Hostname: page-component-7f64f4797f-9h5t2 Total loading time: 0 Render date: 2025-11-04T16:09:17.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Patterns of convection in spherical shells

Published online by Cambridge University Press:  29 March 2006

F. H. Busse
F. H. Busse
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, Los Angeles
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of the pattern of motion realized in a convectively unstable system with spherical symmetry can be considered without reference to the physical details of the system. Since the solution of the linear problem is degenerate because of the spherical homogeneity, the nonlinear terms must be taken into account in order to remove the degeneracy. The solvability condition leads to the selection of patterns distinguished by their symmetries among spherical harmonics of even order. It is shown that the corresponding convective motions set in as subcritical finite amplitude instabilities.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 72 , Issue 1 , 11 November 1975 , pp. 67 - 85
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Busse, F. H. 1962 Das Stabilitätsverhalten der Zellularkonvektion bei endlicher Amplitude. Dissertation, University of Munich. (English trans. by S. H. Davis, Rand Rep. LT-66-19. Rand Corp., Santa Monica, California.)
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Busse, F. H. 1970 Differential rotation in stellar convection zones. Astrophys. J. 159, 629–639.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Gaunt, J. A. 1929 On the triplets of helium. Phil. Trans. A 228, 151–196.Google Scholar
Joseph, D. D. & Carmi, S. 1966 Subcritical convective instability. Part 2. Spherical shells. J. Fluid Mech. 26, 769–777.Google Scholar
Kaula, W. M. 1975 Product-sum conversion of spherical harmonics with application to thermal convection. J. Geophys. Res. 80, 225–231.Google Scholar
Koiter, W. T. 1969 The nonlinear buckling problem of a complete spherical shell under uniform external pressure. Parts I-IV. Proc. Koninklijke Nederland Akad. van Wetenschapen, B 72, 40–123.Google Scholar
Mckenzie, D. P. 1968 The influence of the boundary conditions and rotation on convection in the earth's mantle. Geophys. J. Roy. Astr. Soc. 15, 457–500.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Runcorn, S. K. 1965 Changes in the convection pattern in the earth's mantle and continental drift: evidence for a cold origin of the earth. Phil. Trans. A 258, 228–251.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.Google Scholar
Segel, L. A. 1965 The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below. J. Fluid Mech. 21, 359–384.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. W. 1963 Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech. 18, 513–528.Google Scholar
Spilhaus, A. 1975 Geo-art: plate tectonics and Platonic solids, EOS. Trans. Am. Geophys. Un. 56, 52–57.Google Scholar
Thompson, D. 1942 On Growth and Form. Cambridge University Press.
Walzer, U. 1971 Convection currents in the earth's mantle and the spherical harmonic development of the topography of the earth. Pure Appl. Geophys. 87, 73–92.Google Scholar
Young, R. E. 1974 Finite-amplitude thermal convection in a spherical shell. J. Fluid Mech. 63, 695–721.Google Scholar