Physics
Scientific paper
Jan 1993
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993phdt........40d&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF MARYLAND COLLEGE PARK, 1993.Source: Dissertation Abstracts International, Volume: 54-10, Section:
Physics
3
Dynamos, Fractal Dimension Spectra
Scientific paper
This thesis concerns the fast kinematic dynamo problem. In Part I, we review the concepts we will use in this thesis. First, we introduce fast kinematic dynamos and discuss how they are connected to chaos. Then, we briefly review the concept of fractal dimension spectra. Using the above background, we finally formulate the problem of obtaining dimension spectra for dynamo magnetic fields with and without small diffusivity. In Part II, we consider fractal dimensions of fast dynamo magnetic fields. To gain insight into this problem, we first study a simple toy model of fast kinematic dynamos given by a 4-strip baker's map. Then, we use this insight to obtain results for dynamos in general smooth three-dimensional flows. It will be shown that, when there is no magnetic diffusion, the magnetic field generally exhibits two "pathologies" as the time t goes to infinity: (1) the magnitude of the magnetic field varies on an arbitrarily fine scale, and (2) the orientation of the magnetic field alternates on an arbitrarily fine scale. The first pathology is characterized by a fractal dimension spectrum. We introduce a "cancellation exponent" to characterize the second pathology. When small magnetic diffusion is present, the time-asymptotic magnetic field also exhibits the same two pathologies. The small diffusion does not change the spatial cancellation exponent. However, it does change the dimension spectrum. The change is argued to be the result of the magnetic field cancellation induced by small diffusion. We express the dimension spectrum in terms of the finite time Lyapunov numbers of the underlying flows and the cancellation exponent. Finally, we test our dimension formula on a particular spatially smooth three-dimensional fluid flow. In Part III, we apply our method, gained from our study of fractal dimensions, to obtain the growth rates for fast kinematic dynamos. In particular, we obtain a formula giving the growth rate in terms of finite time Lyapunov numbers of the underlying flows and the cancellation exponent. We also verify our growth rate formula by an analytical example and a numerical example.
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