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Infinite Prandtl number thermal convection in a spherical shell

Published online by Cambridge University Press:  19 April 2006

Abdelfattah Zebib ,
Gerald Schubert  and
Joe M. Straus
Abdelfattah Zebib
Affiliation:
Department of Mechanical, Industrial and Aerospace Engineering, College of Engineering, Rutgers University, Piscataway, New Jersey 08854
Gerald Schubert
Affiliation:
Department of Earth and Space Sciences, University of California, Los Angeles, California 90024
Joe M. Straus
Affiliation:
Space Sciences Laboratory, The Aerospace Corporation, P.O. Box 92957, Los Angeles, California 90009
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Abstract

A Galerkin method is used to calculate the finite amplitude, steady, axisymmetric convective motions of an infinite Prandtl number, Boussinesq fluid in a spherical shell. Convection is driven by a temperature difference imposed across the stress-free, isothermal boundaries of the shell. The radial gravitational field is spherically symmetric and the local acceleration of gravity is directly proportional to radial position in the shell. Only the case of a shell whose outer radius is twice its inner radius is considered. Two distinct classes of axisymmetric steady states are possible. The temperature and radial velocity fields of solutions we refer to as ‘even’ are symmetric about an equatorial plane, while the latitudinal velocity is antisymmetric about this plane; solutions we refer to as ‘general’ do not possess any symmetry properties about the equatorial plane. The characteristics of these solutions, i.e. the isotherms, streamlines, spherically averaged temperature profiles, Nusselt numbers, etc., are given for Rayleigh numbers Ra as high as about 10 times critical for the even solutions and 3 times critical for the general solutions. Linear stability analyses of the nonlinear steady states show that the general solutions are the preferred form of axisymmetric convection when Ra is less than about 4 times critical. Furthermore, while the preferred motion at the onset of convection is non-axisymmetric, axisymmetric convection is stable when Ra exceeds about 1·3 times the critical value.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 2 , 25 March 1980 , pp. 257 - 277
Copyright
© 1980 Cambridge University Press

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