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Planform selection by finite-amplitude thermal convection between poorly conducting slabs

Published online by Cambridge University Press:  20 April 2006

M. R. E. Proctor
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

Three-dimensional finite-amplitude thermal convection in a fluid layer is considered in the case where the boundaries of the layer are much poorer conductors than the fluid. It can be shown that if the conductive heat flux through the layer is not too large, the horizontal scale of motion is much greater than the layer depth. Then a ‘shallow water theory’ approximation leads to a nonlinear evolution equation for the leading-order temperature perturbation, which can be analysed in terms of a variational principle. It is proved that the preferred planform of convection is a square cell tesselation, as found in a rather more restricted parameter range by Busse & Riahi (1980), in contrast to the roll solutions that obtain for perfectly conducting boundaries. It is also shown that the preferred wavelength of convection increases slowly with amplitude.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 113 , December 1981 , pp. 469 - 485
Copyright
© 1981 Cambridge University Press

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References

Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Busse, F. H. & Riahi, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96, 243–256.Google Scholar
Chapman, C. J. 1978 Bénard convection with constant neat flux boundaries. Woods Hole Summer School Fellowship Lectures, vol. 11, pp. 1–17.
Chapman, C. J. 1980 Bénard convection between insulating boundaries. Ph.D. thesis, University of Bristol.
Chapman, C. J., Childress, S. & Proctor, M. R. E. 1980 Long wavelength thermal convection between non-conducting boundaries. Earth & Planet. Sci. Lett. 51, 342–369.Google Scholar
Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh — Bénard convection with poorly conducting boundaries. J. Fluid Mech. 101, 759–782.Google Scholar
Childress, S. & Spiegel, E. A. 1981 Pattern formation in a suspension of swimming microorganisms: nonlinear aspects. (To be published.)
Christopherson, D. G. 1940 Note on the vibration of membranes. Quart. J. Math. 11, 63–65.Google Scholar
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Clever, R. M. & Busse, F. H. 1978 Large wavelength convection rolls in low Prandtl number fluids. Z. angew. Math. Phys. 29, 711–714.Google Scholar
Depassier, M. C. & Spiegel, E. A. 1981a The large-scale structure of compressible convection. Astron. J. 86, 496–512.Google Scholar
Depassier, M. C. & Spiegel, E. A. 1981b Convection with heat flux prescribed on the bound aries of the system I: the effect of temperature dependence on material properties. Geophys. Astrophys. Fluid Dyn. (Submitted.)
Hewitt, J. M., McKenzie, D. P. & Weiss, N. O. 1980 Large aspect-ratio cells in two-dimensional thermal convection. Earth & Planet. Sci. Lett. 51, 370–380.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Hurle, D. T. J., Jakeman, E. & Pike, E. R. 1967 On the solution of the Bénard problem with boundaries of finite conductivity. Proc. R. Soc. Lond. A 296, 469–475.Google Scholar
Krishnamurti, R. 1970 On the transition to turbulent convection. Part I. The transition from two- to three-dimensional flow. J. Fluid Mech. 42, 295–307.Google Scholar
Mckenzie, D., Watts, A., Parsons, B. & Roufosse, M. 1980 Planform of mantle convection beneath the Pacific Ocean. Nature 288, 442–445.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite band width, finite amplitude convection. J. Fluid Mech. 38, 279–303.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Poyet, J.-P. 1980 Time-dependent convection. Ph.D. thesis, Columbia University.
Poyet, J.-P. & Spiegle, E. A. 1982 In preparation.
Schlüter, A., Lortz, E. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Segel, L. A. 1969 Distant side walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. H. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech. 18, 513–528.Google Scholar