Methods for Quantifying Magnetic Field Topology

For the dynamics of a magnetised plasma, the magnetic field line
topology is an important factor. Tangled, linked or knotted fields are not easily broken apart without any violent reconnection events, while topologically trivial fields freely and quickly relax to a force-free state. Magnetic helicity is a long established quantifier of the field line topology. It can be easily used for turbulent and non-turbulent systems. Its presence has been shown to restrict the energy conversion rate from magnetic to kinetic energy. The field-line magnetic helicity is related to the magnetic helicity and can be applied to cases where there is a dominant magnetic field, while being of less use in fully developed turbulence. This is particularly useful for fusion plasmas. The two quadratic helicities are somewhat of an underappreciated pair of helicity quantifiers. While they are ideal invariants, in practical cases they can vanish quickly during reconnection events. Lastly, I am going to show how we can use knot invariants to put numbers on the topology of the field lines.

1. Methods for Quantifying
Magnetic Field Topology
Simon Candelaresi

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

3. Coronal Magnetic Fields
NASA
(Thiffeault et al. 2006)
Field line tangling in solar magnetic fields.
3

4. Magnetic Helicity
Conservation of magnetic helicity:
magnetic resistivity
Realizability condition:
Magnetic energy is bound from
below by magnetic helicity.
4
link twist knot braid

5. AAA (trefoil knot) AABB (Borromean rings)
Magnetic Braid Configurations
5

6. Interlocked Flux Rings
Magnetic helicity rather then actual
linking determines the field decay.
6
(Del Sordo et al. 2010)

7. (Fabian et al. 2000)
galactic disc
Intergalactic Bubbles
hot, under-dense bubble
stratified medium
7
Bubbles’ age is several tens of
millions of years.
Bubbles rise buoyantly through
density difference.

8. Numerical Experiments
8
Full resistive magnetohydrodynamics simulations
with the Pencil Code.
stratified medium
hot, under-dense bubble

9. Initial Condition: Spheromak
9

10. Thermal Emission
10

11. Temperature Iso-Surfaces
11
hydro low helicity high helicity

12. Bubble Coherence
12
Helical magnetic fields can stabilise the bubbles.0.5 1.0 1.5 2.0 2.5 3.0 3.5
zmean
0.6
0.8
1.0
1.2
1.4
dmean
B=0
Hm≈H
0
m/2
Hm≈2H
0
m
Bex=0.8
Bex=0.2

13. Magnetic Fields with a Twist
Helical fields can be made non-
helical by twisting the field lines.
13
Non-helical fields can be made
helical by twisting the field lines.
Simulated twisted knots and links in
MHD (Pencil Code).
(Candelaresi & Beck 2023)

14. Knots and Links
14
trefoil
5-foil IUCAA (8_18)
4-foil
triple rings
Borromean rings

15. Triple Rings
150 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.

16. Knots
160 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.

17. 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.

18. Magnetic Braid
(Wilmot-Smith 2010)
Periodic braid topologically
equivalent to Borromean rings.
Separation into two twisted field
regions.
Conserved invariants like fixed
point index and field line helicity.
18
(Yeates et al. 2011)

19. Fixed Point Index
mapping:
(Yeates et al. 2011)
Trace magnetic field lines from to .
Color coding:
Compare with :
red
yellow
green
blue
fixed points:
19

20. Stability criteria
Woltjer (1958):
Taylor (1974):
constraint equilibrium
constant along field line
= total volume = volume along magnetic field line
Taylor state not reached due to
fixed point conservation.
20
(Yeates et al. 2011)

21. Quadratic Helicities
22
number of mutual linking
magnetic flux
volume of the flux tube
These are not invariant under general diffeomorphisms.
Only under volume preserving ones.

22. Quadratic Helicities
23
Invariant under homogeneously density changing diffeomorphisms.
(Akhmet’ev et al. 2017)

23. Field Line Helicity
24 -05=201=2yxy1=25=2
x
-
05=2
01=2
yxy
1=2
5=2

y
t=0 t=5 t=25 t=125




yxy
2=1
2=5


A
In ideal conditions field line helicity is only being transported.

24. Beyond Magnetic Helicity
25
(Dabrowski-Tumanski et al. (2020))
Describe knots, braids and links using knot polynomials:
Jones polynomials for the trefoil knot:
closure
Use Python package Topoly to
find polynomials.
(Yeates 2011)

25. Knots and Links as Braids
26
Periodic boundaries.

26. Knots and Links as Braids
4-foil knot
need braid representation of knots and links
27

27. MHD Simulations
28
● initial condition: braids
● isothermal compressible gas
● viscous medium
● periodic in z
Pencil Code

28. Link Spectrum
29
Pick a few random field lines and determine the link type.
Time dependent spectrum of links.
Repeat ca. 320,000 times for each snapshot.

29. Trefoil Knot
30
generally simple
generally complex

30. Borromean Rings
31

31. Helical 3 Rings
32

32. Non-Helical 3 Rings
33

33. Vortex Reconnection
34
Field lines untangle into two twisted vortex tubes.

34. Conclusions
[email protected]
● Magnetic helicity restricts the field’s dynamics.
● Twist can induce a significant helicity production.
● Quadratic helicities are ideal invariants, but require field line
tracing.
● Field line helicity ideal for fields with dominant field direction.
● Knot invariants to compute spectra of braids.
[email protected]