Helicity-preserving finite element discretization for magnetic relaxation

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.