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Modal equations for cellular convection

Published online by Cambridge University Press:  29 March 2006

D. O. Gough ,
E. A. Spiegel  and
Juri Toomre
D. O. Gough
Affiliation:
Institute of Astronomy and Department of Applied Mathematics and Theoretical Physics, University of Cambridge
E. A. Spiegel
Affiliation:
Astronomy Department, Columbia University, New York 10027
Juri Toomre
Affiliation:
Joint Institute for Laboratory Astrophysics and Department of Astro-Geophysics, University of Colorado, Boulder
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Abstract

We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possibly useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of ‘modal equations’ is analysed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 68 , Issue 4 , 29 April 1975 , pp. 695 - 719
Copyright
© 1975 Cambridge University Press

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