Proof of Taylor's conjecture on magnetic helicity conservation

TL;DR

This paper proves Taylor's conjecture that magnetic helicity is conserved in 3D ideal MHD in certain domains, and explores the conditions under which this conservation holds, extending the understanding of magnetic field invariants.

Contribution

The paper establishes the conservation of magnetic helicity in 3D MHD for bounded domains and identifies conditions for its invariance in multiply connected domains, also proposing an analogue in 2D.

Findings

Magnetic helicity is conserved in 3D MHD in simply connected domains.

In multiply connected domains, helicity conservation depends on the choice of vector potential.

Mean square magnetic potential is conserved in 2D in the ideal limit.

Abstract

We prove Taylor's conjecture which says that in 3D MHD, magnetic helicity is conserved in the ideal limit in bounded, simply connected, perfectly conducting domains. When the domain is multiply connected, magnetic helicity depends on the vector potential of the magnetic field. In that setting we show that magnetic helicity is conserved for a large and natural class of vector potentials but not in general for all vector potentials. As an analogue of Taylor's conjecture in 2D, we show that mean square magnetic potential is conserved in the ideal limit, even in multiply connected domains.