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

Patrick E. Farrell, Mingdong He, Kaibo Hu, Ganghui Zhang

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.

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

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.
Paper Structure (14 sections, 5 theorems, 52 equations, 5 figures)

This paper contains 14 sections, 5 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: helicity and magnetic topology
  3. Non-conservative scheme
  4. Projection-based mixed finite element scheme
  5. The Lagrange multiplier scheme
  6. Model derivation
  7. Full discretization
  8. Solver
  9. Numerical experiments
  10. Magnetic braids: $E^3$-field
  11. Magnetic knots: Hopf fibration
  12. Local vs. global helicity preservation: physical implications
  13. Conclusion
  14. Woltjer's variational principle

Key Result

Theorem 3.2

\newlabelthm:Hdiv0 Assume $(\bm B_h^{n+1}, \bm E_h^{n+1/2}, \bm j_h^{n+1/2})$ is a solution of eq:nonconservative-scheme. Then the energy is decreasing and the discrete Gauss law holds, that is, and where $\mathcal{E}^n_h=\left( \bm B_h^{n},\bm B_h^{n}\right)$.

Figures (5)

  • Figure 1: Illustration of non-trivial topology of magnetic fields. (a) Magnetic knot with non-zero helicity, (b) Magnetic braid with zero helicity and (c) Borromean ring with zero helicity. Here the thin lines represent the magnetic field lines surrounding the high-intensity core tubes.
  • Figure 1: Magnetic braids ($E^3$-field): evolution of energy and helicity, errors.
  • Figure 2: Magnetic braids: comparison of evolution of stream tubes of the magnetic field under different topological constraints, colored by magnetic field strength $\|\bm B_h\|$.
  • Figure 3: Magnetic knots (Hopf fibration): evolution of energy and helicity, errors.
  • Figure 4: Magnetic knots: comparison of evolution of stream tubes of the magnetic field under different topological constraints, colored by magnetic field strength $\|\bm B_h\|$.

Theorems & Definitions (13)

  • Theorem 3.2
  • Theorem 4.2
  • Proof 1
  • Remark 4.3
  • Theorem 5.1
  • Proof 2
  • Corollary 5.2
  • Proof 3
  • Theorem 5.4
  • Proof 4
  • ...and 3 more