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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 …
Canfield-fest
… advisor extraordinaire
8/11/10
Canfield-fest
Relative helicity
These curves are linked … … are these?8/11/10
Canfield-fest
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
∫
8/11/10
Canfield-fest
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?
8/11/10
yes
Canfield-fest
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
8/11/10
Canfield-fest
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
8/11/10
€
cE × ˆ z = − v × B( ) × ˆ z = vBz − Bvz = v⊥−vzBz
B⊥
⎛
⎝ ⎜
⎞
⎠ ⎟Bz = UBzIdeal induction:
observed motion observed field
reference field
Canfield-fest
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
8/11/10
Canfield-fest
Measured Velocity(Chae 2001)
• Highly variable helicity flux• Up to +3 X 1040 Mx2/day = F2 10-2/day
8/11/10
Canfield-fest8/11/10
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
Canfield-fest8/11/10
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
Canfield-fest8/11/10
Measuring SpinωP01 = −
2π
ΦP012
dH
dtP01
Brown et al. 2003
dH
dt P01
all contributions to dH/dt
P01
Longcope et al. 2007
Canfield-fest8/11/10
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
Canfield-fest8/11/10
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
Canfield-fest8/11/10
Canfield-fest
Can this work?
8/11/10
Malanushenko et al. 2009
17 Different Low & Lou fields
Canfield-fest
Spin +braiding make
twist
8/11/10
braiding
spin+braid
Photospheric flux
Coronal twist
Malanushenko et al. 2010
Malanushenko et al. 2010
Canfield-fest
Twist over time
8/11/10
AR9004
AR9002
But what happens to free energy?
aL/
2
aL/
2
Reconnected flux
Canfield-fest
Twist over time
8/11/10
AR9004
AR9002
aL/
2
aL/
2
EUV loops
Canfield-fest
Helicity in the AR
8/11/10
EG
How twisted is the field within the grey sub-volume?
Ignoring these
e.g. what is the helicity is in a flux rope?
Canfield-fest
Helicity in the AR
8/11/10
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
8/11/10 Canfield-fest
D1
D2
Canfield-fest
Hierarchy of energies
8/11/10
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)
Canfield-fest8/11/10
Malanushenko et al. 2009
Additive helicity drops during instability
Simulations: Fan & Gibson 2003
Canfield-fest
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
8/11/10