Magnetic Helicity in the Sun: Magnetic Helicity in the Sun: The - - PowerPoint PPT Presentation

Magnetic Helicity in the Sun: Magnetic Helicity in the Sun:

The physical concept

Jongchul Chae Jongchul Chae

Seoul National University

2/21/2009 2009 APCTP Plasma Winter School

Solar Magnetic Connection Solar Magnetic Connection

• Solar wind, IMC
• Coronal Mass Ejection

INTER INTERPLANET LANETARY SP SPACE E j

• Flares, prominence

Out Outer Boundar r Boundary Flares, prominence eruptions, coronal loops COR CORONA NA

• Sunspots, magnetic field

measurements Phot Photospheric Boundar

• spheric Boundary
• Generation and transport
• f magnetic field

SOLAR INTERI SOLAR INTERIOR OR

2007-01-25

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Erupting structures are often p g helical!

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Observational studies of helical t t structures

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Hα spirals around sunspots Hα spirals around sunspots

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X ray sigmoids X-ray sigmoids

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X ray sigmoids X-ray sigmoids

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Vector magnetic fields Vector magnetic fields

z z B

/ ) ( B × ∇ = α

z z

) (

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Chirality of Filament Chirality of Filament

dextral sinistral dextral

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dextral filaments in the N- hemisphere

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Hemispheric pattern of solar magnetic helicity

Th S h i h f h S h

• The S-hemisphere of the Sun has more
• f right (+) helical structures such as

– Counterclockwise moving-out spirals around sunspots – S-shaped sigmoids – Photospheric vector magnetic fields with p g positive force-free α – Sinistral filaments

• Why?

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What is magnetic helicity? What is magnetic helicity?

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Helical structures and crossings Helical structures and crossings

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Sign of Crossing Sign of Crossing

2 1

; r r r r e r e r e − ≡ = = = d d

1 2 3 2 2 1 1

; , , r r r e e e − ≡ = = = r ds ds

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Gauss Linking number of Two Curves

1

2 2 1 2 3 2 1 1 12

4 1 r ds ds L e e e ⋅ × = ∫∫ π

1 1 2 2 2 , 1

1 ds ds d d r r r r × − ⋅ − =

∫∫

2 1 3 1 2 2 , 1

| | 4 ds ds ds ds r r × − =

∫∫

π

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Linking number=? One turn consists of two crossings. One crossing corresponds to ½ linking number

3 L

corresponds to ½ linking number

3

12

− = L

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Linking number=?

2 L 2

12

− = L

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Linking number=?

8 L 8

12 =

L

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Linking number of tubes

, 12 2 1 , 12

1 d d L N N L

curve tube = 2 2 1 1 1 1 3 1 2 1 2 2 , 1 2 2

| | 4 1 r r r r r r ds A ds A ds d n ds d n × − − ⋅ − =

∫∫

π

2 3 1 3 1 3 1 2 2 ,

| | 4 1 r r n r r r r n d d × − ⋅ − =

∫∫

π

1 2

| | 4 r r −

∫∫

π

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Magnetic Helicity as a Linking Number of Magnetic Field Lines Magnetic Helicity as a Linking Number of Magnetic Field Lines

1 1 1 1 2 2 1 1

; ; B n B n ⇒ ⇒ Φ ⇒ Φ ⇒ N N

, 12 2 1 curve

L H Φ Φ = | | 4 1

2 3 1 3 1 3 1 2 1 2 2

d d × − − ⋅ − =

∫∫

r r B r r r r B π ) general more in ( | |

1 1 1 2 ij j N i N j i

L Φ Φ =∑∑

1 1 i j = =

Magnetic helicity = sum of linking numbers over all pairs g y f g p

• f flux tubes

2/21/2009 2009 APCTP Plasma Winter School

Magnetic helicity = sum of linking numbers Magnetic helicity = sum of linking numbers

• ver all pairs of field lines

d d H × − − ⋅ − =

∫∫

2 3 1 3 1 3 1 2 1 2 2

| | 4 1 r r B r r r r B π

ij j N N i

L Φ Φ =∑∑

1 2

| | 4 r r π

j j i j

∑∑

= = 1 1

2 2 2

3 2 23 3 1 13 2 1 12 2 3 33 2 2 22 2 1 11

Φ Φ + Φ Φ + Φ Φ + Φ + Φ + Φ = L L L L L L H helicity mutual helicity self 2 2 2

3 2 23 3 1 13 2 1 12 3 33 2 22 1 11

Φ Φ + Φ Φ + Φ Φ + Φ + Φ + Φ L L L L L L H

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M t l h li it li k f ll i f fi ld li Mutual helicity = linkage of all pairs of field lines in two different flux tubes

3 2 23 3 1 13 2 1 12

2 2 2 Φ Φ + Φ Φ + Φ Φ = L L L H m

Φ = Φ = Φ = Φ

3 2 1

1 , 1 , 3

23 13 12

− = − = = L L L

2

Φ =

m

H

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S lf h li it li k f fi ld li i i l fl t b Self helicity = linkage of field lines in a single flux tube ) (

2 1 1 1 11

Φ + = Φ Φ = W T L H s ns twist tur

• f

Number ) (

1

= T turns writhe

• f

Number W axis tube the around lines field

• f

ns twist tur

• f

Number = T itself around axis tube the

• f

turns writhe

• f

Number W =

, < > W T

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Magnetic Helicity: Canonical Definition Magnetic Helicity: Canonical Definition

∫∫

× − − ⋅ − = r r B r r r r B

2 3 1 3 1 3 1 2 1 2 2

| | 4 1 d d H π

∫ ∫

× − − ⋅ = r r B r r B r r

2 3 1 3 1 1 2 2 1 2

] 1 [ | | 4 d d π

∫ ∫ ∫

× − r A B r r B r r B

3 2 1 1 3 1 2 2

] | | 4 [ d d d π

∫ ∫

⋅ = ⋅ = r B A r A B

3 2 2 2

d d

∫

⋅ = r B A d

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∫

⋅ = r B A

3

d H

A B × ∇ =

B J ∇ c B J × ∇ = π 4

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Magnetic Helicity: gauge invariance in a closed volume Magnetic Helicity: gauge-invariance in a closed volume

) C l b ( 1

3 1

B r r A d −

∫

) gauge Coulomb ( | | 4

1 3 1 3 1 1

r B r r A d × − − =

∫

π

φ ∇ ′ A A φ ∇ + = ′ A A

) ( ∇ ⋅ ∇ + ⋅ = ⋅ ′

∫ ∫ ∫ ∫ ∫

dV dV dV dV dV B B A B B A B A φ φ ) ( ) ( ⋅ + ⋅ = ⋅ ∇ + ⋅ =

∫ ∫ ∫ ∫

S d dV dV dV n B B A B B A φ φ if ) ( = ⋅ = + =

∫ ∫ ∫

n

B dV S d dV B A n B B A φ

∫

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How is magnetic helicity defined in How is magnetic helicity defined in an open volume?

? boundary

• n the

if What ≠

n

B

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Introduce potential field Bp satisfying e

• pen volum

the inside × ∇ B boundary

• n the

e

• pen volum

the inside

n pn

B B = = × ∇

p

B field) closed ( = − ≡

cn c

B , B B B

p

) ( ) (

3 3

∫ ∫

⋅ = B B A A r B A d d H ) ( ) (

3 3 3 3 3

∫ ∫ ∫ ∫ ∫

⋅ + ⋅ + ⋅ + ⋅ = + ⋅ + = r B A r B A r B A r B A r B B A A

P c P c

d d d d d ? ? ! !

∫ ∫ ∫ ∫

+ + + = r B A r B A r B A r B A

p p p c c p c c

d d d d

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∫ ∫

The linking of the closed field and the potential field should be equal to that of the potential field and closed field

∫ ∫

⋅ ≡ ⋅ r B A r B A

c p p c 3 3

d d

The helicity of potential field is defined to be zero

3 ≡

⋅

∫

r B A

p p

d

The helicity of potential field is defined to be zero. Relative helicity of an open magnetic field

∫ ∫ ∫ ∫ ∫

⋅ + ⋅ = B A A r B A r B A

c p c c 3 3 3

) 2 ( 2 d d d HR

∫ ∫

− ⋅ + = ⋅ + = r B B A A r B A A

c p c 3 3

) ( ) ( ) 2 ( d d

∫

− ⋅ + = r B B A A

P p

) ( ) ( d

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Concept of relative helicity in an open volume

ref total R

H H d H − = − ⋅ + =∫ r B B A A

P p 3

) ( ) (

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Conservation of magnetic helicity Conservation of magnetic helicity

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Magnetic helicity is conserved in a g y closed volume

∫

dH

∫

⋅ − = V d dt dH σ / 2 B J

) flare solar a for ( 10 3

5 −

× ≈ Δ ≤ Δ

d

t t H H

d

t H

Magnetic helicity is well conserved even while magnetic energy Magnetic helicity is well-conserved even while magnetic energy is dissipated through, for example, magnetic reconnection!

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But in the case of open magnetic field: Magnetic helicity may be transported across the boundary when magnetic field lines move on the boundary

dS B dS v dH

n p t n p t m

) ( 2 ) ( 2 A v A B ⋅ − ⋅ =

∫ ∫

field line passage term shearing flow term

dt

n p t n p t

) ( ) (

∫ ∫

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Magnetic Helicity Injection by Vertical Motion of Flux Rope (a) Horizontal flux rope emergence (b) Vertical flux rope emergence (b) Vertical flux rope emergence

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Magnetic Helicity Injection by Surface Rotational Motion

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Helicity change by horizontal motion of field line footpoints Helicity change by horizontal motion of field line footpoints

dH ⎞ ⎛ ⋅ − =

∫

dS B dt dH

n p t

) ( 2 A v ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Φ Φ + Φ − =

∑ ∑∑

= = = N i N i N j j i ij i i 1 1 1 2

2 1 θ ω π

• 1

2

∑ ∑∑

⎟ ⎟ ⎞ ⎜ ⎜ ⎛ Φ Φ Θ + Φ Ω =

N N N

H 2

2 1 1 1

∑ ∑∑

= = =

Θ Ω ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ Φ Φ Θ + Φ Ω =

N N N i i j j i ij i i

H π h li i l h li i lf 2

1 1 2

∑ ∑∑

= = >

Φ Φ Θ + Φ Ω =

i i i j j i ij i i

π π helicity mutual helicity self

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Braiding and mutual helicity Braiding and mutual helicity

2 1 2 1 2 1

, 2 2 Φ Φ Θ = Φ Φ = Φ Φ = π π π

b a

H H π π

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Braiding and mutual helicity Braiding and mutual helicity

2 1 2 1 2 1 2 1

, Φ Φ Θ + Θ = Φ Φ Θ − Θ =

b

H H

2 1 2 1

, Φ Φ Φ Φ π π

b a

H H

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Conservation of Relative Magnetic Helicity Conservation of Relative Magnetic Helicity

INTER INTERPLANET LANETARY SP SPACE E Out Outer Boundar r Boundary COR CORONA NA

cme inj corona

H H H Δ − Δ = Δ

Phot Photospheric Boundar

• spheric Boundary

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