Global and local helicity-preservation in the finite element discretisation of magnetic relaxation

TL;DR

This paper compares finite element methods for magnetic relaxation, showing that local helicity preservation prevents spurious reconnection and maintains topology, unlike methods enforcing only global helicity conservation.

Contribution

It introduces and compares three finite element formulations with different helicity-preserving properties, highlighting their impact on magnetic topology during relaxation.

Findings

Local helicity preservation prevents spurious reconnection.

Enforcing only global helicity allows further relaxation.

Numerical experiments demonstrate differences in magnetic topology.

Abstract

Magnetic relaxation drives plasma toward lower-energy equilibria under helicity constraints. In ideal magnetohydrodynamics (MHD), helicity is locally conserved, while resistive theories such as Taylor relaxation preserve only global helicity. This distinction has important implications for structure-preserving numerical methods. We compare three finite element formulations: an unconstrained scheme that does not conserve helicity, a mixed method based on finite element exterior calculus that preserves all local helicities, and a Lagrange multiplier approach that enforces only global helicity conservation. Numerical experiments on braided and knotted magnetic fields show that local helicity preservation prevents spurious reconnection and maintains nontrivial topology in ideal MHD or magneto-friction, whereas enforcing only global helicity allows further relaxation through local reconnection. Numerical results on magnetic knots and braids are provided. These results clarify how different levels of discrete helicity constraints influence magnetic relaxation and equilibrium structure in numerical computation.