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 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.

On the proof of Taylor's conjecture in multiply connected domains

“…In this Letter, we bring together the results of Faraco and Lindberg [1] and MacTaggart and Valli [13] to provide a complete proof of Taylor's conjecture for arbitray vector potentials of closed magnetic fields with no restrictions on the magnetic field imposed by gauge invariance. We proceed by only describing extensions and modifications to specific parts of the proof in [1] that are relevant for expanding its scope in the manner described above.…”

“…Equation ( 1) represents the gauge invariant form of helicity in simply connected domains. MacTaggart and Valli [13] showed that the definition of magnetic helicity H needs to be extended for multiply connected domains in order to accommodate arbitrary vector potentials that do not impose any restrictions on the magnetic field. For example, it can be shown that a consequence of imposing gauge invariance with equation (1) in multiply connected domains is that the magnetic flux through internal cuts of the domain must be zero, e.g., the toroidal magnetic flux in a tokamak would be zero.…”

Magnetic winding: what is it and what is it good for?

“…From a geometric perspective, λ measures the rotation of the magnetic field around the direction of the vector B ×F l , a vector normal to both the Lorentz force and magnetic field. If we specify the function λ then the Lorentz force can be written as 12) and λ can be seen to represent the relative magnitude of the Lorentz force to the magnetic field strength. Thus the representation of the magnetic field's varying local geometry through (3.9) has two scalar parameters, λ and α, which represent, respectively, the relative magnitude of the Lorentz force and axial current and hence measure their relative effect on the local geometry of the magnetic field through its curl.…”

“…If around each loop we identify a solid magnetic torus, as in [11], then since the magnetic field is tangential to the toroidal boundary, we can consider the domain to be R 3 , where the magnetic fields in the tori are 'extended by zero' outside of the tori volumes [12]. Thus equation (1.4) can lead to classical helicity formula…”

Magnetic helicity, weak solutions and relaxation of ideal MHD

“…Identical arguments are valid for simply connected magnetically closed domains. For a definition of magnetic helicity in domains with non-trivial topology see [MV19].…”