Abstract
The properties of convection inside a spherical shell heated from within are studied by direct numerical simulations. A pseudo-spectral method is used. Both the compressible and the incompressible (Boussinesq) case are treated.
We consider first a non rotating configuration. It is well known that the solutions of the linear problem are degenerate, due to the spherical symmetry, and that their angular behaviour is in the form of spherical harmonics Y lm with a given l and any m. The degeneracy is removed by taking into account the nonlinear terms. This selects a particular value of the wavenumber m [Bus75]. We observe the expected pattern very near the critical Rayleigh number. However when we increase the Rayleigh number the solution undergoes transitions to other steady configurations.
We then study the transition to chaotic convection by increasing the Rayleigh number, both in a non rotating and in a moderately rotating (Taylor number of 100) configuration. In both cases we observe at first the onset of a periodic behaviour, then the appearance of a second frequency, followed by a chaotic regime. The behaviour of the convection cells in the 1-frequency and 2-frequency regimes is presented.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
D. Bercovici, G. Schubert, G. A. Glatzmaier, and A. Zebib. Three-dimensional thermal convection in a spherical shell. J. Fluid Mech., 206:75, 1989.
F. H. Busse. Patterns of convection in spherical shells. J. Fluid Mech., 72:67, 1975.
U. Frisch, Z. S. She, and P. L. Sulem. Physica D, 28:382, 1987.
P. A. Gilman and G. A. Glatzmaier. Compressible convection in a rotating spherical shell. I. Anelastic equations. The Astroph. J. Suppl. Series, 45:335, 1981.
P. A. Gilman and J. Miller. Nonlinear convection of a compressible fluid in a rotating spherical shell. The Astrophysical Journal Supplement Series, 61:585, 1986.
P. Machetel and M. Rabinowicz. Transitions to a two mode axisymmetrical spherical convection: Application to the earth's mantle. Geophys. Res. Lett., 12:227, 1985.
P. Machetel and M. Rabinowicz. Three-dimensional convection in spherical shells. Geophys. Astrophys. Fluid Dynamics, 37:57, 1986.
P. Machetel and D. Yuen. Infinite Prandtl number spherical-shell convection. Technical report, University of Minnesota Supercomputer Institute, 1986.
M. Steenbeck, F. Krause, and K. H. Rädler. Z. Naturforsch., 21:369, 1966.
P. L. Sulem, Z. S. She, H. Scholl, and U. Frisch. Generation of large-scale structures in three-dimensional flow lacking parity-invariance. J. Fluid Mech., 205:341, 1989.
L. Valdettaro and M. Meneguzzi. Compressible MHD in spherical geometry. In Proceedings from the Workshop on Supercomputing Tools for Science and Engineering, page 573, Pisa, December 1989.
Zhang K.-K. and Busse F. On the onset of convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn., 39:119–147, 1987.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Valdettaro, L., Rieutord, M. (1991). Numerical study of the transition to chaotic convection inside spherical shells. In: Fournier, JD., Sulem, PL. (eds) Large Scale Structures in Nonlinear Physics. Lecture Notes in Physics, vol 392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54899-8_39
Download citation
DOI: https://doi.org/10.1007/3-540-54899-8_39
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54899-7
Online ISBN: 978-3-540-46469-3
eBook Packages: Springer Book Archive