T opics in Fusion and Plasma Studies 2
(459.667, 3 Credits)
Prof. Dr. Yong-Su Na
(32-206, Tel. 880-7204)
Week 1-2. The MHD Model, General Properties of Ideal MHD Week 3. Equilibrium: General Considerations
Week 4. Equilibrium: One-, Two-Dimensional Configurations Week 5. Equilibrium: Two-Dimensional Configurations
Week 6-7. Numerical Solution of the GS Equation Week 9. Stability: General Considerations
Week 10-11. Stability: One-Dimensional Configurations Week 12. Stability: Multidimensional Configurations
Week 14-15. Project Presentation
Contents
Week 1-2. The MHD Model, General Properties of Ideal MHD Week 3. Equilibrium: General Considerations
Week 4. Equilibrium: One-, Two-Dimensional Configurations Week 5. Equilibrium: Two-Dimensional Configurations
Week 6-7. Numerical Solution of the GS Equation Week 9-10. Stability: General Considerations
Week 11-12. Stability: One-Dimensional Configurations Week 13-14. Stability: Multidimensional Configurations Week 15. Project Presentation
3
Contents
Stability: One-Dimensional Configurations
- Results can be unreliable because the geometric effects associated with toroidicity are of the same order as the field ratio: contribution of
toroidicity in δW are of the same order as those of the straight tokamak.
- Providing very useful frame of reference for the understanding of tokamak stability.
- Most dangerous instabilities: internal and external current-driven modes (pressure-driven modes do not play a major role due to the combination
of finite shear and low β)
- Good MHD stability if the total toroidal current is limited and the current profile is peaked: valid for the fully toroidal case
- No important restrictions on β or the pressure profile: not valid for
• Introduction
1
~ ,
~ /
, 1
/ R
0B B q
a
z
~
~ /
2
0p B
z2 2: Ohmically heated tokamak : high-β tokamak5
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
0 2 4
4 /
0 8
8
2
0 2
2
0 2 2 2
0 2
q r q r B
r p q
q r
r p B q
q p r
q q rB
z
z z
or
2~
1
~ / 1
~ ,
~
q q r q
r
q
• Internal Pressure-Driven Modes: localised interchange instabilities Substituting the tokamak expansion into Suydam’s criterion
2 2
0 2
2 0
2 2 2
or
~ ) ~
/ ( 8
~ ,
~
z z
z z
B p q
q rB
p
a p p
a B q
rB q
Over most of the plasma the destabilising term << the stabilising term Suydam’s criterion satisfies over most of the plasma.
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
0 2 4
4 /
0 8
8
0 2
0 0 0
2 2
0 0
0 0
2 2
0 0 0
2
q q r B
p q
q r
r rB p
q q p r
q q rB
z
z z
r q
r q
q r
q q
r q
r p
r p
) 0 ( )
(
) 0 ( 2
/ ) 0 ( )
0 ( )
(
) 0 ( )
(
2
• Internal Pressure-Driven Modes: localised interchange instabilities Exception: near the axis
r q / q ~ 0
1
~ )
2 ( 1/2 1/2
0
0
q a
q a
r
- If the pressure profile is flattened over a distance Δr, then the modes are stabilised.
7
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
- In toroidal case there are important modifications to Suydam’s criterion: Mercier criterion. These corrections can eliminate the need for flattening the pressure profile for stability.
0 ) 1
(
4
22
r q
q q
r
simple, low β circular limit of Mercier criteriontoroidal correction
- Localised interchange modes are not very important in a straight tokamak because β is very small.
- Near the axis, we need either flattening or q(0) > 1 (toroidal).
For q(0)>1, pressure term is stabilising: average curvature is formable.
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
z z
B B
R a
R L
/ /
2
0
0
• Internal Current-Driven Modes
0
0
/ /
2 /
2 k R n k n R
m n q R
mB m
n q R
mB
rB R B m
n R
mB R
nB r
kB mB r
F mB
z
z z
z z
1 1
0 0 0
0 0
0
) (
1 ~
1
20 2
2 2
2 0 2 3
2 2 0 2 2 3 2
2
m aB n q R
B r m
n q R
B m m
r k
f rF
2 2 2
2 2
0 2 2
2 2
2
0
r
m r
m R
n r
k m
k
Substituting the tokamak expansion into δWF
n F
F
W W W W
W
0
2
4 , ~
asymptotic expansion9
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• Internal Current-Driven Modes
r F kB mB
rk rF k
r k
r p k
k
g k
z
4 0 2 2
2 2 0
2 2 0 2 0
0
2
1 2
) 2 (
a B q
m n m
R r B n B
r B R m
n R
B m m R
r F n
r kB mB
rk g k
a B m
n q R
m rB m
n q R
B r m r r
m r r m r rF
k r g k
a p B
R r m p n
m r R p n
k g k
z
2 0 4 2
2 2 2
4 0
3 2 0 2 2
2 0 2
2 2 0 2
2 2
0 2 0 2 4 2 0
3 2 4
0 2 3
2 0 2 2
2 0
2 2 0
2 2
0 2 0 2
2 2
2 2 2
2 2
2 2 0
2 2 0 2
2 0 2 0
2 2 0
2 2 0
2 2 0
0 2 1
1 ~ 2
2 2
1 ~ )
1 1 (
1 1
~ 2 2
) 2 (
< <
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• Internal Current-Driven Modes
a a a F a
rdr m
q r m
n a
W W
rdr m
q r m
n R
B
m dr n q R
m rB m
n q R
B r
dr g
R f W
0
2 2
2 2 2 2
0 2
0
2 2
2 2 2 2
0 2 0 0
2 2 2
0 2 2 0
2 2 2
0 2 0 3 0
2 2
0 0
) 1 1 (
1
) 1 1 (
) 1 1 1 (
) / (
2
m ≥ 2: stable, both terms positive
m = 1: nq(r) > 1 (n = 1 worst)
1 0
1 1 0
1
22
q q
0 0 2 0 2 2
0
2 a B / R
W
11
Stability: One-Dimensional Configurations
m = 1: q(r) < 1 somewhere
• The “Straight” Tokamak
• Internal Current-Driven Modes
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• Internal Current-Driven Modes
) (
0
) (
2 1
1
) 0
( 1
s
s s
s
r r
r r x r
r r
) )(
( 1
) (
) )(
( )
( 1 )
( 1
2
s s
s s s
s
r r r q
r q
r r r q r
q r
q
r
sr x
δWF → 0 as δ → 0
s s
r r F a
q R r
B
dx x
q R r
B
rdr r
x R q
B
rdr m
q r m
n R
B R
W
2 3 2 2 0
2 2
2 3 2 0 2 0
2 2 2 2
0 2 0
0
2 2
2 2 2 2
0 2 0 0
0
6
2 1
) 1
1 (
) 1 1 (
/ 2
- With an m = 1 resonant surface (i.e. if q(0) < 1), marginally stable in leading order
- For this case, stability is determined by the higher-order corrections to δWF .
13
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• Internal Current-Driven Modes
m
n q R
F mB 1
0 0 2
2 0 2 2 2
2 0
2 0 2 3 2
0
2
1
) /
(
n q
R r
n m
R
B m r k
f rF
2 2 2
0 2 2
0
r
m R
k n
Calculate next order δW for m = 1, n = 1 mode
4 2
0 2 0 2 0
0 2 2
2 0
2 0 2 0
0 2 2
4 0
2 2
2 2 0
2 2 0 2 0
0 2
1 ~ 1 3
2
2 2 1
2 1
2 ) 1
2 (
n q q n
R p rB R
r n
n q q n
q n R
p rB R
r n
r F kB mB
rk rF k
r k
r p k
k
g k
zStability: One-Dimensional Configurations
• The “Straight” Tokamak
• Internal Current-Driven Modes
rs
F a
q rdr q
R r r
a W
W
dr g
R f W
0 2
0 2 2
2 0 0
4
0
2 2
0 0
3 1 1 1
) / (
2
- Contributions from both the pressure and parallel current are destabilising
→ unstable
- Often referred to as the m = 1 internal kink mode. But the mode can also be driven by the pressure gradient, particularly in the high-β tokamak regime.
- The nonlinear evolution of the m = 1 internal kink mode, including the
effects of dissipation is believed to be an important component in sawtooth oscillations observed in many tokamaks.
- If the energetic particle effect is included, a modified m = 1 mode thought to be responsible for the so-called fishbone oscillations.
Use the same trial function as before
15
Stability: One-Dimensional Configurations
• Sawtooth
time (s)
0 2 4 6 8
8 6 1 10 1
Ip(MA)
PNBI(MW) PRF(MW)
WMHD(MJ)
Nel x1.1019
Ti(keV) Te(keV)
ASDEX Upgrade pulse 20438
Stability: One-Dimensional Configurations
• Sawtooth
1. T(0) and j(0) rise 2. q(0) falls below 1
→ kink instability grows 3. Fast reconnection event:
T, n flattened inside q = 1 surface q(0) rises slightly above 1
kink stable
• Sawtooth
Stability: One-Dimensional Configurations
• Sawtooth
• Stable m/n=0/1 mode in the initial stage
• m/n=1/1 mode develops as the instability grows (kink or tearing instability)
and reconnection occurs - Tearing mode instability
(slow evolution of the island/hot spot)
- Kink mode instability (sudden crash)
• Reconnection time scale is any different in these two types?
17
Stability: One-Dimensional Configurations
• Sawtooth
• Comparison with the full reconnection model
• Comparison with the quasi-interchange mode model
J. Wesson (1986) B. Kadomtsev (1976)
Stability: One-Dimensional Configurations
• Sawtooth
19
• Comparison with the ballooning mode model
Low field side
Stability: One-Dimensional Configurations
• Sawtooth
• 2-D ECE imaging diagnostic
H.K. Park et al, PRL
21
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• Summary of internal modes
- m ≥ 2 stable for both the ohmically heated and high-β orderings - m = 1 internal mode exhibiting a weak instability (i.e. δW ~ ε4)
which can be prevented if q(0) > 1
Kink mode
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes
- The most dangerous ideal MHD instability in a straight tokamak - Current-driven modes in which the k∙B = 0 singular surface lies
in the vacuum region.
- More dangerous energetically than internal modes:
stability determined by the 2nd order contribution to δW
- Stability boundaries unaffected by the pressure gradient in the straight model
23
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
a a
a a
q m
n q
m m n q
m n
rdr m
q r m
n a
W W
1 1
1
] ) 1 (
1 [ 1
2 0
0
2 2
2 2 2 2
0 2
• External Modes (The m = 1 Kruskal-Shafranov Limit)
- Examining the first nonvanishing contribution to δW of order ε2 - In the limit where the conducting wall moves to infinity
For either ohmically heated or high-β expansion
q
am
n 1
0
necessary condition for instability:Resonance surface for an external mode q = m/n lies in the vacuum region.
m n
q
a0 , 0 ,
Examining the boundary term
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (The m = 1 Kruskal-Shafranov Limit)
a a
a
a
n n q
n q n q
n q W
W 1
1 2 1
1
20 2
0 0
2
Considering the m = 1 mode
- Minimising eigenfunction by ξ(r) = ξa = const (independent of q profile)
→ integral contribution vanished
Kruskal-Shafranov criterion:
stability condition for the m = 1 external kink mode for the worst case, n = 1
1 q
a] MA [ / 5
/
2 a
2B
0 0R
0a
2B
0R
0I
KS 1
2 /
0
R I a
q
aaB
Imposing an important constraint on tokamak operation:
toroidal current upper limit (I < IKS)
25
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (The m = 1 Kruskal-Shafranov Limit)
• Effects of addition of a toroidal field:
- Jz current driving the instability largely flowing parallel to the filed:
The mode described as a current-driven, rather than pressure-driven instability.
- Bending the toroidal magnetic lines: strong stabilising effect
(in order for line bending not to dominate, the modes can only become unstable for sufficiently long wavelengths)
- Tension in the poloidal field increases in the tighter portion of the column when the plasma develops an m = 1 kink perturbation.
- The resulting force is in the direction to enhance the perturbation, thus causing an instability.
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (The m = 1 Kruskal-Shafranov Limit)
- The stability boundary qa = 1 not depending on the existence of
a pressure gradient, the shape of the Jz profile, or the position of the wall - Very dangerous instability but difficult to observe experimentally
because higher m modes, ideal and resistive, posing even stronger requirements on qa (more sensitive to profile and slower growth rates) - Resistive tearing mode being thought to be responsible for the rapid
termination of tokamak discharge by means of a “major disruption”
when qa ~ 2-3, well below the current required to excite the Kruskal-Shafranov mode.
27
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
) 1 )(
2 (
2
2
nq m nq m
q W
W
a a
a
a
• External Modes (m ≥ 2 External Kinks): imposing more stringent conditions on qa and the current profile than m = 1
)
1/ ( )
( r
ar a
m
a a
a a
q m
n q
m m n q
m n
rdr m
q r m
n a
W W
1 1
1
] ) 1 (
1 [ 1
2 0
0
2 2
2 2 2 2
0 2
n q m
n m
a
1
regions of stability boundedFor the uniform current density profile, q(r) = qa = const (n = 1)
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
- Shafranov (1970) pointed out that this pessimistic result is strongly dependent on the shape of the current density profile.
- For more realistic profiles, as m increases, the mode structure for external kinks becomes progressively more localised in the
vicinity of the plasma surface.
- The plasma contribution to W depends sensitively on the local behaviour of the current density near r = a when J(a), J’(a), etc.
are small, the instabilities diminish.
- The m = 1 mode is determined solely by the total current.
• External Modes (m ≥ 2 External Kinks)
29
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
0 )
( )
(
2
2
dr f g f g
a a
a a
q m
n q
m m n q
m n
rdr m
q r m
n a
W W
1 1
1
] ) 1 (
1 [ 1
2 0
0
2 2
2 2 2 2
0 2
1 0 )
1
1
2(
2 23
q m m n
dr r d q
m r n dr
d
The trial function which minimises the plasma term satisfies
m
nq m
nq m a
m nq q
W
W
a aa a a
a
a2
1 1 1
2
0
2
• External Modes (m ≥ 2 External Kinks) Asymptotic Analysis
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (m ≥ 2 External Kinks)
1 0 1
2 2 2
2 2
2
b
dr K d K r
m dr
b d K r
b ( )
Simplifying the analysis by considering the limit of large m
- Instability occurs only when the resonant surface lies just outside the plasma.
- The corresponding plasma displacement is highly localised just inside the plasma, allowing one to perform a high m asymptotic analysis of the Euler-Lagrange equation.
1 0 )
1
1
2(
2 23
q m m n
dr r d q
m r n dr
d
m q
r n r
K 1
)
(
3/231
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
assuming m >> 1
• External Modes (m ≥ 2 External Kinks)
m z a
r
1 2
m z a m
n m z z
m a a
K
m z a m
a n K
a a
a
a a
2 ˆ ˆ
4 3 8
ˆ
2 ˆ ˆ
2 2 2 2
/ 3 2
2 / 3 1
m
m
nq
a1
~ 1
K
1K
2K
0 ≤ z ≤ 1 for localisation of the mode
small if the resonance surface is near the plasma
K1 ~ 1/m, K2 ~ 1/m2 Expand the function K
a a
1 / q ˆ
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
can be more conveniently expressed in terms of the
properties of the current profile at the plasma edge.
• External Modes (m ≥ 2 External Kinks)
) 0 (
4 1
0 2
2
b
z z m dr
b
d
aa a a a
m n i
a z m
a a
a
, 2 ˆ ˆ
ˆ 3 ˆ ˆ 2
1
0 2
J J
a q
aa
a
2 1
ˆ
J q
J a a
a
a a a
a
3 ˆ 2 ˆ
2
0 0 02
0
/ a 2 B ˆ / R
I
J
a ) (
),
( a J J a
J
J
a
a
a a
a a
J J
J m
m nq J z
J J
a , 1 2
1
0J r
rB rB
B
R
0/
0, ( ) /
0ˆ
33
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (m ≥ 2 External Kinks)
J
J m
b b m
m nq m
nq q
W
W
a a aa
a
2 1 2
1 1
1
0 0 2
2
0
2
0 0
0
/ b [( db / dz ) / b ]
zb
) 0 (
4 1
0 2
2
b
z z m dr
b
d
Desired form of the high-m external kink stability problem
The case Ja ≠ 0: calculate the leading-order contribution to b
2
)
/( z e
zb
4 0
2
2
b dr
b
d
solution corresponding toa localised eigenfunction
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (m ≥ 2 External Kinks)
m J
J m
nq m
nq q
m W
W
a a aa
a
1 1
2
2 2
0
2
instability
n q m
J m J
n
aa
1
- The most unstable case occurs for uniform current, Ja/<J> = 1:
neighbouring bands touch and all qa values are unstable.
- As the current at the boundary decreases, the unstable bands become narrower, forming windows of stability.
35
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
b → 0 as z → ∞
• External Modes (m ≥ 2 External Kinks)
b m
b m
nq q
m W
W
aa
a
2 1
1 1
0 0 2
2 2
0
2
U ye
b
y z
z
y 2/ 0
0 1
) 2
2
(
2
U
m dr
y dU dr
U
y d
)
; 2 , / 1
/
(
y m ye
b
y s
The case Ja = 0: 1/m corrections to b(r) taken into account
1 )
/ 1
(
0
1 m nq m
z
a) 0 (
4 1
0 2
2
b
z z m dr
b
d
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
condition for instability
• External Modes (m ≥ 2 External Kinks)
0 ln
1 z
0
n q m
J a
J m m
n
a
a
2 1 exp
The case Ja = 0: 1/m corrections to b(r) taken into account
- Instability can only exist over exponentially narrow bands of qa. - When the current vanishes at r = a the range of unstable qa values
is greatly reduced.
- If Ja’ vanishes, the high-m external kinks can be completely stabilised.
37
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• External Modes (m ≥ 2 External Kinks)
1 1 2 2 0
] ) 1 ( 1 [
) / 1
(
v a
v
u u
q q
a r J
J
2 2 0
, / /
1 q q u r a
v
a
- Unshaded region is stable to both internal and external ideal MHD modes. Most existing tokamaks operate in this regime.
- ν > 2.5 or qa/q0 ≥ 3.5 all m ≥ 2 modes are stable for any qa. - Windows of stability in qa also exist for 1 < ν <2.5.
- Complete external-mode stability is achieved by the additional requirement qa > 1 to stabilise the m =1 mode.
Numerical results by Wesson (1978)
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• Summary
- m ≥ 2 external-kink modes are sensitive to the features of the current density profile near the edge of the plasma.
- A combination of rapidly vanishing current gradient and a limit on the total current is sufficient to completely stabilise all high- and low-m number external modes, respectively.
- The dual requirements of q0 > 1 for internal-mode stability and peaked current density profiles for high-m external-kink stability can only be simultaneously satisfied if qa ≥ 3.
- Therefore, m ≥ 2 kinks impose stronger constraints on than the m = 1 Kruskal-Shafranov mode.
39
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• The Effect of a Resistive Wall
- We have seen that a perfectly conducting wall, placed in close proximity to the plasma can have a strong stabilizing effect on external kink modes.
- In actual experiments, the metallic vacuum chamber surrounding the plasma is a good approximation to a perfectly conducting wall.
- However, its conductivity is not infinite but is finite.
- In fact we do not want the conductivity too high and/or, too thick because it would take too long externally applied feedback fields to penetrate the shell and interact with the plasma.
- Also, higher resistivity, smaller currents are induced in the chamber during transients, alleviating power supply requirements.
- The question raised here concerns the effect of finite resistivity of the wall on external kink stability.
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• The Effect of a Resistive Wall
- There are three possible situations and only one is really interesting.
- In the first case the plasma is stable to external kinks with the wall at ∞.
Here, since the plasma is already stable, a wall, either ideal or resistive does not affect stability. This case is uninteresting.
- In the second case, the plasma is unstable with the wall at ∞ and with the wall at its actual position, assuming the wall is perfectly conducting.
Since the plasma is unstable with a perfectly conducting wall as r = b, making the wall resistive does not help. This case is also uninteresting.
- The interesting case is when the plasma is unstable with the wall at ∞, but stable with a perfectly conducting wall at r = b.
Does the resistivity of the wall destroy wall stabilization?
- To address this issue we investigate the problem in a straight cylindrical geometry. However, the results are valid for a general toroidal geometry as well.
41
Stability: One-Dimensional Configurations
• The “Straight” Tokamak
• The Effect of a Resistive Wall Plan of attack
- The analysis of the resistive wall mode is carried out in four steps.
1. Reference values of δW are calculated for an ideal wall located as r = ∞ (δW∞) and at r = b (δWb)
2. The full eigenvalue problem is solved region by region assuming slow growing modes: on the scale of the wall diffusion time.
3. The fields within the resistive wall are calculated using the BCs.
then wall approximation. This gives rise to a set of jump conditions across the wall.
4. The resulting set of coupled equation and boundary conditions are solved yielding the dispersion relation.
