Ideal Relaxation of the Hopf Fibration

TL;DR

This paper investigates the ideal magnetohydrodynamic relaxation of magnetic fields with Hopf fibration topology, revealing conditions for equilibrium and characterizing the resulting plasma configurations with a toroidal pressure depression.

Contribution

It demonstrates that localized Hopf fibration magnetic fields reach equilibrium only with finite external pressure and cannot attain a Taylor state, identifying a toroidal pressure depression as a key feature.

Findings

Equilibrium requires finite external pressure.

Taylor state is unattainable for these fields.

Equilibrium characterized by a toroidal pressure depression.

Abstract

Ideal MHD relaxation is the topology-conserving reconfiguration of a magnetic field into a lower energy state where the net force is zero. This is achieved by modeling the plasma as perfectly conducting viscous fluid. It is an important tool for investigating plasma equilibria and is often used to study the magnetic configurations in fusion devices and astrophysical plasmas. We study the equilibrium reached by a localized magnetic field through the topology conserving relaxation of a magnetic field based on the Hopf fibration in which magnetic field lines are closed circles that are all linked with one another. Magnetic fields with this topology have recently been shown to occur in non-ideal numerical simulations. Our results show that any localized field can only attain equilibrium if there is a finite external pressure, and that for such a field a Taylor state is unattainable. We find an equilibrium plasma configuration that is characterized by a lowered pressure in a toroidal region, with field lines lying on surfaces of constant pressure. Therefore, the field is in a Grad-Shafranov equilibrium. Localized helical magnetic fields are found when plasma is ejected from astrophysical bodies and subsequently relaxes against the background plasma, as well as on earth in plasmoids generated by e.g.\ a Marshall gun. This work shows under which conditions an equilibrium can be reached and identifies a toroidal depression as the characteristic feature of such a configuration.