Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-28T08:16:10.368Z Has data issue: false hasContentIssue false

Localized convection cells in the presence of a vertical magnetic field

Published online by Cambridge University Press:  14 October 2021

J. H. P. Dawes*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Abstract

Thermal convection in a horizontal fluid layer heated uniformly from below usually produces an array of convection cells of roughly equal amplitudes. In the presence of a vertical magnetic field, convection may instead occur in vigorous isolated cells separated by regions of strong magnetic field. An approximate model for two-dimensional solutions of this kind is constructed, using the limits of small magnetic diffusivity, large magnetic field strength and large thermal forcing.

The approximate model captures the essential physics of these localized states, enables the determination of unstable localized solutions and indicates the approximate region of parameter space where such solutions exist. Comparisons with fully nonlinear numerical simulations are made and reveal a power-law scaling describing the location of the saddle-node bifurcation in which the localized states disappear.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Batiste, O., Knobloch, E., Alonso, A. & Mercader, I. 2006 Spatially localized binary fluid convection. J. Fluid Mech. 560, 149158.CrossRefGoogle Scholar
Blanchflower, S. M. 1999a Magnetohydrodynamic convectons. Phys. Lett. A 261, 7481.CrossRefGoogle Scholar
Blanchflower, S. M. 1999b Modelling photospheric magnetoconvection. PhD thesis, University of Cambridge.Google Scholar
Blanchflower, S. M., Rucklidge, A. M. & Weiss, N. O. 1998 Modelling photospheric magneto-convection. Mon. Not. R. Astron. Soc. 301, 593608.CrossRefGoogle Scholar
Blanchflower, S. M. & Weiss, N. O. 2002 Three-dimensional magnetohydrodynamic convectons. Phys. Lett. A 294, 297303.CrossRefGoogle Scholar
Busse, F. H. 1975 Nonlinear interaction of magnetic field and convection. J. Fluid Mech. 71, 193206.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Coullet, P., Riera, C. & Tresser, C. 2000 Stable static localized structures in one dimension. Phys. Rev. Lett. 84, 30693072.CrossRefGoogle ScholarPubMed
Cox, S. M. & Matthews, P. C. 2001 New instabilities in two-dimensional rotating convection and magnetoconvection. Physica D 149, 210229.CrossRefGoogle Scholar
Cox, S. M. & Matthews, P. C. 2003 Instability and localisation of patterns due to a conserved quantity. Physica D 175, 196219.CrossRefGoogle Scholar
Cox, S. M., Matthews, P. C. & Pollicott, S. L. 2004 Swift–Hohenberg model for magneto-convection. Phys. Rev. E 69, 066314.CrossRefGoogle Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 8511112.CrossRefGoogle Scholar
Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Sandstede, B. & Wang, X. 1997 AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations. Available via FTP from directory pub/doedel/auto at ftp.cs.concordia.ca.Google Scholar
Hoyle, R. B. 2006 Pattern Formation: An Introduction To methods. Cambridge University Press.CrossRefGoogle Scholar
Hunt, G. W., Peletier, M. A., Champneys, A. R., Woods, P. D., Wadee, M. A., Budd, C. J. & Lord, G. J. 2000 Cellular buckling in long structures. Nonlinear Dyn. 21, 329.CrossRefGoogle Scholar
Hurlburt, N. E. & Toomre, J. 1988 Magnetic fields interacting with nonlinear compressible convection. A. J. 327, 920932.CrossRefGoogle Scholar
Knobloch, E., Weiss, N. O. & Da Costa, L. N. 1981 Oscillatory and steady convection in a magnetic field. J. Fluid Mech. 113, 153186.CrossRefGoogle Scholar
Komarova, N. L. & Newell, A. C. 2000 Nonlinear dynamics of sand banks and sand waves. J. Fluid Mech. 415, 285321.CrossRefGoogle Scholar
Julien, K., Knobloch, E. & Tobias, S. M. 1999 Strongly nonlinear magnetoconvection in three dimensions. Physica D 128, 105129.CrossRefGoogle Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130141.2.0.CO;2>CrossRefGoogle Scholar
Matthews, P. C. 1999 Asymptotic solutions for nonlinear magnetoconvection. J. Fluid Mech. 387, 397409.CrossRefGoogle Scholar
Matthews, P. C. & Cox, S. M. 2000 Pattern formation with a conservation law. Nonlinearity 13, 12931320.CrossRefGoogle Scholar
Proctor, M. R. E. & Weiss, N. O. 1982 Magnetoconvection. Rep. Prog. Phys. 45, 13171379.CrossRefGoogle Scholar
Rotermund, H. H., Jakubith, S., von Oertzen, A. & Ertl, G. Solitons in a surface reaction. Phys. Rev. Lett. 66 30833086.CrossRefGoogle Scholar
Sakaguchi, H. & Brand, H. R. 1996 Stable localized solutions of arbitrary length for the quintic Swift–Hohenberg equation. Physica D 97, 274285 CrossRefGoogle Scholar
Strümpel, C., Purwins, H.-G. & Astrov, Y. A. 2001 Spatiotemporal filamentary patterns in a dc-driven planar gas discharge system. Phys. Rev. E 63 026409.CrossRefGoogle Scholar
Swift, J. B. & Hohenberg, P. C. 1977 Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319328.CrossRefGoogle Scholar
Umbanhowar, P. B., Melo, F. & Swinney, H. L. 1996 Localized excitations in a vertically vibrated granular layer. Nature 382, 793796.CrossRefGoogle Scholar
Weiss, N. O. 1966 The expulsion of magnetic flux by eddies Proc. R. Soc. Lond. A 293, 310328.Google Scholar
Weiss, N. O. 1981a Convection in an imposed magnetic field. Part 1. The development of nonlinear convection. J. Fluid Mech. 108, 247272.CrossRefGoogle Scholar
Weiss, N. O. 1981b Convection in an imposed magnetic field. Part 2. The dynamical regime. J. Fluid Mech. 108, 273289.CrossRefGoogle Scholar
Weiss, N. O. 2002 Umbral and penumbral magnetoconvection. Astron. Nachr. 323, 371376.3.0.CO;2-Y>CrossRefGoogle Scholar