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Time-dependent solutions of multimode convection equations

Published online by Cambridge University Press:  20 April 2006

Juri Toomre ,
D. O. Gough  and
E. A. Spiegel
Juri Toomre
Affiliation:
Joint Institute for Laboratory Astrophysics, and Department of Astrophysical, Planetary and Atmospheric Sciences, University of Colorado, Boulder, Colorado 80309, U.S.A.
D. O. Gough
Affiliation:
Institute of Astronomy, and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England
E. A. Spiegel
Affiliation:
Department of Astronomy, Columbia University, New York, New York 10027, U.S.A.
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Abstract

Truncated modal equations are used to study the time evolution of thermal convection. In the Boussinesq approximation these nonlinear equations are obtained by expanding the fluctuating velocity and temperature fields in a finite set of planforms of the horizontal coordinates. Here we report on numerical studies dealing with two or three modes with triad interactions. We have found rich time dependence in these cases: periodic and aperiodic solutions can be obtained, along with various steady solutions. Three-mode solutions reproduce the qualitative appearance of spoke-pattern convection as observed in experiments at high Prandtl numbers. Though the values of the periods of the time-dependent solutions do not agree with those of the experiments, their variation with Rayleigh number compares favourably. Except at the highest Rayleigh number we have considered (107), the theoretical Nusselt numbers agree well with experiment.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 125 , December 1982 , pp. 99 - 122
Copyright
© 1982 Cambridge University Press

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