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Oscillatory and steady convection in a magnetic field

Published online by Cambridge University Press:  20 April 2006

E. Knobloch ,
N. O. Weiss  and
L. N. Da Costa
E. Knobloch
Affiliation:
Department of Physics, University of California, Berkeley
N. O. Weiss
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
L. N. Da Costa
Affiliation:
SEPLAN/PR-CNPq-Observatorio Nacional, Rua General Bruce 586, Rio de Janeiro, Brazil
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Abstract

Two-dimensional convection in a Boussinesq fluid in the presence of an imposed magnetic field is described in terms of a simplified model, which is exact to second order in the amplitude of the motion and appears to be qualitatively correct for larger amplitudes. If the ratio of the magnetic diffusivity to the thermal diffusivity is sufficiently small and the imposed magnetic field is sufficiently large, convection sets in when r = r(o) as overstable oscillations, which grow in amplitude as the normalized Rayleigh number r is increased. There is also a branch of steady solutions that bifurcates from the static equilibrium at r = r(e) < r(o) and stable steady solutions exist for r > rmin. For certain choices of parameters subcritical steady convection, with rmin < r(e), is found and the oscillatory branch ends on the unstable portion of the steady branch, where the period of the oscillations becomes infinite. In some circumstances there may be a bifurcation from symmetrical to asymmetrical oscillations, followed by a sequence of bifurcations at each of which the period doubles. Other choices of parameters allow only supercritical convection with r increasing monotonically on the steady branch; if convection first appears as overstable oscillations the steady branch is then unstable for r(e) < r < rmin and there is a Hopf bifurcation at r = rmin. This complicated pattern of behaviour is consistent with the results of numerical experiments on the full two-dimensional problem.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 113 , December 1981 , pp. 153 - 186
Copyright
© 1981 Cambridge University Press

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References

Busse, F. H. 1975 Nonlinear interaction of magnetic field and convection. J. Fluid Mech. 71, 193–206.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Da Costa, L. N., Knobloch, E. & Weiss, N. O. 1981 Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25–43.Google Scholar
Danielson, R. E. 1961 The structure of sunspot penumbras. II. Theoretical. Astrophys. J. 134, 289–311.Google Scholar
Feigenbaum, M. J. 1978 Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52.Google Scholar
Franceschini, V. 1980 A Feigenbaum sequence of bifurcations in the Lorenz Model. J. Stat. Phys. 22, 397–407.Google Scholar
Franceschini, V. & Tebaldi, C. 1979 Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier — Stokes equations. J. Stat. Phys. 21, 707–726.Google Scholar
Galloway, D. J. & Moore, D. R. 1979 Axisymmetric convection in the presence of a magnetic field. Geophys. Astrophys. Fluid Dyn. 12, 73–105.Google Scholar
Hopf, E. 1942 Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math. Phys. Sächs. Akad. Wiss. 94, 1–22 (trans. in Marsden & McCracken 1976, pp. 163–193).Google Scholar
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Jeans, J. H. 1928 Astronomy and Cosmogony, pp. 179–186. Cambridge University Press.
Knobloch, E. & Proctor, M. R. E. 1981 Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech. 108, 291–316.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Marsden, J. E. & McCracken, M. 1976 The Hopf Bifurcation and its Applications. Springer-Verlag, New York.
Poincaré, H. 1885 Sur l’équilibre d'une masse fluide animee d'un mouvement de rotation. Acta Math. 7, 259–380.Google Scholar
Proctor, M. R. E. & Galloway, D. J. 1979 The dynamic effect of flux ropes on Rayleigh — Bénard convection. J. Fluid Mech. 90, 273–287.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
Sattinger, D. H. 1973 Topics in stability and bifurcation theory. Lecture Notes in Mathematics, 309. Springer-Verlag, Berlin.
Siegmann, W. L. & Rubenfeld, L. A. 1975 A nonlinear model for double-diffusive convection. SIAM J. Appl. Math. 29, 540–557.Google Scholar
Spiegel, E. A. 1972 Convection in stars. II. Special effects. Ann. Rev. Astron. Astrophys. 10, 261–304.Google Scholar
Veronis, G. 1959 Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech. 5, 401–435.Google Scholar
Veronis, G. 1965 On finite amplitude instability in thermohaline convection. J. Marine 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. 1964 Convection in the presence of restraints. Phil. Trans. Roy. Soc. A 256, 99–147.Google Scholar
Weiss, N. O. 1966 The expulsion of magnetic flux by eddies. Proc. Roy. Soc. A 293, 310–328.Google Scholar
Weiss, N. O. 1977 Magnetic fields and convection. Problems of Stellar Convection (ed. E. A. Spiegel and J.-P. Zahn), pp. 176–187. Berlin: Springer-Verlag.
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
Weiss, N. O. 1981c The interplay between magnetic fields and convection. J. Geophys. Res. (in press).