The dominance of the magnetic helicity as the determining topological invariant in magnetohydrodynamics is revisited for helical and non-helical field configurations. For fields with positive z-components we determine the fixed points of the field line mapping. The sum of the Poincare indices at those fixed points, the fixed pints index, gives rise to further restrictions, not captured by the magnetic helicity alone. Various braided field configurations, which correspond to helical and non-helical knots, are simulated in fully resistive magnetohydrodynamics. We observe an evolution which confirms the importance of the fixed point index as additional restricting quantity. Moreover, a topological flux function, the field line helicity, is computed at the fixed points and monitored during turbulent relaxation. Its rate of change gives a proxy for the magnetic reconnection rate.

1. Topological constraints in
magnetic field relaxation
Simon Candelaresi

2. Topologies of Magnetic Fields
Hopf link
twisted field trefoil knot
Borromean rings magnetic braid IUCAA knot
See Ilan Roth's poster for more knots and knot theory.
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3. Twisted Magnetic Fields
Twisted fields are more
likely to erupt (Canfield et al. 1999).
Tuesday's talk by Zhang Mei
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4. 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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5. Interlocked Flux Rings
actual linking vs. magnetic helicity
● initial condition: flux tubes
● isothermal compressible gas
● viscous medium
● periodic boundaries
(Del Sordo et al. 2010)
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6. Interlocked Flux Rings
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7. Interlocked Flux Rings
Magnetic helicity rather then actual
linking determines the field decay.
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8. IUCAA Knot and Borromean Rings
● Is magnetic helicity sufficient?
● Higher order invariants?
Borromean rings IUCAA knot
(Candelaresi and Brandenburg 2011)
IUCAA = The Inter-University Centre for Astronomy and Astrophysics, Pune, India
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9. Reconnection Characteristics
3 rings Twisted ring + 2 twisted rings
interlocked rings
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10. Magnetic Energy Decay
Higher order invariants?
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11. Fixed Point Index
Trace magnetic field lines from to .
mapping:
fixed points:
Color coding:
Compare with :
red
yellow
green
blue
(Yeates et al. 2011)
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12. Fixed Point Index
Sign of fixed point :
Fixed point index: conserved for
Taylor state is not reached
→ is additional constraint
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13. Summary and Outlook
● Braiding increases stability through the realizability condition.
● Turbulent magnetic field decay is restricted by magnetic helicity.
● Fixed point index as additional constraint in relaxation.
● Use fixed point index for knots and links (Yeates A.).
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14. References
Canfield et al. 1999
Canfield, R. C., Hudson, H. S., and McKenzie, D. E.
Sigmoidal morphology and eruptive solar activity.
Geophys. Res. Lett., 26:627, 1999
Del Sordo et al. 2010
Fabio Del Sordo, Simon Candelaresi, and Axel Brandenburg.
Magnetic-field decay of three interlocked flux rings with zero linking number.
Phys. Rev. E, 81:036401, Mar 2010.
Candelaresi and Brandenburg 2011
Simon Candelaresi, and Axel Brandenburg.
Decay of helical and non-helical magnetic knots.
Phys. Rev. E, 84:016406, 2011
Yeates et al. 2011
Yeates, A. R., Hornig, G. and Wilmot-Smith, A. L.
Topological Constraints on Magnetic Relaxation.
Phys. Rev. Lett. 105, 085002, 2010
www.nordita.org/~iomsn 14

15. Equilibrium States
Ideal MHD:
Induction equation:
Task: Find the state with minimal energy.
Constraint: magnetic helicity conservation
constraint equilibrium
Woltjer (1958):
Taylor (1974):
constant along field line
total volume volume along magnetic field line
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16. Magnetic energy decay
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17. Simulations
● mesh point
● Isothermal compressible gas
● Viscous medium
● Periodic boundaries
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18. N-foil Knots
3-foil 4-foil 5-foil 6-foil 7-foil
*
cinquefoil knot
* from Wikipedia, author: Jim.belk
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19. N-foil Knots
Magnetic helicity is approximately conserved.
Self-linking is transformed into twisting after reconnection.
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20. N-foil Knots
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21. N-foil Knots
Realizability condition more important for high .
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22. Linking Number
Number of crossings
increases like
Sign of the crossings
for the 4-foil knot
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23. Helicity vs. Energy
Knot is more strongly
packed with increasing .
Magnetic energy is closer to
its lower limit for high .
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24. Braid Representation
need braid representation of knots and links
4-foil knot
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25. Magnetic Braid Configurations
AAA (trefoil knot) AABB (Borromean rings)
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26. Field Line Tracing
Generalized flux function: Reconnection rate:
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27. Magnetic Reconnection Rate
Classic: look for local maxima of
Partition fluxes 2D:
(Yeates, Hornig 2011b)
Reconnection rate =
magnetic flux through
boundaries (spearatrices):
2D Magnetic field.
Thick lines: separatrices.
(Yeates, Hornig 2011b)
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28. Magnetic Reconnection Rate
Partition reconnection rate 3D:
Yeates, Hornig 2011b
Generalized flux function (curly A):
Fixed points:
Reconnection rate:
invariant in ideal MHD
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