y-k. m. peng and r. a. dory- very small aspect ratio tokamaks

8/3/2019 Y-K. M. Peng and R. A. Dory- Very Small Aspect Ratio Tokamaks

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V E RY S M A L L A S P E C T R A T I O T O K A M A K S '

Y - K . M. P e n gR . A. D o r y

Fus ion Energy D iv is ion

Oak Ridge Na t ion al La bo rato ry

Oak R id ge , Tennessee 37830

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-NOTICE-This report was prepared as an iccount of work

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' R e s e a r c h s p o n s o r e d by the O f f i c e of F u s i o n E n e r g y ( E T M J , U . S . D e p a r t m e n to f E r e r g y u n d e r c o n t r a c t W - 7 4 0 b - e n g - 2 6 w i t h the U n i o n C a r b i d e C o r p o r a t i o n .



C O N T E N T S

A B S T R A C T 1

1 . I N T R O D U C T I O N 2

2 . M H D E Q U I L I B R I U M A N D S T A B I L I T Y 5

2.1 S O L O V ' E V E Q U I L I B R I U M 5

2 .2 N U M E R I C A L E Q U I L I B R IA 6

2 .3 M H D S T A B I L I T Y 7

3 . P O S S I B L E P L A S M A P A R A M E T E R S 9

3.1 E F F E C T S O F S M A L L A A N D H I G H g\ 9

3 .2 A N E X A M P L E W I T H A = 2 . . . . 113 .3 A S M A L L C O N T I N U O US T O K A M A K C O M P R E SS O R 1 23 .4 A D - T I G N I T E D P L A S M A W I T H A = 1 .5 1 43 .5 A C O M P R E S S O M A K F O R I G N I T I O N D E M O N S T R A T I O N 1 5

4 . C O N C L U S I ON S A N D D I S C U SS I O N 1 8

4.1 P O S S I B I L I T I E S O F L O N G P U L S E 1 9

4 .1 .1 N e u tr a l B e a m D r i v e n P l a s m a C u r r e n t 2 04 . 1 .2 B o o t s t ra p C u r r e n t • 2 04 . 1 . 3 R F - D r i v e n C u r r e n t s 21

4 .2 C O M P A R I S O N S W I T H O T H E R T O R O I D A L P L A S M A S 21

A C K N O W L E D G M E N T S * . . . 2 2

A P P E N D IX . S I M P L E T O K A M A K F O R M U L A S . 2 3

R E F E R E N C E S 2 7



1 . I N T R O D U C T I O N

T o k a m a k e x p e r i m e n t s h a v e a c h i e v e d g o o d p l a s m a c o n f i n e m e n t ( n - p \

3 x 1 01 3

c m "3

• s e c ) a n d h i g h t e m p e r a t u r e s ( T -•- T - a b o v e 2 k e V ) .!

E n t h u s i a s m h a s b e e n g e n e r a t e d f o r c a r r y i n g o u t c o n c e p t u a l d e s i g n s t u d i e s2

o f t o k a m a k f u s i o n r e a c t o r s b a s e d o n e x t r a p o l a t i o n o f c o n v e n t i o n a l v a l u e s

o f p l a s m a p a r a m e t e r s . D e p e n d i n g o n t h e a s s u m e d t o r o i d a l f i e l d s t r e n g t h

( B . ) , t h e p l a s m a b e t a v a l u e s (If = 8 i r p / B2, w h i c h i s r o u g h l y s". = 8 T T P / B

2,

f o r l a r g e a s p e c t r a t i o s , A ) u s e d in t h e s e s t u d i e s r a n g e f r o m 3 % t o 7 %

a n d t h e e s t i m a t e d f u s i o n p o w e r d e n s i t i e s f a l l b e l o w 5 M W / m3. A t p r e s e n t ,

i t i s e s t i m a t e d t h a t a r e a c t o r d e v i c e w i l l c o s t a r o u n d $ 0 . 5 - 1 b i l l i o n ;

i t w i l l o p e r a t e i n a p u l s e d m o d e a n d is c o n s i d e r e d c o m p l e x i n t e c h n o l o g y

a n d t o p o l o g y . I t i s h i g h l y d e s i r a b l e to i m p r o v e t o k a m a k s i n t h e s e a r e a s .

T h e c o s t a n d s i z e o f f u s i o n r e a c t o r s c a n b e r e d u c e d b y i n c r e a s i n g

t h e f u s i o n p o w e r d e n s i t y , w h i c h i s p r o p o r t i o n a l t o S2B

1 +f o r a f i x e d

p l a sm a t e m p e r a t u r e d u r i n g b u r n . R e c e n t i n t r o d u c t i o n o f t h e f l u x c o n s e r v -

i ng t o k a m a k ( F C T ) c o n c e p t1

a n d t h e F C T e q u i l i b r i u m c a l c u l a t i o n s3*

1* r e p r e -

s e n t sigrr" ' c a n t a d v a n c e s t o w a r d h i g h p o w e r d e n s i t y i n t o k a m a k s . T h e s e

s t u d i e s s h o w e d t h a t h ig h b e t a ( J > 3 0 % ) , h i g h s a f e t y f a c t o r (q > 1 )

e q u i l i b r i a e x i s t i n c o n t r a s t t o p r e v i o u s e s t i m a t e s5

o f e q u i l i b r i u m

a v e r a g e b e t a l i m i t s o f a f e w p e r c e n t . M o r e r e c e n t s t u d i e s6"

9o f m a g n e t o -

h y d r o d y n a m i c ( M H D ) e q u i l i b r i u m e v o l u t i o n t o h i g h b e t a , b a s e d o n v a r i o u s

m o d e l s o f p l a s m a t r a n s p o r t l o s s e s , h a v e f u r t h e r s h o w n t h a t t h e s e h i g h

b e ta e q u i l i b r i a a r e e x p e c t e d t o b e a c c e s s i b l e t h r o u g h i n t e n s i v e h e a t i n g ,

p r o v i d e d t h a t i n s t a b i l i t i e s d o n o t c a u s e a s e v e r e l o s s o f Diasnici c o n f i n e -

m e n t .



P r e s s u r e d r i v e n i c « al M H D i n s t a b i l i t y i s o n e s u c h c a n d i d a t e t h a t

h a s b e e n s t u d i e d1-

3"

1 3f o r h i g h b e t a F C T e q u i l i b r i a . T h e s e s t u d i e s w i t h

A > 3 s h o w t h a t t h e c r i t i c a l b e t a f o r s t a b i l i t y c a n b e e x p r e s s e d a s

(1 )

w h e r e C d e p e n d s o n t h e s h a p e o f p l a s m a c r o s s s e c t i o n , p r o f i l e s , p o l o i d a l

b e t a ( O > A , a n d c o n d u c t i n g w a l l l o c a t i o n s . S o m e o p t i m i z e d e q u i l i b r i aP

w i t h r e a l i s t i c w a l l l o c a t i o n s f o u n d s o f a r1 0

h a v e r e s u l t e d i n v a l u e s o f

C c l o s e t o 1 . 3 . T h i s g i v e s v a l u e s o f ? c u p t o 0 . 0 8 f o r ( q b / q Q ) =

q ( b o u n d a r y ) / q ( a x i s ) ^ ?. a n d A ^ 3 - 4 . T h e s e r e s u l t s s u g g e s t t h a t t h e

a t t a i n a b l e b e t a v a l u e i n t o k a m a k s m a y b e s e v e r e l y l i m i t e d b y M H D s t a b i l i t y

r e q u i r e m e n t s .

E q u a t i o n ( 1 ) i n d i c a t e s t h a t S c a n a l s o b e i n c r e a s e d b y r e d u c i n g A .

J a s s b y1" s u g g e s t e d S M A R T O R ( S m a l l - A s p e c t - R a t i o T o r u s ) w i t h A ^ 2 b y

a s s u m i n g t h a t t h e i n - b o r e ( i n t h e c e n t e r o f t h e t o r u s ) p r i m a r y o h m i c

S e a t i n g c o i l s c a n b e e l i m i n a t e d o r d r a s t i c a l l y r e d u c e d i n s i z e w i t h t h e

u s e o f t a n g e n t i a l n e u t r a l i n j e c t i o n a s a p r i m a r y d r i v e t o s t a r t u p t h e

p l a s m a c u r r e n t . A n o t h e r a p p r o a c h t o v e r y s m a l l a s p e c t r a t i o s w a s r e c e n t l y

i n t r o d u c e d w i t h t h e c o n c e p t o f c o n t i n u o u s t o k a m a k s .1 5

I n t h i s a p p r o a c h ,

a p l a s m a w i t h l a r g e A i s i n i t i a t e d b y i n - b o r e p r i m a r y c o i l s i n a s t a r t -

u p c h a m b e r . T h e i n d u c t i o n e l e c t r i c f i e l d r e q u i r e d t o m a i n t a i n t h e

t o k a m a k p l a s m a c u r r e n t a f t e r s t a r t u p i s m a n y o r d e r s o f m a g n i t u d e i e s s

t h a n t h a t r e q u i r e d f o r s t a r t u p . . T h e p l a s m a c a n b e s h i f t e d a n d c o m p r e s s e d

t o a b u r n c h a m b e r w h e r e n o i n - b o r e p o l o i d a l f i e l d c o i l s a r e n e e d e d ,

r e d u c i n g t h e a s p e c t r a t i o t o A < 2 .



T h e s e t w o s t u d i e s h a v e a s s u m e d t h a t t h e c r i t i c a l t o r o i d a l b e t a

( ¥ * = 8 T T P " / B ? , w h e r e B . i s t h e v a c u u m t o r o i d a l f i e l d a t t h e c h a m b e rt c t o t o

c e n t e r ) v a r i e s l i k e I /A . T h i s w o u l d g i v e s , ^ 0.1 f o r A ^ 2 . T h i s

a s s u m p t i o n , t o g e t h e r w i t h r e a s o n a b l e v a l u e s f o r B . , h a s l e d t o r e a c t o r

p l a s m a w i t h m i n o r r a d i i o f M m ( R e f s . 1 4 a n d 1 5 ) . H o w e v e r , a s A

d e c r e a s e s t o w a r d 1 , t h e p o l o i d a l f i e l d ( B ^ B t / q A ) b e c o m e s c o m p a r a b l e

t o B t . T h e v a l u e s o f 3 a n d 2 t b e c o m e s i g n i f i c a n t l y d i f f e r e n t , an d t h e

a s s u m p t i o n o f i f ^ I/ A b e g i n s t o b r e a k d o w n . I n S e c t i o n 2 , w e f i r s t

p r e s e n t M H D e q u i l i b r i a w i t h q b = 2 .0 a n d 3 = A / 2 b u t w i t h ~ $ t = 0 . 4 3

(A = 2 . 0 ) a n d 1 .4 0 ( A = 1 . 5 ) . N u m e r i c a l s t a b i l i t y c a l c u l a t i o n s u s i n g

t h e E R A T O s t a b i l i t y c o d e1 6

a r e t h e n c a r r i e d o u t t o s h o w t h a t t h e s e

e q u i l i b r i a h a v e ? . v a l u e s a r o u n d 0 . 1 8 ( A = 2 ) a n d 0 . 2 8 ( A = 1 . 5 ) , w i t h

c o n d u c t i n g w a l l s a t a / 5 f r ^ m t h e p l a s m a .

T h e i m p a c t o f s u c h h i g h $ . v a l u e s o n t h e e c o n o m y o f t o k a m a k f u s i o n

p o w e r w o u l d b e d r a m a t i c . I t i s t h e r e f o r e o f i n t e r e s t t o r o u g h l y c h a r a c t e r -

i z e t h e p a r a m e t e r s o f s u c h p l a s m a s . I f h i g h J . v a l u e s a r e d e m o n s t r a t e d

o n a s m a l l s c a l e , a n i n e x p e n s i v e f u s i o n i g n i t i o n t e s t r e a c t o r c o u l d h a v e

t h e p a r a m e t e r s p r e s e n t e d i n S e c t i o n 3 .

T h i s p a p e r e n d s w i t h a d i s c u s s i o n o f t h e e f f e c t s o f s t r o n g t o r o i d i c i t y

o n t o k a m a k c u r r e n t s d r i v e n b y n e u t r a l b e a m i o n s ,1 7

b o o t s t r a p e f f e c t s 31 8

a nd n e t r f m o m e n t u m .1 3

C o m p a r i s o n s o f t h e ' ' / e r y s m a l l a s p e c t r a t i o

t o k a m a k s w i t h o t h e r m a g n e t i c f u s i o n c o n c e p t s , s u c h as t h e r e l a t i v i s t i c

e l e c t r o n r i n g ,2 0

t h e f i e l d r e v e r s e d t h e t a p i n c h ,2 1

t h e f i e l d r e v e r s e d

m i r r o r ,2 2

t h e s p h e r o m a k ,2 3

a n d l i n e r i m p l o s i o n o f t o r o i d a l p l a s m a s , —

a r e a l s o d i s c u s s e d .



z e r o a s A a p p r o a c h e s u n i t y , c o n s i s t e n t w i t h t h e f i n d i n g s b y C h a r i t o n

a t a l .1 0

t h a t ? t = (A - B p ) .

O n t h e o t h e r h a n d , t h e M e r c i e r c r i t e r i o n w i l l b e s a t i s f i e d a t t h e

m a g n e t i c a x i s i f

1 - 3e (l + e ) L \ 1 + e

(4 )

T h e c r i t i c a l q v a l u e i s 1 . 1 8 i f e = 1 . 6 5 i s a s s u m e d . T h e c o r r e s p o n d i n g

m a x i m u m v a l u e o f s t i s t h e n 0 . 2 5 . W e s e e t h a t i t i s n o t n e c e s s a r i l y

d e s i r a b l e t o h a v e A = 1 f o r m a x i m u m g > . .

2 . 2 N U M E R I C A L E Q U I L I B R I A

G u i d e d b y t h e d i s c u s s i o n i n S e c t i o n 2 . 1 , w e e x a m i n e s o m e n u m e r i c a l

e q u i l i b r i a w i t h A = 2 .0 a n d 1 . 5 . T h e e q u i l i b r i a a r e o b t a i n e d t h r o u g h t h e

F C T a p p r o a c h , ^ w i t h p r e s c r i b e d p ( ^ ) a nd q U ) . P a r a m e t e r s c h a r a c t e r i z i n g

t h e e q u i l i b r i a a r e s h o w n in T a b l e 1 . T h e c o r r e s p o n d i n g p o l o i d a l f l u x

s u r f a c e s a n d c o n s t a n t | B | c o n t o u r s a r e s h o w n i n F i g u r e 1 . T h e p r o f i l e s

o f p ( ' i ) , F2( - d > ) , q ( ^ ) , p ( R , Z = 0 ) , J J R , Z = 0 ) , a n d t he p o l o i d a l f i e l d

( B ) a l o n g t h e p l a s m a b o u n d a r y { i ) a r e s h ow n i n F i g u r e s 2 ( f o r A = 2 . 0 )

a n d 3 ( f or A = 1 . 5 ) .

I t i s s e e n t h a t a l t h o u g h q ( w ) i n c r e a s e s m o n o t o n i c a l l y f r o m 1 a t t h e

m a g n e t i c a x i s t o 2 a t t h e b o u n d a r y , t h e v a l u e s f o r s~. a r e 0 . 43 a n d 1 . 40

f o r A = 2 . 0 a n d 1 . 5 , r e s p e c t i v e l y . T h e s i m p l e d e p e n d e n c e o f e q u i l i b r i u m

3t *

3p /

A 2 i s n o l o n9

e ra p p l i c a b l e h e r e . M o t e t h a t t h e s e h i g h ? t v a l u e s

d o n o t r e q u i r e t h a t 3 a p p r o a c h A , d i f f e r e n t f r o m t h e c a s e o f F C T e q u i -

l i b r i a " w i t h A = 4 . S i n c e 3 = A / 2 f o r t h e c a s e s c o n s i d e r e d , t h e e q u i -

l i b r i a a r e n o t d i a m a g n e t i c o n t h e a v e r a g e .



I t is a l s o s e e n t h a t t h e r e l a t i v e m a g n i t u d e s o f B t a n d B i n t h e s e

e q u i l i b r i a a r e c h a n g e d f r o m t h o s e i n t h e c a s e o f l a r g e r a s p e c t r a t i o s .

• As l i s t e d i n T a b l e 1 , F i g u r e s 2 ( f ) a n d 3 ( f ) s h o w t h a t w h e n A i s r e d u c e d

t o 1 . 5 , B b e c o m e s c o m p a r a b l e t o B t a t R = ( R Q - a ) an d b e c o m e s l a r g e r

t h a n B t a t R = ( R + a ) . | L c a n n o l o n g e r b e a g o o d i n d i c a t o r o f p l a s m a

i \ a l t h o u g h i t c o n t i n u e s t o b e a c o n v e n i e n t e n g i n e e r i n g v a l u e a s l o n g as

B t r e m a i n s u n c h a n g e d d u r i n g t h e t o k a m a k d i s c h a r g e . A p h y s i c s s n o w

m a y b e d e f i n e d w i t h r e s p e c t t o t h e a v e r a g e d B2

a t t h e p o l o i d a l p l a s m a

b o u n d a r y :

f I p ) , ( 5 )

w h e r e V / = ( d V / d ^ ) . . T h e J , v a l u e s a r e f o u n d t o b e 0 . 1 9 a n d 0 .3 1 f o r

A = 2 .0 a nd 1 . 5 , r e s p e c t i v e l y . T h e s e v a l u e s a r e c l o s e t o t h e m a x i m u m

J . v a l u e o f 0 . 2 5 o b t a i n e d f o r t h e S o l o v ' e v e q u i l i b r i u m o f A = V 2~ . I n

a d d i t i o n , b e c a u s e o f t h e d o m i n a n c e o f B a t l a r g e r v a l u e s o f R-» a n

a b s o l u t e m i n i m u m i n |BJ n o w e x i s t s c l o s e t o t h e m a g n e t i c a x i s ( F i g u r e 1 )

e v e n w h e n t h e p l a s m a i s p a r a m a g n e t i c .

2 . 3 M H D S T A B I L I T Y

P r e l i m i n a r y c r i t i c a l s t v a l u e s f o r t h e s e e q u i l i b r i a w i t h a c o n d u c t i n g

w a l l a t 0. 2a a w a y f r o m t h e p l a s m a e d g e h a v e b e e n o b t a i n e d u s i n g t h e

E R A T O c o d e .1 5

B e c a u s e o f t h e s m a l l A v a l u e s , t h e p r e s e n t c a l c u l a t i o n s

r e q u i r e c a r e f u l c o n v e r g e n c e a n a l y s i s , a s i n o u r p r e v i o u s c a l c u l a t i o n s1 0

w h i c h u s e t h e f o l l o w i n g p r o c e d u r e s .



1 . Each equilibrium is scaled by s so that

p 2 - c2 2 ^ + p2 n =b b ' S

q

tS t '

qs(*) =• • & - :

1 / 2

3

P s= s B

P

2 . For g i v e n v a l u e s of s and N ( t he t o r o i d a l m o d e n u m b e r ) , the

c o n v e r g e n t g r o w t h r a t e s ( y ) are d e t e r m i n e d by l i n e a r e x t r a p o l a t i o n s of

Y 2 in the i n v e r s e of t h e y - \ g r i d s i z e ( F i g u r e 4 ) .

3 . U s i n g the c o n v e r g e n t g r o w t h r a t e s o v e r a r a n g e o f - s v a l u e s , the

c r i t i c a l s v a l u e for t a b i l i t y is o b t a i n e d by l i n e a r e x t r a p o l a t i o n s of y

i n 1, g i v i n g T A H ) and q n r ( N ) v a l u e s f o r e a c h N (= 1,2,3) m o d e ( F i g u r e 5 ) .

4 . By l i n e a r e x t r a p o l a t i o n s of "e ( N ) in 1 / N , the c r i t i c a l b e t a

c

va lu es in th e l i m i t of" la r g e N, s (N ->- <=°), are ob ta ined (Figure 6) .

Because the ERATO code conv erges to th e c o r r e c t growth r a te s from

the more uns tab le s ide ,1 6

our presen t g rowth ra tes tend to be pess imis t ic .

The TJ.AU - * • =°) values here may retain some uncertainties from the extra-

p o la ti o n s . C al cu la tio ns of J based on the BALOON co de1 3

have resu l ted

in ? t c > 18%(A = 2) and >30%(A = 1 .5 ) .

The parameters of the eq ui l i b r i a sca led to margina l s ta b i l i t y a resummarized in Tabl e 2. By com paring the T and 'i . va lues (14% and 18%

a t A = 2 .0 , 17% and 28% at A - 1.5 , re sp ec tiv el y) th e dependence of

? . = I/A seems reasonab le and give s si g ni fi ca nt ly higher values for2t c "

p e a '< l : ) e t a v a ^ u e s a t t t i e magnetic axis of the equilibria are of

order unity. Further studies with different q pr of i le s, T > shape, et c.

are needed to determine if f ur th er incre ases in ? t are possible.



A l t h o u g h p r e l i m i n a r y c a l c u l a t i o n s w i t h 5 0 x 5 0 g r i d s i n ip-x h a v e r e s u l t e d

i n h i g h e r ¥ , v a l u e s f o r t h e s m a l l N m o d e s , w e u s e o u r p r e s e n t r e s u l t s

a s t h e b a s i s o f t h e f o l l o w i n g d i s c u s s i o n .

3 . P O S S I B L E P L A S M A P A R A M E T E R S

H i g h b e t a t o k a m a k p l a s m a s w i t h v e r y s m a l l a s p e c t r a t i o s h a v e n o t

y e t b e e n s t u d i e d e x p e r i m e n t a l l y . I t i s i n f o r m a t i v e t o e s t i m a t e t h e

s c a l i n g o f t h e p l a s m a p e r f o r m a n c e w i t h A a n d 3 , i n g e n e r a l a n d t o e x a m i n e

s o m e s p e c i f i c e x a m p l e s o f s u c h p l a s m a s . T h e s i m p l e s c a l i n g f o r m u l a s

u s e d a r e i n c l u d e d i n t h e A p p e n d i x .

3 .1 E F F E C T S O F S M A L L A A N D H I G H |L

I f t h e m a x i m u m t o r o i d a l f i e l d a t t h e c o i l , B , i s f i x e d , a l l o w i n g A

t o d e c r e a s e w o u l d e v e n t u a l l y r e d u c e p". T h i s i s b e c a u s e B . a t t h e p l a s m a

c a n b e w r i t t e n a s

B t Q = B c [ A - (1 + A ) ] / A ,

w h e r e A a is t h e d i s t a n c e b e t w e e n t h e p l a s m a a n d t h e t o r o i d a l f i e l d

c o i l s . T h e r e s u l t s o f S e c t i o n 2 s h o w t h a t , r o u g h l y , 1 L = e / Aa

w i t h

:t > 2 . W e h a v e

p = ( 3 0 B 2 / 3 - 0 [ A - (1 + A ) p / Aa + 2

. ( 7 )

T h e m a x i m u m o f p t h e n o c c u r s w h e n A = [ ( a + 2 ) / a ] ( l + A ) , g i v i n g A <

2 (1 + A ) i f - j . > 2 . T h e c o s t e f f e c t i v e n e s s o f a t o k a m a k c a n b e r o u g h l y



10

measured by (p2/R

2) because th e cos t and the fu s i o n power are rou gh ly

p r o p o r t i o n a l t o R~2

and p2, r e s p e c t i v e l y .

2From Eq. ( 7 ) , we have

2a+sP

2/R

2 = [ ( 2 0 B 2 / M - 7 a ] [ A - (1 + A ) ] V A . (8 )

The maximum o f (p ^/R2) occu rs when A^ = [ ( a + 3 ) / ( a + 1 ) ] (1 + A ) , g i v ing

A' « 1.7(1 + A ) i f a > 2. Thus fo r smal l A , an aspect r a t i o o f a round

2 o r l e s s s h o u l d , i n p r i n c i p l e , be b e n e f i c i a l i f a > 2 . I t i s i n t e r e s t i n g

to note th a t i f a = 1 fo r la rg e va lues o f A , then fT and (p~2/R 2) maximize

when A = 3 (1 + A ) and 2 (1 + A ) , r e sp e c t i v e l y . These a re c on s is t en t w i t h

the c ur re nt low beta re ac to r des igns where A S= 0 . 5 - 1 .

The e f f e c t o f l a r g e 3 t i s to permi t a smal le r p lasma s ize w i th the

same n? and T. I f we assume T « n a2, f us ion power dens i t y PDT <* n

2,

and Jj . <= n / B2, i t can be shown th a t the min or ra diu s a , the p lasma

volume V , and the fusion power P DT s c a l e l i k e

a - l /B " t B2 , V - A/3"3 B6, PDT oc A /s " tB

2 . (9)

I n c o m p a r i s o n to a r e a c t o r w i t h g. ̂ 0 . 0 5 , B t ^ 5.3 T , A ̂ 4 . 2 , a ̂ 1.2 m,

V -̂ 2 3 0 m3, and P D T ^ 1 0 0 0 M W , a p l a s m a w i t h s"t= 0 . 2 8 , B t = 4.0 T , and

A = 1.5 w o u l d h a v e a m i n o r r a d i u s a ̂ 38 cm and V % 2 . 5 m3. T h e s e

v a l u e s are c o i n c i d e n t a l w i t h t h e A = 1.5 p a r a m e t e r s s h o w n in T a b l e 2.

T h e f u s i o n p o w e r Pp.- of the m a l l p l a s m a , a c c o r d i n g to E q . ( 9 ) , is t h e n

1 1 0 M W . It is s e e n t h a t s m a l l A and t h e r e s u l t i n g l a r g e ? . v a l u e s

s h o u l d in p r i n c i p l e a l l o w for d r a m a t i c r e d u c t i o n of t h e u n i t s i z e , c o s t ,

a n d p o w e r l e v e l in a t o k a m a k i g n i t i o n d e v i c e .



11

3 . 2 A N E X A M P L E W I T H A = 2

B a s e d o n ? t = 0 . 1 8 a n d B t = 1 . 2 T , a s s h o w n i n T a b l ^ 2 , t h e

p l a s m a d e n s i t y i s e x p e c t e d t o b e a r o u n d 1 ^ 2 x } Q l k c m " 3 w i t h a t e m p e r -

a t u r e o f a f e w k i l o e l e c t r o n v o l t s . T h e p l a s m a c o l l i s i o n a l i t y [ E q . ( A . 3 ) ]

i s e x p e c t e d t o b e s m a l l e r t h a n 0 . 1 , m a k i n g n e o c l a s s i c a l i o n h e a t c o n d u c t i o n

l o ss i m p o r t a n t . F o r b e s t c o n f i n e m e n t , E q . ( A . 6 ) g i v e s n - 1 . 4 1 x 1 0l l +

c m "3

T h e c o r r e s p o n d i n g v a l u e s f o r T ( a s s u m i n g s = 1 ) , T * , a n d W b a s e d o n

E q s . ( A . 1 ) - ( A . 7 ) a r e 2 . 2 8 k e V , 3 . 2 1 m s e c , a n d 4 . 4 9 k J , r e s p e c t i v e l y .

P o w e r l o s s f r o m t h e p l a s m a t h r o u g h c o n d u c t i o n i s t h e n 1 . 6 M W . P o w e r

l o s s e s f r o m b r e m s s t r a h l u n g a n d c y c l o t r o n r a d i a t i o n a r e i n s i g n i f i c a n t i n

t h i s c a s e .

T h e o h m i c h e a t i n g p o w e r t o t h e p l a s m a a n d . t h e p l a s m a r e s i s t a n c e c a n

b e e s t i m a t e d f r o m E q s . ( A . 1 0 ) - ( A . 1 2 ) t o b e 4 8 k W a n d 6 . 68 x 1 0 "7

a ,

r e s p e c t i v e l y . T h e v a l u e f o r I a c c o r d i n g t o E q . ( A . 1 2 ) a g r e e s w e l l w i t h

t h e s c a l e d v a l u e s h o w n i n T a b l e 2 . T h u s , w i t h o u t a u x i l i a r y h e a t i n g

( w h i c h w o u l d b e a r o u n d 1 .6 M W ) , t h e p l a s m a w i l l c o ol d o w n i n a f e w m i l l i -

s e c o n d s . T h i s i s c o n s i d e r a b l y s h o r t e r t h a n t h e ( L / R ) d e c a y t i m e

( ^2 0 0 m s e c ) o f t h e p l a s m a c u r r e n t i f t h e p l a s m a t e m p e r a t u r e i s s u s t a i n e d

b y a d d i t i o n a l h e a t i n g . T h e s k i n t i m e f o r t h e p o l o i d a l f l u x to d i f f u s e

a c r o s s 2 c m w i t h i n t h e p l a s m a i s a b o u t 1 0 m s e c . T h e p l a s m a p a r a m e t e r s

f o r a d e v i c e w i t h a = 8 c m a r e s u m m a r i z e d i n T a b l e 3 .



1 2

3 . 3 A S M A L L C O N T I N U O U S T O K A M A K C O M P R E S S O R1 5

A s ma l l c o n t i n u o u s t o k a m a k , s h o w n s c h e m a t i c a l l y i n F i g u r e 7 , m a y b e

a n e f f e c t i v e a p p r o a c h t o p r o d u c e t h e p l a s m a j u s t d e s c r i b e d . T h e

p r e c o m p r e s s i o n p l a s m a i s p r o d u c e d w i t h i n d u c t i v e c o i l s e t s a n d i s t h e n

s h i f t e d a n d c o m p r e s s e d s w i f t l y t o a r e g i o n w h e r e o n l y t o r o i d a l f i e l d

c o i l s a r e l i n k e d t o t h e p l a s m a , g i v i n g A = 2 . C o n d u c t i n g s h e l l s c a n b e

u s e d t o g u i d e t h e p l a s m a a l o n g t h e d e s i r e d t r a j e c t o r y a n d t o m i n i m i z e

t h e p o l o i d a l c u r r e n t p r o g r a m m i n g a n d t h e p l a s m a c o n t r o l r e q u i r e d . S i n c e

t h e p o l o i d a l f i e l d c o i l s a r e c l o s e b y , p l a s m a s h a p i n g a n d p o s i t i o n

c o n t r o l s h o u l d n o t b e d i f f i c u l t .

W i t h o h m i c h e a t i n g a l o n e , t h e p r e c o m p r e s s i o n p l a s m a a t R = 4 8 c m ,

a = 1 4 c m , a n d B t = 4 k G c a n h a v e a n e l e c t r o n t e m p e r a t u r e o f 1 1 4 e V ,

a c c o r d i n g t o E q . ( A . 1 3 ) , i f o n e a s s u m e s s = 1 / 2 , a = 0 . 5 , a n d Z --r = 1 . 5 ,

d u e t o l o w - Z i m p u r i t i e s . T h e M u r a k a m i s c a l i n g ,2 9

a c c o r d i n g t o E q . ( A . 1 4 ) ,

p e r m i t s a n e l e c t r o n d e n s i t y r a n g i n g f r o m 1 . 0 - 7 . 2 x 1 01 3

c m "3. . T h e

p r e c o m p r e s s i o n p l a s m a c u r r e n t , a s s u m i n g E q . ( A . 1 2 ) , i s 6 0 . 7 k A .

T o o b t a i n t h e d e n s i t y a n d t e m p e r a t u r e o f 1 .4 1 x ] Ql h

c m "3

a n d

2 . 2 8 k e V a f t e r c o m p r e s s i o n , t h e m i n i m u m p r e c o m p r e s s i o n v a l u e s a r e 1 . 5 7 x

1 01 3

c m "3

a n d 5 2 7 e V . A u x i l i a r y h e a t i n g i s t h e r e f o r e n e e d e d . U s i n g

E q s . ( A . I ) - ( A . 7 ) , o n e f i n d s x , = 1 . 0 m s e c , W = 1 . 2 2 k J , a n d P n u =c o n T p U n

2 1 . 2 k W . T h e r e q u i r e d a u x i l i a r y h e a t i n g p o w e r i s t h e n 1 .2 M W . T h e

p l a s m a g a i n s a b o u t 3 . 2 7 k J t h r o u g h a d i a b a t i c c o m p r e s s i o n i n m a j o r r a d i u s .

T h e p l a s m a a t i s 0 . 0 4 2 a n d s i s 0 . 6 9 b e f o r e c o m p r e s s i o n . A d i v e r t o r

m a y b e n e e d e d t o k e e p Z ^ t o 1 .5 o r b e l o w i n t h e p r e c o m p r e s s i o n p l a s m a .

A c c o r d i n g t o E q . ( A . 1 6 ) , t h e s l o w i n g - d o w n t i m e o f a 2 0 - k e V b e a m i o n

o n t h e p r e c o m p r e s s i o n p l a s m a i s a b o u t 2 9 m s e c . D u r i n g c o m p r e s s i o n , o n e



c a n a s s u m e a c o n s t a n t h e a t i n g b y t h e si o w i n g - d o w n i o n s a n d , i f I * c ,

a c o n s t a n t o h m i c h e a t i n g . T h e c o m p r e s s i o n e f f i c i e n c y t h e n d e p e n d s o n

t h e c o m p r e s s i o n t i m e x a n d t h e c o m p r e s s i o n r a t i o C . F o r a n e f f i c i e n c y

o f (W / W ° ) > ( 1 - ( S j C1* /

3, i t c a n b e s h o w n b y a s i m p l e p o w e r b a l a n c e w i t h

e m p i r i c a l c o n f i n e m e n t s c a l i n g t h a t T m u s t s a t i s f y

- 3 C1/

3+ 2 ) , ( 1 0 )

w h e r e T E i s t h e c o n f i n e m e n t t i m e a f t e r c o m p r e s s i o n a n d w e h a v e a s s u m e d

t h a t V <= C "2. I n o u r c a s e w i t h C = 3 a n d x £ = 3 . 21 m s e c , w e o b t a i n

x c = 0 . 4 3 x T £ = 1 . 4 m s e c i f 6 = 0 . 0 5 ( a 9 5 % e f f i c i e n c y i n W ) i s

a s s u m e d . E q u a t i o n ( 1 0 ) i s s i m i l a r t o w h a t i s i n d i c a t e d b y G r e e n , N o l l ,

a n d S h e f f i e l d .3 0

.

F i n a l l y , t h e c u r r e n t d e n s i t y J i n t h e t o r o i d a l f i e l d co i l i s

a r o u n d 1 .2 k A / c m 2 i n t h e p r e c o m p r e s s i o n s e g m e n t a n d 6 .2 k A / c m 2 i n t h e

c o m p r e s s e d p l a s m a s e g m e n t ( w h i c h i s a b o u t 5 0 c m i n l e n g t h ) . T h e t o t a l

r e s i s t i v e p o w e r t o t h e t o r o i d a l f i e l d c o i l s i s e s t i m a t e d t o b e a r o u n d 1 M W .

S i n c e t h e p o l o i d a l f i e l d s a r e c o m p a r a b l e t o t h e t o r o i d a l f i e l d s , t h e

r e s i s t i v e p o w e r t o t h e f o r m e r s h o u l d a l s o b e o n t h e o r d e r o f 1 M W .

T h e p l a s m a p a r a m e t e r s b e f o r e c o m p r e s s i o n a r e s u m m a r i z e d i n T a b l e 3 .

T h e p e a k v a l u e s o f J c a n b e r e d u c e d s u b s t a n t i a l l y i f w e i n c r e a s eLi

t h e p l a s m a s i z e w h i l e h o l d i n g m ^ , T , i * t, q , a n d A c o n s t a n t . W i t h

e m p i r i c a l s c a l i n g , w e f i n d t h e f o l l o w i n g a p p r o x i m a t e s c a l i n g l a w s :



J c - a-i.3,

n = R = a "1,

B

c *B

t *a

~ ° 'P ( T F c o i l s ) = a

0,

TE ' W i o s s

e Lp "

a'

( 1 1 )

T h e p a r a m e t e r s f o r t h e c a s e w i t h a = 1 6 c m b a s e d o n t h e s e s e a l i n g s a r e

a l s o s h o w n i n T a b l e 3 .

3 .4 A D - T I G N I T E D P L A S M A W I T H A = 1 .5

A s s h o w n i n T a b l e 2 , w e h a v e 3 ~ tc = 0 . 2 8 , B t Q = 4 .0 T . a n d I = 6 . 54 M A .

T h e p l a s m a c o l l i s i o n a l i t y i s e x p e c t e d t o b e s m a l l f o r t e m p e r a t u r e s a r o u n d

1 0 k e V . U s i n g E q s . ( A . 1 ) - ( A . 7 ) , o n e f i n d s t h a t t h e b e s t c o n f i n e m e n t

t i m e , t c o n f = 0 . 5 2 3 s e c , o c c u r s w i t h n = 7 . 8 2 x 1 01 4

c m "3, T = " 7 . 1 1 k e V ,

a n d W = 7 . 2 0 M J . T h e n E q . ( A . 1 5 ) g i v e s a n a l p h a p o w e r ( w i t h s = 2 )

o f P = 2 0 . 2 M W an d a t o ta l f u s i o n o u t p u t o f P D T = 1 01 M W . T h e p l a s m a

r e s i s t a n c e i s e s t i m a t e d t o b e 1 .8 5 x 1 0 "3

Q ( w i t h a = 2 a n d Z „ = 1 . 5 )

a n d P Q H t o b e 0 . 7 9 M W , r e s u l t i n g i n a t o t a l h e a t i n g o f 2 1 . 0 M W . T h e

t ot al p l a s m a p o w e r l o s s f r o m t h e p l a s m a t h r o u g h c o n d u c t i o n , b r e m s s t r a h l u n g ,

a nd c y c l o t r o n r a d i a t i o n [ E q s . ( A . 8 ) a n d ( A . 9 ) ] a m o u n t s t o 1 9 . 2 M W .

A t t h e p l a s m a c e n t e r , t h e g u i d i n g c e n t e r o r b i t e x c u r s i o n f o r t h e

f u s i o n ^ a l ph a s i s e s t i m a t e d t o b e b e t w e e n q a n d 2 q VA~, w h i c h i s a b o u t 2 -5

t i m e s t h e g y r o r a d i u s ( p . ^ 6 c m ) . T h e a l p h a l o s s r e g i o n s3 1

c a n b e

e s t i m a t e d t o s h o w t h a t f o r ( r / a ) > 0 . 8 5 , a ll f u s i o n a l p h a s w i t h a pi t c h



15

a n g l e g r e a t e r t h a n 1 1 0 ° w o u l d b e u n c o n f i n e d . A s s u m i n g a p a r a b o l i c T

p r o f i l e , T w o u l d b e l e s s t h a n 3 . 8 k eV f o r ( r / a ) > 0 . 8 5 . T h e t o t a l a l p h a

p o w e r l o s t w o u l d t h e n b e l e s s t h a n 0 . 0 7 M W .

T h e s e l f - i n d u c t a n c e a n d t h e i n t e r n a l i n d u c t a n c e o f t h e p l a s m a ,

b a s e d o n t h e n u m e r i c a l r e s u l t s s h o w n i n F i g u r e 3 , a r e e s t i m a t e d t o b e

0 . 3 3 3 u H a n d 0 . 2 6 3 u H , r e s p e c t i v e l y . T h e r e s i s t i v e d e c a y t i m e o f t h e

p l a s m a i s t h u s b e t w e e n 1 8 . 0 s e c a n d 1 4 . 2 s e c . T h e s k i n t i m e f o r m a g n e t i c

f l u x t o d i f f u s e a c r o s s 1 0 c m i n p l a s m a i s a b o u t 0 .9 s e c . T h e p a r a m e t e r s

f o r t h i s p l a s m a a r e s u m m a r i z e d i n T a b l e 4 .

3 .5 A C O M P R E S S O M A K F O R I G N I T I O N D E M O N S T R A T I O N

A c o m p a c t d e v i : e t o p r o d u c e t h e h i g h b e t a , A = 1 .5 p l a s m a c o u l d b e

a c o m p r e s s o m a k , s c h e m a t i c a l l y s h o w n in F i g u r e 8 . I t r e p r e s e n t s a n e x t e n -

s i o n f r o m t h e A T C t o k a m a k3 2

t o t h e l i m i t o f l a r g e c o m p r e s s i o n a n d v e r y

s m a l l A b y u s i n g a p o l o i d a l f i e l d c o i l c o n c e p t t h a t d o e s n o t d i s t i n g u i s h

t h e o h m i c h e a t i n g p r i m a r y f r o m v e r t i c a l a n d s h a p i n g f ie l d c o i l s .1 5

'3 3

I t a l s o a s s u m e s t h a t t h e p l a s m a p o s i t i o n , s h a p e , a n d s t a b i l i t y c a n b e

m a i n t a i n e d w i t h p r o p e r p o l o i d a l f i e l d c o i l s a n d c u r r e n t l e s s , l o w p r e s s u r e

p l a s m a t o w a r d t h e o u t s i d e o f t h e p l a s m a r i n g .

A s s u m i n g a c o m p r e s s i o n r a t i o o f C = 4 , t h e i n i t i a l p l a s m a s h o u l d

h a v e R Q = 2 4 0 c m , a = 8 0 c m , B t Q = 1 .0 T , n = 4 . 8 9 x 1 01 3

c m "3, T = 1 . 1 2

k e V , a n d ? t = 0 . 0 4 4 . T h e p l a s m a c u r r e n t a c c o r d i n g t o E q . ( A . 1 2 ) i s

7 7 5 k A . E q u a t i o n s ( A . 2 ) - ( A . 7 ) a n d ( A . 1 0 ) - ( A . 1 2 ) g i v e c o l l i s i o n a l i t y =

0 . 2 , r c o n f = T e m p = 0 . 2 6 2 s e c , W p = 1. 3 1 M J , a n d R p = 2. 8 7 x 1 0 "7

n i f

= 1 . 5 a n d a = 2 a r e a s s u m e d . T h e o h m i c h e a t i n g p o w e r i s 0 . 1 6 5 M W

a n d a n a u x i l i a r y h e a t i n g p o w e r o f 4 . 8 3 M W i s n e e d e d f o r t h e p r e c o m p r e s s i o n

p l a s m a .



16

B e f o r e t h i s a u x i l i a r y h e a t i n g i s a p p l i e d , t h e o h m i c a l l y h e a t e d

p l a s m a t e m p e r a t u r e a n d d e n s i t y c a n b e e s t i m a t e d f r o m E q s . ( A . 1 3 ) a nd ( A . 1 4 )

t o b e T Q = 2 9 1 e V a n d n = 0 . 9 6 - 3 . 6 2 x 1 01 3

c m "3, i f s = 0 .5 a n d a = 0 . 8

a r e a s s u m e d . T h e o h m i c h e a t i n g p o w e r l o s t t h r o u g h p l a s m a c o n d u c t i o n i s

0 . 9 8 M W . A c c o r d i n g t o t h e i m p l i c a t i o n s o f t h e M u r a k a m i s c a l i n g ,2 9

a n

a d d i t i o n a l h e a t i n g o f 4 . 8 3 M W s h o u l d a l l o w a n i n c r e a s e i n t h e l i m i t i n g

p l a s m a d e n s i t y b y a f a c t o r o f f i v e f r o m 1 01 3

c m "3. I f a d e q u a t e i m p u r i t y

c o n t r ol i s p r o v i d e d , t h e p l a s m a d e n s i t y w i l l b e a b o v e t h e m i n i m u m

d e n s i t y r e q u i r e m e n t i n t h e p r e c o m p r e s s i o n p l a s m a .

T h e l i n e d e n s i t y o f t h e p r e c o m p r e s s i c n p l a s m a i s n a s 4 x 1 01 5

c m "2.

F o r g o o d p e n e t r a t i o n w i t h p a r a l l e l n e u t r a l i n j e c t i o n , a n e n e r g y o f

4 0 k eV p e r n u c l e o n i s r e q u i r e d . F o r d e u t e r o n i n j e c t i o n i n t o a D- T p l a s m a ,

t h e b e a m i o n s l o w i n g - d o w n t i m e T i s t h e n a r o u n d 2 3 m s e c [ E q . ( A . 1 6 ) ] .

F o r * m u c h l a r g e r t h a n t h i s T , t h e b e a m c l a m p i n g e f f e c t s a r e t h e n

r e l a t i v e l y i n s i g n i f i c a n t . I t m a y b e n e c e s s a r y t o u s e s e m i p a r a l l e l(^45°)

i n j e c t i o n i n t o t h e p r e c o m p r e s s i o n p l a s m a s o t h a t n e a r l y p a r a l l e l i n j e c t i o n

i s m a i n t a i n e d d u r i n g c o m p r e s s i o n . T h e r e q u i r e d b e a m e n e r g y i s t h e n a r o u n d

7 5 ke V p e r n u c l e o n f o r d o u b l e a n d p a r t i a l p e n e t r a t i o n s o f t h e p r e c o m p r e s -

s i o n a n d c o m p r e s s e d p l a s m a s , r e s p e c t i v e l y . W i t h t h i s a s s u m p t i o n , T

a c c o r d i n g t o E q . ( 1 0 ) c a n b e a s l a r g e a s 2 0 0 m s e c f o r a 5 % d e c r e a s e o f T

f r o m t h e i d e a l c a s e .

A s a n i n d i c a t o r o f f u s i o n p e r f o r m a n c e , w e d e f i n e a s c i e n t i f i c p e r -

p u l s e g a i n f a c t o r ,

0 = f u s i o n e n e r g y o u t p u t , , „»ws c i e n e r g y i n p u t t o p l a s m a '

[



17

I f t h e p l a s m a s t a r t u p , h e a t i n g , a n d c o m p r e s s i o n t a k e 2 s e c a n d t h e b u r n

t a k e s ( L p / 3 R p ) =

i ng g a i n f a c t o r ,

t a k e s ( L / 3 R ) = 6 s e c , w e o b t a i n Q s c i = 4 5 . W e a l s o d e f i n e an e n g i n e e r -

f u s i o n e n e r g y o u t p u te n e r g y t h r o u g h p o w e r s u p p l i e s

( 1 3 ;

W e a s s u m e a r a d i u s R o f 1 8 c m f o r t h e c e n t r a l t o r o i d a l f i e l d

c o n d u c t o r . T h e c oi l c u r r e n t d e n s i t y t h e r e w i l l b e 1 1 . 8 k A / c m2

w i t h a

m a x i m u m f i e l d o f 1 3 . 3 T a n d t e n s i l e s t r e s s b e l o w 1 5 0 0 k g / c m

2

. I f t h e

l e n g t h o f t h e c e n t r a l c o n d u c t o r i s 1 5 0 c m , t h e r e s i s t i v e p o w e r r e q u i r e m e n t

i s t h e n 4 3 . 5 M W . I f t h e r e m a i n i n g t o r o i d a l f i e l d c o i l s c a r r y 2 . 3 6 k A / c m2

f o r a l e n g t h o f 7 5 0 c m , t h e t o t a l t o r o i d a l f i e l d r e s i s t i v e p o w e r w i l l

b e 8 7 M W . S i n c e t h e p o l o i d a l f i e l d i s c o m p a r a b l e t o t h e t o r o i d a l f i e l d ,

w e a s s u m e t h a t t h e r e s i s t i v e p o w e r s u p p l y t o t h e p o l o i d al f i e l d c o i l s i s

a l s o 8 7 M W . I f t h e s t a r t u p , a u x i l i a r y h e a t i n g , a n d c o m p r e s s i o n s u p p l i e s

h a v e a n e n e r g y e f f i c i e n c y o f 1 / 3 , w e th e n f i n d Q - 0 . 4 8 .

F i n a l l y , Q h a s a f a v o r a b l e s c a l i n g w i t h s i z e i f w e a s s u m e th e

c o n d i t i o n s f o r t h e s c a l i n g l a w s s h o w n in E q . ( 1 1 ) . W e f i n d , i n a d d i t i o n

t o E q . ( 1 1 ) , t h a t t h e t o t a l f u s i o n p o w e r p D T s c a l e s l i k e a . I t c a n b e

s h o w n f r o m T a b l e 4 t h a t Q s c a l e s r o u g h l y l i k e

Qe n g

s6 0 0 a

3/ ( 1 2 2 0 a

2+ 4 0 a

3) , ( 1 4 )

w h e r e a = a / 4 0 c m . E q u a t i o n ( 1 4 ) s h o w s t h a t Q c a n b e a b o v e t e n w i t h

i n c r e a s i n g s i z e , b u t t h e e q u a t i o n i s p e s s i m i s t i c w h e n a i s s u f f i c i e n t l y

l a r g e to a l l o w f o r s u p e r c o n d u c t i n g c o i l s a n d t h e i r n e u t r o n s h i e l d ,

s u b s t a n t i a l l y r e d u c i n g t h e f i r s t t e r m i n t h e d e n o m i n a t o r .



18

4 . C O N C L U S I O N S A N D D I S C U S S I O N

F o r v e r y s m a l l a s p e c t r a t i o t o k a m a k s , w e h a v e f o u n d M H D e q u i l i b r i a

w i t h 1 . ^ 1 , q = 1 , q = = 2 , 3 < 1 , a n d A = 2 a n d 1 . 5 . S t a b i l i t y a n a l y s e sx . o a P

b a s e d o n t h e E R A T O ( N = 1,2,3) c o d e s h a v e i n d i c a t e d h i g h v a l u e s o f

I t ( 0 . 1 8 f o r A = 2 . 0 , 0 . 2 8 f o r A = 1 . 5 ) a n d ? . ( p e a k ) ( 0 . 4 0 f o r A = 2 . 0 ,

0 . 5 7 f o r A = 1 . 5 ) in t h e l i m i t o f ( 1 / N ) - 0 . B e c a u s e B ^ B t i n t h e s e

e q u i l i b r i a , ? . b e c o m e s s i g n i f i c a n t l y l a r g e r t h a n 3 ; 3 t a p p a r e n t l y 'no

l o n g e r s c a l e s l i k e I / A. I n t h e s p e c i a l c a s e o f S o l o v ' e v e q u i l i b r i a w e

h a v e s e e n t h a t t h e s L v a l u e , b a s e d o n t h e M e r c i e r c r i t e r i o n , m a x i m i z e s

w h e n A = v 2 ~ a n d b e c o m e s z e r o w h e n A = 1 . M o r e s t u d i e s a r e n e e d e d t o

u n d e r s t a n d t h e A d e p e n d e n c e o f I", w h e n A < 2 .

W i t h t h e s e h i g h a . v a l u e s , w e h a v e e x a m i n e d t w o s p e c i f i c e x a m p l e s

a s s u m i n g t h a t t h e p l a s m a s a r e p r o d u c e d w i t h l a r g e c o m p r e s s i o n i n t h e

f a s h i o n o f c o n t i n u o u s t o k a m a k s . I t h a s b e e n s e e n t h a t a s m a l l e x p e r i -

m e n t a l d e v i c e ( F i g u r e 7 ) w i t h A = 2 , a\ ^ 0 . 1 8 , R = 1 6 c m , a n d B . = 1 . 2 T ,

u n d e r r e a s o n a b l e s c a l i n g a s s u m p t i o n s ( s e e t h e A p p e n d i x ) w o u l d r e q u i r e a

n e u t r a l b e a m p o w e r o f ^1 M W , a r e s i s t i v e c o i l p o w e r o f ^ 2 M W , a n d a

c o m p r e s s i o n e n e r g y i n p u t o f ^ 3 . 3 kJ i n 1 .4 m s e c . T h e e n e r g y c o n f i n e m e n t

t i m e i s a r o u n d 3 m s e c , w h i c h i s t wo o r d e r s o f m a g n i t u d e s m a l l e r t h a n t h e

r e s i s t i v e d e c a y t i m e o f t h e p l a s m a c u r r e n t . T h i s m a y b e a r e l a t i v e l y

i n e x p e n s i v e d e v i c e f o r t h e p u r p o s e o f h i g h 1 L d e m o n s t r a t i o n .

A c a s e h a s a l s o b e e n c o n s i d e r e d o f a n i g n i t i o n d e v i c e w i t h A = 1 . 5 ,

5 t = 0 . 2 8 , R = 6 0 c m , a n d B . = 4 T a f t e r c o m p r e s s i o n ( s e e F i g u r e 8 ) . A t

T = 7.1 k e V a n d n = 7 .8 x 1 0l u

c m " - , t h e p l a s m a w o u l d p r o d u c e a D -T

f u s i o n p o w e r o f 1 0 0 M W w i t h t h e e s t i m a t e d p l a s m a e n e r g y l o s s e s b e l o w t h e



19

a l p h a p a r t i c l e h e a t i n g . T h e r e s i s t i v e d e c a y t i m e o f t h e p l a s m a c u r r e n t

i s e s t i m a t e d t o b e -^18 s e c . A s s u m i n g C = 4 , t h e p r e c o m p r e s s i o n p l a s m a

w o u l d r e q u i r e a n a u x i l i a r y h e a t i n g o f a b o u t 5 M W t o o b t a i n T s 1 .1 k e V ,

n s 4 . 9 x 1 01 3

c m "3, a n d ] L = 0 . 0 4 4 . A c o m p r e s s i o n e n e r g y o f 5 .9 M O

c a n b e s u p p l i e d t o t h e p l a s m a i n 0 .2 s e c i f a p r o p e r i n j e c t i o n a n g l e

i s u s e d . A s s u m i n g a 2 - s e c t i m e f o r s t a r t u p a n d c o m p r e s s i o n a n d a

6-sec b u r n , t h e s c i e n t i f i c e n e r g y g a i n f o r t h e p l a s m a p u l s e ( Q s c i ) i s

e s t i m a t e d t o b e 4 5 .

C o p p e r c o i l s w i t h o u t n e u t r o n s h i e l d s a r e a s s u m e d i n t h e i g n i t i o n

d e v i c e , r e q u i r i n g a c o i l p o w e r s u p p l y a r o u n d 1 7 0 M W . T h e e n g i n e e r i n g

e n e r g y g a i n p e r p u l s e f r o m t h e p o w e r s u p p l i e s o f t h e d e v i c e ( Q e n a ) i s

e s t i m a t e d t o b e a r o u n d 0 , 4 8 . A s s u m i n g a c o n s t a n t n-r, Q D n _ h a s a f a v o r a b l e

s r i Q

s c a l i n g w i t h t h e p l a s m a s i z e a n d a n u l t i m a t e c a p a b i l i t y a b o v e t e n w i t h

l a r g e s i z e . I n c r e a s e d s i z e w o u l d a l s o p e r m i t t h e u s e o f n e u t r o n s h i e l d s .

F u r t h e r s t u d i e s a r e r e q u i r e d t o a s s e s s t h e p o t e n t i a l o f t h e a p p r o a c h o f

v e r y s m a l l a s p e c t r a t i o w i t h r e g a r d t o i n e x p e n s i v e i g n i t i o n t o k a m a k s a n d

t o k a m a k r e a c t o r s u s i n g D - T o r a d v a n c e d f u e l s .

4.1 P O S S I B I L I T I E S O F L O N G P U L S E

O n e o f t h e a p p a r e n t l i m i t a t i o n s o f t h e v e r y s m a l l a s p e c t r a t i o

t o k a m a k s i s i t s s h o r t p u l s e l e n g t h d u e t o r e s i s t i v e d e c a y o f c u r r e n t .

H o w e v e r , s o m e e f f e c t s d u e t o v e r y s ma l l v a l u e s o f A o n t h e l o n g e v i t y o f

t h e p l a s m a c u r r e n t m a y b e s i g n i f i c a n t .



20

4.1.1 Neutral Beam Driven Plasma C u rrent17

A small aspect ratio allow s a shorter toroidal transit length and

hence an increased stacking of the streaming i o n s . For the compressed

plasma listed in Table 3, a neutral beam power of 1.6 MW at 25 keV is

required to maintain the plasma temperature around 2.3 keV and to increase

the pulse length toward the resistive decay time (^200 m s e c ) . The beam

ion slowing-down time is x = 29.2 msec. The total number of the stream-

ing ions is N, = 1.17 x 1 0l s

at an average velocity of roughly 1.1 x

1 03

cm/sec and a stacking factor of s = 1.1 x 106. Since the ion stream-

ing velocity is much less than the electron thermal velocity, the beam ion

current is then17

Ib ^ sNbe [1 - (1/Zeff)] = 670 kA. This result suggests

that the plasma current may be sustained over long times provided that

the plasma purity and density can be maintained.

4.1.2 Bootstrap C u rrent18

*

The plasma temperature in the ignition compressomak (Table 4) is

sustained by the alpha heating. The pressure gradient together with the

small A may enhance the bootstrap current density J, = -jv pj/B VA~.

Immediately after compression and ignition, the bootstrap current can be

estimated to be

1 3

Jb ^ \

{ [ { Jp

+ J.

) 2 +(

4 Bt

3t 0

/ y0

a 2 v^

1 / 2"

( Jp

+ Ja

) }'

( 1 5>

where J is the conduction current and J is the small alp ha seed current.

Equation (15) shows that Jfa is enhanced by high beta and low a and A.

During steady state with J_ = 0 and small J , we obtain J. = 5.43 x

P a b



21

1 06

A / m2 and I. = 4.5 M A . In the c a s e of t h e A = 2 p l a s m a w i t h a s m a l l

n e u t r a l b e a m d r i v e n c u r r e n t , the b o o t s t r a p c u r r e n t e s t i m a t e d w i t h E q . (15)

i s a r o u n d 2 1 0 k A . B o t h v a l u e s are c o m p a r a b l e to the r e s p e c t i v e p l a s m a

c u r r e n t s . It is s e e n t h a t s m a l l A and l a r g e b e t a m a k e it r e l a t i v e l y

e a s y to d r i v e l a r g e c u r r e n t s by n e u t r a l b e a m i n j e c t i o n or b o o t s t r a p e f f e c t .

4 . 1 . 3 R F - D r i v e n C u r r e n t s1 9

A s s u m i n g t h a t n e t rf m o m e n t u m can be i n j e c t e d i n t o the c o m p r e s s e d

p l a s m a , o n e can e s t i m a t e the r e q u i r e d p o w e r to s u s t a i n t h e f u l l p l a s m a

c u r r e n t . S e t t i n g the r f - d r i v e n c u r r e n t d e n s i t y1 9

e q u a l to J of the

c o m p r e s s e d p l a s m a of T a b l e 4,

J r f -v- 6.5 x 1 0 - S n / T f ( w ^ w x = J ,

o n e o b t a i n s Wi = V i / v . = 2 . 5 8 [ h e r e n and T are in ( c e n t i m e t e r s )3

andxe

ele ctron v o l t s , re sp ec t ive ly, and fdw f(w) = 1 ] . The required power can

then be estimated by

/ I + wAP - -v* 5 x 1 0 " 1 7 n 2 T - 1 / 2 f ( w 1 ) £ n l — 1 ^ 8 . 5 x 1 0 7 W/m 3 .

I t is s e e n t h a t b e c a u s e of t h e r a t h e r h i g h c u r r e n t d e n s i t y r e q u i r e d the

r f p o w e r r e q u i r e d is on t h e o r d e r of 2 3 0 M W , w h i l e the t o t a l f u s i o n

p o w e r is a r o u n d 1 0 0 W .

4 . 2 C O M P A R I S O N S W I T H O T H E R T O R O I D A L P L A S M A S

F i n a l l y , it is i n f o r m a t i v e to c o m p a r e s o m e of the d i m e n s i o n l e s s

p a r a m e t e r s of the v e r y s m al l a s p e c t r a t i o t o k a m a k s d e s c r i b e d h e r e w i t h a



2 2

f e w o t h e r t o r o i d a l p l a s m a c o n c e p t s : t h e i m p l o d e d t o r u s ,2 4

s p h e r o m a k s ,2 3

f i e l d r e v e r s e d m i r r o r s ,2 2

f i e l d r e v e r s e d t h e t a p i n c h e s ,2 1

a nd E - l a y e r s .2 0

T h e o r d e r o f m a g n i t u d e s o f ( a / p ) , ( T E / T B ) , q , E g , a n d t h e p r e s e n t l y

p r e d i c t e d v a l u e s o f p e a k b e t a a r e i n d i c a t e d i n T a b l e 5 , w h e r e p i s t h e

g y r o r a d i u s , ~ i s t h e B o h m d i f f u s i o n t i m e , a n d E g i s t h e f r a c t i o n o f t h e

f i e l d e n e r g y o c c u p i e d b y t h e p l a s m a . I t i s s e e n t h a t g i v e n t h e p r e d i c t e d

b e t a v a l u e s , t o k a m a k s w i t h A < 2 h a v e r e l a t i v e l y g o o d c o n f i n e m e n t a n d

e f f i c i e n t u s e o f f i e l d e n e r g y .

A C K N O W L E D G M E N T S

T h e a u t h o r s w i s h t o t h a n k J . D . C a l l e n f o r s u p p o r t i n g t h e w o r k a n d

m a k i n g m a n y v a l u a b l e s u g g e s t i o n s . T h e y a l s o w i s h t o t h a n k L . A . B e r r y ,

T . C . J e r n i g a n , M . M u r a k a m i , J . S h e f f i e l d , a n d D . S t e i n e r f o r u s e f u l

d i s c u s s i o n s . S p e c i a l t h a n k s a r e d u e t o L . A . C h a r l t o n , J . A . H o l m e s ,

D . K . L e e , D . J . S t r i c k l e r , a n d J . K . M u n r o f o r p e r f o r m i n g t h e . c o m p u t e r

w o r k .

T h i s r e s e a r c h w a s s p o n s o r e d b y t h e O f f i c e o f F u s i o n E n e r g y ( E T M ) ,

U . S . D e p a r t m e n t o f E n e r g y u n d e r c o n t r a c t W - 7 4 Q 5 - e n g - 2 6 w i t h t h e U n i o n

C a r b i d e C o r p o r a t i o n .



2 3

A P P E N D I X . S I M P L E T O K A M A K F O R M U L A S

T h e f o r m u l a s u s e d in S e c t i o n 3 to e s t i m a t e t o k a m a k p l a s m a p a r a m e t e r s

a r a l i s t e d b e l o w . T e m p e r a t u r e s (T) are g i v e n in u n i t s of e l e c t r o n v o l t s ,

c u r r e n t s (I) in a m p e r e s , r e s i s t i v i t y (n) in o h m - c e n t i m e t e r s , r e s i s t a n c e (R)

i n o h m s , i n d u c t a n c e (L) in h e n r i e s , p o w e r (P) in w a t t s , and e n e r g y (W) in

j o u l e s . All o t h e r u n i t s are cgs u n i t s .

F o r a p l a s m a w i t h a g i v e n a v e r a g e t o r o i d a l b e t a , st, we h a v e

Jt = 4.03 x 1 0 "u( n e T e + £ n i T . ) / B j = 8.06 x 1 0 "

1 Ln T / B | . (A.I)

B a s e d on r e s u l t s f r o m the I m p u r i t y S t u d y E x p e r i m e n t ( I S X ) ,3^ the e m p i r i c a l

s c a l i n g of e n e r g y c o n f i n e m e n t t i m e is

Temp

= 3-9 6 x

TO "1 9™ ^

1/2

• (A-2)

w h e r e we a s s u m e t h a t the b e n e f i t of c r o s s - s e c t i o n a l e l o n g a t i o n " at f i x e d

i n p u t p o w e r is to i n c r e a s e the p l a s m a d e n s i t y o n l y .

A t s u f f i c i e n t l y h i g h t e m p e r a t u r e , the ion c o l l i s i o n a l i t y (C) may

b e c o m e s m a l l ,

C = 6.0 x 1 0 -1 3

q Z o f f n a A2

-5

/ T2

, (A.3)

w h e r e q^ is the s a f e t y f a c t o r at p l a s m a b o u n d a r y . N e o c l a s s i c a l ion h e a t

c o n d u c t i o n may l i m i t the e n e r g y c o n f i n e m e n t , g i v i n g

t n c = 0 . 2 7 8 T ^ A i / ^ / Z e f f n . (A.4)



24

A t a f i x e d 3. w i t h h i g h d e n s i t y , it may be d e s i r a b l e to d e t e r m i n e the

v a l u e s of n and T ( a s s u m i n g T = T.) for b e s t c o n f i n e m e n t b a s e d on

Te m p

and Tnc'

T h i s d e n s i t y can be s h o w n to o c c u r at

n = /2= 5.22 x 1 0

2 2A i /

2s i /

2B t I ? / a

2q

1/

2, (A.6)

w h e r e the t e m p e r a t u r e T can be o b t a i n e d f r o m Eq. (A.I)

T h e p l a s m a e n e r g y c o n t e n t can be w r i t t e n as

W p = 2.4 x 1(T19(1 + s ) n y p , (A.7)

w h e r e s = T-/T and V is the p l a s m a v o l u m e . The p o w e r l o s t t h r o u g h

c o n d u c t i o n is t h e n W /- -. In the c a s e of h i g h t e m p e r a t u r e and low

Z ,, w i t h l o w - Z i m p u r i t i e s , r a d i a t i o n l o s s e s are p r i m a r i l y due to b r e m s -

s t r a h l u n g and c y c l o t r o n r a d i a t i o n s . The r a d i a t i o n p o w e r d e n s i t i e s can

b e a p p r o x i m a t e d by

P B r = 1.69 x 1 0 " 3

2

n 2 T i /

2

Z 2 f f , (A.8)

and

p _ v = 1.13 x 1 0 -2 S

n1/

2T

2-

1B

2-

5/ a

1/ 2 . (A.9)

c y e

T o e s t i m a t e the i n p u t p o w e r by o h m i c h e a t i n g , one n e e d s the p l a s m a

r e s i s t a n c e and p l a s m a c u r r e n t . The p l a s m a r e s i s t i v i t y can be g i v e n by



25

= 5 . 1 8 x 1 0 - 3 O T i Z e f f x r3/

2, ( A . 1 0 )

w h e r e a ( = 2 ) i n d i c a t e s a n o m a l o u s e n h a n c e m e n t o r t h a t d u e t o f i e l d l i n en

r o t a t i o n a l t r a n s f o r m a n d

x = 2 4 - i n { ^ lJ- n ) . ( A . 1 1 )

T h e p l a s m a c u r r e n t i s d e t e r m i n e d t h r o u g h M H D e q u i l i b r i u m c a l c u l a t i o n s o r

c a n b e a p p r o x i m a t e d b y

3 5

I = 2 . 7 5 a B t ( l + G2) / q A ( l - — ) • ( A . 1 2 )

P t \ A 2 /

T h e p l a s m a o h m i c h e a t i n g p o w e r i s I^R , w h e r e R = 2R n ,, /a2a .

I f t h e p l a s m a i s h e a t e d o h m i c a l l y a n d b a l a n c e d b y e m p i r i c a l l o s s

o n l y , t h e n t h e e l e c t r o n t e m p e r a t u r e c a n b e e s t i m a t e d b y

T h e p l a s m a d e n s i t y i n t h i s c a s e c a n b e e s t i m a t e d b y t h e M u r a k a m i s c a l i n g ,2 9

n = C n ( B t / R Q ) ( l + a2) / 2 , ( A . 1 4 )

w h e r e C n c a n b e 4 . 6 7 x 1 01 1

b a s e d o n t h e I S X r e s u l t s3' ' o r 1 . 2 3 x 1 0

1 1b a s e d

o n t h e O R M A K r e s u l t s .2 5

F o r t e m p e r a t u r e s h i g h e r t h a n t h o s e i n d i c a t e d b y E q . ( A . 1 3 ) , a d d i t i o n a l

h e a t i n g i s r e q u i r e d f o r s t e a d y s t a t e . I n t h e c a s e o f a D -T p l a s m a , t h e

a d d i t i o n a l h e a t i n g c o u l d c o m e f r o m t h e f u s i o n a l p h a p a r t i c l e s ,



26

p = 5 .6 x 1 0 "1 3

S n n n T < c v > as 1 . 4 x 1 0 -1 3

S n2< a v > , ( A . 1 5 )

Ct P U I P

w h e r e < a v > is t h e r e a c t i o n r a t e a v e r a g e d o v e r M a x w e l l i a n d i s t r i b u t i o n a n d

S K 2 ) i s a p r o f i l e e n h a n c e m e n t f a c t o r .3 6

T h e t o t a l f u s i o n p o w e r d e n s i t y

(P n- r) i s f i v e t i m e s p . F i n a l l y , t h e s l o w i n g - d o w n t i m e - o f a h i g h

e n e r g y i o n is g i v e n b y

T S = 3 . 7 9 x 1 07T | /

2m f / m Z

2n , ( A . 1 6 )

w h e r e t h e s u b s c r i p t f d e n o t e s t h e f a s t i o ns a n d m d e n o t e s t h e p l a s m a

i o n m a s s .



27

R E F E R E N C E S

1 . J . R . M c N a l l y , J r . , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t O R N L / T M - 6 3 6 2

( J u n e 1 9 7 8 ) .

2 . D . S t e i n e r e t a ! . , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t O R N L / T M - 5 8 1 3

( 1 9 7 7 ) ; R . H a n c o x a n d J. T . D . M i t c h e l l , i n P l a s m a P h y s i c s a n d

C o n t r o l l e d N u c l e a r F u s i o n R e s e a r c h ( P r o c . 6 t h I n t . C o n f . B e r c h t e s g a d e n ,

1 9 7 6 ) , H _ , I A E A , V i e n n a ( 1 9 7 6 ) 1 9 3 ; D. R . C o h n e t a l . , M a s s a c h u s e t t s

I n s t i t u t e o f T e c h n o l o g y P l a s m a F u s i o n C e n t e r R e s e a r c h R e p o r t R R - 7 8 - 2

( M a r c h 1 9 7 8 ) ; G e n e r a l A t o m i c C o m p a n y R e p o r t G A - A 1 4 6 U ( J a n u a r y 1 9 7 8 ) ;

D . L . J a s s b y , R . A . B o l t o n e t a l . , P r i n c e t o n P l a s m a P h y s i c s L a b o r a -

t o r y R e p o r t P P P L - 1 4 5 3 ( M a y 1 9 7 8 ) .

3 . J . F. C l a r k e a n d D . J . S i g m a r , P h y s . R e v . L e t t . 3 8_, 7 0 ( 1 9 7 7 ) .

4 . R . A . D o r y a n d Y - K . M . P e n g , N u c l . F u s i o n 1 7_, 2 1 ( 1 9 7 7 ) .

5 . J . M . G r e e n e , J . L . J o h n s o n » a n d K . E . W e i m e r , P h y s . F l u i d s 1_4_, 6 7 1

( 1 9 7 1 ) .

6 . R . A . D o r y e t a l . , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t O R N L / T M - 6 4 3 4 ;

R . A . D o r y e t a l . , " H i g h B e t a T o k a m a k s , " p a p e r p r e s e n t e d a t t h e

S e v e n t h I n t e r n a t i o n a l C o n f e r e n c e o n P l a s m a P h y s i c s a n d C o n t r o l l e d

N u c l e a r F u s i o n R e s e a r c h , I n n s b r u c k , A u s t r i a , A u g u s t 2 3 - 3 0 , 1 9 7 8 .

7 . J . T . H o g a n , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t O R N L / T M - 6 0 4 9 ( M a y

1 9 7 8 ) .

8 . D . B . N e l s o n a n d H . G r a d , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t

O R N L / T M - 6 0 9 4 ( 1 9 7 8 ) .

9 . J . A . H o l m e s , Y - K . M. P e n g , a n d R . A . D o r y , " E v o l u t i o n o f F l u x -

C o n s e r v i n g E q u i l i b r i a w i t h P r e s c r i b e d N o n c i r c u l a r C r o s s S e c t i o n , "

t o b e p u b l i s h e d .



28

1 0 . L . A . C h a r l t o n , R . A . D o r y , Y - K . M . P e n g , a n d D . J. S t r i c k l e r ,

" S t a b i l i t y S t u d y o f F C T E q u i l i b r i a , " t o b e p u b l i s h e d .

1 1 .A . T o d d e t a l . , P h y s . R e v . L e t t . 3 8 ., 8 2 3 ( 1 9 7 7 ) .

1 2 . R . G . B a t e m a n a n d Y - K . M . P e n g , P h y s . R e v . L e t t . 3 8 ., 8 2 9 ( 1 9 7 7 ) .

1 3 . D . D o b r o t t e t a l . , P h y s . R e v . L e t t . 3_9, 9 4 3 ( 1 9 7 7 ) .

1 4 . D . L . J a s s b y e t a l . , P r i n c e t o n P l a s m a P h y s i c s L a b o r a t o r y R e p o r t

P P P L - 1 3 7 1 ( S e p t e m b e r 1 9 7 7 ) .

1 5 . Y - K . M . P e n g , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t O R N L / T M - 6 3 1 9

( A p r i l 1 9 7 8 ) .

1 6 . D . B e r g e r , R . G r u b e r , a n d F . T r o y o n , p a p e r p r e s e n t e d a t t h e S e c o n d

E u r o p e a n C o n f e r e n c e o n C o m p u t a t i o n a l P h y s i c s , G a r c h i n g , F e d e r a l

R e p u b l i c o f G e r m a n y , A p r i l 2 7 - 3 0 , 1 9 7 6 .

1 7 . J . W . C o n n o r a n d J . G . C o r d e y , N u c l . F u s i o n 1 4 , 1 8 5 ( 1 9 7 4 ) ; J . G .

C o r d e y e t a l . , " T h e K i n e t i c T h e o r y o f B e a m I n d u c e d C u r r e n t s i n

T o r o i d a l P l a s m a s , " t o b e p u b l i s h e d .

1 8 . D . J . S i g m a r , N u c l . F u s i o n V3 _, 1 7 ( 1 9 7 3 ) ; D . J. S i g m a r a n d P . H .

R u t h e r f o r d , N u c l . F u s i o n J3.»6 7 7

( 1 9 7 3 ) .

1 9 . N . J . F i s c h , " C o n f i n i n g a T o k a m a k P l a s m a w i t h R F - D r i v e n C u r r e n t s , "

s u b m i t t e d t o P h y s . R e v . L e t t .

2 0 . J . D . S e t h i a n e t a l . , B u l l . A m . P h y s . S o c . 2 2 . , 1 1 5 5 ( 1 9 7 7 ) , a l s o

p u b l i s h e d i n P h y s . F l u i d s 2J_, 1 2 2 7 ( 1 9 7 8 ) ; H . A . D a v i s , D . J . R e j ,

a n d H . H . F l e i s c h m a n n , P h y s . R e v . L e t t . 3 9 . , 7 4 4 ( 1 9 7 7 ) .

2 1 . R . K . L i n f o r d , D . A . P l a t t s , a n d E . G . S h e r w o o d , " F i e l d R e v e r s a l

E x p e r i m e n t s , " t o b e p u b l i s h e d .

2 2 . J . L . E d d l e m a n , C . W . H a r t m a n , J . W . S h e a r e r , a n d W . C . T u r n e r ,

p a p e r p r e s e n t e d a t t h e C o n t r o l l e d F u s i o n T h e o r y C o n f e r e n c e , G a t l i n -

b u r g , T e n n e s s e e , A p r i l 2 6 - 2 8 , 1 9 7 8 .



29

2 3 . M . N . B u s s a c e t a l . , p a p e r p r e s e n t e d a t t h e S e v e n t h I n t e r n a t i o n a l

C o n f e r e n c e o n P l a s m a P h y s i c s a n d C o n t r o l l e d N u c l e a r F u s i o n R e s e a r c h ,

I n n s b r u c k , A u s t r i a , A u g u s t 2 3 - 3 0 , 1 9 7 8 .

2 4 . A . G . E s k o v e t a l . , i n P r o c . 7 t h E u r o p e a n C o n f . o n C o n t r o l l e d

F u s i o n a n d P l a s m a P h y s i c s ( L a u s a n n e , 1 9 7 5 ) , _I_, C e n t r e d e R e c h e r c h e

e n P h y s i q u e d e s P l a s m a s , L a u s a n n e ( 1 9 7 5 ) 5 5 .

2 5 . L . S . S o l o v ' e v , S o v . P h y s . - J E T P 26., 4 0 0 ( 1 9 6 8 ) .

2 6 . E . R e b h a n , N u c l . F u s i o n ] S _ t 2 7 7 ( 1 9 7 5 ) .

2 7 . D . B . N e l s o n , p r i v a t e c o m m u n i c a t i o n s .

2 8 . W . K e r n e r , M a x - P l a n c k - I n s t i t u t fiir P l a s m a p h y s i k R e p o r t I P P 6 / 1 3 4

( M a r c h 1 9 7 5 ) .

2 9 . M . M u r a k a m i , J. D . C a l l e n , a n d L . A . B e r r y s N u c l . F u s i o n J_6_, 3 4 7

( 1 9 7 7 ) .

3 0 . B . G r e e n , P . N o l l , a n d J . S h e f f i e l d , P l a s m a P h y s . J 7_ , 1 1 01 ( 1 9 7 5 ) .

3 1 . J . A . R o m e , D . G . M c A l e e s , J . D . C a l l e n , a n d R . H . F o w l e r , N u c l .

F u s i o n 1 6 , 5 5 ( 1 9 7 6 ) .

3 2 . C . C . D a u g h n e y a n d K . B o l , N u c l . F u s i o n V 7 _ , 3 6 7 ( 1 9 7 7 ) .

3 3 . Y - K . M . P e n g e t a l . , O a k R i d g e N a t i o n a l L a b o r a t o r y R e p o r t O R N L / T M -

6 4 8 2 ( to b e p u b l i s h e d ) .

3 4 . M . M u r a k a m i e t a l . , p a p e r p r e s e n t e d a t t h e S e v e n t h I n t e r n a t i o n a l

C o n f e r e n c e o n P l a s m a P h y s i c s a n d C o n t r o l l e d N u c l e a r F u s i o n R e s e a r c h ,

I n n s b r u c k , A u s t r i a , A u g u s t 2 3 - 3 0 , 1 9 7 8 .

3 5 . B . J . G r e e n , J . J a c q u i n o t , K . L a c k n e r , a n d A . G i b s o n , N u c ' I . F u s i o n

1 6 , 5 2 1 ( 1 9 7 6 ) .

3 6 . J . K e s n e r a n d R . W . C o n n , N u c l . F u s i o n Jj 5 , 3 9 7 ( 1 9 7 6 ) .



30

T A B L E 1. P a r a m e t e r s for the q u i l i b r i a s h o w n in F i g u r e s 1-3.T h e d i m e n s i o n l e s s q u a n t i t i e s of the q u i l i b r i a r e m a i n u n c h a n g e d

w h e n B . , B , and I a r e m u l t i p l i e d by a c o n s t a n t .

A s p e c t r a ti o (A)

M a j o r r a d i u s (R )

M i n o r r a d i u s (a)

E l o n g a t i o n (a)

Bt o (R = o }

P l a s m a c u r r e n t (I )

a

I P _•

S" ( = 8T T P /B " 2 )

J. (= 8TTP7 B£

S t 0 ( p eak , =

B t (R = RQ - a )

B t (R = RQ + a )

B p (R = Ro - a )

B p (R = RQ + a )

2.0

1 6 cm

8 cm

1 . 6 5

1 . 3 0 x 1

0.449 x

1.0

2.1

0 . 9 9

0 . 1 9

0 . 2 5

0 . 4 3

0 . 9 5

2 . 6 0 x 1

0 . 8 7 x 1

1 . 1 4 x 1

1 . 0 0 x 1

Qk G

1 06 A

0k G

Qh G

0k G

0k G

1.5

6 0 cm

4 0 cm

1 . 6 5

3 . 7 3 x 10^

1 3 . 6 x 10 5

1.0

2.0

0 . 7 5

0 . 3 1

0 . 5 3

1 . 4 0

2 . 8 5

1 1 . 2 x 10 4

2 . 2 4 x 10 4

7 . 6 0 x 10^

5 . 4 0 x 1 01"

G

A

G

G

G

G



31

TABLE 2. Parameters of m argina l s t a b i l i t y basedon the e q u il ib r ia shown in T able 1 and Figures 1-3

Major radius

Minor radius

(R o )

(a )

Elongat ion (a)

D f D — D \D j . _ l l \ ~ t \ 1

to olasma current (I n )

qo

q b

3 ( = 8 T T ^ / B " 2 )

8 t c ( p e a k )

B t (R - R o -

B t (R = RQ +

Bp (

R=

Ro "

B (R - R o +

VP

r

a)

a)

a)

a)

16

8

1.

1 .

0 .

1 .

3 .

0 .

0 .

0 .

0 .

2.

0 .

0 .

0 .

2.

A s p e c t

2.0

65

20 x lO1* G

268 x 106

A

59

25

99

14

18

40

40 x 101* G

,80 x 101* G

.681 x 101* G

,597 x 1C* G

,91 x 101* cm

3

r a t i o ( A )

60

40

1.65

4 . 0 :

6 . 5 4

2 .08

4 .47

0 . 7 5

0 . 1 7

0 . 2 8

0 . 5 7

12 .0

2.4

3 .65

2 .58

2 .70

1

x 1

X

X

X 1

X

X

X

.5

0kG

1 06 A

101* G

01* G

10k G

101* G

T O6* c m

3



32

TABLE 3. Parameters of a smal l cont inuous tokamak compressor for

a = 8 cm and a = 16 cm

A

3 t

"nK

nD

a

R

B°t o

IpK

n

T

Tconf

auxx (L/R)

^skin ^ Cffl

Af te r

8 cm

16 cm

1.2 x 10" G

2.63 x 105

A

1.41 x 101" cm"

3

2.28 x 103

eV

3.21 msec

4.49 kJ

1.6 x 106

W

200 msec

i)10

msec

J (TF coils)

P (TF or PF co i l s )

T (compression)

R

B c

compression

2.0

0.18

0.99

3.25

16 cm

32 cm

0.98 x 10" G

3.28 x 105

A

1.15 x W* on '3

2.28 x 103

eV

4.8 msec

10.1 kJ

2.4 x 10sW

800 msec

40 msec

a = 8 cm

6.2-1.2 x 103

A/cm3

1.0 MM

1.4 msec

7 cm

2.74 x 101* G

Before

14 cm

48 cm

0.4 x 10u

G

60.7 x 10u

A

1.57 x 1013

cur3

.527 eV

1.0 msec

1.22 kJ

1.2 x 106W

a = 16

3.4-0.65 x 10

1.0 MW

2.3 msec

10.5 cm

2.24 x 101* G

compression

3.46

0.042

0.69

3.25

28 cm

96 cm

6.33 x 1014

G

74.3 x 10u

A

0.70 x TO13

cm"3

527 eV

1.5 msec

2.75 kJ

1.8 x 106W

cm "

•3 A / c m 3



33

TABLE 4. Pa ra m et ers of an i g n i t i o n compressomak

A fte r compression Before compression

A 1 . 5 3R 60 cm 240 cm

a 40 cm 80 cm

B^ 4 . 0 x 10 4 G 1 x 10 4 Gt o

S t 0 . 2 8 0 . 0 4 4

8 0 . 7 5 0 . 4 7

qb 4.47 4 .47

I 6 . 5 4 x 106

A 7.75 x 105

A

n 7 .8 2 x 1 0 1 4 cm "3 4 .89 x 10 1 3 cm "3

T 7.11 x 10 3 eV 1.12 x 10 3 eV

x c o n f 0.5 23 sec 0 .262 sec

W 7 . 20 x 10 5 J 1.31 x 10 5 J

V 2. 7 x 10 5 cm 3 5.0 x 10 7 cm3

p (o r P , ) 2 0 .2 MW 4 .8 3 MWa aux

P Q H 0 . 7 9 MW 0 . 1 7 MW

P L o s s 1 9 . 2 MW 5 . 0 2 MW

T( L /R ) 18 sec

cm) 0. 9 se c

J (T F

P (TF

Tcomp

Rc

B

c

c o i

o r

I s )

PF co i l s)

11.

87

200

18

13.

8 - 2 .

MW

.4 x

msec

cm

3 x TO4

103

A/cm2

G



34

T A B L E 5 . D i m e n s i o n l e s s p a r a m e t e r s ofv a r i o u s t o r o i d a l p l a s m a s

Tokamak (A < 2)

Liner-imploded

torus

Spheromak

Field reversedmirror (plasma gun)

Field reversedtheta pinch

Electron or ion

E-layer

a / p

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> ]

. 1 0

a few

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V

T

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)

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0.6a

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3 5

F I G U R E C A P T I O N S

F i g u r e 1 . C o n s t a n t $ a n d |B j s u r f a c e s f o r M H D e q u i l i b r i a w i t h A = 2 . 0

a n d 1 . 5 a n d p a r a m e t e r s s h o w n i n T a b l e 1 .

F i g u r e 2 . P r o f i l e f u n c t i o n s p U ) , F2( ^ ) , q ( * ) , . p ( R ) , J ( R ) , a n d B p U )

f o r t h e A = 2 .0 e q u i l i b r i u m s h ow n i n F i g u r e 1 a n d T a b l e 1 .

F i g u r e 3 . P r o f i l e f u n c t i o n s p M , FZM , q(i|»). p ( R ) , J ( R ) , a nd B

f o r t h e A = 1 .5 e q u i l i b r i u m s h o wn i n F i g u r e 1 a n d T a b l e 1 .

F i g u r e 4 . E i g e n v a l u e s y2

( x ) o b t a i n e d f r o m t h e E R A T O c o d e w i t h f i n i t e

\ \ i - x a n d s i z e s . T h e c o n v e r g e n t v a l u e s o f y 2 (•) i n t h e l i m i t

o f l a r g e g r i d s i z e a r e e s t i m a t e d b y l i n e a r e x t r a p o l a t i o n .

F i g u r e 5 . . C o n v e r g e n t g r o w t h r a t e s y (•) f o r t h e N = 1 , 2 , an d 3 m o d e s .

T h e c r i t i c a l b e ta v a l u e s (•) a r e e s t i m a t e d b y l i n e a r e x t r a p -

o l a t i o n o f y i n s".

F i g u r e 6 . C r i t i c a l ? c a nd 1 ^ v a l u e s (•) f o r N = 1 , 2 , a nd 3 . T h e

c r i t i c a l b e t a v a l u e s i n t h e l i m i t o f l a r g e N ( ° ) a r e o b t a i n e d

b y l i n e a r e x t r a p o l a t i o n o f F i n ( 1 / N ) .

F i g u r e 7 . S c h e m a t i c d r a w i n g o f a s m al l c o n t i n u o u s t o k a m a k c o m p r e s s o r

f o r d e m o n s t r a t i n g h i g h t o k a m a k b e t a .

F i g u r e 8 . S c h e m a t i c d r a w i n g o f a c o m p r e s s o m a k f o r d e m o n s t r a t i n g

t h e r m o n u c l e a r i g n i t i o n .



36

ORNL/DWG/FED 78-779A

^ - S U R F A C E S , | B | - SURFACES

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45

ORNL/TM-6535D i s t . C a t e g o r y U C-2 0 a , d , g

INTERNAL DISTRIBUTION

1.2.3.4.

5.6.7.8.9.

10.

11.

L . A. B e r r yJ . D. C a l l e nR . A. D a n d lR . A. D o r y

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48

J . S h e f f i e l dD . S t e ine rY-K. M. PengLaboratory Records Depar tmentLaboratory Records, ORNL-RCDocument Reference Sect ionCe nt ra l .Research L i b r ar yF u s io n E ne rg y D i v i s i o n L i b r a r yFus ion E nergy D iv i s ionCommunicat ions Center

ORNL Pate nt O ff ic e

EXTERNAL DISTRIBUTION

4 9 . D. J . Anthony, Energy Systems and Technology D i v i s io n , GeneralE le c t r ic Company, 1 River Road, B ld g . 2 3, Room 290 , Schen ectady,NY 12345

5 0 . B i b l i o t h e k , M a x - P l a n c k I n s t i t u t e f u r P l a s m a p h y s i k , 8 0 4 6Garching bei Munchen, Federal Republic of Germany

5 1 . B ib l i o t h eq ue , S e rv i ce du C on finemen t des Plasmas, C.E.A.,B.P. No. 6, 92, Fontenay-aux Roses (Seine) , France

5 2 . Lung C heung, D epartmen t o f E le c t r o n ic s , U n i ve r s i t y S c ienceCe nter , The Ch inese U n ive rs i t y o f Hong Kong, S h a t in , N .T . ,Hong Kong

5 3 . J. F . C larke , Of f ice o f Fus ion Energy , G-234, Depar tmento f Energy , W as hin gto n, DC 20545

5 4 . D. Cohn, Massachusetts In s t i tu te o f Techno logy , Cambr idge,MA 02139

5 5 . R. W. Conn, Fus ion Technology Program , Nuclear En gin ee r ingDepar tment , U n iv e rs i t y o f W isco ns in , Madison, WI 53706

5 6 . CTR L ib ra ry , c /o Alan F. Haught , Uni te d Te chn ologies ResearchLa bo ra to ry , Eas t H a r t fo rd , CT 06108

5 7 . CTR Reading Room, c/o A ll a n N. Kaufman, Physics De partm ent,U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , CA 94720

5 8 . J . Nar l Dav idson, Schoo l o f Nu c lear En g in ee r ing , Georg iaI n s t i t u t e o f Techno logy , A t l a n t a , GA 30332

5 9 . Docume ntation S . I .G .N . , Depar tment de la Physique du Plasmaet de la Fusion Controle"e, Association EURATOM-CEA sur laFu s ion , Centre d 'E tudes N uc le a i re s , B .P . 85 , Cent re du TR I , 38041Grenoble, Cedex, France

6 0 . W. R. E l l i s , O f f ic e o f Fus ion Energy , G-234, Departmento f Ene rgy, W as hin gto n, DC 20545

6 1 . G. A . Emmer t , Nuc lear Eng ineer ing Depar tment , Un ivers i t yo f W isco ns in, Mad ison, WI 53706

6 2 . Harold K. Forsen, Exxon Nuclear Co. , Inc. , 777 106th Avenue,

N .E . , C-000777, B e l le vu e , WA 98009



46

6 3 . H . P . F u r t h , P r i n c e t o n P l a s m a P h y s i c s L a b o r a t o r y , P r i n c e t o nU n i v e r s i t y , F o r r e s t a l C a m p u s , P . O . B o x 4 5 1 , P r i n c e t o n , N J 0 8 5 4 0

6 4 . R . W . G o u l d , C a l i f o r n i a I n s t i t u t e o f T e c h n o l o g y , M a i l S t o p1 1 6 - 8 1 , P a s a d e n a , C A 9 1 1 2 5

6 5 . C h a r l e s R . H e a d , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n t o fE n e r g y , W a s h i n g t o n , D C 2 0 5 4 5

6 6 . R o b e r t L . H i r s c h , E x x o n R e s e a r c h a n d E n g i n e e r i n g , P . O . B o x 1 0 1 ,F l o r h a m P a r k , N J 0 7 9 3 2

6 7 . R a y m o n d A . H u s e , M a n a g e r , R e s e a r c h a n d D e v e l o p m e n t , P u b l i cS e r v i c e G a s a n d E l e c t r i c C o m p a n y , 8 0 P a rk P l a c e , N e w a r k , N J0 7 1 0 1

6 8 . T . H s u , O f f i c e o f F u s i o n E n e r g y , 6 - 2 3 4 , D e p a r t m e n t o f E n e r g y ,W a s h i n g t o n , D C 2 0 5 4 5

6 9 . V . E . I v a n o v , P h y s i c a l - T e c h n i c a l I n s t i t u t e o f t h e U k r a n i a nA c a d e m y o f S c i e n c e s , S u k h u m i , U . S . S . R .

7 0 . D . L . J a s s b y , P r i n c e t o n P l a s m a P h y s i c s L a b o r a t o r y , P . O . B o x 4 5 1 ,

P r i n c e t o n , N J 0 8 5 4 07 1 . A . K a d i s h , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n t o f E n e r g y ,W a s h i n g t o n , D C 2 0 5 4 5

7 2 . L . M . K o v r i z h n i k h , L e b e d e v I n s t i t u t e o f P h y s i c s , A c a d e m yo f S c i e n c e s o f t h e U . S . S . R . , L e n i n s k y P r o s p e c t 5 3 , M o s c o w ,U . S . S . R .

7 3 . G u y L a v a l , G r o u p e d e P h y s i q u e T h e " o r i q u e , E c o l e P o l y t e c h n i q u e ,9 1 P a l a i s e a u , P a r i s , F r a n c e

7 4 . L i b r a r y , C e n t r e d e R e c h e r c h e s e n P h y s i q u e d e s P l a s m a , 2 1 A v e n u ed e s B a i n s , 1 0 0 7 , L a u s a n n e , S w i t z e r l a n d

7 5 . L i b r a r y , C u l h a m L a b o r a t o r y , U n i t e d K i n g d o m A t o m i c E n e r g yA u t h o r i t y , A b i n g d o n , O x o n , 0 X 1 4 3 D B , U n i t e d K i n g d o m

7 6 . L i b r a r y , F O M - I n s t i t u t v o o r P l a s m a - F y s i c a , - R i j n h u i z e n ,J u t p h a a s , N e t h e r l a n d s

7 7 . L i b r a r y , I n s t i t u t e f o r P l a s m a P h y s i c s , N a g o y a U n i v e r s i t y ,N a g o y a , J a p a n 4 6 4

7 8 . L i b r a r y , I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c s , T r i e s t e ,I t a l y

7 9 . L i b r a r y , L a b o r a t o r i o G a s I o n i z z a t i , F r a s c a t i , I t a l y8 0 . D s u m b e r G . L o m i n a d z e , A c a d e m y o f S c i e n c e s o f t h e G e o r g i a n

S.S.R., 8 D z e r z h i n s k i S t . , 3 8 0 0 4 , T b i l i s i , U . S . S . R .8 1 . O s c a r P . M a n l e y , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n t

o f E n e r g y , W a s h i n g t o n , D C 2 0 5 4 58 2 . D . G . M c A l e e s , E x x o n N u c l e a r C o . , I n c . , R e s e a r c h a n d T e c h n o l o g y

L a s e r E n r i c h m e n t D e p a r t m e n t , 2 9 5 5 G e o r g e W a s h i n g t o n W a y ,R i c h l a n d , W A 9 9 3 5 2

8 3 . J . E . M c C u n e , S c h o o l o f E n g i n e e r i n g , D e p a r t m e n t o f A e r o n a u t i c sa n d A s t r o n a u t i c s , B l d g . 3 7 - 3 9 1 , M a s s a c h u s e t t s I n s t i t u t e o fT e c h n o l o g y , C a m b r i d g e , M A 0 2 1 3 9

8 4 . C l a u d e M e r c i e r , S e r v i c e d u T h e o r i e d e s P l a s m a s , C e n t r e d ' E t u d e sN u c l e a i r e s , F o n t e n a y - a u x - R o s e s ( S e i n e ) , F r a n c e

8 5 . K . G . M o s e s , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n to f E n e r g y , W a s h i n g t o n , D C 2 0 5 4 5

8 6 . M i c h a e l M u r p h y , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n to f E n e r g y , W a s h i n g t o n , D C 2 0 5 4 5



47

8 7 . R . E . P a p s c o , G r u m m a n A e r o s p a c e C o r p . , 1 01 C o l l e g e R o a d ,P r i n c e t o n , N J 0 8 5 4 0

8 8 . D . P f i r s c h , I n s t i t u t e f o r P l a s m a P h y s i c s , 8 0 4 6 G a r c h i n gb ei M ii n c he n , F e d e r a l R e p u b l i c o f G e r m a n y

8 9 . P l a s m a P h y s i c s G r o u p , D e p a r t m e n t o f E n g i n e e r i n g P h y s i c s ,A u s t r a l i a n N a t i o n a l U n i v e r s i t y , P . O . B o x 4 , C a n b e r r a A . C . T .2 6 0 0 , A u s t r a l i a

9 0 . A . R o g i s t e r , I n s t i t u t e f o r P l a s m a P h y s i c s , K F A , P o s t f a c h 1 9 1 3 ,D - 5 1 7 0 , J U l i c h 1 , F e d e r a l R e p u b l i c o f G e r m a n y

9 1 . W . S a d o w s k i , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n to f E n e r g y , W a s h i n g t o n , D C 2 0 5 4 5

9 2 . V . D. S h a f r a n o v , I . V . K u r c h a t o v I n s t i t u t e o f A t o m i c E n e r g y ,4 6 U l i t s a K u r c h a t o v a , P . O . B o x 3 4 0 2 , M o s c o w , U . S . S . R .

9 3 . Y u . S . S i g o v , I n s t i t u t e o f A p p l i e d M a t h e m a t i c s o f t he U . S . S . R .A c a d e m y o f S c i e n c e s , M i u s k a y a , S q . 4 , . Mo s c ow A - 4 7 , U . S . S . R .

9 4 . W . M . S t a c e y , J r . , S c h o o l o f N u c l e a r E n g i n e e r i n g , G e o r g i a

I n s t i t u t e o f T e c h n o l o g y , A t l a n t a , G A 3 0 3 3 29 5 . L . D . S t e w a r t , P r i n c e t o n P l a s m a P h y s i c s L a b o r a t o r y , P . O. B o x

4 5 1 , P r i n c e t o n , N J 0 8 5 4 0

9 6 . J . B . T a y l o r , C u l h a m L a b o r a t o r y , U . K . A t o m i c E n e r g y , A u t h o r i t y ,A b i n g d o n , O x o n , 0 X 1 4 3 D B , U n i t e d K i n g d o m

9 7 . T h e r m o n u c l e a r L i b r a r y , J a p a n A t o m i c E n e r g y R e s e a r c h I n s t i t u t e ,T o k a i , N a k a , I b a r a k i , J a p a n

9 8 . F r a n c i s c o V e r d a g u e r , D i r e c t o r , D i v i s i o n o f F u s i o n , J u n t a d eE n e r g i a N u c l e a r , M a d r i d 3 , S p a i n

9 9 . K . M . Z w i l s k y , O f f i c e o f F u s i o n E n e r g y , G - 2 3 4 , D e p a r t m e n to f E n e r g y , W a s h i n g t o n , D C 2 0 5 4 5

1 0 0 . D i r e c t o r , R e s e a r c h a n d T e c h n i c a l S u p p o r t D i v i s i o n , D e p a r t m e n t

o f E n e r g y , O a k R i d g e O p e r a t i o n s , P . O . B o x E , O a k R i d g e , T N3 7 8 3 0

1 0 1 - 3 3 5 . G i v e n d i s t r i b u t i o n a s s h o w n i n T I D - 4 5 0 0 , M a g n e t i c F u s i o n E n e r g y( D i s t r i b u t i o n C a t e g o r i e s U C - 2 0 a , d , a nd g : P l a s m a S y s t e m s ,F u s i on S y s t e m s , a n d T h e o r e t i c a l P l a s m a P h y s i c s )

y-k. m. peng and r. a. dory- very small aspect ratio tokamaks

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