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Patterns of convection in spherical shells. Part 2

Published online by Cambridge University Press:  20 April 2006

F. H. Busse  and
N. Riahi
F. H. Busse
Affiliation:
Max-Planck-Institut für Astrophysik, Garching near Munich Permanent address: Institute of Geophysics and Planetary Physics, University of California at Los Angeles.
N. Riahi
Affiliation:
Institute of Geophysics and Planetary Physics, UCLA Present address: University of Illinois at Urbana–Champaign.
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Abstract

The analysis by Busse (1975) of preferred patterns of convection in spherical shells is extended to include the case of odd degrees l of spherical harmonics. In the general part of the paper only the property of spherical symmetry of the basic state is used. The results are thus applicable to all bifurcation problems with spherical symmetry. Except in the case 1 = 1 a pattern degeneracy of the linear problem exists, which is partly removed by the solvability conditions that are generated when nonlinear terms are taken into account as perturbations. In each of the cases 1 considered so far, at least 1 physically different solutions have been found. The preferred solution among I existing ones is determined for 1≥2 by a stability analysis. In the case l = 3 emphasized in this paper the axisymmetric solution is found to be always unstable, and the solution of tetrahedronal symmetry appears to be generally preferred. The latter result is rigorously established in the special case of a thin layer with nearly insulating boundaries treated in the second part of the paper.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 123 , October 1982 , pp. 283 - 301
Copyright
© 1982 Cambridge University Press

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References

Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Busse, F. H. 1975 Patterns of convection in spherical shells. J. Fluid Mech. 72, 67–85.Google Scholar
Busse, F. H. & Riahi, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96, 243–256.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Chossat, P. 1979 Bifurcation and stability of convection flows in a rotating or non rotating spherical shell. SIAM J. Appl. Math. 37, 624–647.Google Scholar
Geiger, G. 1977 Ph.D. dissertation, University of Munich.
Golubitsky, M. & Schaeffer, D. G. 1982 Bifurcations with O(3) symmetry including applications to the Bénard problem. Commun. Pure Appl. Math. (in the press).
Joseph, D. D. & Carmi, S. 1966 Subcritical convective instability Part 2. Spherical shells. J. Fluid Mech. 26, 796–777.Google Scholar
Riahi, N., Geiger, G. & Busse, F. H. 1982 Finite Prandtl number convection in spherical shells. Geophys. Astrophys. Fluid Dyn. (to be published).
Sattinger, D. H. 1980 Selection mechanisms in symmetry breaking phenomena. In Approximation Methods for Navier–Stokes Equations (ed. R. Rautmann). Lecture Notes in Mathematics, vol. 771, pp. 494–504.
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Schubert, G. & Zebib, A. 1980 Thermal convection of an internally heated infinite Prandtl number fluid in a spherical shell. Geophys. Astrophys. Fluid Dyn. 15, 65–90.Google Scholar
Young, R. E. 1974 Finite amplitude thermal convection in a spherical shell. J. Fluid Mech. 63, 695–721.Google Scholar
Zebib, A., Schubert, G. & Straus, J. 1980 Infinite Prandtl number thermal convection in a spherical shell. J. Fluid Mech. 97, 257–277.Google Scholar