Ideal Relaxation of the Hopf Fibration (1610.04719v2)

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.

• The paper investigates achieving equilibrium in magnetized plasma structures with Hopf fibration topology using ideal MHD simulations, conserving magnetic topology.
• It shows that equilibrium for localized Hopf fields requires finite external pressure and is not a force-free Taylor state but a Grad-Shafranov equilibrium with pressure variation.
• Numerical simulations validate the existence of these topology-preserving equilibria and provide insights into understanding plasma confinement and astrophysical magnetic structures.

Overview of "Ideal Relaxation of the Hopf Fibration"

The paper "Ideal Relaxation of the Hopf Fibration" by Christopher Berg Smiet, Simon Candelaresi, and Dirk Bouwmeester investigates the process of achieving equilibrium in magnetized plasma structures with the topology of the Hopf fibration. This work focuses on modeling plasmas as perfectly conducting, viscous fluids governed by the principles of ideal magnetohydrodynamics (MHD), conserving magnetic topology while reaching lower energy states. Such relaxation processes are pivotal in understanding plasma equilibria in both fusion devices and astrophysical contexts.

Key Findings

1. Hopf Fibration in MHD: The paper explores magnetic fields with topology derived from the Hopf map, where field lines are closed and nontrivially linked. These fields can be found in astrophysical plasmas or laboratory plasmoids. The notion of topology-conserving relaxation is central, ensuring that initial magnetic linkages or configurations remain throughout the evolution to an equilibrium state.
2. Equilibrium Conditions: It is demonstrated that achieving equilibrium for localized magnetic fields of Hopf topology requires finite external pressure. An unattainability of the Taylor state, a low-energy force-free magnetic configuration, within this setup is shown. The equilibrium state is characterized by a toroidal depression in the plasma's pressure profile, with field lines residing on surfaces of constant pressure, indicating a Grad-Shafranov equilibrium.
3. Numerical Simulations: Utilizing a Lagrangian relaxation technique implemented in GLEMuR, the authors simulate ideal, compressible MHD to discover self-consistent equilibria that maintain field line topology. The results confirm the conservation of helicity and further clarify the nature of the equilibrium states attainable under particular constraints.
4. Extensions and Comparisons: The authors extend their analysis by considering fields with varying poloidal and toroidal winding, observing the influence on resultant equilibria. Comparisons are drawn to Kamchatnov's constructions where dynamics are balanced by parallel fluid motion, underscoring the distinct nature of the static solitons addressed in this work.

Implications and Future Directions

The implications of this paper are twofold. Practically, these equilibrium configurations offer insights into the design and stability analysis of magnetic confinement systems, such as those used in fusion research. Theoretically, the work contributes to the broader understanding of MHD equilibria, particularly in configurations where traditional force-free models are inapplicable.

Future investigations could explore the boundary conditions under which different magnetic topologies influence plasma stability and relaxation dynamics. Additionally, the applicability of these findings to observational astrophysics could provide a deeper understanding of natural magnetic structures, such as those found in stellar environments or the interstellar medium.

In conclusion, this paper offers precise and computationally supported insights into the conditions required for achieving stable, topology-preserving equilibria in magnetized plasmas. The findings advance the field of plasma physics by framing new boundary conditions and configurations for stable magnetic field lines in conducting fluids, providing a concrete step forward in the domain of MHD equilibrium research.