Decay of helical and nonhelical magnetic knots

We present calculations of the relaxation of magnetic field structures that have the shape of particular knots and links. A set of helical magnetic flux configurations is considered, which we call $n$-foil knots of which the trefoil knot is the most primitive member. We also consider two nonhelical knots; namely, the Borromean rings as well as a single interlocked flux rope that also serves as the logo of the Inter-University Centre for Astronomy and Astrophysics in Pune, India. The field decay characteristics of both configurations is investigated and compared with previous calculations of helical and nonhelical triple-ring configurations. Unlike earlier nonhelical configurations, the present ones cannot trivially be reduced via flux annihilation to a single ring. For the $n$-foil knots the decay is described by power laws that range form $t^{-2/3}$ to $t^{-1/3}$, which can be as slow as the $t^{-1/3}$ behavior for helical triple-ring structures that were seen in earlier work. The two nonhelical configurations decay like $t^{-1}$, which is somewhat slower than the previously obtained $t^{-3/2}$ behavior in the decay of interlocked rings with zero magnetic helicity. We attribute the difference to the creation of local structures that contain magnetic helicity which inhibits the field decay due to the existence of a lower bound imposed by the realizability condition. We show that net magnetic helicity can be produced resistively as a result of a slight imbalance between mutually canceling helical pieces as they are being driven apart. We speculate that higher order topological invariants beyond magnetic helicity may also be responsible for slowing down the decay of the two more complicated nonhelical structures mentioned above.

Figure 1

Braid representation of the 4-foil knot. The letters denote the starting position and the numbers denote the crossings.

Figure 2

$xy$ projection of the 4-foil knot. The numbers denote the crossings while the colors (line styles) separate different parts of the curve. The letters denote the different starting positions for the braid representation in Fig. 1. The arrow shows the sense of the knot.

Figure 3

Projection of the 5-foil on the $xy$ plane. The lines show the meaning of the distance $C$, which has to be larger than 1 to make sense.

Figure 4

Isosurface of the initial magnetic field energy for the 4-foil configuration.

Figure 5

Isosurface of the initial magnetic field energy for the IUCAA knot seen from the top (left panel) and slightly from the side (right panel).

Figure 6

Isosurface of the initial magnetic field energy for the Borromean rings configuration.

Figure 7

The apparent self-linking number for $n$-foil knots with respect to $n_{\mathrm{f}}$ (upper panel). The fit is obtained by equating (1) and (8). The length of a $n$-foil knot is plotted with respect to $n_{\mathrm{f}}$ (lower panel), which can be fit almost perfectly by a linear function.

Figure 8

Schematic representation illustrating the sign of a crossing. Each crossing has a handedness which can be either positive or negative. The sum of the crossings gives the number of linking and eventually the magnetic helicity content via equation (8).

Figure 9

The isosurface for the 4-foil knot field configuration. The sign of the crossing is always negative. The rings show the different areas where crossings occur.

Figure 10

Time dependence of the normalized magnetic energy for a given number of foils with periodic boundary conditions. The power law for the energy decay varies between $-2/3$ for $n_{\mathrm{f}}=3$ [solid (blue) line] and $-1/3$ for $n_{\mathrm{f}}=7$ [solid (black)].

Figure 11

Time dependence of the quotient from the realizability condition (2) for $k=2$. It is clear that, for larger $n_{\mathrm{f}}$, the energy approaches its minimum faster.

Figure 12

Time dependence of the normalized magnetic helicity for a given number of foils with periodic boundary conditions.

Figure 13

Time dependence of the normalized magnetic energy for the trefoil and 4-foil knot with periodic and perfect conductor (PC) boundary conditions. There is no significant difference in the energy decay for the different boundary conditions.

Figure 14

Magnetic field lines for the trefoil knot at time $t=0$ (upper panel) and $t=7.76\times {10}^{-2}{t}_{\mathrm{res}}$ (lower panel). Both images were taken from the same viewing position to make comparisons easier. The Lundquist number was chosen to be 1000. The colors indicate the field strength.

Figure 15

Magnetic field lines for the IUCAA knot at $t=0.108t_{\mathrm{res}}$ (upper panel) and at $t=0.216t_{\mathrm{res}}$ (lower panel) for $\text{Lu}=1000$ and $\varphi =(4/3)\pi$.

Figure 16

Normalized magnetic helicity in the northern [green (dashed) line] and southern [red (dotted) line] domain half together with the total magnetic helicity [blue (solid) line] for the IUCAA knot with $\text{Lu}=2000$ and $\varphi =(4/3)\pi$.

Figure 17

Normalized magnetic helicity in the northern [green (dashed) line] and southern [red (dotted) line] domain half together with the total magnetic helicity [blue (solid) line] for the IUCAA knot with $\text{Lu}=2000$ and $\varphi =(4/3+0.2)\pi$.

Figure 18

$xy$-averaged magnetic helicity density profile in $z$ direction for the IUCAA knot with $\text{Lu}=2000$ and $\varphi =(4/3)\pi$. There is an apparent asymmetry in the distribution amongst the hemispheres.

Figure 19

Magnetic energy density in the $xz$ plane for $y=0$ at $t=0$ (upper panel) and $t=5.58\times {10}^{-2}{t}_{\mathrm{res}}$ (lower panel) for the IUCAA knot with $\text{Lu}=2000$ and $\varphi =(4/3)\pi$.

Figure 20

Magnetic energy density in the $xz$ plane for $y=0$ at $t=0$ (upper panel) and $t=5.58\times {10}^{-2}{t}_{\mathrm{res}}$ (lower panel) for the IUCAA knot with $\text{Lu}=2000$ and $\varphi =(4/3+0.2)\pi$. The magnetic field stays centered.

Figure 21

Magnetic energy versus time for the different initial field configurations together with power laws which serve as a guide. The decay speed of the IUCAA knot and Borromean rings lies well in between the helical and nonhelical triple-ring configuration.

Figure 22

Magnetic field lines at $t=0.248t_{\mathrm{res}}$ for the Borromean rings configuration for $\text{Lu}=1000$. In the lower-left corner the interlocked flux rings are clearly visible which differs from the proposed trefoil knot [28]. The flux ring in the opposite corner has an internal twist which makes it helical. The colors denote the strength of the field, where the scale goes from red over green to blue.

Figure 23

Magnetic field lines at $t=0.276t_{\mathrm{res}}$ for the Borromean rings configuration for $\text{Lu}=1000$. The two flux rings in the corners both have an internal twist which makes them helical. The twist is, however, of opposite sign which means that the whole configuration does not contain magnetic helicity. The colors denote the strength of the field, where the scale goes from red over green to blue.