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Finite-amplitude thermal convection in a spherical shell

Published online by Cambridge University Press:  29 March 2006

Richard E. Young
Richard E. Young
Affiliation:
Advanced Study Program, National Center for Atmospheric Research, Boulder, Colorado 80302 Present address: Space Science Division, NASA, Ames Research Center, Moffett Field, California 94035.
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Abstract

The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and non-axisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms.

Axisymmetric solutions become unstable to non-axisymmetric perturbations at values of the Rayleigh number that depend on Prandtl number and shell thickness. However, even when stable, axisymmetric solutions are not a preferred solution to the full equations; steady non-axisymmetric solutions are obtained for the same parameter values. Initial conditions determine the characteristics of the finite-amplitude solutions, including, in the cases of non-axisymmetry, whether or not a steady state is achieved. Transitions in horizontal flow structure can occur, accompanied by a transition in functional dependence of heat flux on Rayleigh number. The dominant modes in the solutions are usually the modes most unstable to the onset of convection, but not always.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 4 , 15 May 1974 , pp. 695 - 721
Copyright
© 1974 Cambridge University Press

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