Bipolar solar magnetic fields Behaviors resulting from a nonlinear force-free equation

Aims. Understanding magnetic fields in the solar corona is closely related to the complex nature of the often nonlinear differential equations describing such structures. Based on the ansatz of force-free fields, a class of solutions is derived and discussed that allows for axisymmetric bipolar magnetic fields. Methods. Allowed dipolar solutions for self-similar axisymmetric force-free magnetic fields use the formalism of a Grad-Shafranov equation involving the vector potential. For separable solutions involving poloidal fields decaying radially as r(-n), there are no dipolar field structures for the decay index n >= 1. Results. In the domain n < 1 dipolar field structures are possible in restricted ranges of the angular coordinate. depending on the value for n. Outside of the restricted domains there are no dipolar solutions, but there can be multipole solutions. The limiting case of the parameter n -> 0 has been discussed previously, so that now the full regime 0 <= n < infinity is covered.