Magnetic fields of active regions and their zero points

TLDR: In this paper, it was shown that the number of singular (zero) points of a magnetic field above the plane of the photosphere is determined by the maximum and minima in the potential of the field, and that the arrangement of the singular points determines the overall topological structure of the magnetic field.

Abstract: The possibility of a potential approximation for the description of magnetic fields in the chromosphere and the corona is discussed. The introduction of a scalar potential allows one to investigate the overall geometrical properties of the field. On the basis of the results of the general theory of differential equations, it is shown that the number of singular (zero) points of a magnetic field above the plane of the photosphere is determined by the number of maxima and minima in the potential at the photosphere. The arrangement of the singular points determines the overall topological structure of the field. The Neumann problem is solved for the fields of active regions. The results of a numerical solution of the boundary problem for the active region McMath No. 11,693 are given as an example of the application of the proposed methods. The structure of the magnetic field of this region and the singular points found for it are described.