Abstract
This book is about the structure of magnetic fields in electrically conducting fluids. Most ordinary fluids, such as air or water at familiar temperatures and pressures, are not good conductors of electricity, and so offer little direct experience of the kind of phenomena we shall be examining. In more extreme environments — the liquid core of the Earth or the atmospheres of stars, for example — electrically conducting fluids or plasmas are common. And there magnetic fields of surprising complexity are observed. Although we shall be concerned with models that are very simple and idealized, the motivation for the work described in the following pages lies in these examples of magnetic fields in astrophysics.
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References
The reader interested in dynamo theories of the Earth’s magnetic field will find extensive discussion in the books by Roberts (1967), Moffatt (1978), Parker (1979) and Krause & Rädler (1980), and may draw on numerous review articles, including Busse (1978), Childress (1984), Cowling (1957a), Ghil & Childress (1987) Hide & Roberts (1961), Roberts (1971, 1987) and Weiss (1971). Other reviews with an emphasis on fast dynamo theory are Roberts & Soward (1992), Bayly (1994) and Soward (1994b). For a readable overview see also Moffatt (1989), reprinted in Childress et al. (1990b).
If the flow is allowed to contract volumes indefinitely, then it is possible to ensure that the STF cycle smoothly maps the torus into itself. This gives a model known as the’ solenoid’ (e.g.), Falconer 1990), which has a strange attractor. Iterating the map leads to flux concentrated on this attractor. However in such a map volumes are contracted indefinitely, leading to unbounded fluid densities; such a map cannot model dynamo action in a physical fluid, even if it is compressible.
We use the term operator here to indicate that Tε acts on initial field, for some particular value of t. We are in fact dealing with a semi-group of operators defined on the interval 0 < t < ∞ (Kato 1966). The term monodromy operator is also used (Yudovich 1989).
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© 1995 Springer-Verlag Berlin Heidelberg
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(1995). The Fast Dynamo Problem. 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_1
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DOI: https://doi.org/10.1007/978-3-540-44778-8_1
Publisher Name: Springer, Berlin, Heidelberg
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