22.615, MDH Theory of Fusion Systems
Prof. Freidberg
Lecture 16: Variational Principle
Variational Principle
ω2 =
δW
K
δW = −
K=
1 *
ξ ⋅ F ξ dr
2
∫
()
2
1
ρ ξ dr
2
∫
Advantages:
1. allows use of trial functions to estimate ω2
2. can be applied to multidimensional systems efficiently
Disadvantages:
1. still somewhat complicated
2. gives more information than minimum required
Energy Principle
1. Sometimes we only want to know whether the system is stable or not
2. No great need to know eigenvalues ω2
3. Growth rate are very fast r2 ∼ v2T a2 ∼ (10 μ sec)
4. Experimental times ∼ 10 msec – sec’s.
5. Since MHD instabilities are very strong, it is more important to know
whether system is stable or not, rather than know the precise growth rate
(which can be easily estimated)
6. In these cases, the variational principle can be simplified further, yielding
the Energy Principle. This is a simpler variational procedure which
accurately gives stability boundaries but only estimates growth rates.
The Energy Principle
1. Variational Principle
ω2 =
δW
K
If all ω2 > 0, the system is stable
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 1 of 9
2. Energy Principle δW ≥ 0 for all
allowable displacements, the system is stable.
Any displacement which makes δW < 0 ⇒ instability
Proof: (based on normal modes) more general proof in text book
1. Assume complete set of normal modes, orthonormal
( ) ∫ ρξ
*
H
−ωn2ρξn = F ξn
⋅ ξmdr = ξmn
2. Arbitrary trial function
∑a
ξ=
n
ξn
3. Evaluate δW
δW = −
1
2
()
∫
1
2
=−
=
1 *
1
ξ ⋅F ξ = −
2
2
∑a a ∫ξ
*
n m
∑ a a ∫ ξ ⋅ ( −ω
*
n
*
n m
∑ω
2
m
am
2
mρξm
*
n
( )
⋅ F ξm dr
)
2
4. If a trial function can be found which makes δW < 0 , then at least one
ω2m < 0 → instability
5. If all trial function make δW > 0 , then all ω2m > 0 → stability
Extended Energy Principle
δW = −
1 *
ξ ⋅ F ξ dr
2
()
∫
()
F ξ = J1 × B0 + J0 × B1 − ∇p1
(
)
(
)
= ⎡⎣∇ × ∇ × ξ × B ⎤⎦ × B + ( ∇ × B ) × ⎡⎣∇ × ξ × B ⎤⎦ × ∇ ⎡⎣ξ ⋅ ∇p + rp∇ ⋅ ξ ⎤⎦
1. Valid with wall on plasma
n⋅ξ
Sp
=0
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 2 of 9
2. Valid with vacuum region
n⋅B
p+
=0
Sp
B2
2μ0
=0
Sp
n ⋅ B1 S = 0
w
3. δW above not convenient because of complicated boundary condition with
wall, and no explicit appearance of Vacuum energy.
4. These are resolved by Extended Energy Principle
Extended Energy Principle
1. Rewrite δW1 introduce natural boundary conditions
2. δW = −
1
B
∇×B
*
dr ⎡⎣∇ × ∇ × ξ × B ⎤⎦ ×
+
× ⎡⎣∇ × ξ × B⎤⎦ + ∇ ξ ⋅ ∇p + rp∇ ⋅ ξ ⎤⎦ ⋅ ξ
2
μ0
μ0
(
∫
)
(
integrate by parts
)
integrate by parts
3. Define Q ≡ B1 = ∇ × ( ξ × B )
⎧ Q2
2
1
⎪
*
δW = +
dr ⎨
+ rp ∇ ⋅ ξ − ξ ⋅ ⎡⎣ J × Q + ∇ ξ ⋅ ∇p
2
μ
⎪⎩ 0
(
∫
)
⎫
⎡
B ⋅ B1 ⎤
⎤ ⎪⎬ − 1 ds n ⋅ ξ* ⎢rp∇ ⋅ ξ −
⎥
⎦
μ0 ⎦
⎣
⎪⎭ 2
∫ (
)
4. Separate ξ⊥ , ξ : ξ = ξ⊥ + ξ b
(
)
5. It is easily shown that b ⋅ ⎣⎡ J × Q + ∇ ξ ⋅ ∇p ⎦⎤ = 0 so that last term becomes
(
)
ξ*⊥ ⋅ ⎡⎣ J × Q + ∇ ξ ⋅ ∇p ⎤⎦
integrate by parts
(
)
(
note Q = ∇ × ξ × B = ∇ × ξ ⊥ × B
)
ξ ⋅ ∇p = ξ⊥ ⋅ ∇p
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 3 of 9
6. δW = δWF + B.T.
⎡ Q2
⎤
2
1
δWF =
− ξ*⊥ ⋅ J × Q + rp ∇ ⋅ ξ + ξ ⊥ ⋅ ∇p ∇ ⋅ ξ*⊥ ⎥
dr ⎢
⎢ μ0
⎥
2
⎣⎢
⎦⎥
∫
Standard form of the fluid energy
BT =
⎡ B ⋅ B1
⎤
1
− rp ∇ ⋅ ξ − ξ ⊥ ⋅ ∇p ⎥
dS n ⋅ ξ*⊥ ⎢
2
⎣ μ0
⎦
∫
7. Introduce natural boundary condition
p+
B2
=0
2μ0
linearize
2
⎡B ⋅ B
⎡
⎛
B ⋅ B1
B2 ⎞ ⎤
B ⎤⎥
1
+ ξ ⋅ ∇ ⎜p +
=⎢
+ ξ⋅∇
⎢p1 +
⎥
⎟
⎜
⎢ μ0
2μ0 ⎟⎠ ⎥⎦
2μ0 ⎥
μ0
⎝
⎣⎢
Sp
⎣
⎦
−rp∇ ⋅ ξ − ξ⊥ ⋅ ∇p
8. Substitute above
BT = δWs +
δWs =
(
)
B ⋅ B1
1
*
n⋅ξ
dS
2
μ0
∫
∫ (
1
*
dS n ⋅ ξ
2
⎛
B2 ⎞
ξ ⋅ ∇ ⎜p +
⎟
⎜
2μ0 ⎟⎠
⎝
)
ξ = ξn n + ξ b + ξ⊥ b × n
only term which contribute
=
2
⎛
1
B2 ⎞
dS n ⋅ ξ n ⋅ ∇ ⎜ p +
⎟
⎜
2
2μ0 ⎟⎠
⎝
∫
Surface term is non-zero only if surface currents flow
9. δW = δWF + δWs +
1
* B ⋅ B1
dS n ⋅ ξ
2
μ0
∫
10. Show last term is related to vacuum energy
11. δWv =
1
2μ0
2
1
∫B
v
dr =
1
2μ0
∫ ∇×A
1
2
dr
0
=
1
2μ0
∫ dr ⎢⎣∇ ⋅ ( A
⎡
*
1
)
⎤
*
× ∇ × A1 − A1 ⋅ ∇ × ∇ × A1 ⎥
⎦
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 4 of 9
=−
1
2μ0
⎛
∫ dS n ⋅ ⎜⎝ A
*
1
S
× B1 ⎞⎟
⎠
12. But: since n ⋅ B1 = n ⋅ ∇ × A1 = n ⋅ ∇ × ξ × B
Choose B1 ⋅ (n × ∇φ ) = 0 as gauge
then A1 = ξ⊥ × B + ∇φ
and δWv = −
=
1
2μ0
1
2μ0
∫ dS n ⋅ ( ξ
*
⊥
∫ dS n ⋅ ξ
*
)
× B × B1
B ⋅ B1
Extended Energy Principle
δW = δWF + δWS + δWV
Boundary Conditions on trial functions
n ⋅ B1
n ⋅ B1
Sw
Sp
=0
(
= n ⋅ B1 S = n ⋅ ∇ × ξ × B
p
(
)
)S
p
(
= − n ⋅ ξ ⎡⎣n ⋅ (n ⋅ ∇ ) B ⎤⎦ +B ⋅ ∇ n ⋅ ξ
)S
p
depends only on n ⋅ ξ
pressure balance conditions not required → natural boundary conditions
Final Step
Intuitive form of δWF
1. Standard form OK
2. Intuitive form gives more insight.
⎧ Q2
⎫
2
1
⎪
⎪
3. δWF =
− ξ*⊥ − J × Q + rp ∇ ⋅ ξ + ξ ⊥ ⋅ ∇p ∇ ⋅ ξ*⊥ ⎬
dr ⎨
2
⎪⎩ μ0
⎪⎭
(
∫
Q
2
= Q⊥
2
+ Q
)
2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 5 of 9
(
)
J⊥ =
b × ∇p
B
ξ*⊥ ⋅ J × Q = ξ*⊥ × b ⋅ Q⊥ J + Q ξ*⊥ ⋅ J⊥ × b
now:
( J × B = ∇p )
(
Q = b ⋅ ∇ × ξ⊥ × B
)
(
= b ⋅ B ⋅ ∇ξ ⊥ − ξ ⊥ ⋅ ∇B − B∇ ⋅ ξ ⊥
(
)
= −B ∇ ⋅ ξ ⊥ + 2ξ ⊥ ⋅ κ +
)
μ0
ξ ⊥ ⋅ ∇p
B
Substitute back
1
2
3
4
5
⎡Q 2
⎤
2
2
1
B2
⎢ ⊥
⎥
*
*
dr ⎢
δWF =
+
∇ ⋅ ξ⊥ + 2ξ⊥ ⋅ κ + rp ∇ ⋅ ξ − 2 ξ ⊥ ⋅ ∇p κ ⋅ ξ ⊥ − J ξ ⊥ × b ⋅ Q⊥ ⎥
2
μ0
μ0
⎥⎦
⎣⎢
(
∫
)(
1. line bending > 0
shear alform wave
2. magnetic compression > 0
compressional alform wave
3. plasma compression > 0
sound wave
)
(
)
4. pressure driven modes + or –
5. current driven modes (kinks) + or –
Summary
Energy Principle: δW = δWF + δWS + δWV
δW ≥ 0 for all allowable displacements → stability
δW < 0 for any allowable displacement → instability
Minimize δW with respect to three components of ξ .
Incompressibility
1. Because of the simple way in which ξ appears in δW , it is possible to
minimize once for all with respect to ξ and eliminate it from the
calculation.
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 6 of 9
2. Only appearance of ξ
δW =
∫ dr
rp ∇ ⋅ ξ
2
3. Let ξ → ξ + δξ
4. Vary δW ξ = ξ ⊥ + ξ
(
B
B
B⎞
⎛
rp ∇ ⋅ ξ ∇ ⋅ ⎜ δξ ⎟
B⎠
⎝
(
) ∫ dr
δ δW =
)
integrate by parts
∫
δξ
∫
δξ
= − dr
= − dr
(
B ⋅ ∇ rp ∇ ⋅ ξ
B
(
rpB ⋅ ∇ ∇ ⋅ ξ
B
)
)
5. Several minimizing condition
(
)
B⋅∇ ∇⋅ξ = 0
6. If B ⋅ ∇ is non-singular then
∇ ⋅ ξ = 0 (obvious)
δW =
2
1
rp ∇ ⋅ ξ → 0
2
∫
7. Two cases where ∇ ⋅ ξ cannot be set to zero
8. Special symmetry
B = Bθ (r ) eθ
Example: Z pinch
B⋅∇
ξ
B
=
ξ ∼ eimθ+ ikz ξ (r )
imξ
Bθ ∂ ξ
=
r ∂θ Bθ
r
For m = 0
B⋅∇
ξ
B
=0
Note: ∇ ⋅ ξ = ∇ ⋅ ξ ⊥ + ∇ ⋅
ξ
B
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
B = ∇ ⋅ ξ⊥ + B ⋅ ∇
ξ
B
Lecture 16
Page 7 of 9
In special symmetry ∇ ⋅ ξ = ∇ ⋅ ξ⊥ and ξn does not appear. The term rp ∇ ⋅ ξ⊥
2
must be maintained for the rest of the minimization.
9. Closed line (periodicity constraints). Choose ξ so ∇ ⋅ ξ = 0
B⋅∇
ξ
B
ξ
B
=−
∫
= −∇ ⋅ ξ⊥ = B
∂ξ B
∂l
∇ ⋅ ξ⊥
dl
B
In general
ξ
B
ξ
(l + L ) ≠
B
(l ) →
no periodicity
Solution
B⋅∇ ∇⋅ξ = 0
∴ ∇ ⋅ ξ = F (p )
homogenous solution
B⋅∇
ξ
B
ξ
B
=−
∫
= −∇ ⋅ ξ ⊥ + F (p )
∇ ⋅ ξ⊥
dl +
B
∫
F (p ) dl
B
=−
∇ ⋅ ξ⊥
dl + F (p )
0
B
∫
l
dl
0 B
∫
l
In periodicity choose
dl
F (p ) = < ∇ ⋅ ξ ⊥ > =
Then δW =
δW =
∫ B ∇⋅ξ
dl
∫B
⊥
2
1
1
rp ∇ ⋅ ξ dr =
rp F2 dr
2
2
∫
∫
2
1
dr rp < ∇ ⋅ ξ ⊥ >
2
∫
Only a function of ξ ⊥
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 8 of 9
Summary of internal modes in a straight tokamak
1. m ≥ 2 stable
2. m = 1, n = 1 must use 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 instability we needed to calculate δW = e2 δW2 + ∈4 δW4
0
must
Consider now external modes
1. Vacuum no force free fields
2. m=1 Kruskal Shafranov limit
3. High m external kinks
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 16
Page 9 of 9