22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
1. Derive δW for general screw pinch
2. Derive Suydams criterion
Screw Pinch Equilibria
'
B
B2
'
μ0p + z + θ (rBθ ) = 0
2
r
'
μ0 Jθ = −B'z
μ0 Jz =
1
(rBθ )'
r
Stability
ξ = ξ (r ) eιmθ+ ιkz
ξ = ξr er + ξθ eθ + ξz ez = ξ⊥ + ξ b
Note: b =
Bθ
B
e + z ez
B θ
B
eη = b × er =
B
Bz
eθ − θ ez
B
B
er , eη ,b → orthogonal unit vectors
1. Carry out calculation in terms of ξ, η, ξ → ξr , ξθ , ξz
ξ = ξθ
Bθ
B
+ ξz z
B
B
ξ = ξ⊥ + ξ b
η = ξθ
B
Bz
− ξz θ
B
B
ξ⊥ = ξer + ηeη
ξ = ξr
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 1 of 12
2. Check Incompressibility
ξ
⎛ξ ⎞
a. ∇ ⋅ ξ = ∇ ⋅ ξ ⊥ + ∇ ⋅ ⎜ B ⎟ = ∇ ⋅ ξ ⊥ + B ⋅ ∇
B
⎝B ⎠
⎛ ιmBθ
⎞
⎛B ∂
∂ ⎞
b. B ⋅ ∇ scalar = ⎜ θ
+ ιkBz ⎟ scalar
+ Bz
⎟ scalar = ⎜
r
z
r
∂θ
∂
⎝
⎠
⎝
⎠
Define F =
mBθ
+ kBz = k ⋅ B ,
r
k=
m
e + kez
r θ
∴B ⋅ ∇ scalar = ιF scalar
c. To make ∇ ⋅ ξ = 0 to minimize δW , we must choose ξ so that
∇ ⋅ ξ ⊥ + ιF
ξ =
ξ
B
= 0 or
ιB
∇ ⋅ ξ⊥
F
d. If k and m are such that F ≠ 0 for 0 < r < a , then ξ is bounded and we
can choose ∇ ⋅ ξ = 0 . This is the usual situation for external modes
e. Suppose k and m are chosen so that F=0 at isolated internal points
0 < r < a . Usual case for internal modes. ξ has the form
At rs , ξ is not bounded (not allowable), but only at one point
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 2 of 12
f.
Resolution: Choose ξ =
ιBF
2
F + σ2
∇ ⋅ ξ1
ξ is now bounded, but ∇ ⋅ ξ is no longer zero.
g. Calculate contribution to δWF
∇ ⋅ ξ = ∇ ⋅ ξ⊥ +
ιFξ
B
ιF ⎛ ιBF ⎞
σ2
∇
⋅
ξ
=
∇ ⋅ ξ⊥
⊥
B ⎜⎝ F2 + σ2 ⎟⎠
F2 + σ2
= ∇ ⋅ ξ⊥ +
Assume now that σ is very small, but finite
Main contribution to γp ∇ ⋅ ξ
2
comes from around r = rs where F ≈ 0
h. Expand about r = rs : F = F (rs ) + F' (rs ) (r − rs ) ≈ F' (rs ) x,
x = r − rs
0
δW =
2
2
1
1
σ4
γ
p
∇
⋅
ξ
dr
=
γ
p
∇
⋅
ξ
⊥
2∫
2∫
F2 + σ2
(
2
= πL ⎡⎢ γp ∇ ⋅ ξ ⊥ r ⎤⎥
⎣
⎦rs
∫ dx
)
2
r dr dθ dz
σ4
2
⎛ σ2 + F'2 x2 ⎞
⎜
⎟
⎝
⎠
2 ⎤
⎡
⎢ γp ∇ ⋅ ξ⊥ r ⎥
= π L⎢
⎥ σ
'
F
⎢⎣
⎥⎦
rs
2
i. For small but finite σ, δW → 0
Conclusion: Even for isolated singular surfaces, the plasma compressibility
term makes no contribution to δW
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 3 of 12
Minimize Remainder of δW
1. Separate terms
(
Q⊥ = ∇ × ξ ⊥ × B
)
= Qr er + Qη eη
⊥
Qr = ιFξ
'
⎛ B' B
Bz ⎛ B θ ⎞ ⎞
z θ
⎜
−r
Q η = ι Fη + ξ
⎜
⎟⎟
⎜ B
B ⎝ r ⎠⎟
⎝
⎠
2. κ = b ⋅ ∇b = −
B2θ
rB2
3. ∇ ⋅ ξ ⊥ + 2ξ ⊥ ⋅ κ =
4.
(ξ
⊥
)(
er
(rξ ) ' − 2B2θ ξ + ιGη
2
r
)
⋅ ∇p ξ*⊥ ⋅ κ = −
B
rB
B2θ
2
rB
p' ξ
G=
mBz
− kBθ
r
= er ⋅ (k × B )
2
1 ⎡ Bz
⎤
(rBθ )' − BθB'z ⎥
⎢
B⎣ r
⎦
5. J = ( J ⋅ B ) B =
(
6. ξ*⊥ × B ⋅ Q⊥ = B Qr η* − Qη ξ*
)
Substitute
{
2
1
δWF = ∫ dr F2 ξ + ιFη + ξ
2
2
+B
+
−
(rξ ) ' − 2B2θ ξ + ιGη
2B2θ
rB2
rB2
r
p' ξ
B
2
'
⎡ B' B
rB ⎛ B ⎞ ⎤
⎢ z θ − z⎜ θ⎟⎥
B ⎝ r ⎠⎥
⎢⎣ B
⎦
2
line bending
2
mag. comp.
pressure driven
'
⎫
⎡
J ⎡
⎛ B ⎞ ⎤⎤
⎢B ⎡ιF ξη* − ξ*η ⎤ − ξ 2 ⎢B'zBθ − rBz ⎜ θ ⎟ ⎥ ⎥ ⎪⎬ kink
⎦
B⎢ ⎣
⎢⎣
⎝ r ⎠ ⎥⎦ ⎥⎦ ⎪
⎣
⎭
(
)
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 4 of 12
Simplify
1. Note that η appears only algebraically. Complete the squares and minimize
with respect to η
η=
i
rk 20B
k 20 =
m2
r2
⎡G (rξ )' + 2kB ξ ⎤
θ ⎥
⎣⎢
⎦
+ k2
2. Remaining δW
2
a
δWF = π (2πR 0 ) ∫ dr ⎡⎢a (r ) ξ' + b (r ) ξξ' + c (r ) ξ2 ⎤⎥
0
⎣
⎦
θ z
(1)
a. integrate (1) by parts
b. lots of algebra, using equilibrium relation
3. Result:
δWF
2π2 R 0 μ0
f =
g=
=
2 2 2
2 2
⎡Fξ'2 + gξ2 ⎤ + ⎡ k r BZ − m Bθ ⎤ ξ2 a
dr
( )
⎢
⎥
∫0 ⎢⎣
⎥⎦
k20r2
⎢⎣
⎦⎥ a
a
rF2
k20
2k2μ0p'
k20
⎛ k2r2 − 1 ⎞
mBθ ⎞
2k2 ⎛
+ ⎜ 0 2 2 ⎟ rF2 + 4 ⎜ kBz −
⎟F
⎜ k r
⎟
r ⎠
rk 0 ⎝
0
⎝
⎠
Complete Calculation by Computing δWs , δWv
1. Assume no surface currents:
2. Vacuum Energy: δWv =
δWs = 0
2
1
B1 dr
∫
2μ0
∇ × B1 = ∇ ⋅ B1 = 0
3. Write B1 = ∇φ1 with ∇2 φ1 = 0
B.C.
a. Wall as r = b → n ⋅ B1
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
b
=0
∂φ1
=0
∂r b
(1)
Lecture 18
Page 5 of 12
b. n ⋅ B a + ξ = 0 → n ⋅ B1
a
(
= n ⋅∇ × ξ ⊥ × B
)
∂φ
= ιFξ ( a)
∂r a
a
(2)
Solution:
φ1 = ⎡⎣c1Im (kr ) + c2Km (kr ) ⎤⎦ eιmθ+ ιkz
∂φ1
= ⎡kc1 I'm + kc2K'm ⎤ eιmθ + ιkz
⎣
⎦
∂r
Choose c1 and c2 so that (1) and (2) are satisfied
Then δWv =
1
2μ0
2
1
∫B
dr =
1
2μ0
∫ ∇φ
*
⋅ ∇φ dr =
1
2μ0
∫ dr ⎡⎣∇ ⋅ ( φ ∇φ ) − φ ∇ φ⎤⎦
*
*
2
0
=
1
2μ0
∫
dS φ* n ⋅ ∇φ = −
2π2R0a ⎡ * ∂φ ⎤
φ
μ0 ⎢⎣ ∂r ⎥⎦ a
Substitute
δWv
2π2R 0
⎡ r2 ΛF2 ⎤ 2
=⎢
⎥ ξ ( a)
μ0 ⎢⎣ m ⎥⎦ a
(
(
'
m K a ⎡⎢1 − KbIa
Λ=−
kaK'a ⎢1 − Kb' I'a
⎣
1 + (a b)
) (I K ) ⎤⎥
) (I K ) ⎥⎦
'
b
a
' '
b a
2m
≈
1 − (a b)
2m
kb
1
≈
m
ka → ∞
ka
kb → ∞
≈1
kb → ∞
ka ∼ 1
Summary
δW for general screw pinch
δW
2π2R 0 μ0
=
∫
a
0
2
2
⎡fξ'2 + gξ2 ⎤ dr + ⎡⎢⎛ krBz − mBθ ⎞ rF + r1 ΛF ⎤⎥ ξ2 a
⎜
⎟
( )
⎢⎣
⎥⎦
⎟
m ⎥⎦
k20r2
⎢⎣⎜⎝
⎠
a
internal modes:
ξ ( a) = 0
external modes:
ξ ( a) ≠ 0
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 6 of 12
Suydam’s Criterion
a. Necessary condition for stability
b. Tests against localized interchanges (external modes)
c. Only necessary, because a special “localized” trial function is used
Mathematical Motivation
a. Choose k such that F (rs ) = 0 for same rs in 0 < rs < a
b. Around this point F ≈ 0, g ≈
2k2
k20
p' < 0 destabilizing
c. A localized mode can still give a finite contribution if ξ' is large.
large
'
fξ
small
Physical Motivation
a. interchange plasma and field: plasma wants to expand, field lines want to
contract
b. interchange is more difficult with shear. As interchange takes place, field lines
are bent from one surface to another.
Derivation
1. look as δWF in the vicinity of x = r − rs
2. assume internal mode so that ξ ( a) = 0
3. assume localized internal mode F (r ) ≈ F (rs ) + F' (rs ) x = F' (rs ) x
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 7 of 12
0
⎡ r3F'2 ⎤
⎥ x2
f ≈⎢ 2 2
⎢ k r + m2 ⎥
⎣
⎦rs
Then
⎡ 2k2r2p'μ0 ⎤
g≈⎢ 2 2
⎥
2
⎢⎣ k r + m ⎥⎦r
s
and
δWF
2π2R 0
⎡ r3F'2 ⎤
⎥
=⎢ 2 2
μ0 ⎢ k r + m2 ⎥
⎣
⎦rs
∫
⎡ ⎛ dξ ⎞2
⎤
− Ds ξ2 ⎥
dx ⎢x2 ⎜
⎟
dx ⎠
⎣⎢ ⎝
⎦⎥
⎡ 2k2p'μ0 ⎤
Ds = − ⎢
⎥
2
⎢⎣ rF'
⎥⎦r
s
mBθ ⎞
⎛
4. Simplify Ds as r = rs , ⎜ kBz +
⎟ = 0 definition
r ⎠r
⎝
s
5. Write q (r ) =
rBz
R 0Bθ
⎛
⎛
mBθ ⎞
m 1⎞
Then F (r ) = kBz ⎜1 +
⎟
⎟ = kBz ⎜ 1 +
krBz ⎠
kR 0 q ⎠
⎝
⎝
but, at r = rs
kR 0 ⎛ R 0Bθ ⎞
1
=⎜
resonant condition
⎟ =
m
rB
q
rs )
(
z ⎠rs
⎝
⎡
q (rs ) ⎤
so that F (r ) = kBz (r ) ⎢1 −
⎥
q (r ) ⎦⎥
⎣⎢
F' (r )
rs
⎡
⎡ q' ⎤
q (rs ) ⎤
= kB'z ⎢1 −
⎥ + kBz (rs ) q (rs ) ⎢ 2 ⎥
q (r ) ⎥⎦
⎣⎢ q ⎦⎥rs
⎣⎢
rs
0
⎛
q' ⎞
= ⎜ kBz ⎟
⎜
q ⎟⎠
⎝
rs
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 8 of 12
∴
Ds = −
2μ0p'q2
2
r2B2z q'
⎛
2 '2
∫ dx ⎜⎝ x ξ
6. δW ∝
only a function of equilibrium quantities (no m’s and k’s)
− Ds ξ2 ⎞⎟
⎠
a. if p' > 0, Ds < 0 → stability
b. assume p' < 0 interesting case, Ds > 0 stability?
7. Vary ξ → ξ + δξ to determine minimizing ξ (r )
⎛
'2
⎡
2 '2
∫ dr ⎜⎝ Fξ
∫ dx ⎢⎣x ξ
+ gξ2 ⎞⎟ →
⎠
− Ds ξ2 ⎤⎥ →
⎦
(Fξ ) − gξ = 0
'
'
(x ξ ) + D ξ = 0
2 '
'
s
8. We can solve Euler–Lagrange equation: assume ξ = xP
p (p + 1) + Ds = 0
p1,2 = −
1
1 1
+ (1 − 4Ds ) 2
2 2
9. Need to distinguish two cases: Ds > 1 4 , Ds < 1 4
⎛
2 '2
∫ ⎜⎝ x ξ
10. Note:
+ Ds ξ2 ⎞⎟ dx = −x2 ξξ' = −px2p +1
⎠
p>−
1
bounded → alternate function
2
p<−
1
unbounded → not allowable
2
p=
1
logarithmic divergence → not bounded
2
11. Consider 1 − 4Ds < 0
ξ=
kr =
1
x
12
⎡c1 sin (kr ln x ) + c2 cos (kr ln x ) ⎤
⎣
⎦
1
( 4Ds − 1)1 2
2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 9 of 12
12. Show oscillatory root always leads to instability by choosing a modified trial
function
a. In I and V, ξ = ξ' = 0 → δWI = δWV = 0
(
b. In II and IV ξ satisfies x2 ξ'
0=
∫ ⎢⎣( x ξ )
⎡
2 '
'
⎤
+ Ds ξ ⎥ ξdx =
⎦
) +D ξ = 0
'
s
⎡
2 '2
∫ dx ⎢⎣−x ξ
+ Ds ξ2 ⎥⎤ + x2 ξξ'
⎦
−δW
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 10 of 12
∴ δWII = x2 ξξ'
r2
r1
δWIV = x2 ξξ'
r4
r3
=0
=0
c. Region III ξ = ξ0 = const, ξ' = 0
δWIII =
⎛
2 '2
∫ ⎜⎝ x ξ
− Ds ξ2 ⎞⎟ dr = −Ds ξ20 Δr
⎠
d. by assumption Ds >
Δr = r3 − r2
1
4
∴δW = −Ds ξ20 Δr < 0 → instability
e. when Ds < 1 4 no oscillatory solutions exist. One root is not allowable, the
other is allowable
Conclusion: when Ds < 1 4 not localized, oscillatory trial functions can be
chosen. System is stable to localized interchanges
when Ds > 1 4 localized treat functions exist which make δW < 0
Ds <
1
Suydams criterion
4
2
⎛ q' ⎞
rB2z ⎜ ⎟ + 8μ0p' > 0
⎜q⎟
⎝ ⎠
for stability
Destabilizing term: 8μ0p' → pressure gradient, bad curvature
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 11 of 12
2
Stabilizing term:
rB2z
q'
q2
→ shear, line bending magnetic energy
Oscillation theorem
If suydams criterion is violated, there is always a zero mode, macroscopic
mode with maximum growth rate.
This is significance of Suydams criterion.
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 18
Page 12 of 12