Abstract
The broad theme of this chapter is the derivation of upper bounds for fast dynamo growth rates. We have already met the line-stretching exponent hline as a plausible upper bound in §2.4 and §3.3; this is discussed further in §6.1. In §6.2 we consider a conjecture of Oseledets (1993), which is related to methods of obtaining fast dynamo growth rates using Fredholm determinants (§6.3). In §6.4 we discuss Arnold’s (1972) suspension of the cat map as a flow on a manifold; this is an example of an Anosov flow, and in §6.5 we develop Bayly’s (1986) analysis of fast dynamo action in such flows. Section 6.6 follows Vishik’s (1988, 1989) construction of an approximate Green’s function, which gives an upper bound on fast dynamo growth rates. In §§6.7, 8 we consider the implementation of diffusion in terms of noisy trajectories, sketch Klapper & Young’s (1995) proof that htop bounds fast dynamo growth rates, and discuss the flux conjecture.
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We shall avoid technicalities in the definition of Liapunov exponents (see the lucid discussion in Ruelle 1989a). Let us just say that ‘almost all’ means with respect to an ergodic invariant measure, which for our purposes is volume restricted to a single connected region of chaos.
Note also that ΛLiap can show complicated behavior when a system parameter is varied, while htop tends to be better behaved; see §9.5 of Ruelle (1989a). It is also easier to measure htop numerically.
Note that Oseledets (1993) has the opposite definition of twisted and untwisted periodic orbits.
Arnold et al. (1981) obtain a separable equation for Bz with the incorrect right-hand side ε(▽2 − 2μ∂z)Bz. Note that these terms are equivalent to ours if the sign of the term −λ−z∂+B+ in (6.4.15c) is changed and ▽ · B = 0 is used.
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© 1995 Springer-Verlag Berlin Heidelberg
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(1995). Upper Bounds. 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_6
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DOI: https://doi.org/10.1007/978-3-540-44778-8_6
Publisher Name: Springer, Berlin, Heidelberg
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