- Email: [email protected]
Numerical simulation of electron-cyclotron heating of a plasma
Volume 32A, number 7
PHYSICS
NUMERICAL ELECTRON-CYCLOTRON
LETTERS
7September1970
SIMULATION HEATING
OF OF A
PLASMA
J. L. S H O H E T *
Service d'lonique G~n~rale , CEN/Saclay , France Received 29 May 1970
A computer model of a plasma has been used to demonstrate electron cyclotron heating. The method first solves Poisson's equation for the self-consistent field and then adds in the external ff field. The particles making up the plasma are then moved by a finite difference scheme.
E x p e r i m e n t a l r e s e a r c h has shown that e f f e c t i v e heating of a p l a s m a is p o s s i b l e by m e a n s of e l e c t r o n c y c l o t r o n r e s o n a n c e [1]. When the rf he a t i ng f i el d is o r i e n t e d p e r p e n d i c u l a r l y to a u n i f o r m dc m a g n e t i c field, highly a n i s o t r o p i c p l a s m a s a r e p ro d u ce d . T h e o r y to e x p la in the he a ti n g m e c h a n i s m is l a r g e l y b a s e d on t h r e e a s s u m p t i o n s . They a r e : (1) At any given instant of t i m e it is equally likely f o r a p a r t i c l e to be d e c e l a r a t e d as it is to be a c c e l e r a t e d . (2) Since a net heating o c c u r s , s o m e s o r t of c o l l i s i o n a l p r o c e s s m u s t " t h r o w " the e l e c t r o n s into the p r o p e r pha se to p r o d u c e a net heating effect. (3) A f t e r h e a t i n g , a n o n - M a x w e l l i a n d i s t r i b u t i o n i s often the r e s u l t , u s u a l l y h a v i n g c o l l e c t i o n s of both "hot" and " c o l d " p a r t i c l e s . R is the p u r p o s e of this note to show that a c o m p u t e r s i m u l a t i o n of a p l a s m a in the r e s o n a n c e (heating) plane will s a t i s f y the above t h r e e a s s u m p t i o n s . The b a s i c method of solution follows the individual m o t i o n s of a c o l l e c t i o n of " s u p e r p a r t i c l e s " in a t w o - d i m e n s i o n a l r e g i o n . S u p e r p a r t i c l e s a r e e s s e n t i a l l y clouds or c o l l e c t i o n s of r e a l p a r t i c l e s , and have the s a m e c h a r g e - t o m a s s r a t i o as r e a l p a r t i c l e s . Both s u p e r - i o n s and s u p e r - e l e c t r o n s will be used. The i n t e r a c t i o n r e g i o n is i m m e r s e d in a dc m a g n e t i c field. The e x t e r n a l ac e l e c t r i c f ie ld ma y be e i t h e r m a d e u n i f o r m o r into a plane s e c tion of the T E l l 1 mode of a c y l i n d r i c a l cavity. Initially, al l p a r t i c l e s a r e d i s t r i b u t e d u n i f o r m l y and e v e n l y o v e r the i n t e r a c t i o n plane. F r o m t h e i r i nit i a l p o s i t i o n s , we m a y obtain the s e l f - c o n s i s t e n t f i e l d of the p a r t i c l e s by s o l v i n g P o i s s o n ' s equation [2]. The i n i t i a l v e l o c i t i e s of the s u p e r * Permanent address: The University of Wisconsin, Madison, Wisconsin, USA.
p a r t i c l e s a r e d i s t r i b u t e d a c c o r d i n g to a Maxw e l l i a n d i s t r i b u t i o n of a given t e m p e r a t u r e . Each s u p e r - p a r t i c l e is a s s u m e d to obey the following finite d i f f e r e n c e equation, obtained f r o m the m o m e n t u m equation mo dv
ot dt - q E
+ q ( v × B o)
which includes a r e l a t i v i s t i c m a s s c o r r e c t i o n in the f a c t o r or: 86+7
Vj+l,x= ~ - ~
-8~+5
and
vj+l,y = 1+82
whe re 2
= vj, X + ~ o A t ~ E j ,
2
x+flVj, y
= Vj, y + - ~ o A t ~ E j , y - f3vj, x m o is the r e s t m a s s , B o is the dc m a g n e t i c f i e l d , c is the v e l o c i t y of light, At is the t i m e s t e p and q is the ch ar g e, v~ x and v~ ,, a r e the known x J, J . ,...7 . and y components of v e l o m h e s . The s u b s c r z p t j + 1 r e f e r s to the v a l u e s of q u a n t i t i e s At units of t i m e l a t e r . The e l e c t r i c field in the above e x p r e s s i o n s is the su m of the s e l f - c o n s i s t e n t and the e x t e r n a l rf e l e c t r i c fields. Once the new v e l o c i t i e s a r e d e t e r m i n e d , the ch an g es in p o s i tion may be c a l c u l a t e d to p r o d u c e the motion. A f t e r e a c h c a l c u l a t i o n , the new a v e r a g e s of p a r 511
Volume 32A, number 7 TOTAL KiI~TIC ENER(Y- JOULES I0"6,
I~.F. CYCLOTI~Ofl HEATING REMOVED
PHYSICS LETTERS
~,
(CASE A)
7September 1970
NUMBER OF PARI"ICLES
R.F, CYCLOTRON HEATIN~ REMOVED
s (CASE S)
10"~ 9.,~
1 0
2
4 6 8 I0 TIME IN CYCLOTRONPEDIOD5~
1'2_
Fig. 1. Upper trace: electrons, lower trace: ions. = 8.39 x 10D Hz, ~ p e = 1.7 x 1 0 7 Hz, ~tDe = 1.2 x °)ce 10-2 M at t= 0, rni= 1836 me, E o = 105 V/m. t i c l e k i n e t i c and p o t e n t i a l e n e r g y a r e c o m p u t e d , and the v e l o c i t y d i s t r i b u t i o n is e x a m i n e d to obs e r v e its changes. The s p e c i f i c p r o b l e m s o l v e d c o n s i s t e d of a c o l l e c t i o n of 900 e a c h , s u p e r - i o n s and s u p e r e l e c t r o n s . The dc m a g n e t i c f ie ld was s e t to 3000 g a u s s and the rf f i e l d a m p l i t u d e was s e t to 105 v o l t / m e t e r . By v a r y i n g the n u m b e r of r e a l p a r t i c l e s p e r s u p e r - p a r t i c l e , the p l a s m a f r e q u e n c y could be changed. R e s u l t s f o r the c a s e when ¢Ope/Wce ~ 0.01 a r e shown in fig. 1. T he total k i n e t i c e n e r g y of the e l e c t r o n s inc r e a s e s with t i m e . The ions do not i n c r e a s e in e n e r g y , and, when the heating is r e m o v e d , the e l e c t r o n k i n e t i c e n e r g y a l s o r e m a i n s constant. The p o s i t i o n s of the ions and e l e c t r o n s m o v e d in a ll d i r e c t i o n s about the i n t e r a c t i o n plane, but no r e g i o n w h e r e p a r t i c l e s s e e m e d to c o l l e c t w a s evident. Fig. 2 shows the e l e c t r o n v e l o c i t y d i s t r i b u t i o n at 3 d i f f e r e n t t i m e s . The solid l i n e s a r e M a x w e l l i a n s , whose t e m j p e r a t u r e s a r e f ix e d by the r e l a t i o n : k T = ½ m (v z} w h e r e (v 2} is the a v e r age of v% C l o s e f i t s to the M a x w e l l i a n s a r e s e e n , but t h e s e b e g i n to d i v e r g e f r o m the c u r v e s as the e l e c t r o n s b e c o m e g r e a t l y heated. The s e l f - c o n s i s t e n t f i el d ( c o l l i s i o n s ) is n e c e s s a r y f o r t h i s r e sult; without it, a d r i f t , but no h e a t in g w a s ob-
512
, ~ 1
9.
3
4
5
~ "~ 6
•
,VELOCITY 7
Fig. 2. Electron velocity distribution, • at time = 0 (T= 10 eV), x after 5½ cycles of heating (T = 24.2 eV), o after 10 cycles of heating and 1 cycle without heating ( T : 56.2 eV). 00~.~= 8.39 x109 Hz, wn, = 1.7 x107 Hz, ~'De=l.2 × 1 0 - 2 M a t t = 0 , m i = l S 3 6 r r ~ . Eo=105V/m. s e r v e d . When the p l a s m a f r e q e n c y was i n c r e a s e d , the v e l o c i t y d i s t r i b u t i o n r a p i d l y b e c o m e s nonMax w el l i an , and the ions a r e heated a s well. U n d e r t h e s e conditions, c o l l i s i o n a l e f f e c t s have begun to d o m i n a t e [3] and a r e o v e r - e m p h a s i z e d f r o m r e a l i t y by about the r a t i o of the n u m b e r of r e a l p a r t i c l e s p e r s u p e r - p a r t i c l e . In o r d e r to o b s e r v e heating at higher p l a s m a f r e q u e n c i e s , it is n e c e s s a r y to i n c r e a s e the n u m b e r of s u p e r p a r t i c l e s , t h e r e b y l o w e r i n g the n u m b e r of r e a l p a r t i c l e s p e r s u p e r - p a r t i c l e f o r a g i v en v al u e of p l a s m a f r e q u e n c y . The m o d e l used r e m a i n s valid as long as the bulk of the v e l o c i t y d i s t r i b u t i o n does not b e c o m e highly r e l a t i v i s t i c . When this o c c u r s , r a d i a t i o n e f f e c t s will d o m i n a t e the p a r t i c l e motion, and it is e x p e c t e d that the c o r r e c t solution will d e v i a t e f r o m that found by this method.
References [1] M. C. Beeker et al., Nuel. Fusion 345 (1962). [2] R.W. Hoekney, Phys. Fluids 9 (1968) 1826. [3] R.W. Hockney, Proc. Conf. of Computational Physics, Culham (1969).
PHYSICS
NUMERICAL ELECTRON-CYCLOTRON
LETTERS
7September1970
SIMULATION HEATING
OF OF A
PLASMA
J. L. S H O H E T *
Service d'lonique G~n~rale , CEN/Saclay , France Received 29 May 1970
A computer model of a plasma has been used to demonstrate electron cyclotron heating. The method first solves Poisson's equation for the self-consistent field and then adds in the external ff field. The particles making up the plasma are then moved by a finite difference scheme.
E x p e r i m e n t a l r e s e a r c h has shown that e f f e c t i v e heating of a p l a s m a is p o s s i b l e by m e a n s of e l e c t r o n c y c l o t r o n r e s o n a n c e [1]. When the rf he a t i ng f i el d is o r i e n t e d p e r p e n d i c u l a r l y to a u n i f o r m dc m a g n e t i c field, highly a n i s o t r o p i c p l a s m a s a r e p ro d u ce d . T h e o r y to e x p la in the he a ti n g m e c h a n i s m is l a r g e l y b a s e d on t h r e e a s s u m p t i o n s . They a r e : (1) At any given instant of t i m e it is equally likely f o r a p a r t i c l e to be d e c e l a r a t e d as it is to be a c c e l e r a t e d . (2) Since a net heating o c c u r s , s o m e s o r t of c o l l i s i o n a l p r o c e s s m u s t " t h r o w " the e l e c t r o n s into the p r o p e r pha se to p r o d u c e a net heating effect. (3) A f t e r h e a t i n g , a n o n - M a x w e l l i a n d i s t r i b u t i o n i s often the r e s u l t , u s u a l l y h a v i n g c o l l e c t i o n s of both "hot" and " c o l d " p a r t i c l e s . R is the p u r p o s e of this note to show that a c o m p u t e r s i m u l a t i o n of a p l a s m a in the r e s o n a n c e (heating) plane will s a t i s f y the above t h r e e a s s u m p t i o n s . The b a s i c method of solution follows the individual m o t i o n s of a c o l l e c t i o n of " s u p e r p a r t i c l e s " in a t w o - d i m e n s i o n a l r e g i o n . S u p e r p a r t i c l e s a r e e s s e n t i a l l y clouds or c o l l e c t i o n s of r e a l p a r t i c l e s , and have the s a m e c h a r g e - t o m a s s r a t i o as r e a l p a r t i c l e s . Both s u p e r - i o n s and s u p e r - e l e c t r o n s will be used. The i n t e r a c t i o n r e g i o n is i m m e r s e d in a dc m a g n e t i c field. The e x t e r n a l ac e l e c t r i c f ie ld ma y be e i t h e r m a d e u n i f o r m o r into a plane s e c tion of the T E l l 1 mode of a c y l i n d r i c a l cavity. Initially, al l p a r t i c l e s a r e d i s t r i b u t e d u n i f o r m l y and e v e n l y o v e r the i n t e r a c t i o n plane. F r o m t h e i r i nit i a l p o s i t i o n s , we m a y obtain the s e l f - c o n s i s t e n t f i e l d of the p a r t i c l e s by s o l v i n g P o i s s o n ' s equation [2]. The i n i t i a l v e l o c i t i e s of the s u p e r * Permanent address: The University of Wisconsin, Madison, Wisconsin, USA.
p a r t i c l e s a r e d i s t r i b u t e d a c c o r d i n g to a Maxw e l l i a n d i s t r i b u t i o n of a given t e m p e r a t u r e . Each s u p e r - p a r t i c l e is a s s u m e d to obey the following finite d i f f e r e n c e equation, obtained f r o m the m o m e n t u m equation mo dv
ot dt - q E
+ q ( v × B o)
which includes a r e l a t i v i s t i c m a s s c o r r e c t i o n in the f a c t o r or: 86+7
Vj+l,x= ~ - ~
-8~+5
and
vj+l,y = 1+82
whe re 2
= vj, X + ~ o A t ~ E j ,
2
x+flVj, y
= Vj, y + - ~ o A t ~ E j , y - f3vj, x m o is the r e s t m a s s , B o is the dc m a g n e t i c f i e l d , c is the v e l o c i t y of light, At is the t i m e s t e p and q is the ch ar g e, v~ x and v~ ,, a r e the known x J, J . ,...7 . and y components of v e l o m h e s . The s u b s c r z p t j + 1 r e f e r s to the v a l u e s of q u a n t i t i e s At units of t i m e l a t e r . The e l e c t r i c field in the above e x p r e s s i o n s is the su m of the s e l f - c o n s i s t e n t and the e x t e r n a l rf e l e c t r i c fields. Once the new v e l o c i t i e s a r e d e t e r m i n e d , the ch an g es in p o s i tion may be c a l c u l a t e d to p r o d u c e the motion. A f t e r e a c h c a l c u l a t i o n , the new a v e r a g e s of p a r 511
Volume 32A, number 7 TOTAL KiI~TIC ENER(Y- JOULES I0"6,
I~.F. CYCLOTI~Ofl HEATING REMOVED
PHYSICS LETTERS
~,
(CASE A)
7September 1970
NUMBER OF PARI"ICLES
R.F, CYCLOTRON HEATIN~ REMOVED
s (CASE S)
10"~ 9.,~
1 0
2
4 6 8 I0 TIME IN CYCLOTRONPEDIOD5~
1'2_
Fig. 1. Upper trace: electrons, lower trace: ions. = 8.39 x 10D Hz, ~ p e = 1.7 x 1 0 7 Hz, ~tDe = 1.2 x °)ce 10-2 M at t= 0, rni= 1836 me, E o = 105 V/m. t i c l e k i n e t i c and p o t e n t i a l e n e r g y a r e c o m p u t e d , and the v e l o c i t y d i s t r i b u t i o n is e x a m i n e d to obs e r v e its changes. The s p e c i f i c p r o b l e m s o l v e d c o n s i s t e d of a c o l l e c t i o n of 900 e a c h , s u p e r - i o n s and s u p e r e l e c t r o n s . The dc m a g n e t i c f ie ld was s e t to 3000 g a u s s and the rf f i e l d a m p l i t u d e was s e t to 105 v o l t / m e t e r . By v a r y i n g the n u m b e r of r e a l p a r t i c l e s p e r s u p e r - p a r t i c l e , the p l a s m a f r e q u e n c y could be changed. R e s u l t s f o r the c a s e when ¢Ope/Wce ~ 0.01 a r e shown in fig. 1. T he total k i n e t i c e n e r g y of the e l e c t r o n s inc r e a s e s with t i m e . The ions do not i n c r e a s e in e n e r g y , and, when the heating is r e m o v e d , the e l e c t r o n k i n e t i c e n e r g y a l s o r e m a i n s constant. The p o s i t i o n s of the ions and e l e c t r o n s m o v e d in a ll d i r e c t i o n s about the i n t e r a c t i o n plane, but no r e g i o n w h e r e p a r t i c l e s s e e m e d to c o l l e c t w a s evident. Fig. 2 shows the e l e c t r o n v e l o c i t y d i s t r i b u t i o n at 3 d i f f e r e n t t i m e s . The solid l i n e s a r e M a x w e l l i a n s , whose t e m j p e r a t u r e s a r e f ix e d by the r e l a t i o n : k T = ½ m (v z} w h e r e (v 2} is the a v e r age of v% C l o s e f i t s to the M a x w e l l i a n s a r e s e e n , but t h e s e b e g i n to d i v e r g e f r o m the c u r v e s as the e l e c t r o n s b e c o m e g r e a t l y heated. The s e l f - c o n s i s t e n t f i el d ( c o l l i s i o n s ) is n e c e s s a r y f o r t h i s r e sult; without it, a d r i f t , but no h e a t in g w a s ob-
512
, ~ 1
9.
3
4
5
~ "~ 6
•
,VELOCITY 7
Fig. 2. Electron velocity distribution, • at time = 0 (T= 10 eV), x after 5½ cycles of heating (T = 24.2 eV), o after 10 cycles of heating and 1 cycle without heating ( T : 56.2 eV). 00~.~= 8.39 x109 Hz, wn, = 1.7 x107 Hz, ~'De=l.2 × 1 0 - 2 M a t t = 0 , m i = l S 3 6 r r ~ . Eo=105V/m. s e r v e d . When the p l a s m a f r e q e n c y was i n c r e a s e d , the v e l o c i t y d i s t r i b u t i o n r a p i d l y b e c o m e s nonMax w el l i an , and the ions a r e heated a s well. U n d e r t h e s e conditions, c o l l i s i o n a l e f f e c t s have begun to d o m i n a t e [3] and a r e o v e r - e m p h a s i z e d f r o m r e a l i t y by about the r a t i o of the n u m b e r of r e a l p a r t i c l e s p e r s u p e r - p a r t i c l e . In o r d e r to o b s e r v e heating at higher p l a s m a f r e q u e n c i e s , it is n e c e s s a r y to i n c r e a s e the n u m b e r of s u p e r p a r t i c l e s , t h e r e b y l o w e r i n g the n u m b e r of r e a l p a r t i c l e s p e r s u p e r - p a r t i c l e f o r a g i v en v al u e of p l a s m a f r e q u e n c y . The m o d e l used r e m a i n s valid as long as the bulk of the v e l o c i t y d i s t r i b u t i o n does not b e c o m e highly r e l a t i v i s t i c . When this o c c u r s , r a d i a t i o n e f f e c t s will d o m i n a t e the p a r t i c l e motion, and it is e x p e c t e d that the c o r r e c t solution will d e v i a t e f r o m that found by this method.
References [1] M. C. Beeker et al., Nuel. Fusion 345 (1962). [2] R.W. Hoekney, Phys. Fluids 9 (1968) 1826. [3] R.W. Hockney, Proc. Conf. of Computational Physics, Culham (1969).