Magnetic helicity in multiply connected domains

Abstract

Magnetic helicity is a fundamental quantity of magnetohydrodynamics that carries topological information about the magnetic field. By `topological information', we usually refer to the linkage of magnetic field lines. For domains that are not simply connected, however, helicity also depends on the topology of the domain. In this paper, we expand the standard definition of magnetic helicity in simply connected domains to multiply connected domains in $\mathbb{R}^3$ of arbitrary topology. We also discuss how using the classic Biot-Savart operator simplifies the expression for helicity and how domain topology affects the physical interpretation of helicity.