Generation of sunspot and polar faculae butterflies using bipolar and quadripolar seed fields

Abstract

The exact solution of the evolution equation for the magnetic field in ideal MHD, Callebaut (2006), with an azimuthal velocity which is function of $r$ and $\vartheta$ only (spherical coordinates) is applied to a bipolar magnetic seed field and to a quadripolar field. Resistivity and $\alpha$-effect are not yet taken into account, but the extensions are possible. From the surface observations we had derived an approximate analytic expression for the differential rotation in order to work fully analytically in the application. Qualitatively the results for a quadripolar field are as for a bipolar seed field. The main features are the same: for some latitudes the field may increase by two orders of magnitude, the separation between sunspots and polar faculae is clearcut, there is, relatively speaking, a too strong amplification in the polar regions (the latter occurs in other models too). The hypothesis that the seed fields are situated at the tachocline is not required: the amplification is active throughout the whole convective zone, albeit with different strengths, and thus during the transit of the flux tubes from tachocline to the solar surface too.