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T opics in Fusion and Plasma

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T opics in Fusion and Plasma Studies 2

(459.667, 3 Credits)

Prof. Dr. Yong-Su Na

(32-206, Tel. 880-7204)

(2)

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

(3)

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)

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

(5)

• 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

(6)

• The q Pinch

Equilibrium: 1-D Configurations

 0

z B

z

r

J B

z

 

0

1

q

m

dr B dp

J

q z

2

0

0

2

  

 

  m

z

p B

dr d

0 2 0 0

2

2

2 m m

B pB

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

(7)

• 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 z

t

rdr

B B prdr a

B

a

0 02

2 0 2

2 0 2

0

2 1

4 m

0  

t

1

0 0

0) (

*

s q

p

B

q

favourable stability against current-driven kinks not available to stabilise MHD instabilities

(8)

• The q Pinch

Equilibrium: 1-D Configurations

2 0 2

2

2

 

dr

dB B

r B

dV d B

W V

z

z

2 0 2 2

0 2 2

0 2

2 0

 

 

 

 m

m m

z

z

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.”

(9)

• 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.

(10)

• 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

(11)

• The Z Pinch

Equilibrium: 1-D Configurations

1  0

 q

q

B r

) 1 (

0

m dr rB

q

d J

z

r

dr B dp

J

z q

 

0 ) (

0

q q

m dr rB d

r B dr

dp 0

2

0

2

0

2

  

 

 

r B p B

dr d

m m

q q

tension 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

(12)

• 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.

(13)

• The Z Pinch

Equilibrium: 1-D Configurations

0 0

0) (

*

s q

B

z

t

poor stability against current-driven kinks Not available to stabilise MHD instabilities

16 1

2 0 0 0

2

 

a

p

prdr

m I

 

1 )

( , )

ˆ (  W r  

dr dB B

r r

W

q

q

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

(14)

• 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

(15)

• 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)

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

q

d J

z

r

dr B dp

J B

J

q z

z q

r

J B

z

 

0

1

q

m

2

0

0

2

0 2

2

   

 

  

r B B

p B dr

d

z

m

m

q

q

- 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.

(17)

• The General Screw Pinch

Equilibrium: 1-D Configurations

16 1

2 0 0 0

2

 

a

p

prdr

m I

 

a

prdr I

0a

B B

z

rdr

0

2 2

0 2

0 0

0

2 2

2 8

 m

 m

a

t

prdr

B a

2 02 0

4 m

0

1   1

t t

p

t

a

0a

 B B

z2

 rdr

0

2

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

(18)

• 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 0

0 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

(

0

r 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 

2

R

0

r

2

(19)

• The General Screw Pinch

Equilibrium: 1-D Configurations

2 0 )

(

2

2

0 2 2

0

    

 

 

B

B p B

dr d B r r

W

q

m 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.

(20)

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

(21)

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/

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