Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-25T17:12:04.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

Published online by Cambridge University Press:  26 April 2006

P. C. Matthews ,
M. R. E. Proctor  and
N. O. Weiss
P. C. Matthews
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
N. O. Weiss
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.

Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.

If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.

In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 305 , 25 December 1995 , pp. 281 - 305
Copyright
© 1995 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

Brandenburg, A., Nordlund, A., Pulkkinen, P., Stein, R. F. & Tuominen, I. 1990 3-D simulation of turbulent cyclonic magneto-convection.Astron. Astrophys. 232, 277–291.
Brownjohn, D. P., Hurlburt, N. E., Proctor, M. R. E. & Weiss, N. O. 1995 Nonlinear compressible magnetoconvection. Part 3. Travelling waves in a horizontal field.J. Fluid Mech. 300, 287–309.
Busse, F. 1972 The oscillatory instability of convection rolls in a low prandtl number fluid. J. Fluid Mech. 52, 97–112.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Cattaneo, F. 1984 Oscillatory convection in sunspots. In The Hydromagnetics of the sun (ed. T. D. Guyenne), pp. 47–50. ESA SP 220.
Cattaneo, F., Brummell, N. H., Toomre, J. Malagoli, A. & Hurlburt, N. E. 1991 Turbulent compressible convection. Astrophys. J. 370, 282–294.Google Scholar
Chandrasekttar, S. 1961 Hydrodynomic and Hydromagnetic Stability. Oxford University Press.
Clune, T. & Knobloch, E. 1994 Pattern selection in three-dimensional magnetoconvection. Physica D 74, 151–176.Google Scholar
Dangelmayr, G. & Knobloch, E. 1987 The Takens-Bogdanov bifurcation with 0(2) symmetry. Phil. Trans. R. Soc. Land. A 322, 243–279.Google Scholar
Drobyshevski, E. M. & Yuferev, V. S. 1974 Topological pumping of magnetic flux by three-dimensional convection. J. Fluid Mech. 65, 33–44.Google Scholar
Golubitsky, M. Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh Bénard convection. PhysicaD 10 249–276.Google Scholar
Howard, L. N. & Krishnamurti, R. 1986 Large-scale flow in turbulent convection: a mathematical model. J. Fluid Mech. 170, 385–410.Google Scholar
Hurlburt, N. E., Matthews, P. C. & Proctor, M. R. E. 1995 Nonlinear compressible convection in oblique magnetic flields. Astrophys. J., in press.
Hurlburt, N. E., Proctor, M. R. E. Weiss, N. O. & Brownjohn, D. P. 1989 Nonlinear compressible magnetoconvection. Part 1. Travelling waves and oscillations. J. Fluid Mech. 207, 587–628.Google Scholar
Hurlburt, N. E. & Toomre, J. 1988 Magnetic fields interacting with nonlinear compressible convection. Astrophys. J. 327, 920–932.Google Scholar
Jones, C. A. & Proctor, M. R. E. 1987 Strong spatial resonance and travelling waves in Bénard convection. Phys. Lett. A 121, 224–227.Google Scholar
Knobloch, E., Weiss, N. O. & Da COSTA, L. N. 1981 Oscillatory and steady convection in a magnetic field. J. Fluid Mech. 113, 153–186.Google Scholar
Lantz, S. R. 1995 Magnetoconvection dynamics in a stratified layer. II. A low-order model of the tilting instability. Astrophys. J. 441, 925–941.Google Scholar
Mannevillf, P. 1991 Dissipative Structures and Weak Turbulence.. Academic press.
Matthews P. C. 1993 Compressible magnetoconvection in three dimensions. In Solar and Planetary dynamos (ed. M. R. E. Proctor, P. C. Matthews & A. M. Rucklidge), pp. 211–218. Cambridge University Press.
Matthews P. C. 1994 Three-dimensional compressible magnetoconvection. In Solar Active Region Evolution: Comparing Models with Observations (ed. K. S. Balasubramaniam & G. W. Simon), pp. 56–61. Astron. Soc. Pacific.
Matthews, P. C., Hurlburt, N. E., Proctor, M. R. E. & Brownjohn, D. P. 1992 Compressible magnetoconvection in oblique fields: linearized theory and simple nonlinear models. J. fluid Mech. 240, 559–569.Google Scholar
Matthews, P. C. Proctor, M. R. E. Rucklidge, A. M. & Weiss, N. O. 1993 pulsating waves in nonliDear magnetoconvection. Phys. Lett. A 183, 69–75.Google Scholar
Matthews, P. C., Proctor, M. R. E. Rucklidge, A. M. & Weiss, N. O. 1994 Nonlinear three-dimensional magnetoconvection in a compressible atmosphere. In Solar Magnetic Fields (ed. M. Schüssler & W. Schmidt), pp. 279–281. Cambridge University Press.
Matthews, P. C. & Rucklidge, A. M. 1993 Travelling and standing waves in magnetoconvection. Proc. R. Soc. Lond. A 441, 649–658.Google Scholar
Matthews, P. C., Rucklidge, A. M., Weiss, N. O. & Proctor, M. R. E. 1996 The three-dimensional development of the shearing instability of convection. Submitted to Phys. Fluids.
Nordlund, A., Brandenburg, A., Jennings, R. L., Rieutord, M., Ruokolainen, J. & Tuominen, I. 1992 Dynamo action in stratified convection with overshool. Astrophys. J. 392 647–652.Google Scholar
Nordlund, A., Galsgaard, K. & Stein, R. F. 1994 Magnetoconvection and magnetoturbulence. In solar Surface magnetism (ed. R. J. Rutten & C. J. Schrijver), pp. 471–498. Kluwer.
Nordlund, A. & Stein, R. F. 1990 Solar magnetoconvection. In Solar Photosphere: Structure, convection and Magnetic Fields (ed. J. O. Stenflo), pp. 191–211. Kluwer.
Prat, J., Massaguer, J. M. & Mercader, I. 1995 Large-scale flows and resonances in 2-D thermal convection. Phys. Fluids 7, 121–134.Google Scholar
Proctor, M. R. E. 1992 Magnetoconvection. In Sunspots: Theory and Observations (ed. J. H. Thomas & N. O. Weiss), pp. 221–241. Kluwer.
Proctor, M. R. E. & Matthews, P. C. 1996 √2 : 1 resonance in non-Boussinesq convection. In Preparation.
Proctor, M. R. E. & Weiss, N. O. 1982 Magnetoconvection. Rep. Prog. Phys. 45 1317–1379.Google Scholar
Proctor, M. R. E., Weiss, N. O., Brownjohn D. P. & Hurlburi, N. E. 1994 Nonlinear compressible magnetoconvection. Part 2. Streaming instabilities in two dimensions. J. Fluid Mech. -280, 227–253.Google Scholar
Rucklidge, A. M. & Matthews, P. C. 1995a Analysis of the shearing instability in nonlinear convection and magnetoconvection. Submitted to Nonlinearity.
Rucklidge, A. M. & Matthews, P. C. 1995b Shear flow instabilities of three-dimensional magne-toconvection. In preparation.
Rucklidge, A. M., Schmidt, H. U. & Weiss, N. O., 1995 The abrupt development of penumbrae in sunspots. Mon. Not. R. Astron. Soc. 273, 491–498.Google Scholar
Silber, M. & Knobloch, E. 1991 Hopf bifurcation on a square lattice. Nonlinearity 4, 1063–1106.Google Scholar
Stein, R. F. & Nordlund, A 1989 Topology of convection beneath the solar surface. Astrophys J.. 342 L95–L98.Google Scholar
Spruit, H. C., Nordlund, A & Title, A. M. 1990 Solar Convection. Ann. Rev. Astron. Astrophys. 28, 263–301.Google Scholar
Swift, J. W. 1988 Hopf bifurcation with the symmetry of the square. Nonlinearity 1 333–377.Google Scholar
Thomas, J. H. & Weiss, N. O. 1992 The theory of sunspots. In Sunspots: Theory and Observations (ed. J. H. Thomas & N. O. Weiss), pp. 3–59. kluwer.
Vainshtein, S. I., Tao, L., Cattaneo, F. & Rosner, R. 1993 Turbulent magnetic transport effects and their relation to magnetic field intermittency. In Theory of solar and Planetary Dynamos (ed. M. R. E. Proctor, P. C. Mathews & A. M. Rucklidge), pp. 311–320. Cambridge University Press.
Weiss, N. O. 1991 Magnetoconvection. Geophys. Astrophys. Fluid Dyn. 62, 229–247.Google Scholar
Weiss, N. O. 1994 Magnetoconvective patterns. In Solar Surface Magnetism (ed. R. J. Rutten & C. J. Schrijver) pp. 287–295. Kluwer.
Weiss, N. O. Brownjohn, D. P. Hurlburt, N. E. & Proctor M. R. E. 1990 Oscillatory convection in sunspot umbrae. Mon. Not. R. Astron. Soc. 245, 434–452.Google Scholar
White, D. B. 1988 The planforms and onset of convection with a temperature-dependent viscosity.J. Fluid. Mech. 191, 247–286.