BORIS ANDREWS

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PhD (DPhil) candidate, University of Oxford (he/him)

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boris.andrews@maths.ox.ac.uk

University of Oxford profile

CV

TOPOLOGY-PRESERVING DISCRETIZATION FOR THE MAGNETO-FRICTIONAL EQUATIONS ARISING IN THE PARKER CONJECTURE

Mingdong He | Patrick Farrell | Kaibo Hu | Boris Andrews

20.JAN.2025 (arXiv)

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[…] 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. […]

FULL 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 discuss extensions to domains with nontrivial topology.

The results of this work are, to me, tremendously exciting. They demonstrate how vital it is, when designing a numerical simulation, that you preserve your conservation and dissipation laws from the continuous level to the discrete.

Magnetic relaxation is the process by which a magnetic or magnetohydrodynamic (MHD) system converges to its equilibrium/steady state. These systems are typically long-duration, large-scale plasmas (e.g. the Sun, in particular its corona visible in the photo below) or liquid metals (e.g. the Earth’s core).

solar_corona

In this work, we develop accurate numerical simulations for a certain magnetic relaxation model: the magneto-frictional equations.

MAGNETO-FRICTIONAL EQUATIONS
\[ \dot{\mathbf{B}} + \mathrm{curl}\,\mathbf{E} = \mathbf{0}, \\ \mathbf{E} + \mathbf{u}\times\mathbf{B} = \mathbf{0}, \\ \mathbf{j} = \mathrm{curl}\,\mathbf{B}, \\ \mathbf{u} = \mathbf{j}\times\mathbf{B}, \] where \(\mathbf{E}, \mathbf{B}\) are the electric and magnetic fields (respectively) and \(\mathbf{u}, \mathbf{j}\) are the fluid's internal velocity and current (respectively).

The magneto-frictional equations conserve a quantity called the helicity, \(\mathcal{H}\), and dissipate a quantity called the energy, \(\mathcal{E}\).

DEFINITIONS FOR \(\mathcal{H}, \mathcal{E}\)
\[ \mathcal{H} \coloneqq \int\mathbf{A}\cdot\mathbf{B}, \qquad \mathcal{E} \coloneqq \int\mathbf{B}\cdot\mathbf{B}, \] where \(\mathbf{A}\) is the magnetic potential satisfying \(\mathbf{B} = \mathrm{curl}\,\mathbf{A}\).

If \(\mathcal{E}\) ever hits \(0\), the system has necessarily relaxed to a trivial steady state, i.e. the magnetic field has simply vanished everywhere. The interest thing however is that this should never be the case.
A simple inequality, the Arnold inequality, says that \(\mathcal{E}\) can not pass below a certain multiple of \(\mathcal{H}\); since \(\mathcal{H}\) is constant, this means \(\mathcal{E}\) can never reach \(0\). In the equilibrium state therefore, the magnetic field should not vanish.

FUN TOPOLOGICAL DIVERSION
This has a neat topological interpretation! (Hence "topology-preserving" in the title.)
The helicity \(\mathcal{H}\) can be interpreted as a continuous analogue of an idea from knot theory: the linking number. This represents the number of times a pair of loops winds around each other (1, 2, 3 in the image below).

linking_numbers

Essentially, \(\mathcal{H}\) quantifies how knotted the intial magnetic field is. The conservation of \(\mathcal{H}\) implies that magnetic relaxation cannot untie these knots.
It should loosen them, but never untie them.

While these structures exist on the continuous level, they are not necessarily preserved in any old simulation. In particular, existing numerical schemes typically do not conserve \(\mathcal{H}\).
In our work, we construct a numerical scheme that conserves \(\mathcal{H}\) exactly; you can compare our scheme (solid red lines) with the typical scheme (dashed blue line) in which \(\mathcal{H}\) dissipates to \(0\) in the figure below. Together with the Arnold inequality, this ensures \(\mathcal{E}\) cannot decay to \(0\)…

sp_laws

…and that the computed magnetic field will not artificially vanish!

field_lines

I’d like to conclude this by emphasising: this is essentially the first magnetic relaxation simulation that does not artificially dissipate to nothing.
This means it’s the first that can be used to investigate these equations’ long-term behaviour. The Parker conjecture supposes that ideal magnetic relaxation may develop tangential discontinuities; our scheme should prove vital for numerical investigations into its validity.

We hope our work can both motivate the use of conservative/structure-preserving integrators, and introduce numerical discretisations as a valid tool for investigations into the Parker conjecture.

CHECK OUT ON ARXIV!

We would all gladly discuss it further!

This scheme can be viewed as a special case of my previous work with Patrick Farrell, on general constructions for conservative finite element integrators.

Co-authors

Mingdong He

Mingdong He

Patrick Farrell

Patrick Farrell

Kaibo Hu

Kaibo Hu

Talks

2025

  • ACOMEN, Ghent University
    • ⬆️ UPCOMING ⬆️
  • METHODS Group seminar, Brown University