Helicity-preserving finite element discretization for magnetic relaxation

TL;DR

The paper tackles Parker's conjecture on current-sheet formation during ideal magnetic relaxation by introducing an energy- and helicity-preserving finite element discretization of the magneto-frictional system. It leverages finite element exterior calculus to construct a structure-preserving scheme that conserves discrete helicity and dissipates energy, and it extends helicity notions to non-contractible domains via a generalized helicity with a discrete Arnold inequality. Through numerical experiments on Hopf fibration and IsoHelix setups, the authors demonstrate that helicity preservation is essential to prevent unphysical collapse and to faithfully capture the topological barrier. The results provide a robust computational tool for investigating Parker's conjecture and highlight the broader importance of structure-preserving discretizations in magnetohydrodynamics. The approach offers a pathway to long-time, physically meaningful simulations without artificial reconnection or loss of topology.

Abstract

The Parker conjecture, which explores whether magnetic fields in perfectly conducting plasmas can develop tangential discontinuities during magnetic relaxation, remains an open question in astrophysics. Helicity conservation provides a topological barrier during relaxation, preventing topologically nontrivial initial data relaxing to trivial solutions; preserving this mechanism discretely over long time periods is therefore crucial for numerical simulation. This work presents an energy- and helicity-preserving finite element discretization for the magneto-frictional system for investigating the Parker conjecture. The algorithm preserves a discrete version of the topological barrier and a discrete Arnold inequality. We also propose extensions of the notion of helicity and the Arnold inequality to certain kinds of topologically nontrivial domains. Numerical experiments demonstrate that helicity preservation is crucial in obtaining physically meaningful simulations of magnetic relaxation, providing an example where structure-preserving schemes are necessary.

Figures (9)

• Figure 1: Discrete energy dissipation \ref{['eq:energy-dissipation']} and helicity conservation \ref{['eq:helicity-conservation']} for a comparable trivial and nontrivial topology, under our structure-preserving scheme \ref{['eq:structure-preserving-scheme']}
• Figure 2: Errors in helicity ($|\mathcal{H}_h - \mathcal{H}_h(0)|$ and $|\tilde{\mathcal{H}}_h(t) - \tilde{\mathcal{H}}_h(0)|$ respectively) and $\|\operatorname{div} \bm B_h\|$ for a comparable domain with trivial and nontrivial topology, under our structure-preserving scheme \ref{['eq:structure-preserving-scheme']}
• Figure 3: Helicity, $\mathcal{H}_h$, and energy, $\mathcal{E}_h$, for our proposed scheme \ref{['eq:structure-preserving-scheme']} and the $H(\operatorname{div})$-conforming scheme \ref{['eqn:without-H']}
• Figure 4: Errors $\|\operatorname{div} \bm B_h\|$ and $|\mathcal{H}_h - \mathcal{H}_h(0)|$ for our proposed scheme \ref{['eq:structure-preserving-scheme']} and the $H(\operatorname{div})$-conforming scheme \ref{['eqn:without-H']}
• Figure 5: Error $||\operatorname{div} \bm B_h||$ and evolution of $\mathcal{E}_h$ for the $H(\operatorname{curl})$-- and $H^1$--conforming schemes

Theorems & Definitions (16)

• Theorem 3.2: Structure-preserving properties of the discretisation
• Proof 1
• Corollary 3.3: Boundedness of the discrete energy
• Theorem 3.4: Invariance of the harmonic component
• Proof 2
• Definition 3.5: Generalized helicity
• Theorem 3.6: Gauge invariance
• Theorem 3.7: Invariance of the generalized helicity
• Proof 4
• Theorem 3.8: Generalized Arnold inequality