Fusion Reactor Technology I

Fusion Reactor Technology I(459.760, 3 Credits)

Prof. Dr. Yong-Su Na

(32-206, Tel. 880-7204)

Week 1. Magnetic Confinement

Week 2. Fusion Reactor Energetics (Harms 2, 7.1-7.5)

Week 3. Tokamak Operation (I):

Basic Tokamak Plasma Parameters (Wood 1.2, 1.3)

Week 4. Tokamak Operation (II): Startup

Week 5. Tokamak Operation (III): Tokamak Operation Mode

Week 7-8. Tokamak Operation Limits (I):

Plasma Instabilities (Kadomtsev 6, 7, Wood 6)

Week 9-10. Tokamak Operation Limits (II):

Plasma Transport (Kadomtsev 8, 9, Wood 3, 4)

Week 11. Heating and Current Drive (Kadomtsev 10)

Week 12. Divertor and Plasma-Wall Interaction

Week 13-14. How to Build a Tokamak (Dendy 17 by T. N. Todd)2

Contents

Week 1. Magnetic Confinement

Week 2. Fusion Reactor Energetics (Harms 2, 7.1-7.5)

Week 3. Tokamak Operation (I):

Basic Tokamak Plasma Parameters (Wood 1.2, 1.3)

Week 4. Tokamak Operation (II): Startup

Week 5. Tokamak Operation (III): Tokamak Operation Mode

Week 7-8. Tokamak Operation Limits (I):

Plasma Instabilities (Kadomtsev 6, 7, Wood 6)

Week 9-10. Tokamak Operation Limits (II):

Plasma Transport (Kadomtsev 8, 9, Wood 3, 4)

Week 11. Heating and Current Drive (Kadomtsev 10)

Week 12. Divertor and Plasma-Wall Interaction

Week 13-14. How to Build a Tokamak (Dendy 17 by T. N. Todd)3

Contents

4

• Neoclassical Transports

Tokamak Transport

- Rarefied plasma at high temperature:trapped particles are the main contributors to transport.Diffusion and thermal conductivity are dominated by the collisions which correspond to transferring the particles from being trapped to transit ones and vice versa.

enl / ,/ LtreffTeff qrxqRv »D»D>>=

( ) ennn // 2|| »» ^ vveff e^vv ~||

- Effective collision frequency:

- Transport coefficients:

( ) 222/32Lefftr rqx nene -=D Trapped particle fraction = ε1/2

- The banana diffusion region is limited by the condition:

1/

//2/3*

2/3

<<=

>>==-

T

TeffTeff

vqR

qRvv

nen

nenele Trapped particle fraction = ε1/2

e/Ldtr qrtvx »»D

5


Tokamak Transport

- In the plateau region, 1 < ν* < ε-3/2

(ε3/2 < νRq/vT < 1 or ε3/2vT/Rq < ν < vT /Rq )- The average collision frequency is less than the mean bounce

frequency → only slow-transit particles contribute to the transport- The relative number of slow-transit particles: ν/vT

qRvrqvv TLTeff /~/ 222nD Slow-transit particle fraction = v/vT

- Transport coefficients:

22 / vvTeff nn »- Effective collision frequency:

- Displacement: vvqr TL /»D

6


Tokamak Transport

- In the Pfirsch-Schlueter region,

- Diffusion flux in a uniform field

drdp

Bnnv ^-= h2

1

drdpBjBBj / 1-^ -=®Ñ=´

rr

evBjejBvE =®=´+ ^^hh rrrr Friction force = Lorentz force (ExB drift

not contributing to diffusion)

- Modified diffusion flux by the additional flux due to longitudinalcurrent, so-called, the Pfirsch-Schlueter current owing to the toroidal effect

- Compared with a uniform magnetic field, the flux in toroidal plasma is enhanced by a factor (1+q2).

÷÷ø

öççè

æ+-=

^^

2||2

211 q

drdp

Bnnv

hh

h in H and D plasmas||2hh »^

D. Pfirsch and A. Schlueter, Der Einflussder elecktrischen Leitfaehigkeit auf das Gleichgewichtsverhalten von Plasmenniedrigen Drucks in Stellaratoren, Max-Planck-Institut, Report MPI/PA/7/62 (1962)

2BkTn

D å^^ =

h

7

• Pfirsch-Schlüter Current

Tokamak Transport

→ charge separation

No charge accumulation on a flux surface (quasineutrality)

A

B

BpJ Ñ

=^

r

BA JJ ^^ >rr

0=×Ñ Jr

J|| needed: Pfirsch-Schlüter current

Diamagnetic current or current needed to establish an equilibrium,

0ˆ22 ¹÷÷

ø

öççè

æ-=

Ñ´=^ y

jddpR

BBI

BpBJ

rrr

pBJ Ñ=´rr

8

• Pfirsch-Schlüter Current (J. Wesson, Tokamaks)

Tokamak Transport

pB

RBddp

BRB

ddp

Bp

Bj pp ¢-=-=Ñ-=Ñ=^ y

yy

11

^--= jBB

jBB

j pp

f||

ppp BFBddFj ¢==y

0 =®¶¶

-=´Ñ ò dsEtBE ps

ffh EBB

EBB

j psp +=||||

p

p

p

p

p

p

p

pps

p

BB

BBE

BB

BpF

BBBpF

BBBBE

EBBB

pFBj

F

/

/

/

/1

//

//

/1

2||

20

20

2||||

220||

hm

mhh

m

ff

ff

+¢=

¢++=

¢+=¢

òò= dsxdsx /

9

Tokamak Transport

÷÷

ø

ö

çç

è

æ-¢= B

BB

BB

pFjp

pPS /

/1120m

BBB

BBEB

BB

BB

pFjp

p

p

p

/

/

/

/112

||20|| h

m ff+÷÷

ø

ö

çç

è

æ-¢=

- This flow is dominant in the SOL region (high collisionality regime).- Pfirsch-Schlüter current removes the main part of the charge

separation caused by the curvature and gradient drifts (but residual charge separation still causes transport)

For the circular CX large aspect ratio toroidal configuration

qef cos10

+=

BBqq

cos12drdp

Rr

BjPS -=

• Pfirsch-Schlüter Current (J. Wesson, Tokamaks)

10

• Shafranov Shift

Tokamak Transport

- The Pfirsch-Schlüter current produces vertical field BZ,0

→ Plasma shifted outwards→ Shafranov shift

0,0

ZBBRi

»D

11

Tokamak Transport

÷÷

ø

ö

çç

è

æ-+

Ñ-=

-Ñ

-=

Ñ-

-=

^^

^^

^^^

fffff

ff

ff

hh

hh

h

EBBB

BBEBB

pjBB

B

BE

Bpj

BBB

Bp

BBEBE

v

p

p

pPS

p

pp

pPS

/

/1

22||

2||||

22

• Pfirsch-Schlüter Diffusion (J. Wesson, Tokamaks)

22 Bp

BBEv Ñ-

´= ^

^h

rrr

BBB

BBEjj

p

pPS /

/2

|||| h

ff+=

ffh EBB

EBB

j psp +=||||

÷÷

ø

ö

çç

è

æ-¢= B

BB

BB

pFjp

pPS /

/1120m

òò ^^ ==G dsRvnRdsvn pp 22

÷÷

ø

ö

çç

è

æ-¢-=^

p

p

pPS BB

BBB

RBpFRv

/

/11220||

fmh

12

Tokamak Transport

For the circular CX large aspect ratio toroidal configuration

• Pfirsch-Schlüter Diffusion (J. Wesson, Tokamaks)

2||

2

0

/2q

hBdrdp

Rr

RRv

PS ÷øö

çèæ-=^

( ) 2||2

20

2/BBE

qBdrdp

RRv qfhh -+-= ^

^

q

f

RBrB

q =

÷÷ø

öççè

æ+=

^

2||21 qDD C

hh

qq

cos12drdp

Rr

BjPS -=

13


Tokamak Transport

TnrTq

nrnD

E

ÑD

-»Ñ-=

ÑD

-»Ñ-=G

tk

t2

2

)(

)(: Fick’s law

: Fourier’s law

Thermal diffusivity

ct

tkc

22

)( aDrn E

E

»®»D

»º

J. Wesson, Tokamaks (2004)

2BkTn

D å^^ =

h

14

• Ware Pinch

Tokamak Transport

- Inward particle transport due to the toroidal electric field

15

• Ware Pinch (J. Wesson, Tokamaks)

Tokamak Transport

- The inward flow occurs for trapped particles and their behaviourfollows directly from the toroidal equation of motion.

( ) ( )[ ]fff BvEevmdtd

jj ´+=

Zero steady state time average of the left-hand side term for trapped particles (the integral between bounces is zero)

( ) ff EBv -=´

q

f

BE

v -=^ ( ) qf BvBv ^=´

q

feBE

n2/1~G Trapped particle fraction = ε1/2

16


Tokamak Transport

- The modified equation of motion along the magnetic field line

j

jb m

Ees

dtsd fw +-= 22

2

mean angle:

2sinbj

jbb m

Eetss

ww f+=

( ) qq rBBs /=rBmEBe

tbj

jbb 2sin

wwqq fq+=

rBmEBe

bj

j2w

q fq=

The effect of the ∇B and curvature drift is not symmetric about the mid-plane

17

Tokamak Transport

rBmEBe

bj

j2w

q fq=

fq

wwqq

q

ErBmBve

tvv

vv

bj

djjbbdjdj

djr

2sin~

sin

--=-

-=

Resulting radial velocity for deeply trapped particles for which θ is small

q

ff

q

ww

BE

ErBBv

vb

djcjr -=-= 2

( )( ) 2/1b

2 2// ,/21 RrqRvRBevmv jjdj ^^ == w

q

f

BE

vr -=


18

Tokamak Transport

- The drift velocity is controlled by the balance of two forces, the electrical field force and the Lorentz force.

- vW ~ 0.2 m/s for E = 0.1 V/m, Bθ = 0.5 T- The effect is much larger (1/ε2) for trapped particles than that

experienced by passing particles. HW: Why?


rBmEBe

bj

j2w

q fq=

q

f

BE

vr -=

19

• Bootstrap current

Tokamak Transport

- Named after the reported ability of Baron von Munchausen to lifthimself by his bootstraps (Raspe, 1785)

- Suggested with ‘Alice in Wonderland’ in mind where the heroine managed to support herself in the air by her shoelaces.

http://en.wikipedia.org/wiki/Bootstrapping

20

• Bootstrap

Tokamak Transport

MEANING:verb tr.: To help oneself with one's own

initiative and no outside help.noun: Unaided efforts.adjective: Reliant on one's own efforts.

ETYMOLOGY:While pulling on bootstraps may help with putting on one's boots, it's impossible to lift oneself up like that. Nonetheless the fanciful idea is a great visual and it gave birth to the idiom "to pull oneself up by one's (own) bootstraps", meaning to better oneself with one's own efforts, with little outside help. It probably originated from the tall tales of Baron Münchausen who claimed to have lifted himself (and his horse) up from the swamp by pulling on his own hair.

http://wordsmith.org/words/bootstrap.html

Baron Münchausen lifting himself up from the swamp by his own hair

Illustrator: Theodor Hosemann

In computing, booting or bootstrapping is to load a fixed sequence of instructions in a computer to initiate the operating system. Earliest documented use: 1891.1

21

• Bootstrap

Tokamak Transport

http://wordsmith.org/words/bootstrap.html

“I was still a couple of miles above the clouds when it broke, and with such violence I fell to the ground that I found myself stunned, and in a hole nine fathoms under the grass, when I recovered, hardly knowing how to get out again. Looking down, I observed that I had on a pair of boots with exceptionally sturdy straps. Grasping them firmly, I pulled with all my might. Soon I had hoist myself to the top and stepped out on terra firma without further ado.”- With acknowledgement to R. E. Raspe, Singular Travels, Campaigns and Adventures of Baron Munchausen, 1786. Edition edited by J. Carswell. London: The Cresset Press, 1948. Adapted from the story on p. 22(???).

22


Tokamak Transport

Nature Physical Science 229 110 (1971)

23


Tokamak Transport

IOH

INB+IOH

INB+IOH+IBS

24

Toroidal direction

Ion gyro-motion

Fast ion trajectory

Poloidaldirection

Projection of poloidallytrapped ion trajectory

R

B

• Neoclassical Bootstrap current

http://tfy.tkk.fi/fusion/research/

ASCOT

Tokamak Transport

25

Currents due to neighbouring bananas largely cancel

orbits tighter where field

stronger

eb

e

q

pBS

BS

II

drdp

BJ

67.0/

2/1

=

-»Tokamak Transport• Neoclassical Bootstrap current

- More & faster particles on orbits nearer the core (green .vs. blue) lead to a net “banana current”.

- This is transferred to a helical bootstrap current via collisions.

26

• Bootstrap Current

Tokamak Transport

M. Kikuch et al, PPCF 37 1215 (1995)

- Trapped-electron orbits and schematics of the velocity distribution function in a collisionless tokamak plasma

Small Coulomb collision smoothes the gap and causes particle diffusion in the velocity space.Collisional pitch angle scattering at the trapped-untrapped boundary produces unidirectional parallel flow/momentum input and is balanced by the collisional friction force between electrons and ions.

27


Tokamak Transport

( ) ( )[ ]( ) 0 ,,

0 ,,,

||||0

0

||||

>D¶

¶-»

>--=+ -+-+

vdrvrrfe

vdvrfrfeJJ

e

gege

vvvvv

( ) 2/12/12 /eLqrrr »D»D

rn

BTq

drn

BTq

vdvvrF

BqmJ

c

Met

¶¶

-»

¶¶

-=

>¶¶

-=

ò

ò-

^-

0

2/1

2/ 2

0

2/1

||||0

2/1

0

cossin23

0 ,

e

qqqe

e

p

q

v

eq 2cos =c

- The trapped electron magnetization current

Assumption:- Uniform temperature- Infinitely massive ions

not depending on collisions, generated solely because of the density (or temperature) gradient of the guiding centers

28


Tokamak Transport

( ) 0 ,,||||

0

0 >D¶

¶-» vdrv

rrfeJ e

p vv

Lqrr »D

rn

BTq

drn

BTq

vdvvrF

BqmJ

c

Mep

¶¶

-»

¶¶

-=

>¶¶

-=

ò

ò ^

0

2/ 2

0

||||00

cossin23

0 ,

p

qqqq

v

eq 2cos =c

- The passing electron magnetization current

29


Tokamak Transport

( )ce

eeeep

eptp

p

peptppep r

nqTJemn

enJ

mnumPwnnnn

¶¶

»-»÷÷ø

öççè

æ-==D ||

( )ce

eeeet

etpt

t

tetpttet r

nqTJemn

enJmnumP

wne

ennn

¶¶

»-»÷÷ø

öççè

æ-==D - 2/1

||

- The collision-driven bootstrap current

( ) ( )tp

PP ||2/1

|| D=D e( ) ( )tp

PP |||| D¹D Collisional momentum balance violated

( ) ( ) ( ) ( )

( ) ( ) ÷÷ø

öççè

æ +-

úúû

ù

êêë

é+D÷

øö

çèæ

¶¶

+»

úúû

ù

êêë

é -+-®÷

÷ø

öççè

æ +-=

^

^

^

^

^

^

2

2||

2

2||

||

||32/3

2

2||

2

32/32

2||

2

32/3

exp211

exp exp,

vvv

vuv

rrn

nvv

vrn

vuvv

vrn

vvv

vrn

rf

T

Bp

T

g

B

T

g

T

ggp

p

ppv Shifted

Maxwellian

- The shift must be in the passing particles since the trapped particles are “trapped” and thus are not allowed to drift toroidally.

30


Tokamak Transport

( ) eeBpp

epun

eJ

mP n÷÷ø

öççè

æ+-»D ||

( ) ( )tp

PP |||| D=Dce

eeeepBe

ce

ee

rnqTnum

rnqT

wnen

wn

¶¶

=+¶¶ - 2/1

úûù

êëé

¶¶

+¶¶

-= -

rT

Tn

rn

BTqJB 04.071.4

0

2/1eLarge aspect ratioCircular CXNon-massive ionsNon-uniform temperature

- A transport driven toroidal plasma current carried by the passingelectrons generated by collisional friction with the trapped electron magnetization current

rn

BTquenJ BpB ¶

¶-»-= -

0

2/1e1/ε and 1/ε1/2 larger than the trapped and passing particle magnetization current, respectively

31


Tokamak Transport

PPB

B GJJrf bebef

2/12/1 ~18.1)( -»º

( ) ( ) ( )¢¢+= qrBTnrG ln/ln04.0ln

- Bootstrap current fraction

- In high-β tokamak, βp ~ 1/ε, implying that fB ~ 1/ε1/2 >>1:The bootstrap current can theoretically overdrive the total current

- No obvious “anomalous” degradation of JB due to micro-turbulence- The bootstrap current is capable of being maintained in steady

state without the need of an Ohmic transformer or external current drive. This is indeed a favourable result as it opens up the possibility of steady state operation without the need for excessive amounts of external current drive power.

- This is critical since bootstrap current fractions on the order of fB > 0.7are probably required for economic viability of fusion reactors.

32

Tokamak Transport• 100% bootstrap discharges

Y. Takase, IAEA FEC 1996, S. Coda, IAEA FEC 2008

33


Tokamak Transport

J. Wesson, Tokamaks (2004)

- May increase D, c up to two orders of magnitude:- ci 'only' wrong by factor 3-5- D, ce still wrong by up to two orders of magnitude!

2BkTn

D å^^ =

h

34

Tokamak Transport• Transport in fusion plasmas is 'anomalous‘.

- Normal (water) flow: Hydrodynamic equations can developnonlinear turbulent solutions (Reynolds, 1883)

- Transport mainly governed by MHD turbulence: radial extent of turbulent eddy: 1 - 2 cmtypical lifetime of turbulent eddy: 0.5 - 1 ms

- Anomalous transport coefficients are of the order 1 m2/s

35

References

- Francis F. Chen, “Introduction to Plasma Physics and Controlled Fusion”, 2nd Edition, Plenum Press, New York (1984)- Acad. M. A. Leontovich et al, “Reviews of Plasma Physics, Volume 1”, Consultants Bureau, New York (1965)- Jeffrey P. Freidberg, “Plasma Physics and Fusion Energy”, Cambridge University Press (2007)- Hartmut Zohm, “Tokamaks: Equilibrium, Stability and Transport”, IPP Summer University on Plasma Physics, Garching, 18 September, 2001- Tim Hender, “Neoclassical Tearing Modes in Tokamaks”, 2009 Korean Physical Society/ Division of Plasma Physics (KPS/DPP) in Daejun, Korea, 24 April 2009- Mitsuru Kikuchi, “Frontiers in Fusion Research”, Springer (2011)

Fusion Reactor Technology I

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