1. Solar magnetic fields: basic concepts and magnetic topology
Anna (Ania) Malanushenko

2. Plasma Is matter… Mass is conserved Momentum is conserved
Energy is conserved
…ionized and in magnetic field
Obeys Maxwell’s equations
Obeys Ohm’s law
e.g., adiabatic gas law,

3. Plasma Is matter… Mass is conserved Momentum is conserved
Energy is conserved
…ionized and in magnetic field
Obeys Maxwell’s equations
Obeys Ohm’s law
e.g., adiabatic gas law,
v<<c
charge-neutral

4. Plasma Is matter… Mass is conserved Momentum is conserved
Energy is conserved
…ionized and in magnetic field
Obeys Maxwell’s equations
e.g., adiabatic gas law,
+ Ohm’s law
+ charge-neutral
+ non-relativistic

5. Plasma: “frozen-in” p. 1, “Ideal” induction equation (1) (2)
“Magnetic Reynolds number” 

6. Plasma: “frozen-in”
p. 2, particle trajectories vs. magnetic field lines
Are the same. To prove:

7. Plasma: “frozen-in”
morale:
For non-resistive plasma (Rm>>1), field lines and particle trajectories are the same. 1) 2)

8. Plasma:  Recall conservation of momentum (a.k.a. Newton’s 2nd law):
(1)
(2)
High density: pressure force dominates
Low density: Lorentz force dominates

9. Plasma: 
(Gary 2001)

10. Plasma:  Solar wind: low density, but field strength is even lower
Corona: low density,
magnetic field determines
“what plasma does”
Chromosphere: both plasma
and field are equally important
Photosphere & below: high
density, plasma motions
determine “what the field does”
(Gary 2001)

11. Concentrate on magnetic field lines
Solves
Unique (where B0 and finite) 
…and flux tubes…
…in low- solar corona:

12. Low- solar corona Most of the corona evolves slowly most of the time
Except for eruptions – catastrophic losses of equilibrium
Coronal field is anchored to the photosphere, where plasma flows “drive” the field
SOHO/EIT

13. Low- solar corona
hint: B k
So:
magnetic pressure
magnetic tension

14. Low- solar corona B k magnetic pressure magnetic tension
…so field lines “want” to be straight – and can’t go through each other – much like rubber bands!
…and – they are “anchored” in the dense photosphere

15. Low- solar corona
…“anchored” in the dense photosphere

16. Low- solar corona …“anchored” in the dense photosphere
If footpoints rotate, flux tubes become twisted
What would happen if one to twist a bundle or rubber bands too much?

17. Low- solar corona Tw=/2 Uniformly twisted: B=krBz,
Putting some math to it: consider a thin flux tube with FL=0
(magnetic pressure + magnetic tension=0)
Uniformly twisted: B=krBz,
so a field line is (z)=kz
or (z)= z/L
Threshold: for Tw> Twcrit the tube is unstable
Twcrit1.65
Tw=/2
– number of turns about the axis
(Hood & Priest, 1979)

18. Low- solar corona
What would happen if one to twist a bundle or field lines too much? That is, Tw>Twcrit?

19. Low- solar corona
What would happen if one to twist a bundle or field lines too much? That is, Tw>Twcrit?

20. Low- solar corona
The problem: what if it is not a thin tube – what is Tw?
Tw: turns about the axis
Twgen?
Solution: via helicity

21. Helicity Has topological meaning!
In general: H=2L12, for untwisted tubes

22. Helicity Has topological meaning!
In general: H=2L12, for untwisted tubes
L=0
L=1
L=2

23. Helicity In a twisted torus: Sum over all ``subtubes’’: H=Tw2
Not an invariant!
Recall: H=2L12 for two untwisted tubes

24. Helicity H=2L12 – L is invariant; HTw=Tw2 – Tw is not invariant!
In general: L=Tw+Wr; Htot=HTw+HWr
(Berger & Field, 1984; Moffatt & Ricca, 1992)

25. Helicity
axis writhing reduces Tw!

26. Helicity H makes sense for closed field lines,
otherwise it is gauge-dependent
What if ?
the change:
…what to do?
Increase the volume and
“close” field lines

27. Helicity H makes sense for closed field lines
Relative helicity: H(B1,B2) =H(B1)-H(B2) – gauge-independent if at the boundary B1n=B2n and A1n=A2n.
Typically: B2 is the potential field: B2=0
=> B2=, 2=0
domain
(Berger & Field, 1984; Finn & Antonsen, 1985)

28. Helicity Proposal: Htot HTw=Twgen2 HTw ``Closing’’ field lines
For a domain: two potential fields, two answers
(Longcope & Malanushenko, 2008)
Proposal:
Htot
HTw=Twgen2
HTw

29. Calculating Twgen
Test case: from Fan & Gibson, 2003

30. Calculating Twgen Test case: from Fan & Gibson, 2003
Identify the domain
(Malanushenko et. al., 2009)

31. Calculating Twgen Test case: from Fan & Gibson, 2003
Identify the domain
Compute the reference field in that domain
(Malanushenko et. al., 2009)

32. Calculating Twgen Test case: from Fan & Gibson, 2003
Identify the domain
Compute the reference field
Calculate Twgen=HTw/2
Compare Twgen with Tw for
a thin subportion
(Malanushenko et. al., 2009)

33. For thin subportion: is Twgen=Tw?

34. For the entire structure
Red:
Blue:

35. So… Thin flux tube => domain Tw => Twgen=HTw/2
Tw=Twgenfor a thin flux tube
Twgen works as predicted for a domain
Twcrit => 1.4 Twgen, crit 1.7
Could now study kink instability on the Sun!
…not yet. This is only a half of the story.
(Hood & Priest, 1979)
(Malanushenko et. al., 2009)
Recall:
Need to know B!
– tomorrow 