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Three-dimensional kinematic dynamos dominated by strong differential rotation

Published online by Cambridge University Press:  26 April 2006

Graeme R. Sarson  and
David Gubbins
Graeme R. Sarson
Affiliation:
Department of Earth Sciences, University of Leeds, Leeds LS2 9JT, UK Present address: Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK.
David Gubbins
Affiliation:
Department of Earth Sciences, University of Leeds, Leeds LS2 9JT, UK
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Abstract

In the kinematic dynamo problem a fluid motion is specified arbitrarily and the induction equation is solved for non-decaying magnetic fields; it forms part of the larger magnetohydrodynamic (MHD) dynamo problem in which the fluid flow is buoyancy-driven. Although somewhat restrictive, the kinematic problem is important for two reasons: first, it suffers from numerical difficulties that are holding up progress on the MHD problem; secondly, for the geodynamo, it is capable of reproducing details of the observable magnetic field. It is more efficient to study these two aspects for the kinematic dynamo than for the full MHD dynamo. We explore solutions for a family of fluid flows in a sphere, first studied by Kumar & Roberts (1975), that is heuristically representative of convection in a rotating sphere such as the Earth's core. We guard against numerical difficulties by comparing our results with well-understood solutions from the axisymmetric (αω) limit of Braginskii (1964a) and with solutions of the adjoint problem, which must yield identical eigenvalues in an adequate numerical treatment. Previous work has found a range of steady dipolar solutions; here we extend these results and find solutions of other symmetries, notably oscillatory and quadrupolar fields. The surface magnetic fields, important for comparison with observations, have magnetic flux concentrated by downwelling flow. Roberts (1972) found that meridional circulation promoted stationary solutions of the αω-equations, preferred solutions being oscillatory when no such circulation was present. We find analogous results for the full three-dimensional problem, but note that in the latter case the ‘effective’ meridional circulation arising from the non-axisymmetric convection (a concept made precise in the asymptotic limit of Braginskii 1964a) must be considered. Thus stationary solutions are obtained even in the absence of ‘true’ meridional circulation, and the time-dependence can be controlled by the strength of the convection as well as by the meridional circulation. The preference for fields of dipole or quadrupole parity is largely controlled by the sign of the velocity: a reversal of velocity from a case favouring a dipole will favour quadrupole parity, and vice versa. For the comparison problem of Proctor (1977b) this symmetry is exact; for the physical problem the boundary conditions make a difference. The boundary effect is first removed by surrounding the dynamo region with a thick layer of quiescent conducting fluid, and then studied numerically by progressively reducing the thickness of this layer to zero. The insulating boundary contributes to the difficulty of obtaining dynamo action, and to the numerical difficulties encountered. The effect of an inner boundary on dynamo action is also considered, but found to be slight.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 306 , 10 January 1996 , pp. 223 - 265
Copyright
© 1996 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Backus, G. E. 1958 A class of self-sustaining dissipative spherical dynamos. Ann. Phys. 4, 372–447.Google Scholar
Barenghi, C. F. 1993 Nonlinear planetary dynamos in a rotating spherical shell: III α2ω models and the geodynamo. Geophys. Astrophys. Fluid Dyn. 71, 163–185.Google Scholar
Braginskiî, S. I. 1964a Self-excitation of a magnetic field during the motion of a highly conducting fluid. Zh. Eksp. Teor. Fiz. 47 1084. English translation, Sov. Phys. JETP 20, 726–735, 1965.Google Scholar
Braginskiî, S. I. 1964b Theory of the hydromagnetic dynamo. Zh. eksp. teor. fiz. 47 2178 (english translation, sov. phys. jetp 20, 1462–1471, 1965).Google Scholar
Braginsky, S. I. 1978 Nearly axially symmetric models of the hydromagnetic dynamo of the Earth II. Geomag. Aeron. 18, 225–231.Google Scholar
Braginsky, S. I. & Roberts, P. H. 1987 A model-Z geodynamo. Geophys. Astrophys. Fluid Dyn. 38, 327–349.Google Scholar
Bullard, E. C. & Gellman, H. 1954 Homogeneous dynamos and terrestrial magnetism. Proc. R. Soc. Lond. A 247, 213–278.Google Scholar
Bullard, E. C. & Gubbins, D. 1977 Generation of magnetic fields by fluid motions of a global scale. Geophys. Astrophys. Fluid Dyn. 8, 43–56.Google Scholar
Conte, S. D. 1966 The numerical solution of linear boundary value problems. SIAM Rev. 8, 309–321.Google Scholar
Dudley, M. L. & James, R. W. 1989 Time dependent dynamos with stationary flows. Proc. R. Soc. Lond. A 425, 407–429.Google Scholar
Gibson, R. D. & Roberts, P. H. 1966 Some comments on the theory of homogeneous dynamos. In Magnetism and the Cosmos (ed.W. Hindmarsh, F. H. Lowes, P. H. Roberts & S. K. Runcorn), pp. 108–120. Oliver and Boyd.
Gubbins, D. 1972 Kinematic dynamos and geomagnetism. Nature 238 PS, 119–121.Google Scholar
Gubbins, D. 1973 Numerical solutions of the kinematic dynamo problem. Phil. Trans. R. Soc. Lond. A 274, 493–521.Google Scholar
Gubbins, D. 1993 Geomagnetism and inferences for core motion. In Flow and Creep in the Solar System: Observations, Modelling and Theory (ed.D. B. Stone & S. K. Runcorn), pp. 97–111. Kluwer Academic.
Gubbins D., Masters, T. G. & Jacobs, J. A. 1979 Thermal evolution of the Earth's core. Geophys. J. R. Astron. Soc. 59, 57–99.Google Scholar
Gubbins, D. & Sarson, G. 1994 Geomagnetic field morphology and kinematic dynamos. Nature 368, 51–55.Google Scholar
Gubbins, D. & Zhang, K. 1993 Symmetry properties of the dynamo equations for paleomagnetism and geomagnetism. Phys. Earth Planet. Inter. 75, 225–241.Google Scholar
Herzenberg, A. & Lowes, F. J. 1957 Electromagnetic induction in rotating conductors. Phil. Trans. R. Soc. Lond. A 249, 507–584.Google Scholar
Hide, R. & Palmer, T. N. 1982 Generalisation of Cowling's theorem. Geophys. Astrophys. Fluid Dyn. 19, 301–309.Google Scholar
Hollerbach, R., Barenghi, C. F. & Jones, C. A. 1993 Taylor's constraint in a spherical αω-dynamo. Geophys. Astrophys. Fluid Dyn. 67, 3–25.Google Scholar
Hollerbach, R. & Jones, C. A. 1993a A geodynamo model incorporating a finitely conducting inner core. Phys. Earth Planet. Inter. 75, 317–327.Google Scholar
Hollerbach, R. & Jones, C. A. 1993b Influence of the Earth's inner core on geomagnetic fluctuations and reversals. Nature 365, 541–543.Google Scholar
Hollerbach, R. & Jones, C. A. 1995 On the magnetically stabilising role of the Earth's inner core. Phys. Earth Planet. Inter. 87, 171–181.Google Scholar
Hutcheson, K. A. 1990 Geomagnetic field modelling. PhD thesis, University of Cambridge.
Hutcheson, K. A. & Gubbins, D. 1994 Kinematic magnetic field morphology at the core mantle boundary. Geophys. J. Intl 116, 304–320.Google Scholar
Kono, M. & Roberts, P. H. 1991 Small amplitude solutions of the dynamo problem: 1. The adjoint system and its solutions. J. Geomagn. Geoelectr. 43, 839–862.Google Scholar
Krause, F. & Rädler, K.-H. 1980 Mean Field Magnetohydrodynamics and Dynamo Theory. Akademie.
Kumar, S. & Roberts, P. H. 1975 A three-dimensional kinematic dynamo. Proc. R. Soc. Lond. A 344 235–238 (referred to herein as KR).Google Scholar
Levy, E. H. 1972 On the state of the geomagnetic field and its reversals. Astrophys. J. 175, 573–581.Google Scholar
Lilley, F. E. M. 1970 On kinematic dynamos. Proc. R. Soc. Lond. A 316, 153–167.Google Scholar
Melchior, P. 1986 The Physics of the Earth's Core — An Introduction. Pergamon.
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids.. Cambridge University Press. Kinematic dynamos with strong differential rotation 265
Nakajima, T. & Kono, M. 1991 Kinematic dynamos associated with large scale fluid motions. Geophys. Astrophys. Fluid Dyn. 60, 177–209.Google Scholar
Parker, E. N. 1969 The occasional reversal of the geomagnetic field. Astrophys. J. 158, 815–827.Google Scholar
Parker, E. N. 1979 Cosmical Magnetic Fields: Their Origin and Their Activity. Clarendon.
Pekeris, C. L., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the earth's magnetic field. Phil. Trans. R. Soc. Lond. A 275, 425–461.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1989 Numerical Recipes. Cambridge University Press.
Proctor, M. R. E. 1977a On the eigenvalues of kinematic α-effect dynamos. Astron. Nachr. 298, 19–25.Google Scholar
Proctor, M. R. E. 1977b The role of mean circulation in parity selection by planetary magnetic fields. Geophys. Astrophys. Fluid Dyn. 8, 311–324.Google Scholar
Roberts, P. H. 1971 In Dynamo theory of geomagnetism. World Magnetic Survey (ed.A. J. Zmuda). IAGA Bull. 28 pp. 123–132. International Union of Geodesy and Geophysics Publication Office.
Roberts, P. H. 1972 Kinematic dynamo models. Phil. Trans. R. Soc. Lond. A 271, 663–697.Google Scholar
Roberts, P. H. 1987 Origin of the main field: dynamics. In Geomagnetism, Vol. 2 (ed. J. A. Jacobs). Academic press.
Roberts, P. H. & Gubbins, D. 1987 Origin of the main field: kinematics. In Geomagnetism, Vol. 2 (ed. J. A. Jacobs). Academic.
Roberts P. H. & Stix, M. 1971 The Turbulent Dynamo: A Translation of a Series of Papers by F. Krause, K. H. Radler & M. Steenbeck. Tech. Note 60. NCAR.
Roberts, P. H. & Stix, M. 1972 α-effect dynamos by the Bullard and Gellman formalism. Astron. Astrophys. 18, 453–466.Google Scholar
Sarson, G. R. 1994 Kinematic dynamo calculations for geomagnetism. PhD thesis, University of Leeds.
Serebrianya, P. M. 1988 Kinematic stationary geodynamo models with separated toroidal and poloidal motions. Geophys. Astrophys. Fluid Dyn. 44, 141–164.Google Scholar
Steenbeck, M., Krause, F. & Rädler, K.-H. 1966 A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces. Z. Naturforsch. 21a, 369–376. English translation in Roberts & Stix (1971).Google Scholar
Weiss, N. O. 1966 The expulsion of magnetic flux by eddies. Proc. R. Soc. Lond. A 293, 310–328.Google Scholar
Zhang, K. & Busse, F. H. 1987 On the onset of convection in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 39, 119–147.Google Scholar
Zhang, K. & Busse, F. H. 1989 Convection driven magnetohydrodynamic dynamos in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 49, 97–116.Google Scholar
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