Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-05T16:36:17.478Z Has data issue: false hasContentIssue false
  • English
  • Français

The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance

Published online by Cambridge University Press:  21 April 2006

M. R. E. Proctor  and
C. A. Jones
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
C. A. Jones
Affiliation:
Department of Applied Mathematics, University of Newcastle-upon-Tyne, NE1 7RU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of two superimposed layers of fluid, heated from below and separated by a thin conducting plate, is investigated. It is shown that when the ratio of the depths of the layers is close to 1:2, two distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:2. The nonlinear evolution of a disturbance consisting of both modes is considered, and it is shown that travelling waves are the preferred mode of nonlinear convection for a wide range of parameter values. Other possible types of behaviour, including modulated waves and an attracting homoclinic trajectory, are also described in detail.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 188 , March 1988 , pp. 301 - 335
Copyright
© 1988 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.)

References

Ahlers, G. & Behringer, R. P. 1978 Phys. Rev. Lett. 40, 772.
Armbruster, D., Guckenheimer, J. & Holmes, P. 1987 Physica D (in press).
Azouni, M. A. & Normand, C. 1983 Geophys. Astrophys. Fluid Dyn. 23, 223–245.
Busse, F. H. & Or, A. C. 1986 Z. Angew. Math. Phys. 37, 608–623.
Busse, F. H. & Whitehead, J. A. 1971 J. Fluid Mech. 47, 305–320.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Chaté, H. & Manneville, P. 1987 Phys. Rev. Lett. 58, 112–115.
Coullet, P. 1986 Cont. Math. 56, 724.
Cross, M. C. & Newell, A. C. 1984 Physica D, 10, 299–328.
Dangelmayr, G. 1986 Dyn. Stab. Systems 1, 159–185.
Dangelmayr, G. & Armbruster, D. 1986 Cont. Math. 56, 53–68.
Fauve, S. 1985 Woods Hole Oceanographic Institution Technical Rep. WHOI-85-36, pp. 55–70.
Jones, C. A. & Proctor, M. R. E. 1987 Phys. Lett. A 121, 224–228.
Knobloch, E. & Guckenheimer, J. 1982 Phys. Rev. A. 27, 408–417.
Knobloch, E. & Proctor, M. R. E. 1981 J. Fluid Mech. 108, 291–316.
Knobloch, E. & Proctor, M. R. E. 1988 Proc. R. Soc. Lond. A 415, 61–90.
Kolodner, P., Passner, A., Surko, C. M. & Walden, R. W. 1986 Phys. Rev. Lett. 56, 2621.
Malkus, W. V. R. & Veronis, G. R. 1958 J. Fluid Mech. 4, 225–260.
Newell, A. C. & Whitehead, J. A. 1968 J. Fluid Mech. 38, 279–303.
Pomeau, Y. 1984 Cellular Structures in Instabilities (ed. J. E. Wesfreid & S. Zaleski). Lecture Notes in Physics, vol. 210, pp. 207–214. Springer.
Rayleigh, Lord 1916 Phil. Mag. (6) 32, 529.
Rosenblat, S., Davis, S. H. & Homsy, G. M. 1982 J. Fluid Mech. 120, 91–138.
Segel, L. A. 1962 J. Fluid Mech. 14, 97–114.
Segel, L. A. 1965 J. Fluid Mech. 21, 359–384.
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer.