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. Stability: General Considerations
Week 10-11. Stability: One-Dimensional Configurations Week 12. Stability: Multidimensional Configurations
Week 14-15. Project Presentation
Contents
4
• Introduction
- The MHD equilibrium problem separates into two parts for most configurations of interest:
a. radial pressure balance: zero order in a/R b. toroidal force balance: first order in a/R
- We examine radial pressure balance in a 1-D geometry:
straight cylinder (no problems with toroidal force balance) - The 1-D radial pressure balance relation is valid for
tokamaks, stellarators, RFP’s, pinches, and EBT’s.
- We introduce toroidal effects as an aspect ratio expansion to see what must be done to achieve toroidal force balance.
This requires 2-D calculations.
Equilibrium: 1-D Configurations
Z pinch
q pinch screw pinch
• The q Pinch
- 1-D analog of the toroidal configuration with purely toroidal field
Equilibrium: 1-D Configurations
externally driven induced diamagnetic
current
- Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0
2. Ampere’s law: m0J = ▽xB
3. The momentum equation: JxB = ▽p
• The q Pinch
Equilibrium: 1-D Configurations
0
z B
zr
J B
z
0
1
q
m
dr B dp
J
q z
2
00
2
m
zp B
dr d
0 2 0 0
2
2
2 m m
B p B
z
- At any local value of r, the sum of the particle pressure and the magnetic pressure is a constant, equal to the externally applied magnetic pressure.
- The plasma is confined radially by the pressure of the externally applied magnetic field.
- Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0
2. Ampere’s law: m0J = ▽xB
3. The momentum equation: JxB = ▽p
• Typical example
2 0 0
0
2 m p / B
Equilibrium: 1-D Configurations
• The q Pinch
2 / 1 2 2 0
0
2 2 0
)]
/ exp(
1 [
) / exp(
a r B
B
a r p
p
z
a a zt
rdr
B B prdr a
B
a
0 022 0 2
2 0 2
0
2 1
4 m
0
t 1
0 0
0) (
*
s q
p
B
qfavourable stability against current-driven kinks not available to stabilise MHD instabilities
• The q Pinch
Equilibrium: 1-D Configurations
2 0 2
2
2
dr
dB B
r B
dV d B
W V
zz
2 0 2 2
0 2 2
0 2
2 0
m
m m
zz
p B dr
d B
r p B
dV d B
W V
at finite - Configuration is neutral w.r.t. magnetic well stabilisation.
- “A plasma cannot dig its own magnetic well by means of its diamagnetic confinement currents.”
• The q Pinch
Equilibrium: 1-D Configurations
- One of the early successes of the fusion program in terms of the performance.
- Ti ~ 1-4 keV, n ~ 1-2x1022 m-3, (0) ~ 0.7-0.9, t ~ 0.05 - No indication of macroscopic instability (neutrally stable) - Severe end loss
- Cannot be bent into a toroidal equilibrium.
• The Z Pinch
Equilibrium: 1-D Configurations
- 1-D analog of the toroidal configuration with purely poloidal field
longitudinal plasma current self field induced
by JZ
- Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0
2. Ampere’s law: m0J = ▽xB
3. The momentum equation: JxB = ▽p
• The Z Pinch
Equilibrium: 1-D Configurations
1 0
q
qB r
) 1 (
0
m dr rB
qd J
z r
dr B dp
J
z q
0 ) (
0
q qm dr rB d
r B dr
dp 0
2
02
0
2
r B p B
dr d
m m
q qtension force by the curvature of the magnetic field lines
particle pressure + magnetic pressure force - Sequence of solution of the MHD equilibrium equations
1. The ▽∙B = 0
2. Ampere’s law: m0J = ▽xB
3. The momentum equation: JxB = ▽p
• The Z Pinch
Equilibrium: 1-D Configurations
- It is the tension force and not the magnetic pressure gradient that provides radial confinement of the plasma.
2 2 0 2
2 0 2
2 0 0
2 2 0 2
2 0 0
2 0 2 0 0
) (
8
) (
2
r r
r p I
r r
r J I
r r
r B I
z
m
m
q
Bennett profiles (Bennett, 1934)
- The plasma is confined by magnetic tension.
- The magnetic pressure and plasma pressure each produce positive (outward forces) near the outside of the plasma.
• The Z Pinch
Equilibrium: 1-D Configurations
0 0
0) (
*
s q
B
zt
poor stability against current-driven kinks Not available to stabilise MHD instabilities
16 1
2 0 0 0
2
ap
prdr
m I
1 )
( , )
ˆ ( W r
dr dB B
r r
W
qq
at finite
- While its high value is desirable for confinement efficiency, the lack of flexibility in achieving small to moderate p is a disadvantage.
- Some classes of potentially dangerous MHD modes can be stabilized if p is sufficiently low (poor stability).
: Bennett Pinch Relation
unfavourable stability properties w.r.t. pressure-driven MHD instabilities
• The Z Pinch
Equilibrium: 1-D Configurations
- A number of linear Z-pinch experiments were constructed during the early years of the fusion program.
- Exhibiting disastrous instabilities, often leading to a complete quenching of the plasma after ~1-2 ms
- Can easily be bent into a toroidal equilibrium.
- Ohmically heated tokamak
• The General Screw Pinch
Equilibrium: 1-D Configurations
- A hybrid combination of Z pinch and θ pinch
- This combination of fields allows the flexibility to optimise configurations w.r.t. toroidal force balance and stability.
- Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0
2. Ampere’s law: m0J = ▽xB
3. The momentum equation: JxB = ▽p
16
• The General Screw Pinch
Equilibrium: 1-D Configurations
- Sequence of solution of the MHD equilibrium equations 1. The ▽∙B = 0
2. Ampere’s law: m0J = ▽xB
3. The momentum equation: JxB = ▽p
1 0
z B B
r q
q z) 1 (
0
m dr rB
qd J
z r
dr B dp
J B
J
q z
z q r
J B
z
0
1
q
m
2
00
2
0 2
2
r B B
p B dr
d
zm
m
qq
- Even though the equations are nonlinear, the forces superpose because of symmetry.
- The screw pinch has many properties of more realistic, multidimensional toroidal configurations.
- The constant pressure contours p(r) = constant are circles
r = constant. The flux surfaces are closed, nested, concentric circles.
• The General Screw Pinch
Equilibrium: 1-D Configurations
16 1
2 0 0 0
2
ap
prdr
m I
aprdr I
0aB B
zrdr
02 2
0 2
0 0
0
2 2
2 8
m
m
at
prdr
B a
2 02 04 m
0
1 1
t t
p
t a
0a B B
z2 rdr
02
2
1
2
t 1
a measure of the energy in the poloidal
magnetic field (poloidal tension) plasma energy
line density
the magnetic energy associated with the diamagnetic part of the
toroidal field
(toroidal diamagnetism)
representing the plasma diamagetism
• The General Screw Pinch
Equilibrium: 1-D Configurations
for circular flux surface
measure of safety against a variety of MHD instabilities
0 0 0
0 2
*
2
I R
B q a
m
2 00 0
R
dz
dz d
d
q
q q
q
( )
) (
, ) 0 (
) (
r rB
r B dz
d r
B r B dz
dr
z z
r
q
q
) (
) ( ) 2
(
0r rB
r B r R
z
q
q
B R
rB r r
q
z)
0( ) 2
( q ( a ) q
*dr dq q r r
s ( ) V 2
2R
0r
2• The General Screw Pinch
Equilibrium: 1-D Configurations
2 0 )
(
22
0 2 2
0
B
B p B
dr d B r r
W
qm m
- Flexibility of making the magnitude of W(r) small by requiring Bθ2<<Bz2. - The destabilising influence of W(r)<0 can be compensated for by shear.
Equilibrium: 1-D Configurations
- Because of its flexibility the general screw-pinch relation describes a wide variety of configurations.
- Capable of a wide range of variation in t and p
- In general, the kink safety factor, the MHD safety factor, the shear, and the magnetic well are nonzero.
- Toroidal equilibrium is in general not difficult to achieve since Bθ is nonzero.
• The General Screw Pinch
References
- http://phys.strath.ac.uk/information/history/photos.php
- http://commons.wikimedia.org/wiki/File:Z-pinch_H-gamma.jpg
-http://www.frascati.enea.it/ProtoSphera/ProtoSphera%202001/6.%20Ele ctrode%20experiment.htm
- http://scienceray.com/technology/industry/where-do-rubber-bands- come-from/