Dynamo and Magnetic Helicity Flux

Hantao JiCMSO & PPPL

CMSO General MeetingPrinceton, October 5-7, 2005

Contributors:Eric Blackman (Rochester)Stewart Prager (Wisconsin)

Outline• Introduction to magnetic relaxation

• Magnetic helicity transport

• The -effect

• Relation between the -effect and helicity transport

• Summary

Reversed Field Pinch (RFP) Plasmas Sustained by a Toroidal

Electric Field• Donut-shaped plasmas is enclosed by electrically conducting “shells”– Cuts in both toroidal and poloidal directions allow flux penetration.

– Rate of flux change corresponds to finite “loop voltages”.

Determining Integrated Quantities

• Directly measured– Total toroidal current (plasma current)

– Total toroidal flux– Toroidal and poloidal loop voltages

• Inferred from equilibrium– Total internal poloidal flux

– Total magnetic helicity– Total magnetic energy

• Periodic “relaxations”

Toroidal flux in MST

€

K = A ⋅BdV∫

Balance of Integrated Quantities

• Between relaxations: dissipation injection

• During relaxations: dissipation rate of change

€

dK

dt= −2 E ⋅BdV − 2φ ⋅B + A ×

∂A

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟⋅dS∫∫

€

dW

dt= − E ⋅ jdV − E × B( ) ⋅dS∫∫

ensemble average:

injectiondissipation

Balance of Mean Profiles: Transport

• K is transported outward

• W is dissipated in the core

€

dK

dt= −2 E ⋅BdV − 2φB + A ×

∂A

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟⋅dS∫∫

€

dW

dt= − E ⋅ jdV − E × B( ) ⋅dS∫∫

Helicity in a Sub-volume of a Torus

• Total helicity K can be split into three parts:

€

K = Kcore + Kedge + K link

b

a

ΦbΨ −Ψa b

€

K link = 2Φb Ψa − Ψb( )

€

d K − Kcore( )dt

=dKedge

dt+

dKlink

dt

€

d K − Kcore( )dt

= −2 E ⋅BdVb

a∫ − 2φB + A ×

∂A

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟⋅dS

a∫ + 2φB + A ×

∂A

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟⋅dS

b∫

€

ΦaVa

€

ΦbVb

€

2Γ

Transport of Magnetic Helicity

• The required magnetic helicity flux at r=b can be determined.

€

dKedge

dt+

dKlink

dt+ 2 E ⋅BdV

b

a∫ − 2ΦaVa + 2ΦbVb = 2 ΓdS

b∫

outward helicity transport

Fluctuation-driven Flux Explains Helicity Transport

€

Γ= ˜ φ ̃ B r +1

2˜ A ×

∂ ˜ A

∂t≈ ˜ φ ̃ B r

Balance of Mean Electric Field Profile: the -Effect

–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||

Requires nonzero

The -effect in MHD• Ohm’s law:

• The mean and turbulent parts:

• The -effect:

€

E + v × B = ηj

€

E0 + v0 × B0 +ε = ηj0

€

˜ E + ˜ v × B0 + v0 × ˜ B + ˜ v × ˜ B −ε = η˜ j

€

=ε⋅B0

Β 02

=˜ v × ˜ B ⋅B0

Β 02

= −˜ v × B0( ) ⋅ ˜ B

Β 02

=˜ E ⋅ ˜ B

Β 02

−η ˜ j ⋅ ˜ B

Β 02

€

ε = ˜ v × ˜ B

€

≈˜ E ⋅ ˜ B

Β 02

€

≈˜ E ⊥⋅ ˜ B ⊥

Β 02

Measured -effect Satisfies Mean Ohm’s Law

Experiment (plasma edge):

Electric Field is Mainly Electrostatic

€

˜ E st = (1− 3)kV /m

€

˜ E = ˜ E st + ˜ E ind

€

˜ E ind = (10 − 20)V /m

Simulation:

Bonfiglio, Cappello & Escande (2005)€

˜ E st ~ 30 ˜ E ind

€

˜ E ≈ −∇ ˜ φ

€

˜ E ⋅ ˜ B ≈ − ∇ ˜ φ ⋅ ˜ B = −∂ ˜ φ

∂z˜ B z +

∂ ˜ φ

∂y˜ B y +

∂ ˜ φ

∂r˜ B r

Averaging over periodic (toroidal & poloidal) directions:

€

˜ E ⋅ ˜ B ≈ −∂ ˜ φ ̃ B z( )

∂z− ˜ φ

∂ ˜ B z∂z

+∂ ˜ φ ̃ B y( )

∂y− ˜ φ

∂ ˜ B y∂y

+∂ ˜ φ ̃ B r( )

∂r− ˜ φ

∂ ˜ B r∂r

€

˜ E ⋅ ˜ B ≈ −∂ ˜ φ ̃ B r

∂r= −

∂Γ

∂r

-effect Closely Related to Helicity Flux

The -effect is propotional to the convergence of helicity flux.

Summary• Magnetic relaxation is accompanied by -effect and magnetic helicity transport.

and helicity flux are related by

– where averaging is taken in the periodic direction(s), and

– helicity flux points towards the un-averaged direction(s).

• Astrophysical implications– Averaging directions important

€

E||

+ α B = η j||

€

≈−1

B 2

∂Γ

∂r