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Axisymmetric convection in a rotating sphere. Part 1. Stress-free surface

Published online by Cambridge University Press:  29 March 2006

A. D. Weir
A. D. Weir
Affiliation:
Department of Aeronautics, Imperial College, London
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Abstract

This paper examines large-scale nonlinear thermal convection in a rotating selfgravitating sphere of Boussinesq fluid containing a uniform distribution of heat sources. Conservative finite-difference forms of the equations of axisymmetric laminar motion are marched forward in time. The surface is assumed to be stress free and at constant temperature. Numerical solutions are obtained for Taylor numbers in the range 0 ≤ Λ ≤ 104 and Rayleigh numbers with \[ R_c \leqslant R\lesssim 10R_c. \] For high Prandtl number (P > 5) the solutions are steady and most of them resemble the solutions of the linear stability equations, though other steady solutions are also found. For P [lsim ] 1, the steady solutions have horizontal wavenumber l = 1 and nearly uniform angular momentum per unit mass, rather than nearly uniform angular velocity. This rotation law seems to be independent of many details of the model and may hold in the convective core of a rotating star.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 75 , Issue 1 , 13 May 1976 , pp. 49 - 79
Copyright
© 1976 Cambridge University Press

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References

Baldwin, P. 1967 Proc. Camb. Phil. Soc. 63, 855.
Batchelor, G. K. 1956 J. Fluid Mech. 1, 177.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bisshopp, F. E. 1958 Phil. Mag. 3, 1342.
Bodenheimer, P. 1971 Astrophys. J. 167, 153.
Brebbia, C. A. & CONNOR, J. J. (eds) 1974 Numerical Methods in Fluid Dynamics. London: Pentech Press.
Busse, F. H. 1970a J. Fluid Mech. 44, 441.
Busse, F. H. 1970b Astrophys. J. 159, 629.
Busse, F. H. 1973 Astron. Astrophys. 28, 27.
Chandrasekhar, S. 1961 Hydrodynamics and Hydromagnetic Stability. Oxford: Clarendon Press.
Durney, B. 1970 Astrophys. J. 161, 1115.
Foster, T. D. 1969 J. Fluid Mech. 37, 81.
Fricke, K. J. & Kippenhahn, R. 1972 Ann. Rev. Astron. Astrophys. 10, 45.
Gough, D. O. & LYNDEN-BELL, D. 1968 J. Fluid Mech. 32, 437.
Graham, E. 1975 J. Fluid Mech. 70, 689.
Hsui, A. T., Turcotte, D. L. & Torrance, K. E. 1972 Geophys. Fluid Dyn. 3, 35.
Krishnamurti, R. 1970 J. Fluid Mech. 42, 295.
Moore, D. R., Peckover, R. S. & Weiss, N. O. 1973 Comp. Phys. Comm. 6, 198.
Moore, D. R. & Weiss, N. O. 1973 J. Fluid Mech. 58, 289.
Morton, K. W. 1971 Proc. Roy. Soc. A, 323, 237.
Oorszag, S. A. & Israeli, M. 1974 Ann. Rev. Fluid Mech. 6, 281.
Parker, E. N. 1963 Astrophys. J. 138, 226.
Peckover, R. S. & Weiss, N. O. 1972 Comp. Phys. Comm. 4, 339.
Roberts, K. V. & Weiss, N. O. 1966 Math. Comp. 20, 272.
Roberts, P. H. 1968 Phil. Trans. Roy. Soc. A, 263, 93.
Rossby, H. T. 1969 J. Fluid Mech. 36, 309.
Schwarzschild, M. 1958 Structure and Evolution of the Stars. Princeton University Press.
Segel, L. A. 1966 In Non-equilibrium Thermodynamics, Variational Techniques, and Stability (ed. R. J. Donnelly, R. Herman & I. Prigogine), p. 165. Chicago University Press.
Simon, G. W. & Weiss, N. O. 1968 Z. Astrophys. 69, 435.
Smith, G. D. 1965 Numerical Solution of Partial Differential Equations. Oxford University Press.
Somerville, R. C. J. & Lipps, F. B. 1973 J. Atmos. Sci. 30, 590.
Tayler, R. J. 1973 Mon. Not. Roy. Astr. Soc. 165, 39.
Veronis, G. 1959 J. Fluid Mech. 5, 401.
Veronis, G. 1966a J. Fluid Mech. 26, 49.
Veronis, G. 1966b J. Fluid Mech. 24, 545.
Veronis, G. 1968 J. Fluid Mech. 31, 113.
Weir, A. D. 1975 Mém. Soc. Roy. Sci. Liège, Sér. 6, 6, 37.
Weiss, N. O. 1964 Phil. Trans. A, 256, 99.
Weiss, N. O. 1966 Proc. Roy. Soc. A, 293, 310.
Young, R. E. 1974 J. Fluid Mech. 63, 695.