22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Ballooning mode equation
δW =
π
μ0
W (ψ) =
∫ dψ W ( ψ )
2
⎡
⎤
2μ0RBp dp 2
⎛ 1 ∂X ⎞
2
⎥
−
κ
−
κ
X
Jdχ ⎢ kn2 + k2t ⎜
k
k
k
⎟
t
n
t
n
t
−∞
dψ
⎢⎣
⎥⎦
B2
⎝ JB ∂χ ⎠
∫
∞
(
)
(
)
Mercier Criterion
1. In using the quasimode representation we have had to assume the solution
Xϕ converges sufficiently rapidly as χ → + ∞ .
2. Whether or not convergence is acceptable depends upon equilibrium profiles
and parameters.
3. Analysis of Euler-Lagrange equation for X indicates that there are two classes
of solutions for large χ depending on profiles
4.
5. Oscillating solutions give rise to unbounded energy: W → ∞ . Strong
convergence gives rise to bounded energy
6. Oscillatory case implies that ballooning mode formation is not valid as χ → ∞ .
However, for this case a trial function of the form
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 1 of 11
leads to δW < 0 (instability)
7. For exponential solutions, one starts with the strongly converging solution as
χ = −∞ and integrates to the right.
8. The condition of oscillatory solutions is known as the Mercier criterion. When
the Mercier criterion is violated, the solutions oscillate for large χ . The
ballooning mode equation is not valid but this does not matter as the system
is already unstable to interchanges.
9. When the solution’s do not oscillate the Mercier criterion is satisfied and the
system is stable to interchanges, and the ballooning mode formalism is solid.
In this case one integrates the equation and looks to see if there is a zerocrossing. If there is one, the system is unstable to ballooning modes
10. Relation of Mercier-Suydam
Suydam: local behavior as x → 0 near regular surface in space
Mercier: behavior as χ → ∞ in pseudo angle
Mercier is actually fourier transform of “Suydam like” analysis
e ιS : S α ( ψ − ψ 0 )
∫
χ
χ0
∂ ⎛ JBθ
⎜
∂ψ0 ⎝ R
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
⎞ '
⎟ dχ
⎠
Lecture 22
Page 2 of 11
≈ ( ψ − ψ0 ) χ
dq ( ψ )
dψ
pseudo angle
radial localization
≈ xk
transform variable
Forms of the Mercier Criterion
1. Exact form DM < 1 4 for stability
μ p' ⎡ RBp κn
Λ
1
DM = 0 2 ⎢2
+ 4 − 2
2
B
B
B
q' ⎢⎣
Λ ⎤
⎥
B2 ⎥⎦
R2Bp2
JB2
⎛
R 2Bp2 ∂q ⎞
⎟
Λ = F ⎜ μ0p'F −
⎜
J ∂ψ ⎟
⎝
⎠
Q =
2π
QB2
0
R2Bp2
∫
J dχ
∫
2π
0
B2
R2BP2
J dχ
2. For tokamaks, pressure is low: β ∼ ∈, ∈2 . As with Suydam criterion, Mercier
criterion is satisfied over most of the discharge because of low β . Only
problem is near the origin.
a. For a βp ∼ 1 tokamak with circular cross section, Mercier becomes
r2q'
2
(
)
+ 4rβ' 1 − q2 > 0
q2
Near r=0 q' is very small and we require
q0 > 1
b. Near the origin for non-circular tokamaks, the criterion becomes
1<
q20
⎧
4
⎪
⎨1 −
1 + 3κ2
⎪⎩
⎡
2
( κ − 1)2 βp0 ⎥⎤ ⎫⎪
⎢ 3 κ − 1 ⎛⎜ κ2 − 2δ ⎞⎟ +
⎬
⎢ 4 κ2 + 1 ⎝
∈⎠
κ ( κ + 1) ⎥ ⎪
⎣
⎦⎭
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 3 of 11
for κ = 1 , triangularity and βp have no effect.
for κ > 1 , βp is destabilizing, +δ stabilizing
good Mercier: elongation and outward triangularity, moderate βp and
q0 > 1
c. Why do toroidal effects introduce such big changes since they are of order
∈ . Compute κn = n ⋅ (b ⋅ ∇b )
Circle: κn ≈ −
Torus: κn ≈ −
B2θ
rB2θ
B2θ
rB20
≈−
κr ≈ −
B2θ
rB20
∈2
∈
eR
−
R
−
1
R0
b ≈ ez +
Bθ
e
B0 θ
b ≈ eθ +
Bθ
e
B0 θ
⎛
⎞
r
cos θ + ...⎟ ( er cos θ − eθ sin θ )
⎜1 −
R0
⎝
⎠
B2θ ⎛
q2 ⎞
−
1
⎜
⎟
2 ⎟⎠
rB20 ⎜⎝
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 4 of 11
Ballooning Modes
1. Simple limit ballooning mode equation for βp ∼ 1 , circular cross section
plasma with gradient in p' . rβp' ∼ 1 ∈
2. χ → θ
J→
R → R0
r
Bθ
κn → −
cos θ
R0
B → B0
Bp → Bθ (r )
dp
1 dp
→
dψ rBθ dr
χ
1
∂ JBθ
dχ' =
∂ψ R
R 0Bθ
and χ =
∫
χ0
⎡ '
⎤
2μ0rqp'
θ
−
θ
sin
sin
(
)
⎢q ( θ − θ0 ) +
⎥
0
R 0B2θ
⎣⎢
⎦⎥
dp ⎞
⎛
F
μ
2μ0r dp
∂ ⎜ 0 dψ ⎟
⎜
⎟=−
sin θ
2
R 0B0Bθ dr
∂χ ⎜ B
⎟
⎜
⎟
⎝
⎠
3. This gives for the Euler-Lagrange equation
(
)
∂ ⎡
∂X ⎤
1 + Λ2
+ α ⎡⎣Λ sin θ + cos θ⎤⎦ X = 0
⎢
∂θ ⎣
∂θ ⎥⎦
Λ = S ( θ − θ0 ) − α ( sin θ − sin θ0 )
S=
rq'
q
α=−
2 μ0r2p'
R 0B2θ
= −q2R 0β'
4. Solved numerically gives S vs α diagram
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 5 of 11
I first region of stability (goes unstable at high α )
II second region of stability (eventually becomes stable at high α )
5. Region I maximum β . Set α ≈ ⋅6 S to determine critical profile for maximum
β . For qa
1, q0 = 1
β t ≤ ⋅3
∈
qa
(circle)
6. Numerical studies by Sykes, Yamazaki
cross sectional area
2B0 A
q* =
μ0R 0I
∈κ
βt < ⋅22
q*
= 0.44
I0
aB0
For optimized profiles with elongation and outer triangularity
7. For κ = 2, q0 = 1.5, ∈= 1 3 → βt ≈ 10%
Second Stability
1. Why does such a region exist
2. Examine local shear
q (ψ ) =
1
2π
∫
2π
0
q ( ψ, χ ) dχ
q=
JBθ
R
local shear
shear
r ∂q
= S − α cos θ
q ∂r
pressure driven modulation
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 6 of 11
3. Note: bad curvature occurs as θ = 0 due to toroidal field. Stabilizing term due
2
⎛ rq' ⎞
to shear α ⎜
⎜ q ⎟⎟
⎝
⎠
External Kinks
1. Consider surface current model. p=const, circular cross section
δW = δWp + δWs + δWv
δWp =
δWv =
∫
p
B12
dr
2μ0
∫
B1
dr
2μ0
v
δWs =
>0
2
>0
⎛
2
1
B2 ⎞
ds n ⋅ ξ n ⋅ ∇ ⎜ p +
⎟
⎜
2
2μ0 ⎟⎠
⎝
s
∫
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 7 of 11
=
⎡
⎤
B2
2
1
dS n ⋅ ξ ⎢2p κn +
κn − κn ⎥
⎢
⎥
2
2μ0
⎣
⎦
(
∫
curvature term
)
=
Q+Q
2
kink term
toroidal curvature
δWs
2π ∈2 B20R 0 μ0
=−
1
2π
∫
2π
0
dθ ξn
2
⎡⎛ B ⎞2 ⎛ β ⎞
⎤
⎢⎜ θ ⎟ + ⎜ t ⎟ cos θ⎥
⎢⎝ ∈ B0 ⎠
⎥
⎝∈⎠
⎣
⎦
high β ballooning effect
kink term
2. Modes have the structure of pressure driven kinks
3. Stability diagram (sharp boundary model)
4. Full numerical studies using optimized profiles and cross-sections
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 8 of 11
βt = ⋅14
and
∈κ
I
= ⋅028
q*
aB
q* > q* min
Note, no dried q* min for following modes.
5. For κ = 1, q* = 1.7
βt = .08 ∈
For optimized Troyon q* min = 1,5 , κ = 1.6
βt = .15 ∈
Set ∈= 1 3 →
βt = 5%
Near the regime of reactor interest!!
Requirements on β
1. There are two basic fusion energy requirements where β enters
2. Ignition and Wall Loading
3. Ignition ( Te = Tι = T )
a. Pf =
β=
b. Pf =
n2 σv
n2 T 2 σv
Qf = Qf
4
4 T2
4μ0nT
B2
Qf σv β2B4
4 T 2 16μ20
as 10 keV, σv = 10−22 m3 sec
c.
Pf = .5 × 106 β2B4 ω m3
d. Pf = Pe
e. Pe =
3nT 3 βB2
βB2
=
= .6
ω m3
τ
τ 4μ0
τ
∴ .5 × 106 β2B4 = .6
β B2
τ
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 9 of 11
βτ =
1.2
B2
4. Wall Loading
.4
a. PE = ηPf = ηPf V = η.5 × 106 β2B4 2πR 0 πa2
3
R
b. PE = 4 × 106 β2 B4 R 0a2 ≈ 4 × 106 β2 B4 0 a3 ≈ 1.2 × 106 β2 B4a3
a
PE = 1.2 × 106 β2 B4a3
c.
Pf = PW A
Pf 2π2 R 0a2 = PW 4π2 aR 0
Pf =
2PW
a
.5 × 106 β2 B4 =
2PW
a
d. Eliminate a from step b.
a=
PE1 3
13
1
(
β2 3B4 3 1.2 × 106
)
13
= 9 × 10
−3
⎛ PE ⎞
⎜⎜ 2 4 ⎟⎟
⎝β B ⎠
13
a = 9 × 10
−3
⎛ PE ⎞
⎜⎜ 2 4 ⎟⎟
⎝β B ⎠
e. Substitute a into (C)
6
2
4
.5 × 10 β B =
βB2 = 3 × 10−3
f.
2PW
9 × 10−3
13
⎛ β2B4 ⎞
⎜⎜
⎟⎟
⎝ PE ⎠
34
PW
PE1 4
For PW = 4 × 106 ω m2 and PE = 109 watts
βB2 ≤ 1.5
g. For B=5T at R = R 0 then
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 10 of 11
β < 6%
5. The Troyon limit
β < .03
I
aB
β (% ) < βN
or
β < ⋅15
IMD
aB
βN = 3
∈κ
1 1.8
≈ .15 × ×
= 6%
q*
3 1.5
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 22
Page 11 of 11