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Nonlinear periodic convection in double-diffusive systems

Published online by Cambridge University Press:  20 April 2006

E. Knobloch  and
M. R. E. Proctor
E. Knobloch
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW Also Institute of Astronomy, University of Cambridge. (Present address: Physics Department, University of California, Berkeley.)
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW
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Abstract

We study two examples of two-dimensional nonlinear double-diffusive convection (thermohaline convection, and convection in an imposed vertical magnetic field) in the limit where the onset of marginal overstability just precedes the exchange of stabilities. In this limit nonlinear solutions can be found analytically. The branch of oscillatory solutions always terminates on the steady solution branch. If the steady solution branch is subcritical this occurs when the period of the oscillation becomes infinite, while if it is supercritical, it occurs via a Hopf bifurcation. A detailed discussion of the stability of the oscillations is given. The results are in broad agreement with the largeramplitude results obtained previously by numerical techniques.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 108 , July 1981 , pp. 291 - 316
Copyright
© 1981 Cambridge University Press

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References

Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37, 289–306.Google Scholar
Caesar, G. J. 51 B.C. De Bello Gallico. (English translation: Penguin Classics.)
Carr, J. 1979 Applications of Centre Manifold Theory, Lecture Notes, Lefschetz Center for Dynamical Systems, Brown University.
Dacosta, L. N., Knobloch, E. & Weiss, N. O. 1981 Oscillations in double-diffusive convection. J. Fluid Mech. (to appear).Google Scholar
Davis, H. T. 1962 Introduction to Nonlinear Differential and Integral Equations. Dover.
Hopf, E. 1942 Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math. Phys. Sächs. Akad. Wiss. 94, 1–22.Google Scholar
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Jordan, D. W. & Smith, P. 1977 Nonlinear Ordinary Differential Equations. Clarendon Press.
Knobloch, E., Weiss, N. O. & Dacosta, L. N. 1981 Oscillatory and steady convection in a magnetic field. J. Fluid Mech. (submitted).Google Scholar
Rubenfeld, L. A. & Siegmann, W. L. 1977 Nonlinear dynamic theory for a double-diffusive convection model. SIAM J. Appl. Math. 32, 871–894.Google Scholar
Takens, F. 1974 Forced Oscillations and Bifurcations, Communication 3, Mathematical Institute, Rijksuniversiteit Utrecht.
Veronis, G. 1959 Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech. 5, 401–436.Google Scholar
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
Veronis, G. 1968a Large-amplitude Bénard convection in a rotating fluid. J. Fluid Mech. 31, 113–139.Google Scholar
Veronis, G. 1968b Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336.Google Scholar
Weiss, N. O. 1964 Convection in the presence of restraints. Phil. Trans. Roy. Soc. A 256, 99–147.Google Scholar
Weiss, N. O. 1977 Magnetic fields and convection. Problems of Stellar Convection (ed. E. A. Spiegel & J.-P. Zahn), pp. 176–187. Springer.
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