This section presents a brief historic account of the ideas behind the dynamo process, followed by a qualitative description of the processes involved in the solar dynamo. More detailed descriptions of both history and dynamo processes can be found elsewhere (e.g. Roberts, 1967; Moffatt, 1978; Parker, 1979, 1987b; Krause and Rädler, 1980; Belvedere, 1985; Stix, 1989; Hoyng, 1992).
The first studies trying to describe the real magnetic field <B> were based on rotating bodies in which both magnetic field and velocity field u are axisymmetric. One of these studies was undertaken by Cowling (1934), who came to the rather disappointing conclusion "that it is impossible that an axially symmetric field shall be self-maintained". This first anti-dynamo theorem says that there exists no flow field that can maintain a steady, axisymmetric magnetic field. Later studies resulted in more anti-dynamo theorems, leading to the qualitative conclusion that there is no simple explanation for the solar cycle: a dynamo features essentially three-dimensional flow fields which interact with three-dimensional magnetic fields, without any symmetry.
A breakthrough came when Parker (1955b) suggested the use of an averaging procedure and to describe the mean magnetic field <B>. This mean field can be steady and axisymmetric since only the real total field B=<B>+dB (where dB are the non-axisymmetric fluctuations in B) is subject to Cowling's theorem. The new element in this approach is what Parker called cyclonic motion: the twisting of magnetic field lines by helical convection. Unfortunately, Parker constructed his model intuitively rather than deductively, which led Roberts (1967, end of his Ch. 3) to belittle Parker's idea by writing only that Parker "has made other interesting qualitative suggestions for possible dynamos".
Then after 1966, Steenbeck, Krause and Rädler published a series of papers (translated in English by Roberts and Stix (1971)), which provided a mathematical bases for Parker's suggestions. The quintessence is that the velocity field is split in two parts: u(r,t) = u0+u1(r,t), where u0=<u> represents the large-scale motion, and u1, having zero average, is the turbulent velocity. These velocities are assumed to act on widely different length scales, while the mean <.> is taken over an intermediate scale. The overall picture of the dynamo that follows from this mean field dynamo theory is given below. Early models that successfully described the transport of magnetic fields on the Sun - i.e. amplification of a toroidal field, the formation of bipolar active regions, Hale's polarity rules, Spörer's law of sunspot latitudes, and polar field reversals - were devised by Babcock (1961) and Leighton (1964, 1969). The basic ingredients of their ideas still hold as basis for all the model computations that have been performed using the ideas of Steenbeck c.s.; some of these models are discussed in Sect. 5.
Magnetic activity seems to take place in all cool stars, irrespective of their mass and radius. All these stars have a convection zone directly beneath their photosphere in common. Dynamo action occurs also in fully convective M-type dwarf stars. A large variety of activity levels has been observed, from much less to much more active than the Sun. The activity level seems to depend mainly on one parameter which is related to the rotation rate: the faster a star rotates, the more active it is (a faster rotation presumably implies a stronger differential rotation and maybe also a stronger alpha-effect). During its lifetime a star loses mass, which streams away along the magnetic field lines, just like the Sun loses mass in the form of the solar wind. Conservation of angular momentum implies that the rotation rate of the star decreases, so that the toroidal field is wound up less and less as the star ages: the older the star, the less active it is. (In binary systems tidal forces can overcome the decrease in rotation, so that the star remains at a high activity level.) As a star evolves into a giant, it expands and the star's rotation rate decreases even further.
Stellar activity does not need to be cyclic: it may also be chaotic. The cycle period tends to be larger towards later spectral type (e.g. Maceroni et al., 1990). The Sun is known to vary in brightness during a cycle (Sect. 2.1) by about 0.1%, in phase with the changes in magnetic activity. Similar variations have been observed at solar-type stars, whereas some stars show almost no activity variations: such stars could be in a phase similar to the Sun's Maunder Minimum (e.g. Baliunas and Jastrow, 1990; see also Giampapa, 1990; Lean et al., 1992). Stellar activity studies may therefore tell us something about possible long-term changes in solar activity.
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