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Topological constraints associated with fast dynamo action

Published online by Cambridge University Press:  20 April 2006

H. K. Moffatt  and
M. R. E. Proctor
H. K. Moffatt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street. Cambridge
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street. Cambridge
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Abstract

The conjecture of Vainshtein & Zel'dovich (1972) concerning the existence of a fast dynamo (i.e. one whose growth rate is independent of magnetic diffusivity η in the limit η → 0) is discussed with particular reference to (i) the stretch–twist–fold cycle which can double the strength of a magnetic flux tube, and (ii) the space-periodic Beltrami flow of maximal helicity, which has been shown to be capable of space-periodic dynamo action with the same period as the velocity field, by Arnold & Korkina (1983) and by Galloway & Frisch (1984). The topological constraint associated with conservation of magnetic helicity is shown to preclude fast dynamo action unless the scale of the magnetic field is almost everywhere of order η½ as η → 0; in this case, the field structure is severely singular in the limit. A steady incompressible velocity field, quadratic in the space variables, is shown to mimic the action of the stretch–twist–fold cycle, and is proposed as a plausible candidate for fast dynamo action.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 154 , May 1985 , pp. 493 - 507
Copyright
© 1985 Cambridge University Press

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