I ntroduction to Nuclear Fusion
Prof. Dr. Yong-Su Na
Tokamak equilibrium
- Consider the axisymmetric torus, the simplest, multi-dimensional configuration
- We shall derive the Grad –Shafranov equation for axisymmetric equilibria.
- This provides a complete description of toroidal equilibrium:
radial pressure balance, toroidal force balance, β limits, q-profiles, etc.
Tokamak Equilibrium
B J p
J B
0
0
B
• The Grad-Shafranov Equation
, sin
0
r cos Z r
R
R
Z
R
e e
e
- Nonlinear
- Partial differential equation
- Grad and Rubin (1958), Shafranov (1960) - Toroidal axisymmetric ∂/∂Φ = 0
Tokamak Equilibrium
d
F dF d
R dp
* 0 2p J B
J B
0
0
B
p B
J
J J
p B B
p
p
Tokamak Equilibrium
• The Grad-Shafranov Equation
Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0
2. Ampere’s law: μ0J = ▽xB
3. The momentum equation: JxB = ▽p
B J p
J B
0
0
B
• Momentum equation
J B p
J J
p B B
p
p
- aim: to express each term with ψ and F
e
B
pR 1
R e
J
*
0
1
2 2 ,
0 p
p
I
F
0
B
J
pB
0
J
B
0
R e
B F F
R e
J
p
1 ,
0
Tokamak Equilibrium
• The Grad-Shafranov Equation
1 0 )
(
1
Z B B
R R
RB R
Z R
0
B
- The ▽ ∙B Equation
In cylindrical coordinates for toroidal axisymmetric fields (∂/∂Φ=0)
R B R
Z
B
RR
Z
1
1 ,
0
B
Ψ: Stream function for the poloidal
magnetic field
0
B Z
B
R R
Z
Magnetic flux surface
Tokamak Equilibrium
• The Grad-Shafranov Equation
- The stream function ψ is closely related to the poloidal flux in the plasma.
Poloidal flux on axis is zero
R B
ZR
1
2 )
0 , ( )
0 , ( 2
2 1
) 0 ,
(
) 0 ,
(
2 0
a R
R
R
R Z
Z p p
R R
R dR R R
Z R RdRB d
dR Rd
Z R B
A d B
a
a
-
∂
∫ ∂
∫
∫
∫
Tokamak Equilibrium
• The Grad-Shafranov Equation
dR
- The ▽ ∙B Equation
1 0 ) (
1
Z B B
R R
RB R
Z
R
0
B
B
pe B
B
R B R
Z
B
RR
Z
1
1 ,
0
B
Ψ: Stream function for the poloidal
magnetic field
0
B Z
B
R R
Z
Magnetic flux surface
dr RB
pd
Rdr B
A d B A
d B A
d B d
d
pA p
A p
A
A p
2 ( ) 2
2
2
Tokamak Equilibrium
• The Grad-Shafranov Equation
∫ B
pd A
p
2
Rdr
dA 2
dr
- Ampere’s law (poloidal component)
J
pB
0
AB d A B d l 2 d ( RB ) 2 d ( RB )
J F
Tokamak Equilibrium
• The Grad-Shafranov Equation
) (
0
F
r F F
dr d R F
J
p
Rdr dA 2
R dr 2
Rdr RB J
d
p
(
) 2
2
0
Rdr J
A d
J
pA
p
0
02
F
dr d
R RB J
p) / (
)
(
0F ( ) RB
B F F
e
J 1
,
R R RB F RB
J
Z
1 1 , ( )
0
0 ) (
0
F
R
B
. - Interpretation of F( ψ)
00
0 0
0 2
0
0 2
0
/ ) 0 , 0 ( )
0 , ( 2
2 1
) 1 (
) 0 ,
(
) 0 ,
(
F R
F
R dR F R R
R RB RdR R
d
Z R RdRJ d
dR Rd Z
R J
A d J I
R
R R
Z Z
p p
Tokamak Equilibrium
• The Grad-Shafranov Equation
B J
0
* 0
2 2
0 0
1
1 1
1 1
R
R R R Z
R R
B Z
J B
R ZR B R
Z
B
RR
Z
1
1 ,
2 2
*
1
Z R
R R R
elliptic operatorTokamak Equilibrium
• The Grad-Shafranov Equation
- Ampere’s law (toroidal component)
- Momentum equation
e
B R F F
B R e
R J
F R e
B R e
J J
R e
B F
R e R e
J
R e B
F R e
J
B B
J J
p
p p
0 0
0 0
0
1 1
1 1
1
1 1
1 1
F R e
J
p
0
1
e
B
pR 1 R B F ( )
*
0
1
R
J
) (
) (
)
(B C B A C C A B A
Symmetry
Tokamak Equilibrium
• The Grad-Shafranov Equation p B
J
- Momentum equation J B p
F
R B R
B J B
J J
p
p
p
0
F R e
J
p
0
1
e
B
pR 1 R B F ( )
* 0 0
0
1
) (
d dF R
F d
R dp
d B dF d
R dp
B F R p
J
*
0
1
R
J
Symmetry
Tokamak Equilibrium
• The Grad-Shafranov Equation
- BCs: provided by the transformer-induced poloidal magnetic field outside the plasma
- In practice, the G-S equation is solved numerically to find the geometrical location of the magnetic surfaces and the radial distribution of the axial
current density in a way that is consistent with the
▽p:
plasma load on a magnetic
flux surface
JpxBΦ: strength
of BΦ JΦxBp:
strength of Bp
d
F dF d
R dp
* 0 2Tokamak Equilibrium
• The Grad-Shafranov Equation
Force balance in a tokamak
▽p:
plasma load on a magnetic
flux surface
JpxBΦ: strength
of BΦ JΦxBp:
strength of Bp
d
F dF d
R dp
* 0 2Tokamak Equilibrium
• The Grad-Shafranov Equation
What kind of forces does a plasma have regarding equilibrium?
18
• Basic Forces Acting on Tokamak Plasmas
- Radial force balance
- Toroidal force balance: Tire tube force
) (
~
R I IINET
e pA pA
F
Tokamak Equilibrium
X Jt Bp
Jt xBp
Magnetic pressure, Tension force
- Toroidal force balance: 1/R force - Toroidal force balance: Hoop force
2 2 2 0
2 2
2 2 2
/ ) (
~ e B A B A a B
F
NET R I I
II II
II II I
I
II I
II I
A B A
B
A A
B B
2 2
,
net
pressure
Tokamak Equilibrium
• Basic Forces Acting on Tokamak Plasmas
0 2
2
) / 2
(
~
R I I II II
NET
e B A B A
F
I I II IIII I
II I
II I
A B A
B
A A
B B
2 2
,
net
pressure
v p
v
BIL R I B
F 2
0v
p
B
I
B B
B
• Basic Forces Acting on Tokamak Plasmas
Tokamak Equilibrium
- External coils required to provide the force balance
How about vertical movement?
Lesch, Astrophysics, IPP Summer School (2008)
