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Abstract
The existence of a weak galactic magnetic field has been repeatedly confirmed by observational data. The origin of this field has not as yet been explained in a fully satisfactory theoretical way and represents one of the main challenges of the astrophysical dynamo theory. In both the galactic dynamo theory and the primordial-origin theory, a major influence is exerted by the small-scale magnetic fluctuations. This thesis is devoted to constructing a systematic statistical theory of such small-scale fields. The statistics of these fields are studied for the case of large Prandtl numbers, which is relevant for the galactic and protogalactic plasma, and in the kinematic approximation. The resulting kinematic-dynamo problem is viewed in the context of the general passive-advection theory. The advecting velocity field is assumed to be Gaussian and short-time correlated. The theory is developed for an arbitrary degree of compressibility and formally in d dimensions, which generalizes some previously known results, elicits the structure of the solutions, and uncovers a number of new effects. In the limit of δ-correlated-in-time velocity field and no diffusion, the statistics are determined solely by the small-scale expansion properties of the velocity correlation tensor. This universality is lost as the assumptions of zero velocity correlation time and no diffusion are relaxed: the statistics become dependent on the specific shape of the time-correlation profile and on the large-scale spatial structure of the velocity field. The thesis consists of three main parts. First, the growth and spreading over scales of the energy spectra of the fluctuating magnetic fields are studied in the limit of zero velocity correlation time and small magnetic diffusivity. Second, a general one-point statistical theory describing all passive tensor fields frozen into the advecting medium (no diffusion) is constructed. The statistics of the passive magnetic field are considered as a particular case of this theory. Third, the effects due to a short, but finite, velocity correlation time are investigated by means of a new expansion method.
