Twist and Current Alignment within non-trivial Magnetic Field Topologies Simon Candelaresi, Celine Beck Twisted magnetic field topologies such as knots and links are studied using numerical simulations. Magnetic helicity provides a measure of magnetic field topology and is an important conserved quantity. Higher order invariants beyond helicity are also considered. A variety of knotted and linked field configurations are generated and their stability and decay properties are examined in relation to magnetic helicity and other topological constraints.

1. Twist and Current Alignment
within non-trivial
Magnetic Field Topologies
Simon Candelaresi, Celine Beck

2. Twisted Magnetic Fields
Twisted fields are more likely to
erupt (Canfield et al. 1999).
Twist increases the stability of
magnetic fields in tokamaks.
2

3. Solar Magnetic Field
(Trace) (Trace)
3
Twisted flux tubes may rise to the
corona. (Prior and MacTaggart 2016).

4. Coronal Magnetic Fields
NASA
(Thiffeault et al. 2006)
Field line tangling in solar magnetic fields.
Study the tangling of solar magnetic field lines.
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5. trefoil knot
twisted field
Topologies of Magnetic Fields
Hopf link
Borromean rings IUCAA knot
magnetic braid
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6. 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.
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7. (Fabian et al. 2000)
galactic disc
Intergalactic Bubbles
hot, under-dense bubble
stratified medium
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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 PencilCode.
stratified medium
hot, under-dense bubble

9. Initial Condition: Spheromak
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10. Thermal Emission
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Magnetic helicity stabilises the bubbles.

11. Temperature Iso-Surfaces
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hydro low helicity high helicity

12. Interlocked Flux Rings
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initial condition: flux tubes
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isothermal compressible gas
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viscous medium
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periodic boundaries
actual linking vs. magnetic helicity
(Del Sordo et al. 2010)
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13. Interlocked Flux Rings
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14. Interlocked Flux Rings
Magnetic helicity rather then actual
linking determines the field decay.
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15. IUCAA knot
Borromean rings
IUCAA Knot and Borromean Rings
IUCAA = The Inter-University Centre for Astronomy and Astrophysics, Pune, India
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Is magnetic helicity sufficient?
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Higher order invariants?
(Candelaresi and Brandenburg 2011)
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16. Magnetic Energy Decay
Higher order invariants?
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17. 3 rings Twisted ring +
interlocked rings
2 twisted rings
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Borromean Rings

18. Magnetic Braid
(Wilmot-Smith 2010)
(Yeates 2011)
Periodic braid topologically
equivalent to Borromean rings.
Separation into two twisted field
regions.
Conserved invariants like fixed
point index and field line helicity.
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19. Magnetic Fields with a Twist
Helical fields can be made non-
helical by twisting the field lines.
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Non-elical fields can be made
helical by twisting the field lines.
Simulated twisted knots and links in
MHD (Pencil Code).

20. Knots and Links
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trefoil
5-foil IUCAA
4-foil
triple rings
Borromean rings

21. Twisted Trefoil Knot
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22. Knots
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23. Triple Rings (non-helical)
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24. Triple Rings (helical)
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25. Triple Rings
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26. Borromean Rings
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27. Borromean Rings
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28. IUCAA Knots
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29. IUCAA Knot
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30. Saffman Invariant
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Non-zero for non-helical turbulence.
(Hosking 2021)
Gauge invariant.
Conserved quantity.
Current results only for isotropic
homogeneous turbulence

31. Conclusions
simon.candelaresi@gmail.com
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Magnetic helicity as constraint on plasma dynamics.
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Magnetic helicity leads to stability at small magnetic energy.
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Non-helical field exhibit intermediate energy decay (sometimes).
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Helicity alone not a good indicator. Consider helicity production.
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Saffman invariant for non-helical fields.
simon.candelaresi@gmail.com