NSF Site VisitMadison, May 1-2, 2006

Magnetic Helicity Conservation and Transport

R. Kulsrud and H. Ji

for participants of the Center for Magnetic Self-

organization

Major Goals

I. Study basic properties of magnetic helicity.

II. Determine role of helicity conservation and transport during magnetic self-organization.

III.Assess applicabilities of the helicity concept to astrophysical circumstances.

• Is magnetic helicity really conserved in reality?• Can the concept of helicity be extended to 2-fluid

plasmas?

• Can the helicity conservation illuminate dynamo process?

• How is the helicity related to solar dynamo, flares, and CMEs (coronal mass ejections)?

Introduction: Magnetic Helicity

• There are many flux conserving quantities (e.g. the flux point mappings for the coronal field). Only one can be expressed simply: magnetic helicity,

where A is the vector potential: A=B.

• Conservation of magnetic helicity is a key element of the concept of magnetic self-organization.

€

K = (A ⋅B)dV∫

Physical Interpretation of Helicity

• Helicity represents the amount of flux linkages between pairs of lines.

• Consider two flux tubes 1 and 2. In first volume

since the integral is the flux in the second tube by Stokes theorem.

• Similarly for the second volume so the total helicity is simply

• If the loops were not linked, K=0.• Gauge invariant definitions exist.

€

A ⋅BdV∫ = (B ⋅dS)(A ⋅dl)∫ = Φ1 A ⋅dl =∫ Φ1Φ2

€

K = 2Φ1Φ2

Major Goals

I. Study basic properties of magnetic helicity.

II. Determine role of helicity conservation and transport during magnetic self-organization.

III.Assess applicabilities of the helicity concept to astrophysical circumstances.

• Is magnetic helicity really conserved in reality?• Can the concept of helicity be extended to 2-fluid

plasmas?

• Can the helicity conservation illuminate dynamo process?

• How is the helicity related to solar dynamo, flares, and CMEs (coronal mass ejections)?

Test Helicity Conservation in Simulation and MST

(Horiuchi & Sato, ‘88)

• MHD simulation:

• MST experiment:

Internal + external measurements

Helicities in 2-fluid Plasmas

• Self-helicities:

• Electron-helicity reduces to magnetic helicity when size is larger than electron skin depth (c/wpe).

• Ion-helicity will be significantly different from magnetic helicity if ion flow is large (e.g. Alfvenic)

• Different relaxed states under conservation of these helicities

€

Kα = Aα ⋅Bα dV∫

€

Aα = A +mα

qαVα

€

Bα =∇ × Aαcanonical momentum: =e,i:

€

Ki = A i∫ ⋅BidV = A ⋅B + A ⋅ ∇ × mie

Vi( ) + m ie

Vi ⋅B + mi2

e2 Vi ⋅ ∇ × Vi( ) ⎡

⎣ ⎢ ⎤

⎦ ⎥∫ dV

(Steinhauer & Ishida, 1998; Mahajan &Yoshida, 1998; Hegna, 1998)

-1.0-0.500.51.01.50102030-10Time (ms)Parallel Velocity (km/s)Core (toroidal)Edge (poloidal)

Parallel Flow Profile Appears to BecomeMore Uniform During Relaxation in MST

• Will be studied also numerically using 2-fluid code and PIC code

Major Goals

I. Study basic properties of magnetic helicity.

II. Determine role of helicity conservation and transport during magnetic self-organization.

III.Assess applicabilities of the helicity concept to astrophysical circumstances.

• Is magnetic helicity really conserved in reality?• Can the concept of helicity be extended to 2-fluid

plasmas?

• Can the helicity conservation illuminate dynamo process?

• How is the helicity related to solar dynamo, flares, and CMEs (coronal mass ejections)?

Helicity Conservation and Dynamo

• Large-scale magnetic field is generated by the -effect.

• Most such large-scale magnetic field contains helicity (e.g. solar magnetic field).

• However, the total helicity should be conserved for timescales shorter than the resistive time.

• Two possibilities:– Transport helicity across scales.– Transport helicity across space.Lab

K Transported Outward While W Dissipated Locally during

Relaxation in MST

Helicity Transport Driven by Fluctuations

• The required helicity flux across the r = b surface can be quantified by evaluating each of three parts:

• Helicity flux due to fluctuations:

outward helicity transport€

≈ ˜ φ ̃ B r€

K = Kcore + Kedge + K link b

a

Fluctuations Also Drive a Nonzero -Effect in Mean Ohm’s

Law

–0.5

0.5

1.0

1.5

2.0

V/m

0

0 0.2 0.4 0.6 0.8 1.0ρ/a

E||

ηneo J ||(Zeff = 2)

€

E||

+ α B = η j||

€

≈˜ E ⋅ ˜ B

Β 02

€

˜ E ≈ −∇ ˜ φ

Electric field mostly electrostatic

€

˜ E ⋅ ˜ B ≈ − ∇ ˜ φ ⋅ ˜ B = −∂ ˜ φ

∂z˜ B z +

∂ ˜ φ

∂y˜ B y +

∂ ˜ φ

∂r˜ B r

€

˜ E ⋅ ˜ B = −∂ ˜ φ ̃ B r

∂r= −

∂Γ

∂r

-effect Closely Related to Helicity Flux

€

≈−1

B 2

∂Γ

∂r

• Averaging is taken in the periodic direction(s)• Helicity flux points towards the un-averaged direction(s).

Thus, limiting helicity transport can affect -effect

Major Goals

I. Study basic properties of magnetic helicity.

II. Determine role of helicity conservation and transport during magnetic self-organization.

III.Assess applicabilities of the helicity concept to astrophysical circumstances.

• Is magnetic helicity really conserved in reality?• Can the concept of helicity be extended to 2-fluid

plasmas?

• Can the helicity conservation illuminate dynamo process?

• How is the helicity related to solar dynamo, flares, and CMEs (coronal mass ejections)?

Motivations to Study Helicity in Solar Physics• Most large-scale magnetic field contains

helicity (e.g. “S” and “reversed S” loops).

• Why solar flares are sporadic in space and intermittent in time with constant shearing motion of magnetic field on the surface?

– Related to helicity flux?

• Helicity piling up is a reason for CME?

• Closely related to the solar dynamo process

Correlations found between helicity flux and X-ray flux (Kusano et al.)

Major Goals : Summary and Plans

I. Study basic properties of magnetic helicity.

II. Determine role of helicity conservation and transport during magnetic self-organization.

III.Assess applicabilities of the helicity concept to astrophysical circumstances.

• Assessments of helicity concept, its conservation, and its extensions to 2-fluid plasmas.

• More data expected from experiments and simulations

• Experimental verification of relation between magnetic helicity transport and -effect.

• Further study by “AC helicity injection” experiments

• Evaluation of role of magnetic energy and helicity during solar flares and CMEs.

END