Two general alternative models for the origin of magnetic fields in galaxies have been proposed;- the primodial model where the fields came through the compression of a relict field or the dynamo scenario where the field is generated through the amplification of a seed field in a galactic rotation. The primodial field model was developed in some detail by Piddington, (1964, 1978, 1981). This seemed at first feasible when the primodial intergalactic field was put at 10-8 Gauss or more. A compression by a factor of 100 would bring the fields into the observed range of a few µG. However as a result of more recent observations values for the magnetic field were estimated to be 5 < Bt < 50 µG. Also more recent estimates of the intergalactic magnetic field suggest that the upper limits are 10-9 - 10-10 G. Also it was pointed out by Parker (1979) that in presence of turbulence the primodial field would be dissipated in 107 years. Clearly these two arguments do not allow sufficient room for the primodial field hypothesis.
All the recent theoretical investigations have been directed towards the interpretation of the magnetic fields in galaxies in the context of the turbulent hydromagnetic dynamo theory. Originally the explanation of the magnetism of the Earth, the Sun and Planets determined the development of the dynamo theory. (e.g. Parker, 1955, 1971, 1979; Steenbeck et al., 1966; Krause and Rädler 1980). The application of the dynamo theory to galaxies was investigated by Parker (1971) and Vainshtein and Ruzmaikin (1971). This idea was reinvestigated in some detail in view of the new observational data by Ruzmaikin and Shukurov (1981). The difficulty at first was that the solution of the dynamo equation was for the basic mode only. Since the observations showed magnetic fields with a spiral-like structure this dynamo interpretation was obviously inapplicable. In addition investigations of the Faraday rotation in M51 by Tosa and Fujimoto (1978) suggested that the field was 'bisymmetric', [BSS], i.e. with the field orientation going along a spiral arm into the nucleus and coming out of the nucleus with unchanged field direction. This is in contrast to the axisymmetric [ASS] configuration, where the field lines point in or out of the nucleus.
The basic dynamo equation is given by (e.g. Krause and Rädler, 1980):
the first term of the dynamo equation describes the large-scale velocity field (usually given by u = × r), the second term gives the mean helicity of the turbulence and the third one the turbulent magnetic diffusity.
Parker (1971) has suggested a concept of the 'aw-dynamo', where a mean toroidal magnetic field is generated by the non-uniform (differential) rotation w from an original poloidal magnetic field. The poloidal field is regenerated from the toroidal field by the effects of cyclonic convection (the a-effect). Parker described the dynamo in the 'slab' geometry, i.e. a thin layer of infinite extended electrically conducting gas in cyclonic turbulent motion subject to a large shear.
A spherical dynamo without differential rotation, (which is particularly applicable to the Earth and the Planets), has been investigated by Krause and Steenbeck (1967). This dynamo with constant a leads to an 'a2-dynamo'. A detailed study of the stability of a2 dynamos was given by Krause and Meinel (1988).
In fact solutions of the dynamo equation in other geometries are few. An exception, which may be applicable to galaxies, is the solution of the oblate spheroid by Stix (1975). More recently a partial solution for a slender torus was given by Grosser (1988) which may apply to the situation in the nucleus of a galaxy.
Since the observations of magnetic fields in galaxies suggest a dominance of the BSS field structure numerous theoretical papers were published to explain this observational fact. Also the problem of the fields above the plane of the disc (in the halo) were investigated. The solution of the dynamo equation for many modes (Ruzmainin et al., 1985; Baryshnikova et al., 1987; Krasheninnikova et al., 1989) showed that the co-existence of the BSS and ASS modes was possible in the context of the dynamo theory. A special investigation of the dynamo solution leading to the BSS case was given by Sawa and Fujimoto (1986) and Fujimoto and Sawa (1987). The extension to three dimensions was discussed by Sawa and Fujimoto (1987). Further attempts to model three dimensional situations are given by Strachenko and Shukurov (1989). The question of the growth rates of different modes and the stability of the nonlinear dynamo was treated by Brandenburg et al. (1989).
In spite of the great activity in the understanding of the dynamo many questions are open. There is still some lingering hope that compression of the primodial field could be used to explain some of the observed phenomena. The role of reconnections, a perennial discussion point, is still unclear. The applicability of equipartition, so often used to explain observational results, is still not universally accepted. The role of the local magnetic fields (stellar fields, pulsars, supernovae, bi-polar sources, molecular clouds, etc.) in relation to the global magnetic fields is also unclear. The reason for the dominance of the BSS field structure in some galaxies has not been as yet explained. The interplay of theory and observations is doing a lot of good in advancing our understanding but; 'all is not well in the house of magnetic fields'.