7.2 Flux-tube models

High-latitude spots on very active stars can be explained by non-linear models for flux-tube instability (Schüssler and Solanki, 1992 ; Schüssler et al., 1996 ). In a rapid rotator with a dynamo operating near the base of the convection zone, the effect of the Coriolis force makes flux tubes to emerge nearly parallel to the rotation axis, thus producing high-latitude activity even if the dynamo amplifies the field only at low latitudes. Such a model is capable to reproduce the sunspot behaviour as well (Caligari et al., 1995 ). In addition to rapid rotation, the depth of the convection zone, stratification and magnetic field strength play an equally important role in the poleward deflection of rising flux-tubes. In particular, the size of the stellar core affects the magnetic curvature force and the rise time, while gravity determines the strength of the buoyancy force.

Calculations for zero-age main-sequence stars show that for increasing rotation rate the emergence pattern shifts to higher latitudes, while an equatorial zone of avoidance grows (Granzer et al., 2000 ). The concentration to higher latitudes is stronger for stars with lower mass and deeper convection zones because of geometrical reasons and the larger rise time. It is remarkable that once the Coriolis effect has enforced a nearly axis-parallel rise of the flux tube, further increase of the rotation rate has no effect on the emergence latitudes, indicating a saturation effect. Truly polar flux emergence was, however, not found for main-sequence stars. Also, low-latitude activity in rapid rotators cannot be described by such a model. For T Tauri-type stars Granzer et al. (2000 ) find a different pattern of behaviour. Because of the small size of the stellar core, the unstable flux tubes detach completely from the overshoot layer and rise through the convection zone as free-floating rings. They can emerge in a broad range of latitudes, almost from pole to equator.

A similar perturbation study of rising flux-tubes have been made for close synchronised binary stars with main-sequence components in order to examine an influence of tidal effects on the formation of observed active longitudes (Holzwarth and Schüssler , 2003a , 2003b ). It was found that although the magnitude of tidal effects is rather small, they nevertheless lead to the formation of clusters of flux tube eruptions at preferred longitudes on opposite sides of the star. The longitude distribution of the clusters depends on the initial parameters of flux-tubes in the overshoot region, such as magnetic field strength and latitude. This implies that there is no preferred orientation of the active longitudes and agrees with most of the observations. The extention of the model to post-main-sequence stars revealed difficulties in recovering starspot properties frequently observed on evolved binary components (Holzwarth, 2004 ). This points out an insufficiency of the applied flux tube model and implies that additional flux transport and possibly amplification mechanisms have to be taken into account.

Note that the flux-tube concept, when applied to heavily spotted stars, implicitly assumes that large starspots represent clusters of smaller, more sunspot-like magnetic structures, to which the thin flux-tube approximation is applicable during most of their rise through the convection zone (Schüssler, 2002 ). Further, such an approach gives only the information about potential emergence locations. The actual prediction of emergence patterns would require a coupling of the emergence calculations with a full dynamo model providing the flux distribution as functions of latitude, longitude and time.