7.1 Dynamo models

Mean field dynamo models assume a dynamo action which is extended over the entire convection envelope. In the Sun, a shell dynamo based on the $a_O_$ mechanism seems to be able to explain the main features of the 11- $yr$ cycle. However, in the most active stars it is difficult to reproduce the observed latitudinal range of starspots with the thin-shell dynamo models. It appears that in addition to a solar-like dynamo, probably working in the overshoot zone, a distributed $a2_O_$ dynamo is likely to be present in very active stars (Brandenburg et al., 1989

; Moss et al., 1995

). Such a distributed dynamo is able to explain, at least qualitatively, a threshold for light modulation, active longitudes and flip-flops (see below). Also, it is a good candidate for explaining the observed orbital period modulation in active close binaries (Lanza et al., 1998b ).

The geometry and behaviour of solar and stellar magnetic fields are globally determined by the stability of dynamo modes with different symmetry (Brandenburg et al., 1989 ). For instance, the sunspot cycle can be explained by an axisymmetric mean-field dynamo mode of $A0$ type, which is antisymmetric with respect to the equator (dipole-like). Similarly, spot cycles in other stars can be also associated with an axisymmetric mode of $S0$ type which is symmetric with respect to the equator (quadrupole-like).

Persistent active longitudes separated by $o 180$ on the Sun and cool active stars clearly indicate the presence of non-axisymmetric dynamo modes. They can be either symmetric with respect to the equatorial plane, e. g., a dipole-like $S1$ mode (Moss et al., 1991 , 1995 ), or antisymmetric such as a quadrupole-like $A1$ mode (Tuominen et al., 2002 ). The magnetic field configuration in such modes consists of magnetic spots of opposite polarities (active longitudes) $180o$ apart. It appears that such modes can be excited at lower dynamo numbers than axisymmetric modes in the case when the differential rotation is not large. The extention of these models to synchronised binaries reveals that the maxima of the mean magnetic field appears to concentrate near the line joining the centres of the binary components (Moss and Tuominen, 1997 ).

Beside the symmetry of the modes, their oscillatory properties are important. The mean-field dynamo theory favours oscillating axisymmetric modes with a clear cyclic behaviour and sign changes (as in the sunspot cycle), while non-axisymmetric modes appear to be rather steady.

The alternating active longitudes and flip-flop cycles observed on the Sun and other active stars imply, however, the existence of apparently oscillating non-axisymmetric fields. As is suggested by Berdyugina et al. (2002 ), perhaps the coexistence of oscillating axisymmetric and steady non-axisymmetric modes results in the appearance of flip-flop cycles. Then, the relative strengths of the two dynamo modes and the period of the oscillations of the axisymmetric mode should define the amplitudes and lengths of observed cycles. The possibility of such a mechanism was first demonstrated by the mean-field dynamo calculation of Moss (2004 ) who obtained a stable solution with an oscillating $S0$ type mode and a steady, mixed-polarity non-axisymmetric mode. In this case flip-flops are quasi-periodic and as frequent as sign changes of the $S0$ mode, which is reminiscent of the behaviour observed in some RS CVn stars. A similar mechanism involving an oscillating $A0$ mode and a steady $S1$ mode is discussed by Fluri and Berdyugina (2004 ).

More frequent flip-flops, compared to the sunspot-like cycle in single stars and the Sun, suggest a more complex field configuration. As shown by Fluri and Berdyugina (2004 ) flip-flops could also occur due to alternation of relative strengths of non-axisymmetric $S1$ and $A1$ modes without sign changes of any involved modes. If in addition a co-existing axisymmetric mode were changing its sign with a different frequency, it would result in the behaviour observed in solar-type stars. The stability of such a solution should however be tested by dynamo calculations.

Differential rotation is a key parameter for stellar dynamos, and a theoretical study of its dependence on the rotation rate and spectral type is important for interpreting observations. A noticeable progress in such modelling was achieved during the recent decade (Kitchatinov and Rüdiger, 1995 ; Rüdiger et al., 1998 ). In particular, the following results are important. Simulations of global circulation in outer stellar convection zones for spectral classes G2 and K5, rotating at the same rate, demonstrate that differential rotation for G2 is larger compared to K5 (Kitchatinov and Rüdiger, 1999 ). However, as the rotation period decreases, the differential rotation first decreases as well but starts to increase for the shortest periods. It appears that rapid rotation can explain the rather strong total surface differential rotation of the observed very young solar-type stars, as it creates an equatorward meridional flow at the stellar surface which accelerates the equatorial rotation (Rüdiger and Küker, 2002 ). A possibility for stars to have an anti-solar differential rotation was discussed by Kitchatinov and Rüdiger (2004 ).

The latest developments in the solar and stellar dynamo theory are reviewed by Brandenburg and Dobler (2002 ) and Rüdiger and Hollerbach (2004 ).