & Weiss, N. O. 1991 Symmetry breaking in stellar dynamos. Mon. Not. R. Astron. Soc. (in press).Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions II. Springer.
Knobloch, E., Deane, A. E., Toomre, J. & Moore, D. R. 1986a Doubly diffusive waves. Contemp. Maths 56, 203–216.Google Scholar
Knobloch, E. & Moore, D. R. 1990 Minimal model of binary fluid convection.. Phys. Rev. A 42, 4693–4709.Google Scholar
Knobloch, E., Moore, D. R., Toomre, J. & Weiss, N. O. 1986b Transitions to chaos in two-dimensional double-diffusive convection. J. Fluid Mech. 166, 409–448 (referred to herein as I).Google Scholar
Knobloch, E. & Proctor, M. R. E. 1981 Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech. 108, 291–316.Google Scholar
Leibovich, S., Lele, S. K. & Moroz, I. 1989 Nonlinear dynamics in Langmuir circulations and in thermosolutal convection. J. Fluid Mech. 198, 471–511.Google Scholar
Lennie, T. B., McKenzie, D. P., Moore, D. R. & Weiss, N. O. 1988 The breakdown of steady convection. J. Fluid Mech. 188, 47–85.Google Scholar
Mckenzie, D. R. 1988 The symmetry of convection transitions in space and time. J. Fluid Mech. 191, 287–339.Google Scholar
Moore, D. R., Peckover, R. S. & Weiss, N. O. 1973 Difference methods for time-dependent two-dimensional convection. Comput. Phys. Commun. 6, 198–220.Google Scholar
Moore, D. R., Toomre, J., Knobloch, E. & Weiss, N. O. 1983 Period-doubling and chaos in partial differential equations for thermosolutal convection. Nature 303, 663–667.Google Scholar
Moore, D. R. & Weiss, N. O. 1990 Dynamics of double convection.. Phil. Trans. R. Soc. Lond. A 332, 121–134.Google Scholar
Moore, D. R., Weiss, N. O. & Wilkins, J. M. 1990a Symmetry-breaking in thermosolutal convection.. Phys. Lett. A 147, 209–214.Google Scholar
Moore, D. R., Weiss, N. O. & Wilkins, J. M. 1990b The reliability of numerical experiments: transitions to chaos in thermosolutal convection. Nonlinearity 3, 997–1014.Google Scholar
Nagata, M., Proctor, M. R. E. & Weiss, N. O. 1990 Transactions to asymmetry in magnetoconvection. Geophys. Astrophys. Fluid Dyn. 51, 211–241.Google Scholar
Proctor, M. R. E. & Weiss, N. O. 1982 Magnetoconvection. Rep. Prog. Phys. 45, 1317–1379.Google Scholar
Proctor, M. R. E. & Weiss, N. O. 1990 Normal forms and chaos in thermosolutal convection. Nonlinearity 3, 619–637.Google Scholar
Sattinger, D. H. 1978 Group representation theory, bifurcation theory and pattern formation. J. Fund. Anal. 28, 58–101.Google Scholar
Shi, A. & Orszag, S. A. 1987 Order and disorder in two-dimensional double-diffusive convection. Preprint.
Tuckerman, L. S. & Barkley, D. 1988 Global bifurcations to traveling waves in axisymmetric convection. Phys. Rev. Lett. 61, 408–411.Google Scholar
Veronis, G. 1968 Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336.Google Scholar
Weiss, N. O. 1981 Convection in an imposed magnetic field Part 2. The dynamical regime. J. Fluid Mech. 108, 273–289.Google Scholar
Weiss, N. O. 1987 Dynamics of convection.. Proc. R. Soc. Lond. A 413, 71–85.Google Scholar
Weiss, N. O. 1990 Symmetry breaking in nonlinear convection. In Nonlinear Evolution of Spatio-temporal Structures in Dissipative Continuous Systems (ed. F. Busse & L. Kramers), pp. 359–374. Plenum.
Wiggins, S. 1988 Global Bifurcations and Chaos. Springer.