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The nonlinear evolution of magnetic instabilities in a rapidly rotating annulus

Published online by Cambridge University Press:  26 April 2006

Kenneth A. Hutcheson  and
David R. Fearn
Kenneth A. Hutcheson
Affiliation:
Department of Mathematics, University Gardens, University of Glasgow, Glasgow G12 8QW, UK
David R. Fearn
Affiliation:
Department of Mathematics, University Gardens, University of Glasgow, Glasgow G12 8QW, UK
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Abstract

A numerical investigation of the stability of an axisymmetric magnetic field is discussed. The magnetic field permeates a finitely conducting fluid contained within a rapidly rotating cylindrical annulus. The fluid is incompressible and viscid. The evolution of a non-axisymmetric perturbation to the axisymmetric magnetic field is governed by the momentum and induction equations which are integrated using a spectral timestep method. We follow the growth of the perturbation to finite amplitude and find that the character of the solution is dominated by the most unstable axially dependent mode found from the linear theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 291 , 25 May 1995 , pp. 343 - 368
Copyright
© 1995 Cambridge University Press

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References

Acheson, D. J. 1972 On the hydromagnetic stability of a rotating fluid annulus. J. Fluid Mech. 52, 529–541.Google Scholar
Acheson, D. J. 1973 Hydromagnetic wave-like instabilities in a rapidly rotating stratified fluid. J. Fluid Mech. 61, 609–624.Google Scholar
Acheson, D. J. 1978 Magnetohydrodynamic waves and instabilities in rotating fluids. In Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 315–349. Academic Press.
Acheson, D. J. 1983 Local analysis of thermal and magnetic instabilities in a rapidly rotating fluid. Geophys. Astrophys. Fluid Dyn. 27, 123–136.Google Scholar
Barenghi, C. F. 1991 Computations of transitions and Taylor vortices in temporally modulated Taylor-Couette flow. J. Comput. Phys. 95, 175–194.Google Scholar
Barenghi, C. F. & Jones, C. A. 1989 Modulated Taylor-Couette flow. J. Fluid Mech. 208, 127–160.Google Scholar
Bloxham, J., Gubbins, D. & Jackson, A. 1989 Geomagnetic secular variation. Phil. Trans. R. Soc. Lond. 329, 415–502.Google Scholar
Fearn, D. R. 1983 Hydromagnetic waves in a differentially rotating annulus I. A test of local stability analysis. Geophys. Astrophys. Fluid Dyn. 27, 137–162.Google Scholar
Fearn, D. R. 1984 Hydromagnetic waves in a differentially rotating annulus II. Resistive instabilities. Geophys. Astrophys. Fluid Dyn. 30, 227–239.Google Scholar
Fearn, D. R. 1985 Hydromagnetic waves in a differentially rotating annulus III. The effect of an axial field. Geophys. Astrophys. Fluid Dyn. 33, 185–197.Google Scholar
Fearn, D. R. 1988 Hydromagnetic waves in a differentially rotating annulus IV. Insulating boundaries. Geophys. Astrophys. Fluid Dyn. 44, 55–75.Google Scholar
Fearn, D. R. 1989 Differential rotation and thermal convection in a rapidly rotating hydromagnetic system. Geophys. Astrophys. Fluid Dyn. 44, 55–75.Google Scholar
Fearn, D. R. & Weiglhofer, W. S. 1991a Magnetic instabilities in rapidly rotating spherical geometries I. From cylinders to spheres. Geophys. Astrophys. Fluid Dyn. 56, 159–181.Google Scholar
Fearn, D. R. & Weiglhofer, W. S. 1991b Magnetic instabilities in rapidly rotating spherical geometries II. More realistic fields and resistive instabilities. Geophys. Astrophys. Fluid Dyn. 60, 275–294.Google Scholar
Fearn, D. R. & Weiglhofer, W. S. 1992a Resistive instability and the magnetostrophic approximation. Geophys. Astrophys. Fluid Dyn. 63, 111–138.Google Scholar
Fearn, D. R. & Weiglhofer, W. S. 1992b Magnetic instabilities in rapidly rotating spherical geometries III. The effect of differential rotation. Geophys. Astrophys. Fluid Dyn. 67, 163–185.Google Scholar
Gubbins, D. & Roberts P. H. 1987 Magnetohydrodynamics of the earth's core. In Geomagnetism (ed. J. A. Jacobs), Vol. 2, pp. 1–183. Academic Press.
Hollerbach, R. & Ierley, G. I. 1991 A modal α2-dynamo in the limit of asymptotically small viscosity. Geophys. Astrophys. Fluid Dyn. 56, 133–158.Google Scholar
Ierley, G. I. 1985 Macrodynamics of α2 dynamos. Geophys. Astrophys. Fluid Dyn. 34, 143–173.Google Scholar
Jones, C. A. 1981 Nonlinear Taylor vortices and their stability. J. Fluid Mech. 102, 249–261.Google Scholar
Jones, C. A. 1985 Numerical methods for the transition to wavy Taylor vortices. J. Comput. Phys. 61, 321–344.Google Scholar
Lan, S., Kuang, W. & Roberts, P. H. 1993 Ideal instabilities in rapidly rotating MHD systems that have critical layers. Geophys. Astrophys. Fluid Dyn. 69, 133–160.Google Scholar
Knobloch, E. 1994 Bifurcations in rotating systems. In Lectures on Solar and Planetary Dynamos (ed. M. R. E. Proctor & A. D. Gilbert). Cambridge University Press.
McFadden, P. L. & Merrill, R. T. 1993 Inhibition and geomagnetic reversals. J. Geophys. Res. 98, 6189–6199.Google Scholar
Taylor, J. B. 1963 The magneto-hydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc. R. Soc. Lond. A 274, 274–283.Google Scholar
Zhang, K. & Fearn, D. R. 1993 How strong is the invisible component of the magnetic field in the earth's core. Geophys. Res. Lett. 20, 2083–2086.Google Scholar
Zhang, K. & Fearn, D. R. 1994 Hydromagnetic waves in rapidly rotating spherical shells, generated by magnetic toroidal decay modes. Geophys. Astrophys. Fluid Dyn. 77, 123–147.Google Scholar