Solar magnetic fields basic concepts and magnetic topology
Solar magnetic fields: basic concepts and magnetic topology Anna (Ania) Malanushenko
Plasma Is matter… • Mass is conserved • Momentum is conserved • Energy is conserved e. g. , adiabatic gas law, …ionized and in magnetic field • Obeys Maxwell’s equations • Obeys Ohm’s law
Plasma Is matter… • Mass is conserved • Momentum is conserved • Energy is conserved e. g. , adiabatic gas law, …ionized and in magnetic field • Obeys Maxwell’s equations • Obeys Ohm’s law v<<c chargeneutral
Plasma Is matter… • Mass is conserved • Momentum is conserved • Energy is conserved e. g. , adiabatic gas law, …ionized and in magnetic field • Obeys Maxwell’s equations + 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 2 nd 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 => 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 So: magnetic pressure magnetic tension k
Low- solar corona magnetic pressure magnetic tension B k …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 Putting some math to it: consider a thin flux tube with FL=0 (magnetic pressure + magnetic tension=0) • Uniformly twisted: B =kr. Bz, so a field line is (z)=kz or (z)= z/L Tw= /2 – number of turns about the axis • Threshold: for Tw> Twcrit the tube is unstable Twcrit 1. 65 (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 Solution: via helicity Twgen?
Helicity • Has topological meaning! L 2, 2 L 1, 1 • In general: H=2 L 1 2, for untwisted tubes
Helicity • Has topological meaning! • In general: H=2 L 1 2, for untwisted tubes L=0 L=1 L=2
Helicity • In a twisted torus: Recall: H=2 L 1 2 for two untwisted tubes • Sum over all ``subtubes’’: H=Tw 2 • Not an invariant!
Helicity • H=2 L 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 domain • Relative helicity: H(B 1, B 2) =H(B 1)-H(B 2) – gauge-independent if at the boundary B 1 n=B 2 n and A 1 n=A 2 n. • Typically: B 2 is the potential field: B 2=0 => B 2= , 2 =0 (Berger & Field, 1984; Finn & Antonsen, 1985)
Helicity • ``Closing’’ field lines • For a domain: two potential fields, two answers (Longcope & Malanushenko, 2008) Proposal: Htot HTw=Twgen 2
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 (Hood & Priest, 1979) (Malanushenko et. al. , 2009) • Could now study kink instability on the Sun! • …not yet. This is only a half of the story. Recall: Need to know B! – tomorrow
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