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Three-dimensional thermal convection in a spherical shell

Published online by Cambridge University Press:  26 April 2006

D. Bercovici
Affiliation:
Department of Earth and Space Sciences, University of California, Los Angeles, CA 90024, USA
G. Schubert
Affiliation:
Department of Earth and Space Sciences, University of California, Los Angeles, CA 90024, USA
G. A. Glatzmaier
Affiliation:
Earth and Space Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
A. Zebib
Affiliation:
Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA

Abstract

Independent pseudo-spectral and Galerkin numerical codes are used to investigate three-dimensional infinite Prandtl number thermal convection of a Boussinesq fluid in a spherical shell with constant gravity and an inner to outer radius ratio equal to 0.55. The shell is heated entirely from below and has isothermal, stress-free boundaries. Nonlinear solutions are validated by comparing results from the two codes for an axisymmetric solution at Rayleigh number Ra = 14250 and three fully three-dimensional solutions at Ra = 2000, 3500 and 7000 (the onset of convection occurs at Ra = 712). In addition, the solutions are compared with the predictions of a slightly nonlinear analytic theory. The axisymmetric solution is equatorially symmetric and has two convection cells with upwelling at the poles. Two dominant planforms of convection exist for the three-dimensional solutions: a cubic pattern with six upwelling cylindrical plumes, and a tetrahedral pattern with four upwelling plumes. The cubic and tetrahedral patterns persist for Ra at least up to 70000. Time dependence does not occur for these solutions for Ra [les ] 70000, although for Ra > 35000 the solutions have a slow asymptotic approach to steady state. The horizontal and vertical structure of the velocity and temperature fields, and the global and three-dimensional heat flow characteristics of the various solutions are investigated for the two patterns up to Ra = 70000. For both patterns at all Ra, the maximum velocity and temperature anomalies are greater in the upwelling regions than in the downwelling ones and heat flow through the upwelling regions is almost an order of magnitude greater than the mean heat flow. The preferred mode of upwelling is cylindrical plumes which change their basic shape with depth. Downwelling occurs in the form of connected two-dimensional sheets that break up into a network of broad plumes in the lower part of the spherical shell. Finally, the stability of the two patterns to reversal of flow direction is tested and it is found that reversed solutions exist only for the tetrahedral pattern at low Ra.

Type
Research Article
Copyright
© 1989 Cambridge University Press

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References

Ali-Kahn, M. A. 1982 Three-dimensional thermal convection of an infinite Prandtl number fluid in a spherical shell. Dissertation, Rutgers University, The State University of New Jersey.
Baumgardner, J. R. 1985 Three-dimensional treatment of convective flow in the Earth's mantle. J. Stat. Phys. 39, 501511.Google Scholar
Bercovici, D., Schubert, G. & Zebib, A. 1988 Geoid and topography for infinite infinite Prandtl number convection in a spherical shell. J. Geophys. Res. 93, 64306436.Google Scholar
Bloxham, J. & Gubbins, D. 1985 The secular variation of the Earth's magnetic field. Nature 317, 777781.Google Scholar
Bowin, C. 1986 Topography at the core-mantle boundary. Geophys. Res. Lett. 13, 15131516.Google Scholar
Busse, F. H. 1975 Patterns of convection in spherical shells. J. Fluid Mech. 72, 6785.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 19291967.Google Scholar
Busse, F. H. 1981 Transition to turbulence in Rayleigh-Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Springer.
Busse, F. H. 1983 Quadrupole convection in the lower mantle? Geophys. Res. Lett. 10, 285288.Google Scholar
Busse, F. H. & Riahi, N. 1982 Patterns of convection in spherical shells. Part 2. J. Fluid Mech. 123, 283301.Google Scholar
Busse, F. H. & Riahi, N. 1988 Mixed-mode patterns of bifurcations from spherically symmetric basic states. Nonlinearity 1, 379388.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Creager, K. C. & Jordan, T. H. 1984 Slab penetration into the lower mantle. J. Geophys. Res. 89, 30313049.Google Scholar
Creager, K. C. & Jordan, T. H. 1986 Aspherical structure of the core-mantle boundary from PKP travel times. Geophys. Res. Lett. 13, 14971500.Google Scholar
Dziewonski, A. M. 1984 Mapping the lower mantle: Determination of lateral heterogeneity in P velocity up to degree and order 6. J. Geophys. Res. 89, 59295952.Google Scholar
Dziewonski, A. M. & Woodhouse, J. H. 1987 Global images of the Earth's interior. Science 236, 3748.Google Scholar
Glatzmaier, G. A. 1984 Numerical simulations of stellar convective dynamos I. The model and method. J. Comp. Phys. 55, 461484.Google Scholar
Glatzmaier, G. A. 1988 Numerical simulations of mantle convection: time-dependent, three-dimensional, compressible, spherical shell. Geophys. Astrophys. Fluid Dyn. 43, 223264.Google Scholar
Gubbins, D. & Richards, M. 1986 Coupling of the core dynamo and mantle: thermal or topographic? Geophys. Res. Lett. 13, 15211524.Google Scholar
Hart, J. E., Glatzmaier, G. A. & Toomre, J. 1986 Space laboratory and numerical simulations of thermal convection in a rotating hemi-spherical shell with radial gravity. J. Fluid Mech. 173, 519544.Google Scholar
Hsui, A. T., Turcotte, D. L. & Torrance, K. E. 1972 Finite amplitude thermal convection within a self-gravitating fluid sphere. Geophys. Fluid Dyn. 3, 3544.Google Scholar
Machetel, P. & Rabinowicz, M. 1985 Transitions to a two mode axisymmetrical spherical convection: Application to the Earth's mantle. Geophys. Res. Lett. 12, 227230.Google Scholar
Machetel, P., Rabinowicz, M. & Bernadet, P. 1986 Three-dimensional convection in spherical shells. Geophys. Astrophys. Fluid Dyn. 37, 5784.Google Scholar
Machetel, P. & Yuen, D. A. 1986 The onset of time dependent convection in spherical shells as a clue to chaotic convection in the Earth's mantle. Geophys. Res. Lett. 13, 14701473.Google Scholar
Machetel, P. & Yuen, D. A. 1987 Chaotie axisymmetrical spherical convection and large-scale mantle circulation. Earth Planet. Sci. Lett. 86, 93104.Google Scholar
Machetel, P. & Yuen, D. A. 1988 Infinite Prandtl number convection in spherical shells. In Mathematical Geophysics (eds. N. J. Vlaar, G. Nolet, M. J. R. Wortel & S. A. P. L. Cloetingh), Reidel.
Morelli, A. & Dziewonski, A. M. 1987 Topography of the core-mantle boundary and lateral homogeneity of the liquid core. Nature 325, 678683.Google Scholar
Olson, P. 1981 Mantle convection with spherical effects. J. Geophys. Res. 86, 48814890.Google Scholar
Olson, P., Schubert, G. & Anderson, C. 1987 Plume formation in the D"-layer and the roughness of the core-mantle boundary. Nature 327, 409413.Google Scholar
Oxburgh, E. R. & Turcotte, D. L. 1978 Mechanisms of continental drift. Rep. Prog. Phys. 41, 12491312.Google Scholar
Roberts, P. H. 1987 Convection in spherical systems. In Irreversible Phenomena and Dynamical Systems Analysis in Geosciences (ed. C. Nicolis & G. Nicolis). Reidel.
Schubert, G., Stevenson, D. & Cassen, P. 1980 Whole planet cooling and the radiogenic heat source contents of the Earth and moon. J. Geophys. Res. 85, 25312538.Google Scholar
Schubert, G. & Zebib, A. 1980 Thermal convection of an internally heated infinite Prandtl number fluid in a spherical shell. Geophys. Astrophys. Fluid Dyn. 15, 6590.Google Scholar
Silver, P. G., Carlson, R. W. & Olson, P. 1988 Deep slabs, geochemical heterogeneity, and the large-scale structure of mantle convection: investigation of an enduring paradox. Ann. Rev. Earth Planet. Sci. 16, 477541.Google Scholar
Stacey, F. D. 1977 Physics of the Earth. Wiley.
Turcotte, D. L. & Schubert, G. 1982 Geodynamics. Wiley.
Williams, Q., Jeanloz, R., Bass, J., Svendson, B. & Ahrens, T. J. 1987 The melting curve of iron to 250 Gigapascals: A constraint on the temperature at the Earth's centre. Science 236, 181182.CrossRefGoogle Scholar
Woodhouse, J. H. & Dziewonski, A. M. 1984 Mapping the upper mantle: Three dimensional modeling of earth structure by inversion of seismie wave-forms. J. Geophys. Res. 89, 59535986.Google Scholar
Young, R. E. 1974 Finite-amplitude thermal convection in a spherical shell. J. Fluid Mech. 63, 695721.Google Scholar
Zebib, A., Schubert, G. & Straus, J. M. 1980 Infinite Prandtl number thermal convection in a spherical shell. J. Fluid Mech. 97, 257277.Google Scholar
Zebib, A., Schubert, G., Dein, J. L. & Paliwal, R. C. 1983 Character and stability of axisymmetric thermal convection in spheres and spherical shells. Geophys. Astrophys. Fluid Dyn. 23, 142.Google Scholar
Zebib, A., Goyal, A. K. & Schubert, G. 1985 Convective motions in a spherical shell. J. Fluid Mech. 152, 3948.Google Scholar