3.1 A c c e s s ib i l it y o f slo w waves t o th e lo w e r h ybr id resonance In th e c o ld plasma a p p r o x im a tio n , two waves p ro p a g a te , one o f w h ic h , th e s lo w mode, has a resonance a t p o in t s where plasma p a ra m e te rs are such t h a t
fT Q.
= r n r ^ “lh
pe e
In a c o n s id e r ab le number o f e x p e rim e n ts th e slow mode i s used f o r h e a tin g and f o r c u r r e n t d r i v e .
As i s w e l l known, ( S t i x , 1962, p 64; G o la n t, 1972) t h e r e i s a c r i t i c a l v a lu e , n ^ , o f th e p a r a l l e l r e f r a c t i v e in d e x f o r w h ic h , i n s lab ge o m e try, th e f a s t and slo w waves have th e same p e r p e n d ic u la r r e f r a c t i v e in d e x a t some p o i n t between th e plasma edge and the reso nan ce . For v a lu e s o f n „ c lo s e t o n^ we may e x p e c t p a r t i a l mode c o n v e r s io n , w it h e nergy i n c i d e n t i n one mode be in g s p l i t between th e two modes a f t e r p a s s in g th ro u g h th e r e g io n where th e two r o o ts o f h | a re a lm o s t c o i n c i d e n t . T h is phenomenon d e te rm in e s th e s h a rpn ess o f th e t r a n s i t i o n between v a lu e s o f n„ f o r w hich th e lo w e r h ybr id resonance i s a c c e s s ib le t o th e slow wave and th o s e f o r w hich an i n c i d e n t slow wave i s r e f l e c t e d on th e f a s t mode.
Our aim i n t h i s c h a p te r i s t o e s tim a te th e range o f n „ o v e r w hich such an e f f e c t i s s i g n i f i c a n t . F i r s t l y , th e a c c e s s ib i l it y c r i t e r i o n p e r t a i n i n g t o a plasma i n a u n ifo r m m a g n e tic f i e l d i s
r e - d e r i v e d . N e x t, we d is c u s s th e g e n e ra l n a tu re o f th e d is p e r s io n r e l a t i o n i n th e w -k ^ p la n e and f i n d t h a t t h i s mode c o u p lin g problem i s d i f f e r e n t i n c h a r a c te r from cases p r e v io u s ly c o n s id e re d by C a irn s and Lashm ore-D avies (1 9 8 3 ). An e x p a n s io n about a sa d d le p o i n t o f th e d is p e r s io n r e l a t i o n i s then used t o c o n s t r u c t a model second o rd e r O .D .E ., W eber's e q u a tio n , d e s c r ib in g th e c o u p lin g p ro c e s s . The e q u a tio n has a f i r s t i n t e g r a l w hich i s i n t e r p r e t e d as a c o n s e r v a tio n la w , a llo w in g us to i d e n t i f y a m p litu d e s a p p e a rin g i n th e c o n n e c tio n fo rm u la w ith i n c i d e n t , t r a n s m it t e d and c o n v e rte d wave a m p litu d e s . In t h i s way, tr a n s m is s io n and c o n v e rs io n c o e f f i c i e n t s a re o b ta in e d . F i n a l l y we a p p ly th e g e n e ra l th e o r y t o cases i n v o l v i n g v a r io u s s p a t i a l g r a d ie n t s . The r e s u l t f o r a plasma w it h a l i n e a r d e n s ity p r o f i l e i n a u n ifo r m f i e l d i s g iv e n , w h ile th e i n c l u s i o n o f a
m a g n e tic f i e l d g r a d ie n t i n th e c a l c u l a t i o n s h i f t s th e mode c o n v e rs io n re g io n tow ard s th e plasma edge.
The q u e s tio n o f o u r somewhat ad hoc a r r i v a l a t a model second o rd e r O.D.E. f o r t h i s pro ble m i s addressed i n C hapter 4.
We c o n s id e r a t o k a m a k - lik e c o n f i g u r a t i o n i n w hich th e s p a t i a l g r a d ie n ts a re take n to be p e r p e n d ic u la r t o th e e x t e r n a l m agn etic f i e l d d i r e c t i o n . Then l o c a l l y , n „ , th e p a r a l l e l r e f r a c t i v e in d e x a lo n g th e f i e l d d i r e c t i o n may be ta k e n t o be c o n s ta n t w h ile n, v a r i e s . Ray t r a c i n g s t u d ie s i n t o r o i d a l plasma ( B o n o li and O t t , 1982) have shown t h a t n „ changes a lo n g th e ra y due to t o r o i d a l e f f e c t s , b u t t h i s can be n e g le c te d i n th e s tu d y o f th e l o c a l i s e d p ro c e s s o f mode c o n v e r s io n , rem em bering, how ever, t h a t th e n „ o f a wave when i t undergoes c o n v e r s io n may n o t be t h a t w it h w hich i t was la u n c h e d .
- 37.