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T r a n s p o r t E q u a t i o n s I n Axisyrnrnet'ric T o r o i d a l C o o r d i n a t e s

Samuel L. Gra ln i ck

P r i n c e t o n Plasma P h y s i c s Laboratory - - - -

1

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ABSTRACT

T h i s a r t i c l e p r e s e n t s a d e r i v a t i o n o f t h e

c o n s e r v a t i o n law form of t he ' s i n g l e e n e r g y . group t r a n s p o r t

equa t ion i n an . axisymrnetric t o r o i d a l c o o r d i n a t e system

formed , by r o t a t i n g a n e s t of smooth, s imp ly c l o s e d , p l a n e

c u r v e s of a r b i t r a r y p a r a m e t r i c d e s c r i p t i o n about an a x i s

which does n o t i n t e r s e c t t h e n e s t . 'l'his g e n e r a l e q u a t i o n

may be used f o r g e n e r a t i n g . e q u a t i o n s s p e c i f l c t o p a r t i c u l a r

c r o s s s e c t i o n g e o m e t r i e s , . o r a s t h e b a s i s o f a f i n i t e

d i f f e r e n c e e q u a t i o n f o r t h e g e n e r a l ca se .

The e f f e c t o f bo th t h e t o r o i d a l . .and . p o l o i d a l

c u r v a t u r e s o f t h e sys tem are i n v e s t i g a t e d , and c r i t e r i a f o r

t h e v a l i d i t y . of c y l i n d r ' i c a l . and . p l a n a r app rox ima t ions . a r e

e s t a b l i s h e d .

The d i f f u s i o n e q u a t i o n f o r t h i s g e o m e t r y ' i s d e r i v e d ,

and it is shown t o be . f o r m a l l y . homologus t o t h e "r-8"

c y l i n d r i c a l d i f f u s i o n , equa t ion i f t h e c o o r d i n a t e sys tem is

o r t h o g o n a l and i f t h e az imu tha l c o o r d i n a t e , . $ ; is i g n o r a b l e .

! \ I ; {$fi!--T? ! 3 ! ~ ~ ~ ~ " T { Q N Oi THIS EOCUhJENT 1s ". -. , . . .,-&

I. INTRODUCTION

The s o l u t i o n o f n e u t r o n and photon t r a n s p o r t proljlems

1-4 i n t h e a n a l y s i s and d e s i g n o f proposed f u s i o n r e a c t o r s ,

and t h e a p p l i c a t i o n o f t h e t e c h n i q u e s o f t r a n s p o r t t h e o r y t o

t h e problems a s s o c i a t e d w i t h t h e t r a n s p o r t of n e u t r a l

p a r t i c l e s i n f u s i o n r e a ' c t o r p lasmas . .

5-7 r e q u i r e t h a t w e

co,ncern otirsel-ves w i t h ax isymmetr lc tdYolda l geoiiie'rrfes.

E x i s t i n g , d i s c r e t e o r d i n a t e t y p e , t r a n s p o r t .codes t h a t have

been t a i l o r e d t o p l a n a r , c y l i n d r i c a l o r s p h e r i c a l geome t r i e s

can , a t b e s t , . b e u sed t o g i v e approximate answers which are

sometimes v a l i d b u t f r e q u e n t l y a r e ex tended beyond t h e i r

r ange o f a p p l i c a b i l i t y . I n f a c t , a r e a s o n a b l e knowledge o f

t h e e r r o r i n t r o d u c e d by u s i n g t h e s e codes can o n l y b e

gues sed a t i n ' t h e absence of t o r o i d a l a n a l y s e s . The

a l t e r n a t i v e , a v a i l a b l e is: t o . u s e Monte C a r l o . s i m u l a t i o n . .

t e c h n i q u e s which are i n h e r e n t l y s u i t a b l e t o compl ica ted

geomet r i e s . T h i s approach , however, ha s no t proven t o b e

u n i v e r s a l l y a c c e p t a b l e and t h e need f o r a ' t o r o i d a l d i s c r e t e

o r d i n a t e t r a n s p o r t code s t i l l e x i s t s .

I n t h i s a r t i c l e w e w i l l u s e a t e n s o r t r a n s f o r m a t i o n

formal i sm deve loped p r e v i o u s l y 8 t o t r a n s f o r m t h e . s t r e a m i n g

t e r m of a s i n g l e ' energy group. t r a n s p o r t equa t ion t o a system

of ax isymmetr ic , non-or thogonal t o r o i d a l c o o r d i n a t e s . These

c o o r d i n a t e s a r e g e n e r a t e d by r o t a t i n g a n e s t of s imply

c l o s e d smooth c u r v e s about an a x i s which does n o t i n t e r s e c t

t h e n e s t . S e v e r a l p a r t i c u l a r c a s e s have been t r e a t e d

p r e v i o u s l y by o t h e r . a u t h o r s The d ive rgence form of

t h e t r a n s p o r t e q u a t i o n f o r one o f t h e s e w i l l b e d e r i v e d a s

an i l l u s t r a t i v e example; i t .is r e a d i l y . o b t a i n e d from t h e

g e n e r a l e q u a t i o n which we w i l l d e r i v e .

' R ~ c a s t i n g t h e g e n e r a l e q u a t i o n i n a s l i g h t l y a l t e r e d

c y l i n d r i c a l form, i t is p o s s i b l e : to i l l u m i n a t e t h e r o l e s of

bo th t h e t o r o i d a l and p o l o i d a l c u r v a t u r e s of t h e system and

t o demons t r a t e t h e c o n d i t i o n of a p p l i c a b i l i t y o f t h e

c y l i n d r i c a l approximat ion i n which t h e t o r o i d a l c u r v a t u r e is

ignored . S i m i l a r l y i f bo th t h e t o r o i d a l and p o l o i d a l

c u r v a t u r e s a r e i g n o r e d . a p l a n a r ' app rox ima t ion w i l l r e s u l t

which w i l l b e adequa te f o r some pu rposes .

F ina l ly , i t h a s been no ted t h a t t w o of t h e p r e v i o u s l y

t r e a t e d c o o r d i n a t e sys tems ',lo erijoy t h e p r o p e r t y of

y i e l d i n g d i f f u s i o n e q u a t i o n s w h i c h are formal homologs of

t h e c y l i n d r i c a l , "r-0" , d i f f u s i o n e q u a t i o n when t h e t o r o i d s l

c o o r d i n a t e is i g n o r a b l e . By u s i n g t h e same t e n s o r fo rmal i sm

t h a t we have a p p l i e d t o t h e t r a n s p o r t e q u a t i o n w e have

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

e q u a t i o n . We f i n d t h a t t h e p r o p e r t y of b e i n g homologous t o

t h e c y l i n d r i c a l "r-8" d i f f u s i o n e q u a t i o n is s h a r e d by a l l

d i f f u s i o n equa ' t ions d e r i v e d f o r o r t h o g o n a l , axisymrnetric

t o r o i d a l c o o r d i n a t e sys tems i n which t h e t o r o i d a l c o o r d i n a t e

is i g ~ i u r a b l e .

TOROIDAL COORDINATES

Cons ider a c o o r d i n a t e system i n which t h e c o o r d i n a t e

s u r f a c e s are formed by r o t a t i n g a n e s t o f smooth, s imply

c l o s e d , p l a n e c u r v e s about an a x i s which does n o t i n t e r s e c t

t h e n e s t . ( s e e F i g . 1.) The e q u a t i o n s o f t r a n s f o r m a t i o n

bctwoon t h o . ( y , ) C a r t e ~ i a n c o o r d i n a t e . ~ y s t e m and t h e

( p , 8 , @ ) t o r o i d a l c o o r d i n a t e s a r e

( I n d i c i a 1 n o t a t i o n w i l l b e , u s e d th roughout w i t h r e p e a t e d

1 2 i n d i c i e s summed. ) I n e q u a t i o n s (1) x =x , x =y a n d x3=Z.

4 is t h e t o r o i d a l o r az imutha l ang le . We may chose p t o be

t h e s q u a r e r o o t o f t h e a r e a c o n t a i n e d w i t h i n a member of t h e

n e s t o f c u r v e s and 8 t o b e ' t h e . l e n g t h of a r c a long t h e curve

no rma l i zed t o t h e p e r i m e t e r ,of th.e cu rve ' alth'ough t h i s i s

n o t e s s e n t i a l , o t h e r s e l e c t i o n s o f p ' and 8 b e i n g a c c e p t a b l e .

2 The f u n c t i o n s s a n d X g i v e a p imamet r ic d e s c r i p t i o n of t h e

elements of t h e n e s t - they may be a r b i t r a r i l y specf f ied ,

and i n f a c t , need n o t b e g iven by a n a l y t i c a l fo rmulas ,

numer i ca l f u n c t i o n s s u f f i c i n g . We w i l l r e s t r i c t t h e c h o i c e

o f X' and %such t h a t t h e n e s t so d e s c r i b e d may have a t most

one s i n g u l a r p o i n t a t which t h e J acob ian of T van i shes . 1

\Ve now w r i t e th.e s i n g l e energy ,g roup t r a n s p o r t

equa t ion as a c o n s e r v a t i o n law i n a 5-dimensional Riemannian

R 1 space , hav ing c o o r d i n a t e s . x =x, X2=y, X 3 = ~ , 5 '

4 -1 x = t a n (XZ/hx), X5'XyY (where A x , A Y and X Z a r e t h e x , y ,

and z C a r t e s i a n components of - Q , t h e p ropaga t ion v e c t o r ) ,

and hav ing t h e m e t r i c t e n s o r ,

Ig ( d e n o t e s t h e de t e rminan t o f t h e m e t r i c t e n s o r , Y i s t h e

a n g u l a r f l u x , a is t h e t o t a l c r o s s s e c t i o n and S is t h e

s o u r c e . f u n c t i o n .

~ o l l o w i n ~ t h e ' p r e s c r i p t i o n g iven i n r e f e r e n c e (8 ) we.

apply t h e t r a n s f o r m a t i o n . TI t o e q u a t i o n ( 3 ) . Under t h i s

t r a n s f o r m a t i o n ,

and

I n e q u a t i o n s (4). and ( 5 ) t h e unbar red x ' s r e f e r t o t h e

C a r t e s i a n c o o r d i n a t e s and t h e b a r r e d x ' s r e f e r t o t h e

-i j t o r o i d a l c o o r d i n a t e s . ' The c o n t r a v a r i a n t t e n s o r g is found

. . ,by s o l v i i i g

'where is a Kronecker d e l t a . The s o l u t i o n o f e q u a t i o n s 6k

( 6 ) and t h e e v a l u a t i o n of e q u a t i o n s ( 4 ) and ( 5 ) y i e l d t h e

f o l l o w i n g r e s u 1 . t ~ : . .

axL ax' + -- a , ~ . a e

ax"

where w Z x 4 , yzX5 and

Also ,

The a p p r o p r i a t e o r t h o g o n a l r e p r e s e n t a t i o n of - 52 f o r

t h i s c o o r d i n a t e sys t em w i l l b e i n t e rms of i ts components i n

t h e $ d i r e c t i o n , i n t h e d i r e c t i o n normal t o t h e t o r o i d a l

s u r f a c e and i n t h e d i r e c t i o n o r t h o g o n a l t o t h e s e two which

is t h e d i r e c t i o n o f i n c r e a s i n g 8 . These components are . .

formed by t a k i n g t h e i n n e r p roduc t of R - 'and t h e u n i t v e c t o r

i n t h e p a r t i c u l a r d i r e c t i o n . We deno te t h e u n i t v e c t o r s

( s e e F i g . 1) by f o r t h e u n i t v e c t o r normal to'. t h e t o r o i d a l

s u r f a c e , afid by 8 and f o r t h e u n i t v e c t o r s i n t h e 0 and $ -

d i r e c t i o n s . The components o f - 52 w e w i l l c a l l X,, X g and $ -.

-1' I n t h e . X c o o r d i n a t e sys t em,

a n d consequen t ly ,

Using t h i ' s or thogon.al r e p r e s e n t a t i o n of 2 w e i n t r o d u c e

a n g u l a r c o o r d i n a t e s F4 and 55 by t h e t r a n s f o r m a t i o n .

=4 . = tan

Three sys t ems of c o o r d i n a t e s i n R g have been used.

The o r i g i n a l c o o r d i n a t e sys tem denoted by Xi, (w i thou t

o v e r b a r s ) c o n s i s t s of t h e x , y and z C a r t e s i a n s p a t i a l

c o o r d i n a t e s and w and y as t h e a n g u l a r v a r i a b l e s . The

c o o r d i n a t e sys t em E~ ( s i n g l e o v e r b a r ) r e s u l t s when

t r a n s f o r m a t i o n TI is a p p l i e d t o t h e s p a t i a l v a r i a b l e s .

T h i s c o o r d i n a t e sys t em .has p , 8 and + f o r s p a t i a l v a r i a b l e s

b u t uses t h e o and y a n g u l a r v a r i a b l e s . F i n a l l y when T i s 2

a p p l i e d t h e Ti c o o r d i n a t e system (doub le ' o v e r b a r s ) r e s u l t s .

I n t h i s c o o r d i n a t e s y s t e m , t h e s p a t i a l v a r i a b l e s a r e p , 8

and @ and t h e a n g u l a r v a r i a b l e s a r e

and

The v , 8 and @ components of 2 i n t h e ;i c o o r d i n a t e sys tem

are

The 5 components o f zi ( i . e . w r i t t e n ' i n ;i c o o r d i n a t e s ) a r e

t h e n ,

\ ax1 a2x2 ax2 a2x1 ) : [($i2 + (a?j2] - - - - - . - ( a e a p a e a e a p a e

1 2 2

( ax a . x - - - - - ax2 i a e a e 2 a e a e 2

. -

= ax1 "v

. . 2 ax1 - - - ax x .X - + kg

;5 = - A. vae x1 1 2 2 2 1 /2

[(%) +(%) 1, . . . . - .. . . - - - . . . . . S i n c e t h e ' c h o i c e o f a n g u l a r c o o r d i n a t e s u s e d g i v e s an

o r t h o g o n a l r e p r e s e n t a t i o n o f $2, t h e J a c o b i a n - o f .8 t r a n s f o r m a t i o n T is . u n i t y . C o n s e q u e n t l y ,

2

We now s u b s t i t u t e i n . t h e d ive rgence e x p r e s s i o n of

e q u a t i o n ( 3 ) w r i t t e n i n t h e c o o r d i n a t e s t o g i v e t h e

s i n g l e energy group t r a n s p o r t equa t ion i n ax isymmetr ic

t o r o i d a l c o o r i n d a t e s . (Note: Th i s e q u a t i o n and t h e

r ema in ing c o n t e n t o f t h i s p a p e r w i l l be w i t h r e f e r e n c e t o

t h e fi system. The o v e r b a r s w i l l now be dropped. )

1

1 2 -1/2

a 1 x~ [i iX 1 + (",'I [ - ( ax ' ax"' ax2 ax2)] 1 +

- - +-- a e a' e a e A~ K A a e ap

. .

1 2 2 a x a x 2 2 1 1 2 ( - - - - a x an) + - a e a p a e a e a p a e C

2 1 2 2 a x 2 a x 3X 3 X - - - C\

a e a e " a e a e "

111. A PARTICULAR COORDINATE SYSTEM OF INTEREST

Here we w i l l d i s p l a y t h e form of t h e t r a n s p o r t

e q u a t i o n f o r a p a r t i c u l a r c o o r d i n a t e system of i n t e r e s t .

Such a c o o r d i n a t e s y s t e m i s s p e c i f i e d when ~ ' ( p , 9 ) and

2 X ( p , 8 ) a r e chosen. The geomet r i c f a c t o r s i n t h e te rms of

1 t h e t r a n s p o r t e q u a t i o n ( 2 1 ) i n v o l v e X , x2 and d e r i v a t i v e s

of t h e s e f u n c t i o n s up t o and i n c l u d i n g o r d e r two. For

1 a n a l y t i c a l l y d e s c r i b e d X . and x2, t h e s e f a c t o r s a r e

e v a l u a t e d and s u b s t i t u t e d i n t o t h e e q u a t i o n . For p u r e l y

numer ica l a p p l i c a t i o n s . i t i s p d s s i b l e t o s p e c i f y X' and x2

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

may then be used i n t h e . s o l u t i o n of t h e e q u a t i o n . The

geomet r i c f a c t o r s . t h a t must. be. e v a l u a t e d a r e :

1 . 1 - ax ax . ~ ~ ~ ~ ' ( p ~ 9 ) = -- ax2 ax2 +. - -

a p . ? 9 a p a 9

a x "

S e v e r a l g e o m e t r i e s .have been cons ide red p r e v 2 o u s l y by

o t h e r a u t h o r s and t h e p r e s e n t writer. Pomraning and

s t evens ' d e r i v e d t h e g r a d i e n t form of t h e t r a n s p o r t e q u a t i o n

and an a p p r o p r i a t e d i f f u s i o n e q u a t i o n f o r n e s t e d c i r c u l a r

t o r o i d s . The d i v e r g e n c e form f o r t h i s sys tem o f c o o r d i n a t e s

h a s been d e r i v e d by t h e p r e s e n t w r i t e r 8 and independen t ly by 4

who h a s a l s o d e r i v e d a set a p p r o p r i a t e f i n i t e

d i f f e r e n c e e q u a t i o n s . Pomraning lo h a s cons ide red two

sys t ems of e l l i p t i c a l - t o r o i d a l c o o r d i n a t e s . We will nnw 1j.ae . .

t h e . g e n e r a l e q u a t i o n .(21) t o d e r i v e t h e d ive rgence form f o r

t h e l a t t e r o f t h e t w o i l l u s t r a t i n g i ts u t i l i t y f o r t h i s

t a s k .

[sinh p s i n e

The sca le f a c t o r , s , u s e d i n r e f e r e n c e (10). i s set t o 1 h e r e

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

and g are - n o t t o b e i n t e r p r e t e d as t h e s q u a r e r o o t o f t h e

area a n d f r a c t i o n a l arc l e n g t h . IVe f i n d t h a t f o r t h i s

geomet ry ,

2. 2 G.i = [sinh2 p cos 8 + cosh2 p sin 91

2 2 2 Gii = - [sinh p cos 9 + cosh2 p sin 91

- Giv

- - sinh p cosh p

[sinh p, sin6 + R ] /tanhL p + tanL9

- - tan 8 1 Gvii

I 7 9 , [sinh' p sin9 + R] JtanhL p . +.

and substituting into the transport equation,

-1 X

2 3 2 2 (sinhp sin8 + R) (sinh p cos'8 + cosh p sin 8)

2 2 2 2 { {[Av(slnb p ios 0 - 1 cosh p sin

a - (sinhp sine. + R) 2 2 a e 2 2

Fxe(sinh p cos 8 + cosh2 sin2@) Y ] ) + { (sinh2p cos 8 + cosh p sin28)1/2

a 2 2 2 - - {\ (sin" p "0s 8. + cosh p sin28) Y} + a +

2 2 2 [- (iinhb sin8 + R) (sinh p cos + cosh p sin28.) x a x .

1. . . 3...2, [A (sin8 cos0)' - X8 (sinho coshp) 1 2 2 2 ( (sinh p cos 8 + cosh p 'sin 8) . -

2 ( A + ) (,!@tan8 - Avtanhp),'

+ 2 2 1/2 (Av) + (Ae) * (sinhp sine + R ) (tanh p + tan 8 )

- . ..

. .

2 2 2 2 (Avtan8 + he & {A+(sinh p cos 8 +cash p sin 8) 2 2 .1/2 .(tanh p + tan 8)

T h i s r e s u l t may be compared t o t h a t of Pomraning by n o t i n g

t h a t ,

and forming t h e . g r a d i e n t s t r e a m i n g o p e r a t o r ,

I V . LII.4ITING CASES - THE EFFECT O F CURVATURE

To d a t e , approximate a n a l y s e s u s i n g " r - 8 " c y l i n d r i c a l J

t r a n s p o r t .codes have been employed ' i n t h e s o l u t i o n of

t o r o i d a l f u s i o n , r e a c t o r . problems. These s o l u t i o n s i g n o r e

e f f e c t s i n t r o d u c e d by t h e n o n - c i r c u l a r c h a r a c t e r i s t i c o f . t h e

minor cross s e c t i o n and a r e on ly v a l i d when t h e p h y s i c a l

s c a l e l e n g t h of t h e problem, A, is s m a l l compared t o t h e

i n v e r s e of t h e t o r o i d a l cu rva tu re , . K T' a t t h e p o i n t i n

q u e s t i o n . (We d e f i n e . the t o r o i d a l c u r v a t u r e a t a p o i n t ,

Y ( p , 6, $1, as the c u r v a t u r e of t h e r i g h t c i r c u l a r c y l i n d e r

p a s s i n g th rough tha t p o i n t and hav ing t h e p r i n c i p a l a x i s of

t h e t o r u s a s i t s a x i s . R e f e r r i n g t o F i g . 1, t h e t o r o i d a l

c u r v a t u r e i s e q u a l t o t h e -cur'vature~of~~the~cir-cle l a b e l l e d

curve o i consLant 8 ) . The s t a t e m e n t r e g a r d i n g t h e r a t i o of

X and 1 / r T , t o o u r knowledge, h a s no t been. r i g o r o u s l y

demons t ra ted . We o f f e r a r i g o r o u s proof of i ts v a l i d i t y

h e r e . We . a l s o show t h a t , when i n a d d i t i o n , X i s s m a l l

compared t o t h e i n v e r s e of K t h e p o l o i d a l c u r v a t u r e a t P, P '

a p l a n a r approximat ion i s v a l i d . ( K ~

i s t h e c u r v a t u r e a t a

p o i n t o f t h e c u r v e l a b e l l e d , "curve o f c o n s t a n t (I" i n

F i g . 1 ) . F i n a l l y , we n o t e , that a s b e f o r e we c o n c i d e r

a r b i t r a r i l y chosen c r o s s - s e c t i o n shapes impusiilg o n l y tho , sc

l i m i t a t i o n s d i scussed p r e v i o u s l y .

To b e g i n w i t h , f o r comparison, w e w i l l need t h e

d i v e r g e n c e form of t h e t r a n s p o r t e q u a t i o n f o r a c y l i n d r i c a l

c o o r d i n a t e sys t em of a r b i t r a r y c r o s s s e c t i o n a l shape. . T h i s '

e q u a t i o n is r e a d i l y ' d e r i v e d by i n t r o d u c i n g ' t h e

t r a n s f o r m a t i o n ,

and employing t h e s a m e . t e c h n i q u e a s used p r e v i o u s l y . We

find,

1 2 2 2 1 2 2 a x a x a x a x a e a p a e a e a p a e +

Although c e r t a i n s i m i l a r i t i e s between t h i s e q u a t i o n and t'he

t o r o i d a l e q u a t i o n (,21) a r e a p p a r e n t , a more s t r i k i n g

s i m i l a r i t y can b e ach ieved by p u t t i n g t h e t o r o i d a l e q u a t i o n

i n t o what we s h a l l c a l l c y l i n d r i c a l form. We accomplish

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

Under t h i s t r a n s f o r m a t i o n t h e t o r o i d a l e q u a t i o n becomes,

Equat ion ( 30) i s t h e same a s equa t ion ( 2 8 ) s a v e f o r t h e

appearance of a d d i t i o n a l t e rms due t o t h e e f f e c t of t o r o i d a l

c u r v a t u r e . To f a c i l i t a t e o u r d i s c u s s i o n of t h e s e t o r o i d a l

e f f e c t s w e s t a t e t h e f o l l o w i n g in fo rma t ion about t h e s u r f a c e

p=const . ( s e e F ig . 1 ) : The c u r v e s 9=cons t . and $=cons t . a r e

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

d e f i n e d by t h e u n i t t a n g e n t v e c t o r s t o t h e s e c u r v e s a r e t h e

p r i n c i p a l d i r e c t i o n s of normal c u r v a t u r e of t h e s u r f a c e .

The p r i n c i p a l . . . c u r v a t u r e s a r e ,

and

We n o t e t h a t ,

The geodes i c c u r v a t u r e of the B=cpnst.. cu rve i s ,

F i n a l l y and i m p o r t a n t l y w e p o i n t o u t t h a t ,

A f t e r s c a l i n g a l l l e n g t h s t o A , t h e p h y s i c a l s c a l e l en ' g th of

t h e problem, w e s e e . f r o m e q u a t i o n (35) which c o n t a i n s a sum

o f s q u a r e t e r m s on t h e l e f t hand s i d e t h a t t h e n e c e s s a r y and

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

i s t h a t , '

If i n a d d i t i o n ,

a l l of ' t h e e f f e c t s of c u r v a t u r e v a n i s h and a p l a n a r problem

r e s u l t s .

V. . DIFFUSION.EQUATIONS

The g e n e r a l form o f t h e d i f f u s i o n e q u a t i o n f o r an

ax i symmet r i c t o r o i d a l geometry may be found by s u b s t i t u t i n g

t h e a p p r o p r i a t e q u a n t i t i e s i n t o ,

where Jj,. a r e - t he c o v a r i a n t components of t h e c u r r e n t ,

. r e l a t e d t o t h e s c a l a r . . f lux . , T,, by,

'a i s t h e a b s o r p t i o n c r o s s s e c t i o n and t h e metric . t e n s o r d i j

i s formed by a p p l y i n g TI, t o t h e m e t r i c f o r C a r . t e s i a n t h r e e

s p a c e . I n terms o f t h e ( p , 8 , @ ) c o o r d i n a t e s t h e e q u a t i o n is

J d e f i n e d a s b e f o r e in e q u a t i o n (:9 1. W e : now t u r n o u r

a t t e n t i o n t o t h e ( 2 x . 2 ) s u b m a t r i x ,

of t h e metric dij F i r s t w e n o t e that ,

and now assume t h a t it is p o s s i b l e , by a s u i t a b l e change of

v a r i a b l e s , t o w r i t e .

where 6ij i s a Kronecker d e l t a . T h i s can a lways be

accompl i shed f o r an o r t h o g o n a l c o o r d i n a t e sys tem, ( i . e . one ~.

f o r wh ich . dI2 - - d 2 1 = 0 ) , by i n t r o d u c i n g . v a r i a b l e s p 1 and € I 1

d e f i n e d by

and

The e lement of a r c f o r t h e s e c o o r d i n a t e s is

, f r o m which.we see t h a t

For s u c h a sys tem of o r t h o g o n a l c o o r d i n a t e s t h e d i f f u s i o n

e q u a t i o n is

2 . Noting t h a t J = v ' and e l i m i n a t i n g t h e independent v a r i a b l e

p 1 i n f a v o r o f v . b y a l o g r i t h m i c t r a n s f o r m a t i o n ,

where w e have used D 1 (1/30). ~ o r n r a n i n ~ ~ ' ~ ~ h a s no ted t h a t '

i f t h e 4 d e r i v a t i v e te rm i s dropped and i f t h e d i f f u s i o n

c o e f f i c i e n t a b s o r p t i o n c o e f f i c i e n t , and s o u r c e f u n c t i o n a r e

r e d e f i n e d as .

and

t h a t a fo rmal ly i d e n t i c a l s t r u c t u r e t o . t h e "r-8" c y l i n d r i c a l

d i f f u s i o n equa t ion i s . o b t a i n e d f o r t h e two cases of.

c o n c e n t r i c c i r c u l a r . and o r t h o g o n a l - e l l i p t i c a l t o r o i d a l

c o o r d i n a t e s . . \Ye have proven t h a t t h i s homology i s ob ta ined

f o r a l l orthogon.al , axisyrnmetri'c t o r o i d a l c o o r d i n a t e systems

p r o v i d i n g t h e t o r o i d a l c o o r d i n a t e i s ignorab le i n t h e

, problem of i n t e r e s t .

. .

V I . CONCLUDING REMARKS

In t h i s paper w e have presente 'd s e v e r a l r e s u l t s of

importance i n t h e a p p l i c a t i o n of d i s c r e t e o r d i n a t e methods

t o the. s o l u t i o n of t r a n s p o r t problems ' i n t o r o i d a l

geometries. Such problems a r i s e i n t h e design and a n a l y s i s

of t h e b l a n k e t s t r u c t u r e s of proposed f u s i o n r e a c t o r s , a s . .

w e l l as i n a t t e m p t i n g t o apply t h e s e techniques t o t h e

a n a l y s i s of t h e - ' t r a n s p o r t of n e u t r a l p a r t i c l e s i n a , tokamak . .

plasma.

The axisymmetr ic t o r o i d a l c o o r d i n a t e s system i n which

we have w r i t t e n t h e s i n g l e energy group t r a n s p o r t e q u a t i o n ,

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

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

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

need t o model t o r o i d a l r e g i o n s w i t h compl ica ted combina t ions

of non - to ro ida l geome t r i c e l emen t s . Equa t ion ( 2 1 ) h a s been

w r i t t e n ' i n c o n s e r v a t i o n law form a p p r o p r i a t e t o t h e

d e r i v a t i o n o f f i n i t e d i f f e r e n c e e q u a t i o n s by i n t e g r a t i o n .

1 2 The g e n e r a l n a t u r e of , t h e e q u a t i o n , X ( p ,8 ) and X ( p ,8 ) a r e .

u n s p e c i f i e d , a l l o w s one t o c o n s i d e r t h e des ign of a g e n e r a l

purpose t o r o i d a l t r a n s p o r t code which would a c c e p t a s i n p u t

2 d a t a t h e s p e c i f i c a t i o n of X1 and X . A l t e r n a t i v e l y

equa t ion (21) 'can be used t o . . q u i c k l y d e r i v e t r a n s p o r t

equa t ion ' s f o r p a r t i c u l a r geomet r i e s of i n t e r e s t . T h i s has

bee11 i l l u s t r a t e d . i n S e c t i o n 1'11.

I t i s c l e a r from t h e a n a l y s i s g iven i n S e c t i o n I V .

t h a t many c i r cums tances can b e env i s iongd i n which t h e

n e g l e c t of t h e e f f e c t s of o n e , o r bo th o f t h e c u r v a t u r e s of

th.e sys tem would l e a d t o s e r i o u s e r r o r s . Fo r example,

4 - Fig . 2 is a s chema t i c r e p r e s e n t a t i o n o f a s e t of

supe rconduc t ing t o r o i d a l f i e l d c o i l s . f o r a f u t u r e tokamak .-

dev ice . Accura t e d e t e r m i n a t i o n s w i l l have t o be made of t h e

.. --. .. . . ' - .radiat . ion ..and. n e u t r o n . f l u x e s . . t o .. dete rmine ... t h e . .hea t l o a d on

t h e c r y o s t a t s , e s p e c i a l l y i n t h e c e n t r a l c o r e r e g i o n where

t h e s p a c e a v a i l a b l e f o r s h i e l d i n g is a t a premium.

P a r t i c u l a r l y . h e r e , where x1 is s m a l l , (comparable t o

c h a r a c t e r i s t i c , mean-free p a t h s ) . Accura te answers . w i l l

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

E q u a t i o n . ( 2 1 ) is s u i t e d t o t h e s o l u t i o n o f t h i s - t y p e of

problem.

In 3 e c t i o n V: a n impor t an t c o n c l u s i o n about d i f f u s i o n

e q u a t i o n s is reached . The d i f f u s i o n e q u a t i o n f o r any

o r t h o g o n a l , axisyrnmetr ic t o r o i d a l c o o r d i n a t e system i n which

t h e r e - a r e no v a r i a t 5 o n s . i n t h e t o r o i d a l d i r e c t i o n , can be

w r i t t e n i n a form which is f o r m a l l y homologous t o t h e "r-0"

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

s t r u c t u r e w a s f i r s t p o i n t e d o u t by Pornraning and S tevens 9

f o r a t o r o i d a l . sys tem' hav ing c i r c u l a r c r o s s s e c t i o n s and

l a t e r ~omraning' ' found the, same ' r e s u l t f o r an o r thogona l

e l l i p t i c a l t o r o i d a l sys tem. Our a n a l y s i s e x t e n d s ' t h i s

r e s u l t t o a g e n e r a l c l a s s o f t o r o i d a l c o o r d i n a t e systems. \

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h. G. M i l l s , e d . , "A Fus ion Power P l a n t " , P r i n c e t o n

Plasma P h y s i c s Labora tory Report MATT-1050 (1974) .

2 ~ . Badger e t . a l . , "Wisconsin Tokamak Reac tor

Design", Vol. 1, Report UWFDM-68, U n i v e r s i t y of isc cons in,

Madison,'. Wisconsin (1974) .

1 . 1. l . Stacey , Jr. e t . a 1 . , Tbkamak Exper imenta l Power

Reac tor S tud ie s .Argonne N a t i o n a l Labora to ry . CTR. T e c h n i c a l , .

Memorandum-75-2 (1975) .

4 ~ e n e r a l Atomic Eng inee r ing S t a f f , Exper imenta l Power,

Reac tor Conceptual Design S tudy , ( P r o g r e s s Report f o r

J u l y 1, 1974 - June 30, 1975) GA-A13534 (1975) .

.p

5 ~ . Greenspan, C a l c u l a t i o n of t h e T r a n s p o r t of N e u t r a l

Atoms i n Highly I o n i z e d ~ l a s m a s Using Neutron T r a n s p o r t

Methods, Nuclear Fusion -9 1 4 771-778. (1974) .

6 ~ . G. 6 1 1 L i g n . n , S, T.!. C r r l n i c k , W. Price, Jr. , '

T r a n s p o r t of N e u t r a l Impur i ty ~ t o m s th rough a Plasma, B u l l .

A m e r . Phys. Soc. 1 1 , - 1 9 853, (1974) .

7 ~ . L. Miller and G. H. Mlley, Monte Ca r lo S imula t ion

of N e u t r a l Beam I n j e c t i o n I n t o Fusion R e a c t o r s , Second

I n t e r n a t i o n a l Conference on Plasma S c i e n c e (Ann Arbor ,

l , f ichigan, May 1975) , t o be p u b l i s h e d .

8 ~ . L. G r a l n i c k , A Tensor Transformat ion Technique f o r

t h e T r a n ~ p o r t E q u a t i o n , (Submi t ted f o r p u b l i c a t i o n i n

Nuc lea r ,Sc ience & E n g i n e e r i n g ) ; P r i n c e t o n Plasma P h y s i c s

k!a..boratory Repor t .MATT-1154 (1975) .

'G. C. Pomraning a n d C. A . S t e v e n s , T r a n s p o r t and - .

D i f f u s i o n E q u a t i o n s i n ' T o r o i d a l Geometry, N u c l e a r s c i e n c e

and E n g i n e e r i n g , - 55, 359-367 (1974) .

'OG. C. Pomraning , E l l i p t i c a l T o r o i d a l Geometry - A

Geometr ic T r a n s p o r t C o r r e c t i o n , . Nuclear Scfence and

E n g i n e e r i n g -9 5 7 188-195 (1975) .

"J. Jung , F i n i t e D i f f e r e n c e ~ ~ u a t i d n s Fo r T r a n s p o r t

Equa t ion I n T o r o i d a l G e u n ~ e t ~ y , Argonne N a t i o n a l Labora tory

CTR T e c h n i c a l EJemorandum .No. 52 (1975) .

1 2 ~ . W. Drawbaugh, The Tensor Form ' o f t h e

Nedtron-Transpui:t Equa t ion w i t h A p p l i c a t i nn t o F i n i t e

D i f f e r e n c i n g , Nuc lea r S c i e n c e and Eng inee r ing "44 58-65 -9

The a u t h o r w i s h e s t o t h a n k h i s many c o l l e a g u e s a t t h e 'i

P r i n c e t o n Plasma P h y s i c s L a b o r a t o r y f o r t h e i r c o n t i n u i n g

1 a s s i s t a n c e . I n , p a r t i c u l a r , I w i s h t o t h a n k Dr ' . W i l l i a m G.

P r i c e , Jr. f o r many h e l p f u l and i l l u m i n a t i n g d i s c u s s i o n s .

T h i s r e s e a r c h . w a s s u p p o r t e d by t h e Energy R e s e a r c h and

Development A d m i n i s t r a t i o n u n d e r c o n t r a c t E(11-1)-3073.

curves of constont 8

curves of constont

Fig. 1. Geometry of the p , 0 , $. axisymmetric~ toroidal coordinate system generated by rotating a nest of smooth curves about the Y axis.

/ . .: -- _-_ _ . . SEGMENTED SUPPORTING CY,LINDER-

. . .:. . . , ". , . ,. ... ., .. ' --- . . , . .

Fig. 2. Toroidal Field Coils proposed for a future Tokamak power reactor (see Reference 1). :