Generation of Solar Magnetic Fields Using a Quadripolar Seed Field

Abstract

The exact solution for the kinematic dynamo problem in spherical coordinates r, #, ' is given in Ref. 1. The velocity is supposed to be azimuthal and to be an arbitrary function of r and # only. Using a bipolar seed field yielded a qualitative agreement with the sunspot butterfly diagram and the polar faculae butterfly diagram. Here we investigate the case that a quadripolar seed field resides in the Sun (maybe in the whole convective zone or rather only in the tachocline at the bottom of the convective zone). In fact some observations reveal a quadripolar contribution to some surface phenomena. A combination of a bipolar and a quadripolar field yields better agreement for a suitable choice of their amplitudes. The separation between the sunspot region and the polar faculae, although both are generated by the same mechanism, is manifest: the region where the radial variation of the angular frequency of the rotation vanishes. Here the dynamo is considered in the ideal magnetohydrodynamic (MHD) approximation: no resistivity and no �-effect. The basic equations are the equation for the evolution of the magnetic field H and the conservation of the magnetic flux @tH = rot(v × H), (1) divH = 0, (2) where v is the velocity (SI units). The velocity field in the solar case may be assumed as being mainly a rotation around the solar axis (there is quite a good symmetry, sometimes a perfect anti-symmetry, with respect to the equator). Superposed on this main rotation are some turbulent motions, usually of much smaller amplitude. (Sometimes the turbulent speeds may be much higher, but they usually do not last long and alternate their direction.) They are not considered here, although their azimuthal component may be included in the treatment.