22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10: The High Beta Tokamak Con’d and the High Flux Conserving
Tokamak
Properties of the High β Tokamak
1. Evaluate the MHD safety factor:
ψ0
2
a B0
=
(
)
1 ⎡ 2
ρ − 1 + ν ρ 3 − ρ cos θ ⎤
⎦
2q* ⎣
Bθ
1
=
∈ B0
q*
ν
⎡
⎤
2
⎢ ρ + 2 3 ρ − 1 cos θ ⎥
⎣
⎦
(
)
2. The safety factor on axis is given by
−1 2
(exact)
a. q0 = Δ0 Bφ ⎣⎡ψ rr ψθθ ⎦⎤
12
⎡
⎤
3
b. q0 = q* ⎢
⎥
⎣⎢η (2 + η ) ⎦⎥
(
η = 1 + 3ν 2
)
12
c. Note q0 < q*
3. The safety factor at the plasma edge is given by
a. qa =
1 ⎛ rBφ
⎜
2π ∫ ⎜⎝ RBθ
⎞
aB0
∈ B0
1
dθ =
⎟⎟ dθ ≈
∫
2π RBθ ( a, θ )
2π
⎠S
b. qa =
∈ B0 q*
2π ∈ B0
∫0
c.
qa =
2π
2π
∫0
dθ
Bθ ( a, θ )
dθ
1 + ν cos θ
q*
(1 − ν )
2
12
4. Note that
a. qa > q*
b. for ν → 0
qa → q* ∼
1
I
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 1 of 10
c. as ν → 1
qa → ∞ ?
d. as ν → 1
q* ∝
1
by definition: qa ≈ 1 I
I
5. What is the significance of ν → 1. Clearly ν ≤ 1 for real solutions
6. As ν → 1
a. ∈ β p → 1
b.
c.
βt
∈
→
1
q*2
⎛ βt
ν ⎞
= 2⎟
⎜
⎜∈
q* ⎟⎠
⎝
Δa
1
→
3
a
7. In the high β tokamak there is an equilibrium β limit
βt
∈
<
1
q*2
8. The significance of ν → 1 can be understood by solving the Grad-Shafranov
equation outside the plasma
9. Outside the plasma we solve
2
1 ∂ ∂ψ0
1 ∂ ψ0
r
+ 2
=0
r ∂r
∂r
r ∂θ 2
(no current, no pressure)
ψ0 ( a, θ ) = 0
(continuity of flux)
Bθ ( a, θ ) = Bθ ( a, θ ) = (∈ B0 q* ) ⎡⎣1 + ν cos θ ⎤⎦
(no surface currents)
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 2 of 10
10. The solution is given by
ψ = c1 + c2 ln r + c3r cos θ +
ψ (r,θ )
2
a B0
=
c4
cos θ
r
⎤
1 ⎡
ν⎛
1⎞
⎢ln ρ + ⎜ ρ − ⎟ cos θ ⎥
ρ⎠
q* ⎣
2⎝
⎦
I
Bv
Dvam.
⎤
Bθ
1 ⎡1 ν ⎛
1 ⎞
=
⎢ + ⎜⎜1 + 2 ⎟⎟ cos θ ⎥
∈ B0 q* ⎢⎣ ρ 2 ⎝
ρ ⎠
⎦⎥
Br
1 ν ⎛
1 ⎞
=
⎜⎜ 1 − 2 ⎟⎟ sin θ
∈ B0
q* 2 ⎝
ρ ⎠
11. The vacuum field has a separatrix: B r ( rs , θ s ) = Bθ ( rs , θ s ) = 0
12. Choose θ = π or 0. This makes B r = 0
a. Only θ = π has the possibility of a real solution for rs, satisfying
Bθ ( rs , θ s ) = 0
b. At θ s = π
B θ ( rs , θs ) =
1
q*
⎡1 ν ⎛
1 ⎞⎤
⎢ − ⎜1 + 2 ⎟ ⎥ = 0
ρs ⎟⎠ ⎥⎦
⎢⎣ ρs 2 ⎜⎝
c. Solve for ρs
ρs =
(
1⎡
1 + 1 −ν2
ν ⎢⎣
)
12⎤
radius of the separatrix X point
⎦⎥
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 3 of 10
13. For low β (ν
1) , ρs ≈ 2 ν : the X point is far from the plasma
For ν ∼ 1, ρs ∼ 1 : the X point is near the plasma
For ν = 1, ρs = 1 : the X point moves onto the plasma surface
14. Physical picture of the separatrix and X point
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 4 of 10
15. The equilibrium β limit corresponds to the situation where the separatrix
moves onto the plasma surface
16. At fixed I, the β limit given by βt ≤ ∈2 q*2
17. At fixed I, the only way to hold higher pressure is to increase the vertical
field. Eventually, the separatrix moves onto the plasma surface
18.
19. Calculation of the vertical field
a. Bθ =
Br =
∈ B0
q*
⎡1 ν ⎛
⎤
1 ⎞
⎢ + ⎜⎜1 + 2 ⎟⎟ cos θ ⎥
ρ ⎠
⎢⎣ ρ 2 ⎝
⎥⎦
∈ B0 ν ⎛
1 ⎞
⎜⎜1 − 2 ⎟⎟ sin θ
q* 2 ⎝
ρ ⎠
b. Far from the plasma
c.
Bθ =
∈ B0 ν
cos θ
q* 2
Br =
∈ B0 ν
sin θ
q* 2
Bv = Bθ cos θ + Br sin θ =
∈ B0ν
2q*
d. Note: Bv increases with ν
Bv =
μ0 I
4πR0
βp
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
(high β )
Lecture 10
Page 5 of 10
Bv =
μ0 I ⎡
βp +
4πR0 ⎢⎣
8R ⎤
li − 3
+ ln 0 ⎥
2
a ⎦
(ohmic)
dominates at high β p ∼
1
∈
Summary of the High β Tokamak
1. Ordering
q ∼1
βt ∼ ∈
βp ∼ 1 ∈
Δa a ∼ 1
2. There is an equilibrium βt limit when the separatrix moves onto the plasma
surface
3. This will always occur at fixed I and βt increases
Flux Conserving Tokamak
The Equilibrium β Limit
1. Is there really an equilibrium βt limit in a tokamak?
2. A more realistic treatment shows that such a limit need not exist
3. This corresponds to the flux conserving tokamak equilibrium (FCT)
4. Paradoxically, the FCT is a special case of the HBT equilibrium just
discussed
What is Flux Conservation?
1. Consider a tokamak with a large external heating source (rf, neutral
beams)
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 6 of 10
2. a. The plasma absorbs energy
b. The temperature rises
βt rises
c.
e. Poloidal currents are induced
3. Assume the heating time is slow compared to the ideal MHD inertial time
MHD:
τ M ∼ a v ti
τH
τM
Heating: τH ∼ T ( ∂T ∂t )
4. The plasma evolution can be thought of as a series of quasistatic
equilibria, each one satisfying the Grad-Shafranov equation
ρ
dv
= J × B − ∇p
dt
neglect when τH
τM
5. Assume the heating time is fast compared to the resistive diffusion time
Resistive time τD ∼
τD
a2 μ0
η
τH
6. If we neglect resistive diffusion, then during the heating process the
plasma behaves electrically, like a perfect conductor
7. The FCT assumptions τD
τH
τM imply that the free functions
p ( ψ ) , F ( ψ ) must satisfy certain constraints
8. a. In general p, F are determined by the transport evolution
b. For the FCT p, F are determined by the FCT “transport prescription”
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 7 of 10
FCT Prescription for p ( ψ )
1. Assume we start with an ohmically heated tokamak before auxiliary power
is added
p ( ψ, t = 0 ) = pΩ ( ψ ) initial pressure distribution
2. At any time later in the heating sequence
a.
p ( ψ, t ) = W ( ψ, t ) pΩ ( ψ )
modeled from heating calculations
b. Often W ( ψ, t ) = W ( t ) , corresponding to a slow increase in the
magnitude of p due to heating
FCT Prescription for F ( ψ ) (The Critical Issue)
1. Since the plasma acts like a perfect conductor, the toroidal and poloidal
fluxes must be conserved. This is the FCT constraint
2. Consider a given poloidal flux surface ψ p initially and at a later time
3. For flux conservation, the toroidal flux contained within the surface
ψ p = const must remain the same as the plasma evolves. There is no
diffusion of flux. This is the FCT constraint. We must choose F ( ψ ) so this
property is preserved.
4. Calculate ψt = ψt ( ψ, t ) , ψ p = 2πψ
ψt =
∫ Bφ ( r , θ ) rdrdθ
5. Let us write ψ t as a function of F ( ψ, t )
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 8 of 10
6. Change variables
(
)
a. r , θ → ψ' r , θ ' , θ '
θ' = θ
ψ = ψ (r, θ )
b. d ψ'dθ ' =
∂ψ'
∂r
∂ψ'
∂θ
∂θ '
∂r
∂θ '
∂θ
dr dθ =
∂ψ'
dr dθ = RBθ dr dθ
∂r
7. Then
a. ψt ( ψ, t ) =
ψ
2π
'
∫0 dψ ∫0
⎛ rBφ
dθ ' ⎜⎜
⎝ RBθ
(
ψ
= 2π∫ d ψ'q ψ' , t
0
b.
⎞
⎟⎟
⎠ ψ' , θ '
)
∂ψt
= 2πq ( ψ, t )
∂ψ
8. If ψt ( ψ, t ) is to remain unchanged during the heating sequence
∂ψt
=0
∂t
then q ( ψ, t ) must be the same for each quasistatic equilibrium
9. Thus, we must choose F ( ψ, t ) so that
q ( ψ, t ) = qΩ ( ψ )
initial ohmic q profile
10. We can now relate F ( ψ, t ) to qΩ ( ψ )
q ( ψ, t ) = qΩ ( ψ ) =
1
2π
⎛ rBφ ⎞
∫ dθ ⎜⎜ RBθ ⎟⎟
⎝
⎠S
=
F ( ψ, t )
2π
2π
∫0
rdθ
R ( ∂ψ ∂r )
11. Solving for F we find that FCT Grad-Shafranov equation becomes
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 9 of 10
⎡
⎢
d
1 d ⎢
Δ*ψ = −R2
( μ0WpΩ ) −
dψ
2 dψ ⎢ 1
⎢ 2π
⎣
∫
⎤
⎥
qΩ
⎥
rdθ
⎥
R ( ∂ψ ∂R ) ⎥⎦
2
This is an exact form, using no expansions
12. It is a nonlinear partial-integro-differential equation
13. In general, it must be solved numerically
14. It can be solved approximately by variational techniques
15. In class we shall calculate an “industrial strength” solution to the FCT
equation
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 10
Page 10 of 10