Magnetic HelicityMagnetic Helicity• Magnetic helicity measures intrinsic properties of flux tubes (shear, linking, kinking twisting & handedness)

• It is defined by a volume integral where A is the vector potential

and B is the magnetic field

• Magnetic helicity is a conserved quantity even in resistive MHD (Berger, 1984)

• Methods to measure magnetic helicity on the Sun are only 3-4 years old. We are one of the pioneer groups, who can do it!

∫ ⋅=V

dVH BA A B×∇=

Why is helicity interesting?Why is helicity interesting?

• Helicity can be traced as the magnetic field – emerges through the photosphere

– fills the corona

– erupts as a coronal mass ejection (CME)

– reaches the Earth as a magnetic cloud

• Helicity accumulation is the key to CMEs – reaching a stability threshold of helicity is emerging as an underlying cause of CMEs

• Helicity output is the key to the solar dynamo – puts constraints on the dynamo models taking into account

the conservation of helicity

How can helicity change?How can helicity change?

In a given volume, change of magnetic helicity can be written: [ ]dSB)v(Av)BA 00 (2∫ ⋅−⋅−=

S

r

dtdH

horizontal photospheric motions twisted flux emergence (increase) ejection via CMEs (decrease)

twisting

shearing

braiding

emergence

advection termshear term

(Berger & Field, 1984)

Helicity injection to the coronaHelicity injection to the corona

Photospheric helicity injection:Local Correlation Tracking methodsLCT has intrinsic problems…

∑∑= = +

=x

x

y

y

yx

N

n

N

n yx

nn

rCorona kkl

BH

1 122

2,

)(

||2α

Green et al, 2002

Chae, 2003

Coronal helicity:Computed from magneticextrapolation models matched with observed loops

Our projectOur project

• To develop new methods for measuring the – inflow of helicity through the photosphere using SOHO/MDI magnetograms – helicity increase in the corona using coronal models and images

• To develop methods for magnetic helicity measurements using vector magnetograms – present methods use only the line-of-sight component of the magnetic field, since Blos ± 5 G, while Btrans ± 200 G

• To get prepared for Solar-B, which will provide us with greatly improved magnetic field measurements and coronal images

Helicity & Solar-BHelicity & Solar-B

SOT/FPP (0.25”, 164”x164”, Blong= ± 1-5 G, Btrans= ± 30-50 G)• Measurements of helicity inflow through the photosphere using full vector magnetograms of high resolution

XRT + EIS (+ STEREO): • Modelling of coronal loops, of a large range of T• Measuring the increase of helicity in the corona establishing a the limit of stability eventually CME forecast (CME identification with STEREO)

Solar-B: InstrumentsSolar-B: Instruments

• Solar Optical Telescope (SOT)– 0.25” spatial resolution optical telescope– Focal Plane Package (FPP) vector

magnetograph & spectrograph

fov: 164”x164”, Blong=1-5 G, Btrans=30-50 G

temp. res.=5 min.; detailed Stokes profiles

• X-ray telescope (XRT)

-- 2” spatial resolution, 2 sec. cadence; full-disc and partial frame • EUV imaging Spectrometer (EIS) -- in imaging mode: temporal res.~ 3-10 s; fov: 4’x4’

Helicity injection via footpoint motionsHelicity injection via footpoint motions

• Helicity generation rate by footpoint motions can be understoodas the summation of the rotation rate of all the individual elementary flux pairs weighted by their magnetic flux (Berger, 1986).

• We can separate the helicity generated by the differential rotation (a shear flow) into two terms:

Twist and writhe helicity have opposite signs, while their magnitudes are similar they partially cancel

(Démoulin et al, 2002).

'')()(21

'

dSdSrBrBdtd

dtdH

nn

S S

r ⋅= ∫∫θπ

=⋅Δ+⋅Δ=Δ ∫∫∫∫ <⋅>⋅''

0''

0 '' 21

21)( dSdSBBdSdSBBtH n

BBnn

BBnr

nnnnθ

πθ

πwrithertwistr tHtH )()( Δ+Δ=

Helicity & Solar B Helicity & Solar B

• Please use this template to produce your RG panel slides– Use sub-bullets for effect – Don’t alter the style

• Be concise

Add your own text boxes if needed

For example, as

labels

We consider the first term (effect of horizontal motions):

which shows that the helicity generation rate can be understood as the summation of the rotation rate of all the individual elementary flux pairs weighted by their magnetic flux (Berger, 1986).

'')()(21

'

dSdSrBrBdtd

dtdH

nn

S S

r ⋅= ∫∫θπ

Computation of magnetic helicity generated by differential rotation

Differential rotation: time-independent shear flow.We can separate the helicity generated by the differential rotation into two terms:

Twist and writhe helicity have opposite signs, while their magnitudes are similar they partially cancel (Démoulin et al, 2002, SP).

=⋅Δ+⋅Δ=Δ ∫∫∫∫ <⋅>⋅''

0''

0 '' 21

21)( dSdSBBdSdSBBtH n

BBnn

BBnr

nnnnθ

πθ

π

writhertwistr tHtH )()( Δ+Δ=

Relative magnetic helicityRelative magnetic helicity

• Computation of helicity is physically meaningful when B is fully contained inside V (since A A + ), when field lines cross the boundary (Bn 0 on S around V, solar case) a relative magnetic helicity can be computed:

B0 has the same Bn distribution on S as B.

• The helicity is measured relative to that of the potential field. Hr is gauge invariant (Berger & Field, 1984; Field & Antonsen 1985).

dVdVHV V

r 00 BABA ⋅−⋅=∫ ∫ 00 AB ×∇=