On the relationship between the topology of magnetic field lines and flux surfaces

Abstract

Abstract We consider the topological relationship between magnetic field lines and magnetic flux surfaces. Magnetic helicity provides the most elementary description of the topology of magnetic field lines in terms of their linkage. In a simply-connected volume, a sufficient but not necessary condition for the total magnetic helicity to vanish is that there exist two independent families of globally-extendable flux surfaces (given by the level surfaces of Euler potentials). In contrast, the existence of two distinct global Euler potentials for multiply-connected volumes is insufficient to guarantee that the total magnetic helicity vanishes. These well-known results are discussed in the context of Frobenius’ theorem as applied to the differential equations describing magnetic lines of force; and the notion of Euler potentials is extended by introducing an analogy to the Hopf map between the three-sphere and the two-sphere.