Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-23T07:48:24.482Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite Prandtl number thermal convection in a spherical shell

Published online by Cambridge University Press:  19 April 2006

Abdelfattah Zebib ,
Gerald Schubert  and
Joe M. Straus
Abdelfattah Zebib
Affiliation:
Department of Mechanical, Industrial and Aerospace Engineering, College of Engineering, Rutgers University, Piscataway, New Jersey 08854
Gerald Schubert
Affiliation:
Department of Earth and Space Sciences, University of California, Los Angeles, California 90024
Joe M. Straus
Affiliation:
Space Sciences Laboratory, The Aerospace Corporation, P.O. Box 92957, Los Angeles, California 90009
Get access Rights & Permissions [Opens in a new window]

Abstract

A Galerkin method is used to calculate the finite amplitude, steady, axisymmetric convective motions of an infinite Prandtl number, Boussinesq fluid in a spherical shell. Convection is driven by a temperature difference imposed across the stress-free, isothermal boundaries of the shell. The radial gravitational field is spherically symmetric and the local acceleration of gravity is directly proportional to radial position in the shell. Only the case of a shell whose outer radius is twice its inner radius is considered. Two distinct classes of axisymmetric steady states are possible. The temperature and radial velocity fields of solutions we refer to as ‘even’ are symmetric about an equatorial plane, while the latitudinal velocity is antisymmetric about this plane; solutions we refer to as ‘general’ do not possess any symmetry properties about the equatorial plane. The characteristics of these solutions, i.e. the isotherms, streamlines, spherically averaged temperature profiles, Nusselt numbers, etc., are given for Rayleigh numbers Ra as high as about 10 times critical for the even solutions and 3 times critical for the general solutions. Linear stability analyses of the nonlinear steady states show that the general solutions are the preferred form of axisymmetric convection when Ra is less than about 4 times critical. Furthermore, while the preferred motion at the onset of convection is non-axisymmetric, axisymmetric convection is stable when Ra exceeds about 1·3 times the critical value.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 2 , 25 March 1980 , pp. 257 - 277
Copyright
© 1980 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.)

References

Busse, F. H. 1967 On thestability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46, 140–150.Google Scholar
Busse, F. H. 1975 Patterns of convection in spherical shells. J. Fluid Mech. 72, 67–85.Google Scholar
Cassen, P. & Young, R. E. 1975 On the cooling of the Moon by solid convection. The Moon 12, 361–368.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, cla. II and VI. Clarendon.
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625–645.Google Scholar
Davis, P. J. & Polonsky, I. 1964 Numerical interpolation, differentiation, and integration. In Handbook of Mathematical Functions (ed. M. Abramowitz & I. A. Stegun), pp. 875–924. Washington: National Bureau of Standards.
Hsui, A. T., Turcotte, D. L. & Torrance, K. E. 1972 Finite amplitude thermal convection within a self-gravitating fluid sphere. Geophys. Fluid Dyn. 3, 35–44.Google Scholar
Oxburgh, E. R. & Turcotte, D. L. 1978 Mechanisms of continental drift. Rep. Prog. Phys. 41, 1249–1312.Google Scholar
Richter, F. M. 1978 Mantle convection models. Ann Rev. Earth Planet. Sci. 6, 9–19.Google Scholar
Schubert, G. 1979 Subsolidus convection in the mantles of terrestrial planets. Ann. Rev. Earth Planet. Sci. 7, 289–342.Google Scholar
Schubert, G. & Young, R. E. 1976 Cooling the Earth by whole mantle subsolidus convection: A constraint on the viscosity of the lower mantle. Tectonophys. 35, 201–214.CrossRefGoogle Scholar
Schubert, G., Young, R. E. & Cassen, P. 1977 Solid state convection models of the lunar internal temperature. Phil. Trans. Roy. Soc. A 285, 523–536.Google Scholar
Smythe, W. R. 1968 Static and Dynamic Electricity, 3rd edn. McGraw-Hill.
Tozer, D. C. 1967 Towards a theory of thermal convection in the mantle. In The Earth's Mantle (ed. T. F. Gaskell), pp. 325–353.
Turcotte, D. L., Hsui, A. T., Torrance, K. E. & Oxburgh, E. R. 1972 Thermal structure of the Moon. J. Geophys. Res. 77, 6931–6939.Google Scholar
Turcotte, D. L. & Oxburgh, E. R. 1972 Mantle convection and the new global tectonics. Ann. Rev. Fluid Mech. 4, 33–68.Google Scholar
Young, R. E. 1974 Finite-amplitude thermal convection in a spherical shell. J. Fluid Mech. 63, 695–721.Google Scholar
Young, R. E. & Schubert, G. 1974 Temperatures inside Mars: Is the core liquid or solid? Geophys. Res. Lett. 1, 157–160.Google Scholar