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 INTER INTERPLANET LANETARY SP SPACE E • Coronal Mass Ejection j Out Outer Boundar r Boundary • Flares, prominence Flares, prominence eruptions, coronal loops COR CORONA NA • Sunspots, magnetic field Phot Photospheric Boundar ospheric Boundary measurements • Generation and transport SOLAR INTERI SOLAR INTERIOR OR of magnetic field 2007-01-25 2

Erupting structures are often p g helical! 2/21/2009 2009 APCTP Plasma Winter School

2/21/2009 2009 APCTP Plasma Winter School

2/21/2009 2009 APCTP Plasma Winter School

2/21/2009 2009 APCTP Plasma Winter School

2/21/2009 2009 APCTP Plasma Winter School

Observational studies of helical structures t t 2/21/2009 2009 APCTP Plasma Winter School

Hα spirals around sunspots Hα spirals around sunspots 2/21/2009 2009 APCTP Plasma Winter School

X ray sigmoids X-ray sigmoids 2/21/2009 2009 APCTP Plasma Winter School

X ray sigmoids X-ray sigmoids 2/21/2009 2009 APCTP Plasma Winter School

Vector magnetic fields Vector magnetic fields α = ∇ × ( ( ) ) / B z B z z z 2/21/2009 2009 APCTP Plasma Winter School

Chirality of Filament Chirality of Filament dextral dextral sinistral 2/21/2009 2009 APCTP Plasma Winter School

dextral filaments in the N- hemisphere 2/21/2009 2009 APCTP Plasma Winter School

Hemispheric pattern of solar magnetic helicity • The S-hemisphere of the Sun has more Th S h i h f h S h of 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? 2/21/2009 2009 APCTP Plasma Winter School

What is magnetic helicity? What is magnetic helicity? 2/21/2009 2009 APCTP Plasma Winter School

2/21/2009 2009 APCTP Plasma Winter School

Helical structures and crossings Helical structures and crossings 2/21/2009 2009 APCTP Plasma Winter School

Sign of Crossing Sign of Crossing d r d r r = = = = = = ≡ ≡ − − 1 2 , , ; ; e e e e e e r r r r r r 1 2 3 2 1 ds ds r 1 2 2/21/2009 2009 APCTP Plasma Winter School

Gauss Linking number of Two Curves = ∫∫ 1 1 × ⋅ e e e L ds ds π 12 1 3 2 1 2 2 4 r 1 1 , 2 2 − 1 r r r r d d ∫∫ ∫∫ = = − ⋅ × × 2 2 1 1 ds ds ds ds π − 1 2 3 4 | | ds r r ds 2 1 1 , 2 2/21/2009 2009 APCTP Plasma Winter School

Linking number=? One turn consists of two crossings. One crossing corresponds to ½ linking number corresponds to ½ linking number = − 3 3 L L 12 2/21/2009 2009 APCTP Plasma Winter School

Linking number=? = − 2 2 L L 12 2/21/2009 2009 APCTP Plasma Winter School

Linking number=? 12 = 8 8 L L 2/21/2009 2009 APCTP Plasma Winter School

Linking number of tubes tube = L N N L 12 , 1 2 12 , curve − 1 1 r r r r d d d d ∫∫ = − ⋅ × 2 2 1 1 n n A ds A ds π − 2 1 1 1 2 2 3 4 | | ds r r ds 2 1 1 , , 2 − 1 r r ∫∫ ∫∫ = − ⋅ × 3 3 2 1 n n r r d d π π r − 2 1 1 2 3 4 4 | | | | r r r 2 1 2/21/2009 2009 APCTP Plasma Winter School

Magnetic Helicity as a Linking Number of Magnetic Field Lines Magnetic Helicity as a Linking Number of Magnetic Field Lines ⇒ Φ ⇒ Φ ; N N 1 1 2 2 ⇒ ⇒ ; n B n B 1 1 1 1 = Φ Φ H L 1 2 12 , curve − 1 r r ∫∫ = − ⋅ × 3 3 2 1 B B d r d r π − 2 1 1 2 3 4 | | | | r r 2 2 1 1 N N = ∑∑ Φ Φ ( in more general ) L i j ij = = 1 1 1 1 i i j j Magnetic helicity = sum of linking numbers over all pairs g y f g p of flux tubes 2/21/2009 2009 APCTP Plasma Winter School

Magnetic helicity = sum of linking numbers Magnetic helicity = sum of linking numbers over all pairs of field lines − 1 r r ∫∫ = − ⋅ × 3 3 2 1 H B B d r d r π π − 2 1 1 2 3 4 4 | | | | r r r r 2 2 1 1 N N = ∑∑ ∑∑ Φ Φ L i j j ij j = = 1 1 i j = Φ Φ + + Φ Φ + + Φ Φ + + Φ Φ Φ Φ + + Φ Φ Φ Φ + + Φ Φ Φ Φ 2 2 2 2 2 2 2 2 2 H H L L L L L L L L L L L L 11 11 1 1 22 22 2 2 33 33 3 3 12 12 1 1 2 2 13 13 1 1 3 3 23 23 2 2 3 3 self helicity mutual helicity 2/21/2009 2009 APCTP Plasma Winter School

M t Mutual helicity = linkage of all pairs of field lines l h li it li k f ll i f fi ld li in two different flux tubes = Φ Φ + Φ Φ + Φ Φ 2 2 2 H m L L L 12 1 2 13 1 3 23 2 3 = = − = − Φ = Φ = Φ = Φ 3 , 1 , 1 L L L 12 13 23 1 2 3 = Φ 2 H m 2/21/2009 2009 APCTP Plasma Winter School

S lf h li it Self helicity = linkage of field lines in a single flux tube li k f fi ld li i i l fl t b = Φ Φ H s L 11 1 1 = + Φ 2 ( ( ) ) T W 1 1 = Number Number of of twist tur twist tur ns ns T T of field lines around the tube axis W = = W Number Number of of writhe writhe turns turns of the tube axis around itself > < 0 , 0 T W 2/21/2009 2009 APCTP Plasma Winter School

Magnetic Helicity: Canonical Definition Magnetic Helicity: Canonical Definition − 1 r r ∫∫ = − ⋅ × 3 3 2 1 H B B d r d r π π − 2 1 1 2 3 4 4 | | | | r r r r 2 2 1 1 − 1 r r ∫ ∫ ∫ ∫ = ⋅ − × × 3 3 2 1 [ [ ] ] B B B B d d r r d d r r π − 2 2 1 1 1 1 2 2 3 4 | | r r 2 1 ∫ ∫ = ⋅ 3 B B A A r r d d 2 2 2 ∫ ∫ = = ⋅ ⋅ 3 A A B B d d r r 2/21/2009 2009 APCTP Plasma Winter School

∫ = ⋅ 3 A B r H d = ∇ × B A c = π ∇ ∇ × J J B B 4 2/21/2009 2009 APCTP Plasma Winter School

Magnetic Helicity: gauge invariance in a closed volume Magnetic Helicity: gauge-invariance in a closed volume − 1 r r ∫ ∫ = − × 3 3 1 1 ( ( C Coulomb l b gauge ) ) A A B B r d d π − 1 1 3 4 | | r r 1 ′ ′ = + ∇ ∇ φ φ A A A A ∫ ∫ ∫ ′ ⋅ = ⋅ + ∇ φ ⋅ A B dV A B dV B dV ∫ ∫ ∫ ∫ = ⋅ + ∇ ∇ ⋅ φ φ ( ( ) ) A A B B dV dV B B dV dV ∫ ∫ ∫ ∫ = = ⋅ + + φ φ ⋅ ( ( ) ) A A B B B B n n dV dV d d S S ∫ ∫ = ⋅ = if 0 A B dV B n 2/21/2009 2009 APCTP Plasma Winter School

How is magnetic helicity defined in How is magnetic helicity defined in an open volume? ≠ What if 0 on the boundary ? B n 2/21/2009 2009 APCTP Plasma Winter School

Introduce potential field B p satisfying ∇ ∇ B × × = 0 0 inside inside the the open volum open volum e e B p = on the boundary B B pn n ≡ − = 0 ( closed field) B B B , B c p cn ∫ = ⋅ 3 H A B d r ∫ ∫ = + ⋅ + 3 3 ( ( ) ) ( ( ) ) A A A A B B B B d d r c P c P ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ = = ⋅ + + ⋅ + + ⋅ + + ⋅ 3 3 3 3 A A B B d d r r A A B B d d r r A A B B d d r r A A B B d d r r c c p c c p p p ! ! ? ? 2/21/2009 2009 APCTP Plasma Winter School

The linking of the closed field and the potential field should be equal to that of the potential field and closed field ∫ ∫ ∫ ∫ ⋅ ≡ ⋅ 3 3 A B d r A B d r c p p c The helicity of potential field is defined to be zero The helicity of potential field is defined to be zero. 3 ≡ ∫ ⋅ 0 A B r d p p Relative helicity of an open magnetic field ∫ ∫ ∫ ∫ = ⋅ + ⋅ 3 3 2 A B r A B r H R d d c c p c ∫ ∫ = + ⋅ 3 3 ( ( 2 2 ) ) A A A A B B r d d c p c ∫ ∫ = = + + ⋅ ⋅ − − 3 ( ( ) ) ( ( ) ) A A A A B B B B d d r r p P 2/21/2009 2009 APCTP Plasma Winter School

Concept of relative helicity in an open volume = ∫ + ⋅ − = − 3 ( ) ( ) H A A B B d r H H R p P total ref 2/21/2009 2009 APCTP Plasma Winter School

Conservation of magnetic helicity Conservation of magnetic helicity 2/21/2009 2009 APCTP Plasma Winter School

Magnetic helicity is conserved in a g y closed volume dH dH ∫ ∫ = − ⋅ σ 2 / J B d V dt Δ Δ H t ≤ ≈ × − 5 3 10 ( for a solar flare ) H H t t d d 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! 2/21/2009 2009 APCTP Plasma Winter School