Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T21:02:34.026Z Has data issue: false hasContentIssue false
  • English
  • Français

Magnetohydrodynamic convectons

Published online by Cambridge University Press:  07 November 2011

David Lo Jacono ,
Alain Bergeon  and
Edgar Knobloch
David Lo Jacono*
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France CNRS; IMFT; F-31400 Toulouse, France
Alain Bergeon
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France CNRS; IMFT; F-31400 Toulouse, France
Edgar Knobloch
Affiliation:
Department of Physics, University of California, Berkeley CA 94720, USA
*
†Email address for correspondence: david.lojacono@imft.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical continuation is used to compute branches of spatially localized structures in convection in an imposed vertical magnetic field. In periodic domains with finite spatial period, these branches exhibit slanted snaking and consist of localized states of even and odd parity. The properties of these states are analysed and related to existing asymptotic approaches valid either at small amplitude (Cox and Matthews, Physica D, vol. 149, 2001, p. 210), or in the limit of small magnetic diffusivity (Dawes, J. Fluid Mech., vol. 570, 2007, p. 385). The transition to standard snaking with increasing domain size is explored.

JFM classification

Nonlinear Dynamical Systems: Bifurcation MHD and Electrohydrodynamics: Magneto convection Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 595 - 605
Copyright
Copyright © Cambridge University Press 2011

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

1. Assemat, P., Bergeon, A. & Knobloch, E. 2008 Spatially localized states in Marangoni convection in binary mixtures. Fluid Dyn. Res. 40, 852–876.CrossRefGoogle Scholar
2. Batiste, O., Knobloch, E., Alonso, A. & Mercader, I. 2006 Spatially localized binary fluid convection. J. Fluid Mech. 560, 149–158.CrossRefGoogle Scholar
3. Bergeon, A., Burke, J., Knobloch, E. & Mercader, I. 2008 Eckhaus instability and homoclinic snaking. Phys. Rev. E 78, 046201.CrossRefGoogle ScholarPubMed
4. Bergeon, A. & Knobloch, E. 2008 Spatially localized states in natural doubly diffusive convection. Phys. Fluids 20, 034102.CrossRefGoogle Scholar
5. Blanchflower, S. 1999 Magnetohydrodynamic convectons. Phys. Lett. A 261, 74–81.CrossRefGoogle Scholar
6. Blanchflower, S. & Weiss, N. 2002 Three-dimensional magnetohydrodynamic convectons. Phys. Lett. A 294, 297–303.CrossRefGoogle Scholar
7. Burke, J. & Knobloch, E. 2007 Homoclinic snaking: structure and stability. Chaos 17, 037102.CrossRefGoogle ScholarPubMed
8. Cox, S. M. & Matthews, P. C. 2001 New instabilities in two-dimensional rotating convection and magnetoconvection. Physica D 149, 210–229.CrossRefGoogle Scholar
9. Cox, S. M., Matthews, P. C. & Pollicott, S. L. 2004 Swift–Hohenberg model for magnetoconvection. Phys. Rev. E 69, 066314.CrossRefGoogle ScholarPubMed
10. Dawes, J. H. P. 2007 Localized convection cells in the presence of a vertical magnetic field. J. Fluid Mech. 570, 385–406.CrossRefGoogle Scholar
11. Dawes, J. H. P. 2008 Localized pattern formation with a large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst. 7, 186–206.CrossRefGoogle Scholar
12. Dawes, J. H. P. & Lilley, S. 2010 Localized states in a model of pattern formation in a vertically vibrated layer. SIAM J. Appl. Dyn. Syst. 9, 238–260.CrossRefGoogle Scholar
13. Dias, F. & Iooss, G. 1996 Capillary–gravity interfacial waves in infinite depth. Eur. J. Mech. (B/Fluids) 15, 367–393.Google Scholar
14. Firth, W. J., Columbo, L. & Scroggie, A. J. 2007 Proposed resolution of theory-experiment discrepancy in homoclinic snaking. Phys. Rev. Lett. 99, 104503.CrossRefGoogle ScholarPubMed
15. Ghorayeb, K. & Mojtabi, A. 1997 Double diffusive convection in a vertical rectangular cavity. Phys. Fluids 9, 2339–2348.CrossRefGoogle Scholar
16. Houghton, S. M. & Bushby, P. J. 2011 Localized plumes in three-dimensional compressible magnetoconvection. Mon. Not. R. Astron. Soc. 412, 555–560.CrossRefGoogle Scholar
17. Houghton, S. M. & Knobloch, E. 2011 The Swift–Hohenberg equation with broken cubic–quintic nonlinearity. Phys. Rev. E 84, 016204.CrossRefGoogle ScholarPubMed
18. Iooss, G. & Pérouème, M. C. 1993 Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J. Differ. Equ. 102, 62–88.CrossRefGoogle Scholar
19. Knobloch, E. 2008 Spatially localized structures in dissipative systems: open problems. Nonlinearity 21, T45–T60.CrossRefGoogle Scholar
20. Knobloch, E., Weiss, N. O. & Da Costa, L. N. 1981 Oscillatory and steady convection in a magnetic field. J. Fluid Mech. 113, 153–186.CrossRefGoogle Scholar
21. Lo Jacono, D., Bergeon, A. & Knobloch, E. 2010 Spatially localized binary fluid convection in a porous medium. Phys. Fluids 22, 073601.CrossRefGoogle Scholar
22. Matthews, P. C. & Cox, S. M. 2000 Pattern formation with a conservation law. Nonlinearity 13, 1293–1320.CrossRefGoogle Scholar
23. Mercader, I., Batiste, O., Alonso, A. & Knobloch, E. 2009 Localized pinning states in closed containers: homoclinic snaking without bistability. Phys. Rev. E 80, 025201(R).CrossRefGoogle ScholarPubMed
24. Mercader, I., Batiste, O., Alonso, A. & Knobloch, E. 2011 Convectons, anticonvectons and multiconvectons in binary fluid convection. J. Fluid Mech. 667, 586–606.CrossRefGoogle Scholar
25. Schneider, T. M., Gibson, J. F. & Burke, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.CrossRefGoogle ScholarPubMed
26. Weiss, N. O. 1966 The expulsion of magnetic flux by eddies. Proc. R. Soc. A 293, 310–328.Google Scholar
27. Weiss, N. O. 1981 Convection in an imposed magnetic field. Part 1. The development of nonlinear convection. J. Fluid Mech. 108, 247–272.CrossRefGoogle Scholar
28. Woods, P. D. & Champneys, A. R. 1999 Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate Hamiltonian–Hopf bifurcation. Physica D 129, 147–170.CrossRefGoogle Scholar