22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6: The Safety Factor and the Ohmic Tokamak
Calculate q ( ψ ) in an Axisymmetric Torus
1. Introduce toroidal coordinates
R = R0 + r cos θ
Z = r sin θ
2. Calculate the field line trajectory by means of the tangent curve
dr
rdθ
Rdφ
=
=
Br
Bθ
Bφ
3. As the magnetic line wraps exactly once around the poloidal cross section (i.e.
Δθ = 2π) it travels along an angle Δφ in the toroidal direction given by
Δφ = ∫
Δφ
0
=
2π
∫0
dφ =
dθ
Δθ = 2π
∫0
dφ
dθ
dθ
rBφ
RBθ
4. The rotational transform is the average Δθ per single transit in the toroidal direction
Δφ = 2π .
5. Because of symmetry, the pattern of field line motion repeats once Δθ = 2π .
Thus, we simply have to take proportions:
average Δθ per toroidal transit
Δθ 2π
ι
:
=
Δφ Δφ 2π
one toroidal transit Δφ = 2π
6. Safety factor:
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 1 of 8
⎛ ι ⎞
q=⎜
⎟
⎝ 2π ⎠
q (ψ) =
−1
=
Δφ
2π
1 2π ⎛ rBφ ⎞
⎜
⎟
2π ∫0 ⎜⎝ RBp ⎟⎠
dθ =
s
dl p
F
∫
2π R ∇ ψ
7. This integral is difficult to calculate in general because the integrand must be
evaluated on the flux surface
a. ψ ( r , θ ) = ψ0 = const. → r = r ( ψ0 , θ ) projection of a field line trajectory
b. Note that on a flux surface
Bφ = Bφ ⎡⎣ r (θ, ψ 0 ) , θ ⎤⎦
Bθ = Bθ ⎡⎣ r (θ, ψ 0 ) , θ ⎤⎦
Br = Br ⎡⎣ r (θ, ψ 0 ) , θ ⎤⎦
c. We find r (θ, ψ0 ) as follows
rB ( r , θ )
dr
= r
dθ
Bθ ( r , θ )
r ( 0 ) = r0 ( ψ0 )
d. This allows us to calculate q = q ( ψ0 )
Basic Procedure for Solving the Grad–Shafranov Equation
1. We assure r R
1 but not zero
2. The equilibrium problem separates into two parts
a. radial pressure balance
b. toroidal force balance
3. Applications
a. ohmic tokamak, circular, conducting shell
b. ohmic tokamak, circular, vertical field
c. high β tokamak, circular
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 2 of 8
d. flux conserving, tokamak, circular
e. noncircular tokamak
f.
spherical tokamak
Ohmic Tokamak
1. This is the simplest configuration: low β , circular cross section, simple boundary
conditions
2. Mathematical statement of the problem
Δ* ψ = H ( ψ , R )
Grad–Shafranov equation
ψ is regular inside the plasma, (no wires, current sources in plasma)
ψ ( b, θ ) = const. ↔ n ⋅ B = Br =
1 ∂ψ
r ∂θ
=0
r =b
perfect conducting boundary condition
3. Solution: first transform from R, φ', Z to r , θ , φ . This step is exact; no expansions are
used
R = R0 + r cos θ
eR = er cos θ − eθ sin θ
Z = r sin θ
eZ = er sin θ + eθ cos θ
φ' = φ
4. Then
∇ψ =
∂ψ
1 ∂ψ
1 ∂ψ
er +
eθ +
eφ
∂r
r ∂θ
R ∂φ
∇⋅V=
1 ∂rVr 1 ∂Vθ
1 ∂Vφ 1
+
+
+ (Vr cos θ - Vθ sin θ )
r ∂r
r ∂θ
R ∂φ
R
Δ*ψ = ∇ ⋅ ∇ψ −
=
2 ∂ψ
R ∂R
⎞
1 ∂r ∂ψ 1 ∂2 ψ 1 ⎛ ∂ψ
1 ∂ψθ
r
cos θ −
sin θ ⎟
+ 2
− ⎜
2
r ∂r ∂r r ∂θ
R ⎝ ∂r
r ∂θ
⎠
∇2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 3 of 8
5. The Grad–Shafranov equation becomes
∇2 ψ = −μ0 ( R0 + r cos θ )
2
dp
dF 1 ⎛ ∂ψ
1 ∂ψ
⎞
cos θ −
sin θ ⎟
−F
+ ⎜
dψ
d ψ R ⎝ ∂r
r ∂r
⎠
6. In the infinite aspect ratio limit
R→∞
r R→0
ψ ( r, θ ) → ψ ( r ) cylindrically symmetric
7. We show the Grad–Shafranov equation reduces to radial pressure balance in this
limit
Bφ =
F
Fψ
→
R0 + r cos θ
R0
Bp =
1
1 ∂ψ ⎤
1 dψ
⎡ ∂ψ
eθ −
er ⎥ →
eθ
⎢
R0 + r cos θ ⎣ ∂r
r ∂θ
R0 dr
⎦
∇2 ψ =
1 ∂ ∂ψ 1 ∂2 ψ
1 d
+ 2
→
r
r ∂r ∂r r ∂θ
r dr
⎛ dψ ⎞
⎜ r dr ⎟
⎝
⎠
8. Thus the Grad–Shafranov equation becomes
1 d ⎛ dψ ⎞
dp
dF
r
= −μ R02
−F
⎜
⎟
r dr ⎝ dr ⎠
dψ
dψ
T1 =
radial pressure balance?
R d
1 d ⎛ dψ ⎞ 1 d
r
rR0 Bθ = 0
rBθ
=
⎜
⎟
r dr ⎝ dr ⎠ r dr
r dr
T2 = −R02
T3 = −F
dp
dp ⎛ d ψ ⎞
= −R02
dψ
dr ⎜⎝ dr ⎟⎠
dF
⎛ dψ ⎞
= −⎜
⎟
dψ
⎝ dr ⎠
−1
−1
=−
R0 dp
Bθ dr
2
2
R2 d Bφ
R d Bφ
d F2
=− 0
=− 0
dr 2
R0 Bθ dr 2
Bθ dr 2
B2 ⎞ B d
d ⎛
⎜p + φ ⎟ + θ
rBθ = 0
dr ⎜
2μ0 ⎟ μ0 r dr
⎝
⎠
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 4 of 8
Ohmic Tokamak Expansion
1. Radial pressure balance
2. Expand in terms of ∈= a R0 . Assume ∈
1.
a. Assume q ∼ 1 for stability
rBφ
q∼
RBθ
→
Bθ
r
∼
∼∈
Bφ
R
3. Assume the plasma is confined in radial pressure balance by the poloidal field (due to
ohmic current)
p∼
βp ∼
β ∼
Bφ2
μ0
μ0 p
Bθ2
∼1
μ0 p
Bφ2 + Bθ2
∼
μ0 p
Bφ2
∼
μ0 p Bθ2
Bθ2 Bφ2
∼ ∈2
4. Assume the toroidal field does not confine much plasma. It is actually paramagnetic
in practice in experiments with no auxiliary heating
p∼
δBφ
Bφ
Bφ δBφ
μ0
∼
μ0 p
Bφ2
∼ ∈2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 5 of 8
Summary of Assumptions
1. The ohmically heated tokamak is dominated by a large toroidal field. The plasma
pressure and current are small but the safety factor is of order unity. The toroidal
field is required only for stability. The small toroidal current provides both radial
balance and toroidal force balance.
2. The ohmically heated tokamak expansion is given by
Bp
Bφ
∼∈
q ∼1
βt ∼
2μ0 p
2
Bφ
∼ ∈2
β ∼
βp ∼
δBφ
Bφ
2μ0 p
Bp2
2μ0 p
2
Bφ +
Bp2
∼
2μ0 p
Bφ2
∼ βt
∼1
∼ ∈2
Solution of Grad–Shafranov Equation
1. Expand
ψ ( r, θ ) → ψ0 ( r ) + ψ1 ( r, θ ) + ....
ψ1 ψ 0 ∼ ∈
ψ0 ∼ rR0 Bθ
2. Expand
F ( ψ ) = RBφ = R0 ⎡⎣ B0 + B2 ( ψ ) ⎤⎦
⎡
⎤
dB2
= R0 ⎢ B0 + B2 ( ψ0 ) +
ψ1 + ....⎥
dψ0
⎣
⎦
B2
∼ ∈2
B0
(small dia/paramagnetism)
3. Expand p ( ψ ) = p ( ψ0 ) +
dp
ψ1 + ...
dψ0
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 6 of 8
⎛
⎞
r
R = R0 ⎜ 1 +
cos θ ⎟
R0
⎝
⎠
4. Expand
5. Fields:
Bθ =
⎤
1 d ψ0 ⎡ 1 ∂ψ1
r d ψ0
+⎢
− 2
cos θ...⎥
R0 dr
R0 dr
⎣⎢ R0 ∂r
⎦⎥
Br = −
1 ∂ψ1
R0 r ∂θ
⎡
⎤
B (ψ )
r
r2
Bφ = B0 ⎢1 −
cos θ + 2 cos2 θ + 2 0 + ...⎥
R0
B0
R0
⎣⎢
⎦⎥
6. Expansion of the Grad–Shafranov equation
1 ∂ ⎛ ∂ψ ⎞ 1 ∂2 ψ
d F 2 1 ⎛ ∂ψ
sin θ ∂ψ ⎞
2 dp
+
=
−
+
−
+ ⎜
μ
θ
r
R
r
cos
cos θ −
(
)
0
0
⎜
⎟
2
2
r ∂r ⎝ ∂r ⎠ r ∂θ
dψ dψ 2
R ⎝ ∂r
r ∂θ ⎠⎟
∈0:
d
1 d ⎛ d ψ0 ⎞
2 dp
−
R02 B0 B2
⎜r
⎟ = − μ0 R0
r dr ⎝ dr ⎠
d ψ0 d ψ0
∈1:
1 ∂ ⎛ ∂ψ1 ⎞ 1 ∂2 ψ1
dp
d2 p
d2
1 d ψ0
2
=
−
2
R
r
cos
−
R
ψ
−
ψ
R02 B0 B2 +
cos θ
μ
θ
μ
⎜r
⎟+ 2
0
0
0
0
1
1
2
2
2
r ∂r ⎝ ∂r ⎠ r ∂θ
d ψ0
R0 dr
dψ 0
d ψ0
(
)
∈0 consequences
1.
B0 B2 ⎞ B0 d
d ⎛
rB = 0
⎜p +
⎟+
dr ⎝
μ0 ⎠ μ0 r dr θ
2.
q (r ) =
βt =
βp =
rB0
+ O (∈)
R0 Bθ
4μ0
a2 B02
16π2
μ0 I 2
∫ pr dr
all are determined in leading order
∫ pr dr
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 7 of 8
3. Equilibrium relation for a tokamak
a. From the definitions we can find a relation between βt and β p
μ0 I
2πa2 B0
μ0 R0 I
b.
Bθ ( a) =
c.
4μ0
⎛ μ I ⎞
βt = 2 2 ∫ pr dr = 2 2
β = ⎜ 0 ⎟ βp
2 p
a B0
a B0 16π
⎝ 2πaB0 ⎠
d.
a2 ⎛ μ R I ⎞
βt = 2 ⎜ 0 20 ⎟ β p
R0 ⎜⎝ 2πa B0 ⎟⎠
2πa
→ qa =
4μ0 μ0 I 2
2
2
e.
∈2
βt =
qa2
βp
4. q on axis
a. q0 =
rB0
R0 Bθ
r =0
Bθ ( r ) ≈ Bθ ( 0 ) + Bθ' ( 0 ) r + Bθ'' ( 0 ) r 2 2 ≈ Bθ' ( 0 ) r
b. q0 =
B0
R0 Bθ' 0
μ J r
μ J
1
( rBθ )' = μ0 J0 → Bθ ≈ 0 0 → B'θ 0 = 0 0
r
2
2
c.
q0 =
2B0
μ0 R0 J0
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 6
Page 8 of 8