Abstract
The purpose of this chapter is to link the discussion and examples of Chaps. 1–3 to the subsequent analysis and modelling. We reexamine a number of topics which were introduced in Chap. 1. Aided by the examples of fast dynamos given in Chaps. 2 and 3, we reconsider in §4.1 the formulation of the fast dynamo problem and the role of diffusion. In §4.2 we prove several anti-dynamo theorems, some of which have already been mentioned and used. These can be used to exclude classes of flows from consideration and to help determine possible fast dynamo mechanisms. Finally in §4.3 we discuss a result of Moffatt & Proctor (1985) concerning the non-existence of smooth eigenfunctions when ε = 0.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
The fast dynamos we classify as diffusive are termed ‘intermediate’ by Molchanov, Ruzmaikin & Sokoloff (1985). We prefer the former adjective as suggestive of the source of fast dynamo action in small-scale diffusive structures of the magnetic field.
In terms of the discussion of §9.1.1 the corresponding eigenvalue is likely to be in the approximate spectrum of T in L2, and the corresponding sequence of near-eigenfunctions in L2 would tend to the distribution d in a weak sense (de la Llave 1993).
This point was emphasized by H.K. Moffatt in a summarizing talk at the 1992 NATO Advanced Study Institute on Solar and Planetary Dynamos; see also Kraichnan (1979).
The functions f and g are sometimes called Clebsch potentials or Clebsch coordinates from the form of vorticity used by Clebsch; see Lamb (1932), p. 248, and Roberts (1967).
Moffatt & Proctor (1985) point out that an alternative approach makes use of a construction of Ince (1926), provided (x, y, z) are taken to be complex-valued, with an eventual restriction to the real lines. The calculation relies on assumptions of analyticity which are significantly stronger than are actually needed in Frobenius’ theorem.
This idea is developed by Soward (1994a) who, in a study of pulsed Beltrami waves, discusses how one might define ε = 0 eigenfunctions with singularities localized at each periodic orbit. The fast dynamo growth rate and the structure of the Floquet matrix on each orbit determine the form of the field locally; see also Núñez (1994).
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(1995). Dynamos and Non-dynamos. In: Stretch, Twist, Fold: The Fast Dynamo. Lecture Notes in Physics Monographs, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44778-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-44778-8_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60258-3
Online ISBN: 978-3-540-44778-8
eBook Packages: Springer Book Archive