Twisted magnetic knots and links and their current alignment

In magnetohydrodynamics, astrophysics and plasma physics, the most used quantifier of the magnetic field line topology is the magnetic helicity. We know that its presence restricts the dynamics of the fluid which would not be the case for a non-helical magnetic field of similar energy. Various magnetic field geometries contribute to the magnetic helicity content, such as twisting, knotting, braiding and linking. Here I will present recent progress on relaxing magnetic field and their dynamics in plasmas, including work on twisted structures. The latter show a strong helicity creation and annihilation that can only be explained by taking into account the alignment of the magnetic field and the electric current density, which has implications on our understanding on the topology of large-scale magnetic structures.

1. Twisted Magnetic Knots and
Links and their Current Alignment
Simon Candelaresi, Celine Beck

2. Solar Magnetic Field
(Trace) (Trace)
2
(Prior and MacTaggart 2016) (Yamasaki et al. 2021)

3. Magnetic Helicity
Measure for the topology:
number of mutual linking
Conservation of magnetic helicity:
magnetic resistivity
Realizability condition:
Magnetic energy is bound from
below by magnetic helicity.
3

4. trefoil knottwisted field
Topologies of Magnetic Fields
Hopf link
Borromean rings IUCAA (8_18) knotmagnetic braid
4

5. Interlocked Flux Rings
● initial condition: flux tubes
● isothermal compressible gas
● viscous medium
● periodic boundaries
actual linking vs. magnetic helicity
(Del Sordo et al. 2010)
5

6. Interlocked Flux Rings
Magnetic helicity rather then actual
linking determines the field decay.
7

7. Magnetic Fields with a Twist
Helical fields can be made non-
helical by twisting the field lines.
8
Non-helical fields can be made
helical by twisting the field lines.
Simulated twisted knots and links in
MHD (Pencil Code).

8. Knots and Links
9
trefoil
5-foil IUCAA (8_18)
4-foil
triple rings
Borromean rings

9. Triple Rings
100 25 50 75 100 125 150
t
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
hA·Bi
linked,h=0,tw=0
linked,h=0,tw=1
linked,h=2,tw=0
linked,h=2,tw=1
linked,h=2,tw=−1 10
−1
10
0
10
1
10
2
t
10
−3
10
−2
10
−1
10
0
hB
2
i/hB
2
0i
t
−1
linked,h=0,tw=0
linked,h=0,tw=1
linked,h=2,tw=0
linked,h=2,tw=1
linked,h=2,tw=−1 0 5 10 15 20 25
t
0:015
0:010
0:005
0:000
0:005
0:010
0:015
h
J

B
i
linked;h=0;tw=0
linked;h=0;tw=1
linked;h=2;tw=0
linked;h=2;tw=1
linked;h=2;tw=1
Helicity restricts decay.
Small helicity production in
twisted non-helical field.

10. Knots
110 5 10 15 20 25
t
0:000
0:005
0:010
0:015
0:020
0:025
0:030
h
J

B
i
n=3;tw=0
n=3;tw=1
n=3;tw=2 0 25 50 75 100 125 150
t
0:0002
0:0001
0:0000
0:0001
0:0002
h
A

B
i
n=3;tw=0
n=3;tw=1
n=3;tw=2 10
1
10
0
10
1
10
2
t
10
2
10
1
10
0
h
B
2
i
=
h
B
20
i
t
1
n=3;tw=0
n=3;tw=1
n=3;tw=2
Significant helicity production.
Initial helicity is not a good
predictor on dynamics.

11. Low Resistivity Twisted Trefoil Knot0 5 10 15 20 25
t
0:000
0:005
0:010
0:015
0:020
0:025
0:030
h
J

B
i
tw=2;==1e3
tw=2;==5e4 0 25 50 75 100 125 150
t
0:0001
0:0000
0:0001
0:0002
h
A

B
i
tw=2;==1e3
tw=2;==5e4 10
1
10
0
10
1
10
2
t
10
2
10
1
10
0
h
B
2
i
=
h
B
20
i
t
1
tw=2;==1e3
tw=2;==5e4
Stronger alignment of J and B.
Lower resistivity partially
compensated by stronger
alignment.

12. Conclusions
[email protected]
● Helicity alone not a good indicator.
●Consider helicity production (current – magnetic field alignment)
●
● Increased turbulent effects at lower resistivity leads to stronger J-B
alignment and significant helicity production.
Candelaresi and Beck
Physics of Plasmas, 30, 8 (2023)
doi.org/10.1063/5.0148156
BlenDaViz:
github.com/SimonCan/BlenDaViz