Post on 27-Mar-2015
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