22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
1. Stability of the straight tokamak
1. pressure driven modes (Suydams Criterion)
2. internal modes
3. external modes
2. Tokamak Ordering
∈ ≡ a R0
Bθ
∼∈
Bz
1
2μ0p
q∼1
B2z
∼ ∈2 or ∈
3. Suydams Criterion
p' ∼
p
a
2
rB2z
∴
⎛ q' ⎞
B2
⎜ ⎟ ∼ z
⎜q ⎟
a
⎝ ⎠
8μ0p'
ιB2z
( q q)
'
2
∼
μ0p
B2z
∼∈, ∈2
1. Over most of the plasma the destabilizing term in Suydams criterion is much
smaller than the stabilizing contribution.
∴ Suydams criterion satisfies over most of the plasma
2. Exception:
near r=0
p' (r ) ≈ p" ( 0 ) r
q (r ) ≈ q ( 0 ) + q" ( 0 )
2
r
≈ q (0)
2
q' (r ) ≈ q" ( 0 ) r
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 1 of 10
2
rB2z
q'
q2
+ 8μ0p' > 0
2
⎡⎛
"⎞ ⎤
⎢⎜ Bz q ⎟ ⎥ r3 + 8μ ⎡p" ⎤ r > 0
0 ⎣ ⎦
0
⎢⎜⎝ q ⎟⎠ ⎥
⎣
⎦
0
dominates near r=0
3. Resolutions: straight case
4. Resolution: Toroidal Case
a. In toroidal case there are important modifications to Suydams criterion:
Mercier criterion. These corrections can eliminate the need for flattening
the p profile
b. Simple, low β circular limit of Mercier criterion
2
⎛ q' ⎞
rB2z ⎜ ⎟ + 8μ0p' 1 − q2 > 0
⎜q⎟
⎝ ⎠
(
)
toroidal correction
c. For q ( 0 ) > 1 , pressure term is stabilizing: average curvature is formable.
5. Conclusion:
Localized interchange modes are not very important in a straight tokamak
because β is very small. Near r=0, we need either flattening (straight) or
q ( 0 ) > 1 (toroidal)
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 2 of 10
Internal Modes in a Straight Tokamak
m ∼ 1 poloidal wave number
L z = 2πR 0
2π 2πR 0
n
=
→k = −
k
n
R0
a R0 ≡ ∈
λ=
Bθ Bz ∼ ∈
n ∼ 1 toroidal wavenumber
1. Use this ordering to simplify f and g
a. f =
rF2
k20
k20 = k2 +
m2
r2
F = kBz +
=
mBz
R0
=
n2
R 20
+
m2
r2
≈
m2
r2
mBθ
nBz mBθ mBz
=−
+
=
r
R0
r
R0
⎡ 1 n ⎤ mB0
⎢q − m⎥ ≈ R
⎣
⎦
0
⎡1 n ⎤
⎢q − m⎥
⎣
⎦
2
∴f =
b. g1 =
g3 =
g2 =
⎡ n BθB0 ⎤
+
⎢−
⎥
rBz ⎦
⎣ m
2
2 2
r3B20 ⎛ 1 n ⎞
r3 m B0 ⎛ 1 n ⎞
2
2
−
=
⎜
⎟
⎜ − ⎟ ∼ ∈ aB0
m2 R20 ⎝ q m ⎠
R20 ⎝ q m ⎠
2k 2μ0p'
k20
=
2 2
2 2 3
2k2 ⎛ 2 2 m Bθ ⎞ 2n B0r
k
B
−
=
⎜
⎟
z
rk 04 ⎜⎝
r2 ⎟⎠
R 04m2
k20r2 − 1
k20r2
)
2
⎛ n r ⎞ ' ∈2 βB20
'
p
=
2
⎜
⎟ p ∼
a
m2
⎝ m R0 ⎠
2n2 r2
R 20
(
(small)
4 2
⎛ n2
1 ⎞ ∈ B0
(small)
⎜⎜ 2 − 2 ⎟⎟ ∼
a
q ⎠
⎝m
2
rB2 ⎛ 1 n ⎞
∈4 B20
rF ≈ m − 1 20 ⎜ − ⎟ ∼
a
R0 ⎝ q m ⎠
2
(
2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
)
Lecture 19
Page 3 of 10
2. Therefore
δWF
2π2R 0 μ0
≈
B20
R20
∫
2
(
)
2
⎛ n 1⎞
r dr ⎜ − ⎟ ⎡⎢r2 ξ' + m2 − 1 ξ2 ⎤⎥
⎦
⎝m q⎠ ⎣
Stability of Internal Modes
1. m ≥ 2 → stable, both terms positive.
2. m = 1
nq (r ) > 1
1−
(n=1 worst)
1
≠0
q
2
⎛1
⎞ 2
I ∝ ⎜ − 1 ⎟ ξ' > 0
⎝q
⎠
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 4 of 10
3. m=1 q(r) < 1 somewhere
use the following trial function
a. ξ = 1
=
1⎛
x⎞
1− ⎟
2 ⎜⎝
δ⎠
=0
0 < r < rs − δ
rs − δ < r < rs + δ
q' (rs ) (r − rs )
1
1
=
−
q (r ) q (rs )
q2 (rs )
= 1 − q' (r − rs )
r > rs + δ
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 5 of 10
b.
δWF
=
2
2 π R 0 μ0
=
B20
R 20
2
δ
∫
2
r dr ⎡1 − 1 + q'x ⎤ r2 ξ'
⎣
⎦
−δ
B20 ⎛ 3 '2 ⎞
⎜r q ⎟
⎠rs
R 20 ⎝
2
1 ⎞
⎟ dx
⎝ 2δ ⎠
2 ⎛
∫ (x) ⎜ −
1
δ
6
=
c.
B20 ⎛ 2 '2 ⎞
⎜r q ⎟ δ
⎠rs
6R 20 ⎝
δWF → 0 as δ → 0
d. with an m=1 resonant surface in the plasma, the system is marginally
stable in leading order; i.e. if q ( 0 ) < 1
e. to test stability for this case we must calculate δW to next order for the
m=1 mode.
Calculate Next Order δW for m=1, n=1 Mode
1.
f =
2.
g=
3.
(
2
r3m2B2z
R 20 m2 + n2r2 R 20
m2 − 1 + k2r2
k20r2
)
⎛
1⎞
⎜n − ⎟
q⎠
⎝
rF2 +
2k 2μ0p'
k20
+
mBθ ⎞
2k2 ⎛
kBz −
⎟F
4 ⎜
r ⎠
rk 0 ⎝
≈
rB20 ⎛ 1
⎞⎛1
n2r2 ⎡
2⎞
'
+
− n ⎟ ⎜ − n − 2n − ⎟
μ
2p
⎢
0
2
2 ⎜q
q⎠
R 0 ⎢⎣
R0 ⎝
⎠⎝q
=
rB20
n2r2 ⎡
'
μ
−
2
p
⎢
0
R 20 ⎢⎣
R20
δWF
2
2π R 0 μ0
=
⎛
'2
∫ dr ⎜⎝ fξ
⎛1
⎞⎛
1⎞
4
⎜ − n ⎟ ⎜ 3n + ⎟ ∼ ∈
q
q
⎝
⎠⎝
⎠
+ gξ2 ⎞⎟
⎠
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 6 of 10
Use same trial function as before
Summary of Internal Modes in a Straight Tokamak
1. m ≥ 2 stable
2. m=1, n=1 worst case for n=1, requires q ( 0 ) > 1 for stability
3. internal modes do not limit β , or I ( q ( a) ) , but clamp q ( 0 ) ≈ 1 by sawtooth
oscillations
4. To show stability we needed to calculate δW = ∈2 δW2 + ∈4 δW4
0
const.
Consider now External Modes
1. Vacuum is force free fields
2. m=1 Kruskal Shafranov limit
3. High m external kinks
Subtle Issues For External Modes
Vacuum as force free Plasma
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 7 of 10
1. cold, but highly conducting plasma surrounds core - more realistic than
vacuum
2. Is there any difference in stability in these 2 cases.
I σ=0
might anticipate big difference
II σ = ∞
1
2
3. But! Vac. δWv =
∫
2
B1 dr
(
)
1
2
⎡ 2
⎤
dr ⎢ B1 + γp ∇ ⋅ ξ + ξ*⊥ ⋅ J × B1 + ( ξ ⋅ ∇p ) ∇ ⋅ ξ*⊥ ⎥
2
⎣
⎦
∫
FFP δWFFP =
in FFP J = p = 0 in equilibrium
δWFFP =
2
1
dr B1
2
∫
Thus, FFP same as Vac.
might anticipate no difference in stability since
δW ’s are the same for each.
4. How do we calculate δWv , δWFFP . Minimizing condition is
∇ × B1 = ∇ ⋅ B1 = 0 “vacuum” fields
Vac: BC. n ⋅ B1
FFP n ⋅ B1
Sw
Sw
=0
=0
(
n ⋅ B1
and B1 = ∇ × ξ × B
n ⋅ B1
Sp
Sp
= n ⋅ ∇ × ξ⊥ × B
= n ⋅ ∇ × ξ⊥ × B
Sp
Sp
)
5. In the FFP we must check that a well behaved B1 always gives rise to well
behaved ξ . This is an additional constraint that can make the FFP more
stable
6. Example: cylindrical screw pinch
B1r + ιFξ → ξ = −
ιB1r
F
a. if k, m are such that F ≠ 0 in FFP region then ξ is well behaved and
δWv = δWFFP
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 8 of 10
b. Usually, however F = 0 in FFP for external modes. Then, ξ is
unbounded
leads to infinite energy. This is not an allowable
displacement
c. Calculation must be redone with new boundary condition B1r (rs ) = 0 .
Thus is an additional constraint, which is equivalent to placing a
conducting wall at r = rs
external
internal mode with wall at singular surface.
d. ∴ FFP is more stable than Vac if F (rs ) = 0 in FFP region.
7. But !! most realistic case is neither vacuum nor FFP, but a plasma with a
small resistivity
In that case δWη =
and
1
2
∫
2
B1 dr
∂B1
ιη
= ∇v ( v × B − ηJ) → B1 = ∇ × ξ × B −
∇ × ∇ × B1
∂t
ω
(
)
Careful analysis choose that ξ is bounded at the resonant surface.
∴ Stability boundary is the same as Vacuum case, but growth rate is smaller,
depending upon resistivity
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 19
Page 9 of 10
Summary
Vacuum: certain stability boundary, growth rate ∼ ν T R
Ideal FFP: same stability boundary, growth rate if k ⋅ B ≠ 0
much more stable ( γ = 0 ) if k ⋅ B = 0
Resistive FFP: same boundary as vacuum but
⎛τ
⎞
γ ∼ γMHD ⎜ MHD ⎟
⎝ τRES ⎠
ν
0< ν <1
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
10
Lecture 19
Page 10 of