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The breakdown of steady convection

Published online by Cambridge University Press:  21 April 2006

T. B. Lennie
Affiliation:
BP Exploration Co. Ltd, Britannic House, London EC2Y 9BU, UK
D. P. Mckenzie
Affiliation:
Department of Earth Sciences, University of Cambridge, Cambridge CB3 0EZ, UK
D. R. Moore
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, UK
N. O. Weiss
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK

Abstract

Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased.

Type
Research Article
Copyright
© 1988 Cambridge University Press

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