22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5: The Screw Pinch and the Grad-Shafranov Equation
Screw Pinch Equilibria
1. A hybrid combination of Z pinch and θ pinch
2. This combination of fields allows the flexibility to optimize configurations with
respect to toroidal force balance and stability.
3. Field components: B = Bθ (r )eθ + Bz (r )ez , p = p ( r )
a. ∇ ⋅ B = 0
1 ∂Bθ ∂Bz
+
= 0 automatically satisfied
r ∂θ
∂z
b.
μ0 J = ∇ × B = -B'Zeθ + ⎡⎢( rBθ ) r ⎤⎥ ez
⎣
⎦
c.
J × B = ∇p
'
∇p = p'er
J × B = ( Jθ Bz - Jz Bθ ) er
=−
1 ⎡ Bθ
μ0 ⎢⎣ r
⎤
( rBθ )' + Bz Bz' ⎥
⎦
Even though the equations are nonlinear, the forces superpose because of
symmetry
d
dr
⎛
B2 + Bθ2
⎜p + z
⎜
2μ0
⎝
⎞ Bθ2
=0
⎟+
⎟ μ0 r
⎠
screw pinch pressure balance relation
4. The screw pinch has many properties of more realistic, multidimensional
toroidal configurations
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 1 of 11
a. There are two free functions, say Bθ (r ), Bz (r ). (The θ pinch, Z pinch are
special degenerate cases)
b. The constant pressure contours p(r) = constant are circles r = constant.
The flux surfaces are closed, nested, concentric circles.
c.
β can be varied over a wide range if Bz (0) ≠ 0
d. The magnetic lines wrap around the plasma giving a nonzero rotational
transform
General Equilibrium Relation for 1-D Configurations
1. This is a useful relation for defining β
d
dr
⎛
B2 + Bz2 ⎞ Bθ2
= 0 > ∫ 2πr 2dr
⎜p + θ
⎟+
⎜
⎟
μ
μ
2
r
0
0
⎝
⎠
a
'
⎛
⎛
⎛
B2 ⎞
B2 ⎞
B2 ⎞
2. T1 = 2π∫ dr r ⎜ p + z ⎟ = −4π ∫ rdr ⎜ p + z ⎟ + 2πr 2 ⎜ p + z ⎟
⎜
⎜
⎜
2μ0 ⎟⎠
2μ0 ⎟⎠
2μ0 ⎟⎠
⎝
⎝
⎝
0
2
(
B02 − Bz2
⎛ B2 ⎞
⎛
B2 ⎞
= 2πa2 ⎜ 0 ⎟ = −4π∫ rdr ⎜ p + z ⎟ = −4π∫ rdrp + 4π∫ rdr
⎜
⎜ 2μ ⎟
2μ0 ⎟⎠
2μ0
⎝
⎝ 0⎠
(
⎡ r 2 B2
θ
2π
'
2 Bθ
3. T2 =
dr r
( rBθ ) = ∫ dr ⎢⎢
∫
r
μ0
μ0
2
⎣
2π
) ⎤⎥
⎥
⎦
'
=
πr 2 Bθ2
μ0
)
a
0
2
=
μ0 I 2
πa2 ⎛ μ0 I ⎞
⎜
⎟ =
4π
μ0 ⎝ 2πa ⎠
4. The general equilibrium relation is given by
2π∫ prdr =
plasma energy line density
μ0 I 2
8π
+ 2π∫
poloidal tension
B02 − Bz2
r dr
2μ0
toroidal diamagnetism
5. This suggests the following cylindrical definitions
βp =
16π2 ∫ pr dr
μ0 I 2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
poloidal β
Lecture 5
Page 2 of 11
βT =
4μ0 ∫ pr dr
a2 B02
toroidal β
Calculate the Rotational Transform
1. Note that Δθ is independent of θ0 because of cylindrical symmetry.
2. The average value of Δθ is just the change θ as the magnetic line moves
one length along the torus
3. Calculate the field line trajectories
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 3 of 11
B
dr
= r =0
dz Bz
(1)
B
dθ
= θ
dz rBz
(2)
4. Equation (1) implies that r(z) = const. The magnetic lines lie on circles. This
is not surprising since the p(r) = const. surfaces are circles.
5. Solve Eq. (2)
r = const.
B (r )
dθ
= θ
dz rBz (r )
dθ =
Δθ
∫0
Bθ
dz
rBz
dθ =
2 πR0
∫0
Bθ
dz
rBz
6. The angle Δθ is by definition just equal to ι
ι(r ) =
2πR0 Bθ
rBz
7. The safety factor is defined as
q(r ) =
2π
ι(r )
q(r ) =
rBz
R0 Bθ
8. Note ι ( r ) = 0 for a θ pinch and ι ( r ) = ∞ for a Z pinch
9. The 1-D radial pressure balance relation for a screw pinch accurately describes
radial pressure balance in all fusion configurations of interest
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 4 of 11
Examples
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 5 of 11
Toroidal Force Balance
1. Consider the axisymmetric torus, the simplest, multi-dimensional configuration
2. We shall derive the Grand–Shafranov equation for axisymmetric equilibria
3. This provides a complete description of toroidal equilibrium
a. radial pressure balance
b. toroidal force balance
c.
β limits
d. q profiles
e. magnetic well, etc….
4. It applies to the following configurations
a. RFP
b. ohmic tokamak (circular and noncircular)
c. high β tokamak (circular and noncircular)
d. flux conserving tokamak (circular and noncircular)
e. spherical tokamak (circular and noncircular)
f.
spheromak (circular and noncircular)
g. toroidal multipole (circular and noncircular)
5. Grad–Shafranov equation
a. exact (no expansion)
b. axisymmetric ∂ ∂φ = 0
c. 2-D
d. nonlinear
e. partial differential equation
f.
elliptic characteristics
6. Plan of action
a. Derive the exact Grad–Shafranov equation
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 6 of 11
b. Solve by means of an asymptotic expansion in a R
c. Zero order: (a/R)0 → 1-D screw pinch radial pressure balance
d. First order: (a/R)1 → toroidal force balance
Derivation
1. Geometry
2. Axisymmetry
∂
=0
∂φ
Q ( R, φ, Z ) → Q ( R, Z )
3. We solve in this order
∇ ⋅B = 0
μ0 J = ∇ × B
J × B = ∇p
4. ∇ ⋅ B = 0
a.
1 ∂
1 ∂Bφ ∂BZ
=0
RBR +
+
R ∂R
R ∂φ
∂Z
=0
b. Bφ is arbitrary as of now
c.
BZ =
1 ∂ψ
R ∂R
introduce “flux” function ψ
1 ∂ψ
B =−
R
R ∂Z
d. These results can be summarized as follows
B = Bφ eφ + B p
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 7 of 11
Bp =
1
∇ψ × eφ
R
ψ = ψ ( R, Z )
5. Why is ψ the “flux” function?
(
)
a. B p = ∇ × A = ∇ × Aφ eφ =
∂Aφ
1 ∂
RAφ eZ −
eR
R ∂R
∂Z
ψ = RAφ
b. ψ p = ∫ B p ⋅ dA
=
=
c.
R
∫R
e
R
∫R
dR ∫ Bz ( R, Z = 0 )Rdφ
2πR
e
1 ∂ψ
dR
R ∂R
ψ p = 2π ⎡⎣ ψ ( R, 0 ) − ψ ( Ra , 0 ) ⎤⎦ ψ ( Ra , 0 ) is arbitary → set to zero
ψ p = 2πψ
d. We usually label the flux surfaces with ψ values rather than p values
6. μ0 J = ∇ × B
a.
⎡
eφ
⎣
R
μ0 J = ∇ × ⎢ RBφ
(
)
eφ
)
eφ
= ∇ RBφ ×
R
+
⎤
1
∇ψ × eφ ⎥
R
⎦
+ RBφ ∇ ×
eφ
R
+
=0
eφ
R
=0
⋅ ∇ ( ∇ψ ) − ∇ψ ⋅ ∇
eφ
R
−
eφ
R
∇ ⋅ ∇ψ + ∇ψ∇ ⋅
eφ
R
=0
(
= ∇ RBφ ×
R
−
eφ
R
∇2 ψ +
1
∂
∂
R φ
2
∂ψ
1 ∂ψ
⎛ ∂ψ
⎞
⎜ ∂R eR + ∂Z ez ⎟ → 2 ∂R eφ
R
⎝
⎠
=0
e
∂ψ ∂ ⎞ φ
1 ∂ψ
⎛ ∂ψ ∂
−⎜
+
⎟ R → 2 ∂R eφ
R
R
Z
Z
∂
∂
∂
∂
R
⎝
⎠
= ∇RBφ ×
eφ
R
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
−
eφ ⎡ ∂2 ψ 1 ∂ψ ∂2 ψ 2 ∂ψ ⎤
+
+
−
⎢
⎥
R ⎢⎣ ∂R2 R ∂R ∂Z 2 R ∂R ⎥⎦
Lecture 5
Page 8 of 11
μ0 J = ∇RBφ ×
eφ
R
−
eφ ⎡ ∂ ⎛ 1 ∂ψ ⎞ ∂2 ψ ⎤
+
⎢R
⎥
R ⎣⎢ ∂R ⎜⎝ R ∂R ⎟⎠ ∂Z 2 ⎦⎥
b. These results can be summarized as follows
μ0 J = μ0 Jφ eφ + μ0 Jp
μ0 Jp =
1
∇RBφ × eφ
R
μ0 Jφ = −
1 *
Δ ψ
R
Δ* ψ ≡ R
∂ 1 ∂ψ ∂2 ψ
⎛ ∇ψ ⎞
+
= R2∇ ⋅ ⎜ 2 ⎟
∂R R ∂R ∂Z 2
⎝R ⎠
7. Force balance J×B = ∇p
Decompose this relation into three components along B, J, ∇ψ
8. B component
a. B ⋅ ∇p = 0
b.
Bφ ∂p ∇ψ × eφ
+
⋅ ∇p = 0
R ∂φ
R
c.
eφ ⋅ ∇ψ × ∇p = 0
d.
p = p (ψ )
p is an arbitrary free function of ψ .
e. There is no way to determine p ( ψ ) from ideal MHD. We need transport
theory or some other simple physical model.
f.
Note: p ( ψ ) is more of a constraint than p ( r, θ )
9. J component
a.
J ⋅ ∇p = 0
b.
Jφ ∂p
dp
1
+ Jp ⋅ ∇p = 0 → ∇RBφ : × eφ ⋅ ∇ψ
=0
dψ
R ∂φ
R
c.
1 dp
⎡eφ ⋅ ∇ψ × RBφ ⎦⎤ = 0
R dψ ⎣
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 9 of 11
d. RBφ = F ( ψ ) F is an arbitrary free function
10. Interpretation of F ( ψ )
a.
Ip =
R
2π
∫ Jp ⋅ dA = ∫0 dR∫0
R
= 2π ∫ dR
0
eφ ⎞
⎛
Rdφ ⎜⎜ ∇RBφ ×
⎟z
R ⎟⎠
⎝
∂F
= 2π ⎡⎣ F ( R, 0 ) − F ( 0, 0 ) ⎤⎦
∂R
= RBφ
0
=0
b.
I p = 2πF ( ψ )
c.
Ip ( ψ ) is the total poloidal current passing through the circle ψ (R, 0)=
const.
11. ∇ψ component
∇ψ ⋅ ( J × B - ∇p ) = 0
a. T1 = −∇ψ ⋅ ∇p = −
(
dp
( ∇ψ )2
dψ
) (
)
b. T2 = Jp + Jφ eφ × B p + Bφ eφ ⋅ ∇ψ
(
)
= ⎡ Jp × B p + Jp × eφ Bφ + eφ × B p Jφ + eφ × eφ Jφ Bφ ⎤ ⋅ ∇ψ
⎣
⎦
c.
Ta =
1 ⎡1
1
⎤
∇F × eφ × ∇ψ × eφ ⎥ ⋅ ∇ψ
⎢
μ0 ⎣ R
R
⎦
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 10 of 11
=
d. Tb =
1
dF ⎡
∇ψ × eφ × ∇ψ × eφ ⎤ ⋅ ∇ψ = 0
⎦
μ0 R d ψ ⎣
F
dF
μ0 R2 d ψ
=−
)
( ∇ψ × eφ ) × eφ ⋅ ∇ψ
F
dF
( ∇ψ )2
μ0 R d ψ
2
∇ψ × eφ ⎛
1 * ⎞
Δ ψ ⎟ ⋅ ∇ψ
⎜−
R
⎝ μ0 R
⎠
e. Tc = eφ ×
f.
) (
⎤
1 ⎡⎛ 1
⎞
⎢⎜ ∇F × eφ ⎟ × eφ ⎥ ⋅ ∇ψBφ
μ0 ⎣⎝ R
⎠
⎦
=
=−
(
2
1
Δ*ψ ( ∇ψ )
2
2
μ0 R
Combine terms
⎡ dp
⎤
1 d F2
1
−
−
Δ
ψ
*
⎥=0
2
μ0 R2
⎢⎣ dψ μ0 R d ψ 2
⎥⎦
( ∇ψ )2 ⎢ −
g. The Grad–Shafranov equation is given by
Δ*ψ = −μ0 R2
dp
dF
−F
dψ
dψ
where
p = p (ψ)
free functions
F = F (ψ )
B=
1
F
∇ψ × eφ + eφ
R
R
μ0 J =
1 dF
1
∇ψ × eφ − Δ * ψeφ
R dψ
R
and ψ p = 2πψ , Ip = 2πF
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 5
Page 11 of 11