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

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,

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

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

Plasma: “frozen-in” p. 1, “Ideal” induction equation (1) (2) “Magnetic Reynolds number” 108 - 1012

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

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

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

Plasma:  (Gary 2001)

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)

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

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

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

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

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

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?

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)

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

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

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

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

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

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

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)

Helicity axis writhing reduces Tw!

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

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)

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

Calculating Twgen Test case: from Fan & Gibson, 2003

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

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

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)

For thin subportion: is Twgen=Tw?

For the entire structure Red: Blue:

So… Thin flux tube => domain Tw => Twgen=HTw/2 Tw=Twgenfor a thin flux tube Twgen works as predicted for a domain Twcrit1.65 => 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 