22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17: Stability of Simple Function
Memory of the Energy Principle
δW ≥ 0 for all displacements implies stability
The potential energy is given by
δW = δWF + δWS + δWV
1
δWF =
2
⎡ Q2
⎤
2
*
*⎥
⎢
−
ξ
⋅
×
+
∇
⋅
ξ
+
ξ
⋅
∇
∇
⋅
ξ
dr
J
Q
rp
p
⊥
⊥⎥
∫ ⎢ μ0 ⊥
P
⎣⎢
⎦⎥
(
δWS =
1
2
c ⎛
2
B2 ⎞fg
d
dS
n
n
p
⋅
ξ
⋅
∇
+
d
⎜
⎟g
⊥
∫
d ⎜
2μ0 ⎟⎠ghg
ed ⎝
δWV =
1
2
∫ dr
V
12
B
= ∇ ⋅B
=0
, ∇×B
1
1
μ0
n⋅B
1
Sp
)
n⋅B
1
(
SW
=0
) (
)
⋅ ∇ n ⋅ ξ − n ⋅ ξ n ⋅ (n ⋅ ∇ ) B
=B
1
⊥
⊥
Only Appearance of ξ&
δW1 =
2
1
γp ∇ ⋅ ξ
∫
2
(
)
a. Minimizing condition : close ξ& as ∇ ⋅ ξ = 0 ⎡⎣B ⋅ ∇ ∇ ⋅ ξ = 0⎤⎦
b. Possible if operator B ⋅ ∇ can be inserted.
c. Not possible : symmetry B ⋅ ∇ ≡ 0
∇ ⋅ ξ = ∇ ⋅ ξ⊥ + B ⋅ ∇
ξ&
B
= ∇ ⋅ ξ⊥
d. Not possible : closed line case: ∇ ⋅ ξ = F (p ) , ∇ ⋅ ξ = ∇ ⋅ ξ ⊥ =
φ
μ
∇ ⋅ ξ⊥
Β
μ
φ
Β
Final Step: Intuitive Form of δWF
1
δWF =
2
⎡ Q2
⎤
2
*
*⎥
⎢
dr
J
Q
rp
p
−
ξ
⋅
×
+
∇
⋅
ξ
+
ξ
⋅
∇
∇
⋅
ξ
(
)
⊥
⊥
∫ ⎢ μ0 ⊥
⎥
⎢⎣
⎥⎦
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
(
)
standard
Lecture 17
Page 1 of 12
a.
Q
2
= Q⊥
2
+ Q&
2
(
)
(
)
b. ξ*⊥ ⋅ J × Q = ξ*⊥ × b ⋅ Q⊥ J& + ξ*⊥ ⋅ J⊥ × b Q&
c.
J⊥ =
b × ∇p
B
( J × B = ∇p )
(
d. Q& = b ⋅ ∇ × ξ ⊥ × B
)
(
= b ⋅ B ⋅ ∇ξ ⊥ − ξ ⊥ ⋅ ∇B − B ∇ ⋅ ξ ⊥
(
)
= −B ∇ ⋅ ξ ⊥ − 2ξ ⊥ ⋅ κ +
)
μ0
ξ ⊥ ⋅ ∇p
B
κ = b ⋅ ∇b
2. Substitute task
1
1
δWF =
2
2
3
4
5
⎡Q 2
⎤
2
2
B2
⎢ ⊥
⎥
*
*
∫ dr ⎢ μ0 + μ0 ∇ ⋅ ξ⊥ + 2ξ⊥ ⋅ κ + γp ∇ ⋅ ξ − 2 ξ⊥ ⋅ ∇p κ ⋅ ξ⊥ − J × ξ⊥ × b ⋅ Q⊥ ⎥
⎢⎣
⎥⎦
(
)(
)
(
)
1. line bending > 0
2. magnetic compression > 0
3. plasma compression > 0
4. pressure driven modes + or 5. current driven modes + or Classes of MHD Instability
1. Internal or fixed boundary: plasma surface is held fixed during perturbation:
n ⋅ ξ⊥
= 0 , same as a conducting wall
Sp
2. External or free boundary: plasma surface is allowed to move: n ⋅ ξ⊥ ≠ 0 . Often
the most severe stability criteria
3. Current driven modes: also called kink modes. J& is the most dominant
destabilizing term. Modes driven by parallel current. Important in tokamaks,
RFP: (K-S limit, saw tooth oscillations, disruptions). In general modes have
long wavelength, low m, n. Cures: tight aspect ratio, low current, packed
current profiles, conducting wall.
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 2 of 12
4. Pressure driven modes: κ∇ p dominant destabilizing term. Special cases
interchange or flute, following mode, sausage instability. Kaydoms criterion,
mercuir criterion. Important in tokamak, RFP, stillaratio, EBT, mirror. In general
long & wavelength, short ⊥ wavelength. Cures: low β , shear, average
formable curvature (min B, magnetic wall)
Applications Today
1. θ pinch
2. Z pinch
Procedure
1. Sine equilibrium J0 ,B0 ,P0
2. Test in compressibility condition for ξ&
3. Minimize δW with respect to ξ⊥
4. If δWmin > 0 stable
δWmin < 0 unstable
δWmin = 0 marginally stable
θ Pinch
1. Equilibrium: p (r ) ,Bz (r ) , Jθ (r )
μ0 Jθ = −B'z
p (r ) +
B2z (r )
2μ0
=
B20
2μ0
2. Stability: ξ (r ) = ξ (r ) eιmθ+ιkξ : Fourier analyze analog θ and z
3. Check compressibility
(
)
∇ ⋅ ξ = ∇ ⋅ ξ ⊥ + ∇ ⋅ ξ & e x = ∇ ⋅ ξ ⊥ + ι kξ &
Set ∇ ⋅ ξ = 0 : ξ& = −
ι
∇ ⋅ ξ⊥
k
( ξ& = ξ z )
OK if k ≠ 0
4. Evaluate terms in δWF
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 3 of 12
a.
κ = b ⋅ ∇b =
∂
e =0
∂z x
no pressure driven terms
b. J& = J ⋅ b = J0 eθ ⋅ ex = 0 no current driven terms
5. Conclusion:
δWF ≥ 0
sum of positive terms
δWs = 0
no surface currents
δWv ≥ 0
positive term
θ pinch is stable at any value of β
worst case: δW → 0 as k → 0
Z Pinch
1. Equilibrium: ρ (r ) ,Bθ (r ) , Jz (r )
μ0 Jz
ρ' +
'
rBθ )
(
=
r
Bθ
(rBθ )' = 0
μ0r
2. Stability: ξ (r ) = ξ (r )
ιmθ+ ιkz
3. Check incompressibility: & → θ
∇ ⋅ ξ = ∇ ⋅ ξ ⊥ + ∇ ⋅ ξ& e θ = ∇ ⋅ ξ ⊥ +
But ∇ ⋅ ξ = 0
ξ&r −
ι
r∇ ⋅ ξ ⊥
m
ιmξ&
r
ξ& = ξ θ
ok if m ≠ 0
4. Evaluate terms in δWF
a.
J& = J ⋅ b = Jz ez ⋅ eθ no current driven terms
b.
κ = b ⋅ ∇b = +eθ ⋅ ∇eθ =
e
1 ∂
eθ = − r
r ∂θ
r
⎛ ξ* ⎞ 2p'
ξr
−2 ξ⊥ ⋅ ∇p ξ*⊥ ⋅ κ = −2ξrp' ⎜ − r ⎟ =
⎜ r ⎟
r
⎝
⎠
destabilizing term
(
)(
) (
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
)
2
< 0 if p' < 0
Lecture 17
Page 4 of 12
c.
(
)
B1 = ∇ × ξ ⊥ × B = B ⋅ ∇ξ ⊥ − ξ⊥ ⋅ ∇B − B∇ ⋅ ξ ⊥
B1r =
ιmBθ
B ξ
B ξ
ιmBθ
ξr − θ θ + θ θ =
ξr
r
r
r
r
B1⊥ =
B1z =
B1 ⊥
2
ιmBθ
ξ⊥
r
ιmBθ
ξz
r
=
m2B2θ ⎡ 2
2
ξr − ξz ⎤⎥
2
⎢
⎣
⎦
r
'
d. ∇ ⋅ ξ ⊥ + 2ξ ⊥ ⋅ κ =
2
2ξ
1
⎛ξ ⎞
(rξr )' + ιkξz − r = r ⎜ r ⎟ + ιkξz
r
r
⎝ r ⎠
B ∇ ⋅ ξ⊥ + 2ξ⊥ ⋅ κ
( for
e. ∇ ⋅ ξ = 0
'
rξr )
(
=
r
γp ∇ ⋅ ξ
2
2
B2θ
=
2
⎡
'
⎢ r ⎛ ξr ⎞ + k2 ξ 2 + ιkrξ
z
z
⎢ ⎝⎜ r ⎠⎟
⎢⎣
'
'⎤
⎛ ξr* ⎞
* ⎛ ξr ⎞ ⎥
⎜ ⎟ − ιkrξz ⎜ ⎟ ⎥
⎜ r ⎟
⎝r ⎠
⎝ ⎠
⎥⎦
m ≠ 0)
( for
+ ιkξz
⎡
'
rξr )
(
⎢
= γp ⎢
r
⎢⎣
m = 0)
2
2
+ k ξz
2
ιkξz
+
rξr*
r
( )
'
⎤
ιkξ*z
'⎥
−
(rξr ) ⎥
r
⎥⎦
(m = 0 )
Examine m ≠ 0
2 2
1
⎪⎧ m Bθ
δWF = ∫ dr ⎨
2
2
⎩⎪ μ0r
⎡
⎢⎣ ξr
2
+ ξr
2p'
+
ξr
⎥⎦
r
2⎤
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
2
2
'
⎡
'
' ⎤⎫
⎛ ξr* ⎞
B2θ ⎢ ⎛ ξr ⎞
2
2
* ⎛ ξr ⎞ ⎥ ⎪
+
r ⎜ ⎟ + k ξz + ιkrξz ⎜ ⎟ − ιkrξz ⎜ ⎟ ⎬
⎜ ⎟
μ0 ⎢ ⎝ r ⎠
⎝ r ⎠ ⎥⎪
⎝ r ⎠
⎥⎦ ⎭
⎣⎢
Lecture 17
Page 5 of 12
Minimize δWF
1. Note that ξz only appears algebraically: complete squares
ξz terms:
k2 (r )
0
⎡ ⎤
'⎥
*⎞
⎛
⎞
B2θ ⎢⎛ m2
ξ
ξ
⎛
⎞
2
⎢⎜
+ k2 ⎟ ξz + ιkrξz ⎜ r ⎟ − ιkrξ*z ⎜ r ⎟ ⎥
⎟
⎜ ⎟
μ0 ⎢⎝⎜ r2
⎝r ⎠⎥
⎠
⎝ r ⎠
⎢
⎥
⎣
⎦
a.
2
2
⎡
'
' ⎤
B2θk20 ⎢ ιkr ⎛ ξr ⎞
k2r2 ⎛ ξr ⎞ ⎥
ξz 2 ⎜ ⎟ − 4 ⎜ ⎟
b.
μ0 ⎢
k0 ⎝ r ⎠
k0 ⎝ r ⎠ ⎥
⎣⎢
⎦⎥
'
c. Choose ξz =
ikr ⎛ ξr ⎞
⎜ ⎟ minimizing condition
k20 ⎝ r ⎠
2. Then ( ξr ≡ ξ )
⎡
⎤
⎢
⎥
2
⎢
'
2
2
'
2 ⎥
1
⎢ ξ 2 ⎛⎜ m Bθ + 2μ0p ⎞⎟ + B2 r ⎛ ξ ⎞ ⎛⎜ 1 − k ⎞⎟ ⎥
dr
δWF =
θ
⎜ ⎟
⎜ r2
2μ0 ∫ ⎢
r ⎟⎠
k20 ⎟⎠ ⎥
⎝ r ⎠ ⎜⎝
⎝
⎢
⎥
⎢
⎥
m2
⎥
m2 + k2r2 ⎦
⎣⎢
1
δWF =
2μ0
2
⎡
' ⎤
2 2
2μ0p' ⎞
m2B2θ
2 ⎛ m Bθ
⎛ξ⎞ ⎥
⎢
∫ dr ⎢ ξ ⎜⎜ r2 + r ⎟⎟ + m2 + k2r2 r ⎜⎝ r ⎟⎠ ⎥
⎝
⎠
⎢⎣
⎥⎦
only appearance of k
3. k appears only in a satisfying term this term is minimized by choosing
k2 → ∞
4. Then
δWF =
1
2μ0
∫ dr ξ
2
⎛ m2B2 2μ p' ⎞
⎜ 2θ + 0 ⎟
⎜ r
r ⎟⎠
⎝
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 6 of 12
5. Stability condition
−rp ' <
m2B2θ
2μ0
6. Simplify using equil relation μ0p '+
'
Bθ
(rBθ ) ' = 0
r
'
⎛ B ⎞
⎛B ⎞
Bθ (rBθ ) ' = Bθ ⎜ r2 θ ⎟ = r2Bθ ⎜ θ ⎟ + 2B2θ
r ⎠
⎝
⎝ r ⎠
'
'
⎛ B2 ⎞
⎛ rB2 ⎞ B2
= r ⎜ θ ⎟ + B2θ = ⎜ θ ⎟ + θ
⎜ 2 ⎟
⎜ 2 ⎟
2
⎝
⎠
⎝
⎠
or
7. Then
'
r2 ⎛ Bθ ⎞
1
m2 − 4
⎜
⎟ <
Bθ ⎝ r ⎠
2
(
)
(1)
or
(rB )
2
θ
B2θ
'
<
(
)
1
m2 − 1
2
(2)
8. Typical profile
From (1)
stability for m ≥ 2
From (2)
instability for m=1 near small r, k
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
∞
Lecture 17
Page 7 of 12
9. Physical Mechanism
Examine m=0
1
1. δWF =
2μ0
⎧
⎪ 2
dr
∫ ⎨Bθ
⎪
⎩
⎡
(rξ )'
+μ0 γp ⎢⎢ r
r
⎢⎣
2
'
⎡
'
'⎤
*
⎢ r ⎛ ξr ⎞ + k2 ξ 2 + ιkrξ ⎛⎜ ξr ⎞⎟ − ιkrξ* ⎛ ξr ⎞ ⎥
⎟
z ⎜
z
z ⎜
⎢ ⎜⎝ r ⎟⎠
⎟
⎝r ⎠⎥
⎝ r ⎠
⎢⎣
⎥⎦
2
+ k 2 ξz
2
+
ιkξz
rξr*
r
( )
'
−
⎤
2μ p'
ιkξr*
(rξr )' ⎥⎥ + 0 ξr
r
r
⎥⎦
2⎫
⎪
⎬
⎪⎭
Note again that ξz only appears algebraically: complete squares.
2. ξz terms:
a. k
b.
(
2
(
B2θ
)
+ μ 0 γp ξ z
2
( ) ⎥⎤ ( ιkξ
'
⎡
rξr*
⎛ ξr* ⎞
2
⎢
+ B θ r ⎜ ⎟ + μ 0 γp
⎜ r ⎟
⎢
r
⎝ ⎠
⎣
'
⎡
(rξ )' ⎤
⎛ξ ⎞
ι ⎢B2θr ⎜ r ⎟ + μ0 γp r ⎥
r ⎥
⎢
⎝ r ⎠
⎦
B2θ + μ0 γp kξz − ⎣
B2θ + μ0 γp
)
⎥
⎦
z
) + c.c.
2
−
1
B2θ + μ0 γp
B2θr
(rξr )
⎛ ξr ⎞
⎜ ⎟ + μ0 γp
r
⎝r ⎠
'
'
2
3. Only appearance of ξz is in a stabilizing term. δW is minimized by choosing
'
'
⎡
rξr ) ⎤
(
2 ⎛ ξr ⎞
⎢
⎥
B θ r ⎜ ⎟ + μ 0 γp
δz =
r ⎥
⎝ r ⎠
k B2θ + μ0 γp ⎣⎢
⎦
(
ι
)
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 8 of 12
4. δWF becomes ( ξi ≡ ξ )
1
δWF =
2μ0
⎧⎪ 2μ0p' 2
2
∫ dr ⎨⎪ r ξ + Bθ
⎩
(
'
⎡
2
ξ'ξ* + ξ* ξ
⎢ '2 ξ
⎢ξ + 2 −
r
r
⎢
⎣
) ⎤⎥
⎥
⎥
⎦
(
'
⎡
2
ξ'ξ* + ξ* ξ
⎢ '2 ξ
+γμ0p ⎢ ξ + 2 −
r
r
⎢
⎣
−
1
=
2μ0
⎧
2
⎪ 2μ0p' 2 ξ
∫ dr ⎨ r ξ + r2
⎪
⎩
B2θ
B2θ
)
'
+ μ 0 γp ξ −
0
(
(
⎥
⎥
⎦
B2θ
)
ξ
− μ 0 γp
r
2⎫
⎪
⎬
⎪⎭
)
2⎤
⎡
B2θ − μ0 γp ⎥
⎢ 2
⎢Bθ + μ0 γp −
⎥
B2θ + μ0 γp ⎥
⎢
⎣
⎦
'
+
(
+ μ γp
1
) ⎤⎥
ξ'ξ* + ξ* ξ ⎡
μ γp − B2θ + B2θ − μ0 γp ⎤
⎣ 0
⎦
r
+ ξ'
2
(
(
)
)}
⎡B2 + γμ p − B2 − μ γp ⎤
θ
0
0
⎣ θ
⎦
5. Thus:
δWF =
1
2μ0
∫ dr
ξ
2
r2
⎡
4μ0 γpB2θ ⎤
'
⎢2μ0rp + 2
⎥
Bθ + μ0 γp ⎦⎥
⎣⎢
6. Stability condition
−
2γB2θ
rp'
<
p
μ0 γp + B2θ
Instability criterion usually violated in experiments. For Benneth profiles we
require γ > 2 for stability
Instability:
a. competition between increased magnetic pressure and increased particle
pressure
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 9 of 12
b. sausage instability
plasma pushes back less (3 degrees of freedom) than magnetic
pressure compresses plasma (2 degrees of freedom)
c. Stability boundary is independent of k
d. Mode is catastrophic experimentally.
e. Can be stable theoretically if p’ is weak enough. However, reliance on “ γ ” is
suspicious. Not easily stabilized experimentally
Conclusion
θ pinch stable
z pinch unstable
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 10 of 12
Single Particle Picture
why does curvature enter?
Consider m=0 mode
1. Calculate drifts
V∇B =
Vκ = −
2Bθ
v2⊥ B × ∇B
mv2⊥ 1
mv2⊥ B'θ
B
e
=
−
⋅
=
−
e
θ
2ωc B2
2e B3θ
2r z
2e B2θ z
2
2
v2⊥ κ × B mv& er × ez mv&
=
=
e
ωc B
er
Bθ
erBθ z
2. Assume isotropic plasma v2& =
v0 =
v2⊥
= v2
2
'
2
⎞
mv2 ⎛ Bθ
mv2 ⎛ Bθ ⎞
'
B
r
−
=
−
⎜
⎟
⎜
⎟
θ⎟
eB2θ ⎜⎝ r
eB2θ ⎝ r ⎠
⎠
'
⎛B ⎞
In most profiles ⎜ θ ⎟ < 0 : vD is in same direction as curvature drift.
⎝ r ⎠
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 11 of 12
Curvature drift creates E × B drift which enhances perturbation
If the curvature drift is in the opposite direction, E × B drift would oppose the
perturbation → stability
Summary
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 17
Page 12 of 12