Self-Organizing Knotted Magnetic Structures in Plasma

We perform full-magnetohydrodynamics simulations on various initially helical configurations and show that they reconfigure into a state where the magnetic field lines span nested toroidal surfaces. This relaxed configuration is not a Taylor state, as is often assumed for relaxing plasma, but a state where the Lorentz force is balanced by the hydrostatic pressure, which is lowest on the central ring of the nested tori. Furthermore, the structure is characterized by a spatially slowly varying rotational transform, which leads to the formation of a few magnetic islands at rational surfaces. We then obtain analytic expressions that approximate the global structure of the quasistable linked and knotted plasma configurations that emerge, using maps from $S^3$ to $S^2$ of which the Hopf fibration is a special case. The knotted plasma configurations have a highly localized magnetic energy density and retain their structure on time scales much longer than the Alfvénic time scale.

Figure 1

Simulated configurations and evolution of magnetic fields. (a),(b) Initial condition and state at $t=22.5t_A$ for the $n=3$ and $T=1.8$ simulation. (c),(d) The same for the $n=6$ and $T=0$ simulation. In (a) and (c) a magnetic isosurface of $|\mathbf{B}|=0.1{\mathcal{B}}_0$ is shown to indicate the boundary of the flux tube. In (b) and (d) the lines represent the magnetic field lines. The outer field lines are partially transparent to not obstruct the view of the central configuration. (e) Decay of magnetic energy for the simulations with $T=0$ and $n$ ranging from 1 to 6, and (f) the simulations with $n=3$ and $T$ ranging from 0 to 4.4. The shaded region indicates reconfiguration, after that resistive decay takes over. (g) The evolution of the average of the magnetic energy $⟨B^2⟩/⟨B_0^2⟩$, normalized helicity $⟨\mathbf{A}·\mathbf{B}⟩/⟨B^2⟩$, helicity $⟨\mathbf{A}·\mathbf{B}⟩/⟨B_0^2⟩$, and velocity $⟨|\mathbf{v}|⟩$ for the simulation with $n=3$ and $T=1.8$.

Figure 2

The simulation with $n=3$ and $T=1.8$ at time $t=22.5t_A$. (a) The magnetic field contains invariant tori. Every surface is a single integral curve of the field of length $1000l_0$ colored differently for clarity. The surfaces are clipped to show the nested configuration. (b) Surfaces of constant magnetic field strength. A single torus is shown in black to indicate the scale of the magnetic structure.

Figure 3

(a) The radial component of minus the gradient of the averaged pressure field, and (b) the radial component of the averaged Lorentz force, taken in the $x,y$ plane (top view). The geometrical center of the tori is taken as the origin, and is marked by the blue dot. (c) The pressure field in the $x,z$ half-plane, showing a lowered pressure in the center of the magnetic surfaces.

Figure 4

Poincaré plot and properties of the force-balanced toroidal structure. (a) Poincaré plot of the magnetic field. (b) The value of the rotational transform for each magnetic surface. Rational values are indicated by the labeled horizontal lines, and the positions where they cross are shown by vertical lines. (c) Value of the magnetic field strength, and the components at each position. (d) The magnetic field strength and components of the magnetic field for the analytical expression of a field with the same energy and rotational transform.