Canfield-fest

Helicity: where it comes from and what it tells us

Dana Longcope, MSUAnna Malanushenko, MSU/LMSAL

8/11/10

Thanks: Graham Barnes, CoRA B. Ravindra, IIA and …

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… advisor extraordinaire

8/11/10

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Relative helicity

These curves are linked … … are these?8/11/10

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Relative helicity

Are these coronal field lines twisted about each other?

Are they linked with one another?

What about gauge-dependence?€

H = A⋅B d3xz>0

∫

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Relative helicity

Need reference field B0(x) for comparison

€

HR (BB0) = (A + A 0)⋅ (B − B0) d3xz>0

∫

Relative helicityBerger & Field 1984Finn & Antonsen 1985

Does HR dependon choice of B0?

So what can it tell us?

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yes

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Potential field reference

€

∇×B0 = 0One choice of reference: potential magnetic field

Advantages:• Potential field is unique (from normal)• Potential field has minimum magnetic energy: HR≠0 => free magnetic energy• minimizing magnetic energy subj. to HR:

• can be computed

€

∇×B = α B constant from Lagrange multiplier imposing constraint

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Helicity flux: observableUsing LCT velocities (Demoulin & Berger 2003)

• Track features in LOS m-gram:

• Compare to Faraday’s Law

€

dHR

dt= −2 (cE × ˆ n )⋅A 0 ds

∂V

∫

€

∂Bz∂t

= −∇⋅ UBz( )

zE ˆ

ct

Bz

(Berger & Field 1984)

V

zR dsB

dt

dH02 AUHelicity Flux:

feature velocity:measured

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€

cE × ˆ z = − v × B( ) × ˆ z = vBz − Bvz = v⊥−vzBz

B⊥

⎛

⎝ ⎜

⎞

⎠ ⎟Bz = UBzIdeal induction:

observed motion observed field

reference field

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Measured Velocity(Chae 2001)

AR8011• F = 1.7 X 1021 Mx2

• S6o latitude

• Bz measured: LOS mg• U measured: LCT of Bz

• A0 extrapolated

V

zR dsB

dt

dH02 AU

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Measured Velocity(Chae 2001)

• Highly variable helicity flux• Up to +3 X 1040 Mx2/day = F2 10-2/day

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Measuring SpindH

dt= −

Φa2

2πωa

⎛

⎝⎜⎞

⎠⎟a∑ −

ΦaΦb

2πdθ ab

dta,b∑

spin braiding

1

2

2

12

dH

dt=−2 v⋅A 0( )Bz dxdy

z=0

dH/dt < 0 for clockwise spin (w<0)

dH/dt < 0 for clockwise motion(if FaFb < 0 ) Internal motion: spin

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Source of braiding

Braiding: deformation of flux tube axis

Source of spin

Natural time scaleFor twist injection:

(Longcope & W

elsch 2000)

propagation of current (twist) within flux tube:torsional Alfven wave Twist moves to balance torques

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Measuring SpinωP01 = −

2π

ΦP012

dH

dtP01

Brown et al. 2003

dH

dt P01

all contributions to dH/dt

P01

Longcope et al. 2007

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separation d(t)

Longcope & Welsch model

Hbr

HspH

• Helicity dominated by braiding• Northern AR: H > 0 Hsp < 0

fit:

q = -0.67 x 10-8 m-1

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separation d(t)Longcope & Welsch model

Hbr

Hsp

H

• Helicity dominated by spin• Southern AR: H > 0 Hsp > 0

fit:

q = +2.3 x 10-8 m-1

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Can this work?

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Malanushenko et al. 2009

17 Different Low & Lou fields

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Spin +braiding make

twist

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braiding

spin+braid

Photospheric flux

Coronal twist

Malanushenko et al. 2010

Malanushenko et al. 2010

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Twist over time

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AR9004

AR9002

But what happens to free energy?

aL/

2

aL/

2

Reconnected flux

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Twist over time

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AR9004

AR9002

aL/

2

aL/

2

EUV loops

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Helicity in the AR

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EG

How twisted is the field within the grey sub-volume?

Ignoring these

e.g. what is the helicity is in a flux rope?

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Helicity in the AR

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Which reference field to use?

€

∇×B0 = 0

€

B0 = 0

unconfined additive

JJ

Longcope & M

alanushenko 2008

Advantages of the additive self-helicity

Constraint on self-helicity of Di

Lagrange multiplier

Mag. energy

Constrained minimum: piece-wise constant-a

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D1

D2

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Hierarchy of energies

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NLFFF

Piecewiseconst.-a

Domains:• flux• shape• self-helicity

Every field line:• footpoint connectivity

Piecewisepotential

Domains:• flux• shape

MCC Domains:• flux

Field: Constraint:*

potential Ø

≥

≥

≥

≥

*other than p-spheric field BC

B(x)

Ba(x)

P(x)

BFCE(x)

B0(x)

€

HR (BPi)

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Malanushenko et al. 2009

Additive helicity drops during instability

Simulations: Fan & Gibson 2003

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Summary

• Relative helicity depends on reference field

• Choosing pot’l field: helicity is proxy for free magnetic energy

• Can observe injection of rel. helicity into corona: injection of free energy

• Additive rel. helicity leads to piece-wise constant-a field: bound on energy

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