..... - -,, ,

INSTITUTE Felt FUSION STUDIES

T H E U N I V E R S I T Y OF T E X A S

MASTER .

DO NOT MlCROFl COVER

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

- - - - -- - - - - - - -.- - - - - - J DOE/ET-53088-68 IFSR #68

IMPLICIT-PARTICLE SIMULATION OF MAGNETIZED PLASMAS

D. C. Barnes, T. Kamimura, J.-N. Leboeuf; T. Tajima

Institute for Fusion Studies The University of Texas at Austin

Austin, Texas 78712

September 1982

1 . 'DIECLAIMER \

PORTIONQF THIS RXFORT ARF. ILL)?GIsLE. 2% bas been reproduoed from the b=Gilablg '

ooPY t o I?emt Ehe broadest pos s ib l e avail- e b l l i t , ? ~ +b .

DISTRIBUTION OF THIS DOCUMENT 15 UNLIMITED %$

Implicit-Particle Simulation of

Magnetized Plasmas

* D. C. Barnes, T. Kamimura , J-N. Leboeuf,-To Tajima

Institute for Fusion Studies University of Texas Austin, Texas 78712

* Permanent Address: Institute of Plasma Physics

Nagoya Urilversi ty Nagoya, 464 JAPAN

Manuscript Pages : 43 Figures: 9 Tables: NONE

P r o p o s e d Running H e a d e r :

Please send proofs to:

I m p l i c i t P a r t i c l e S i m u l a t f o n

Dr. Daalel. C. Barnes I inst i t i~te c u r P i i s ion S t ~ l d i e s RLM 11.222 U n i v e r s i t y o f Texas A u s t i n , TX 78712

111t r o d r ~ c t ion

SYMBOLS

( i n o r d e r o f o c c u r r e n c e )

S e c t i o n I1

. .

S e c t i o n 111

Appendix

Greltk heti:i

Greek omega

G r e e k . c a p i t a 1 omega

Greek a l p h a

Greek c a p i t a l d e l t a , t

. p o s i t i o n x , sub ' j

p o s i . t i o n v , s u b j . . .

Greek p h i . . ' .

i n v e r t e d Greek c a p i t a l d e l t a

G-reek rlio

d , e n s i t y ' n , s u b Greek a l p h a .

Greek c a p i t a l d e l t a

Greek gamma

s u b p e r p e n d i c u l a r

s u b p a r a l l e l

Greek d e l t a

v e l o c i t y v , s u b c a p i t a l T

Greclc lambda , s u b c a p i t a l D

Greek E p s i l o n

Greek x i

Greek p i

C , s u b S

k , s u b c a p i t a l H

Greek c a p i t a l t h e t a

Greek p s i

Greek t a u

Abstract

A second-order accurate, direct method for the simulation of

magnetized, multi-di~nensional plasmas is developed. A time dccentered

particle push is combined with the direct method for implicit plasma

simulation to include finite sized particle effects in an absolutely

stable algorithm. A simple iteration (renormali~ed Poisson equation)

i s used to solve the field corrector equation. Details of the

two-dimensional, electrostatic, constant magnetic field, periodic case

are glven. Numerical results for ion-acoustic fluctuations and for an

unstable gravitational interchange confirm the accuracy an11 efficacy of

the method applied to low-frequency plasma phenomena.

-3-

I . I n t r o d u c t i o n -- C u r r e n t plasma conf inement e x p e r i m e n t s p rov ide many examples of

low-frequency phenomena i n which k i n e t i c e f f e c t s a r e a l l i m p o r t a n t .

D r i f t f r equency f l u c u a t i o n s i n a low-B plasma a r e c o r r e c t l y d e s c r i b e d

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

, e l e c t r o n motion. A s i m i l a r s i t u a t i o n o c c u r s i n c o n s i d e r i n g

c o l l i s i o l l l e s s t e a r i n g modes a t ' e i t h e r macroscdp ic o r m i c r o s c o p i c

s c a l e s . For s l i g h t l y h i g h e r f r e q u e n c y ' modes, ' t h e i n t e r a c t i o n of

k i n e t i c e f f e c t s w i t h MHD i n s t a b i l i t i e s [nust be ' c o n s i d e r e d t o . s t u d y

s t a b i l i t y of t h e Bumpy Torus o r ~ a n d e m . 0 ~ F i e l d Keversed N i r r o r plasma.

These m o d i f i c a t i o n s t o MHD behav io r may a l s o be i m p o r t a n t i n Tokamak

conf inement where k i n e t i c e f f e c t s o n - b a l l o o n i n g modes may a l l o w h i g h e r

B o p e r a t i o n . . .

I n t h i s paper a r e c e n t l y . d e v e l o p e d a l g o r i t h m f o r multi-d:mensional

plasma s i m u l a t i o n of such low-f requency phenomena is d e s c r i b e d . These

phenomena a r e t h o s e whose c h a r a c t e r i s t i c f r e q u e n c y (mode f r e q u e n c y ) w

. i s s m a l l e r t h a n e i t h e r . t z h e plasma f r e q u e n c y wa o r c y c l o t r o n f requency

Ra f o r one o r . s e v e r s 1 s p e c i e s a (w<<w;,R,).

The , a l g o r i t h m d e s c r i b e d , d i f f e ' r s from d o n v e n t i o n a i p a r t i c l e

s i m u l a t i o n methods [ I ] , which a c c u r a t e l y a n d e f f i c i e n t l y r e p r e s e n t

s h o r t t ime s c a l e plasma phenomena, i n t h a t t ime s t e p s A t much l a r g e r

t h a n o i l o r a r e employed. I n t h i s way a p r a ~ t i e a l m~mber of t ime

s t e p s can r e p r e s e n t t h e ' l o n g . t ime s c a l e s of i n t e r e s t

(w,At, R a A t > > l , w A a l ) .

I n c o n t r a s t t o s i n g l e o r m u l t i p l e f l u i d plasma s i m u l a t i o n s [ Z ] ,

which e f f e c t . i v e l y r e p r e s e n t phenomena o f conf ined plasmas on v e r y long

time s c a l e s by moment e q u a t i o n s , t h e a l g o r i t l i ~ n d e s c r i b e d h e r e r e t a i n s .

a l l low f r e q u e n c y k i n e t i c e f f e c t s by a c c u r a t e l y f o l l o w i n g s i n g l e

p a r t i c l e o r b i t s .

The p o s s i b i l i t y of u s i n g i m p l i c i t f i e l d computa t ions , i n which t h e

p a r t i c l e s a r e a c c e l e r a t e d i n on e l e c t roloagne t i c f i e l d de te rmined .by

t i m e advanced p a r t i c l e d a t a , was p o i n t e d o u t by Langdon 131. R e c e n t l y ,

s e v e r a l a u t l l o r s [4-71 have d e s c r i b e d i n g e n i o u s p r a c t i c a l a l g o r i t h m s f o r

i m p l i e i t f i e l d plasma s i m u l a t i o n f o r t h e one-dimensional ~ l ~ c t r ~ g t a t i c

plasma and p r e s e n t e d r e s u l t s f o r w,A t>>l . B r a c k b i l l atid Fors1,vnd [ 8 ] :

have ex tended t h e s e methods t o t h e two-dimensional e l e c t r o m a g n e t i c '

c a s e , u s i n g t h e moment e q u a t i o n method [ 4 1 .

The a l g o r i t h m d e s c r i b e d h e r e d i f f e r s f rom e a r l i e r work i n , s e v e r a l

i m p o r t a n t r e s p e c t s . F i r s t , t h e d i f f e r e n c e e q u a t i o n s of motion a r e

second-order a c c u r a t e f o r small wbt , and t h e numer ica l damping of sucl~

1 .0~ - f requency modes i .s much s m a l l e r ' t l ian t h a t f o r . f i r s t -o rde r a c c u r a t e

methods. Second, t h e i m p l i c i t e q u a t i o n s are Eormula,ted usil'lg the

d i r e c t method [6,71 w i t h s i l n p l i f i e d d i f f e r e n c i n g . T h i s a l l o w s t h e

c o n s i s t e n t i n c l u s i o n of f i n i t e s i z e p a r t i c l e s s o t h a t l o n g wavelength

~nadcs . a r e v e r y a c c u r a t e l y r e p r e s e n t e d with r e l a t i v e l y f e i ~ p a r t i c l e s ,pe r

c e l l . T h i r d , a s i m p l e method [9] f o r s l m u l i ~ t i o n of s t r o r ~ g l y magnetized

p lasmas (R,A t>>l) i s combined w i t h the i m p l i c i t e l e c t r o s t a t i c f i e l d

method. F i n a l l y , a v e r y s i m p l e t e c h n i q u e f o r i t e r a t i v e l y s o l v i n g t h e

i m p l i c i t f i e l d c o r r e c t o r e q u a t i o n based on t h e renormal ized p a r t l c l e

code method [ l o ] i s implemented f o r t h e two-dimensional magnet ized

plasma .

-5-

A.lthough the method described here may be applied to fullly

electroinagnetic plasma simulation [ i l l and may include firiite

gyroradius'effects for arbitrarily large time steps 19,121, the present

paper is limited to a description of the simplest multi-dimensional

algorithm. For this purpose, the electrostatic approximation is made,

the static magnetic field is taken.to be uniform, and a11 effects of

finite gyroradius .are neglected. In addition, the spatial boundaries

are taken as periodic throughout.

The plan of 'the remainder of the paper is as follows., In

Section 2 the implicit plasma method is developed, and several issues

associated with accuracy, numerical damping, and implementation of this

algorithm are addressed. In Section 3 numerical results for

ion-acoustic fluctua,tions in a therma1,plasma and for a gravitationally

driven interchange mode are presented and the accuracy'of these results

discussed. Finally, in Section 4 conclusions are drawn and future

directions for extension are outlined.

11. Numerical Method

a . Normal ized D i f f e r e n c e qua t i o n s

Fo l lowing t h e f i n i t e s i z e p a r t i c l e method [ 1 , 3 ] , a c o l . l i s i o n l e s s

plasma is r e p r e s e n t e d by t h e ' m o t i o n of a l a r g e number of

s u p e r p a r t i c l e s , which move i n accordance w i t h t h e e q u a t i o n s o f ' motion

(rationalized MKS u n i t s a r e employed h e r e )

f o r j=1,. .. , N o , a n d t h e s p e c i e s i n d e x a=e,i . I n Eq. 1 t h e s u p e r d o t

i n d i c a t e s . t h e u s u a l tirue d e r i v a t i v e and % i s tlie e l e c t r i c a c c e l e r a t i o r ~

F e l t by t h e p a r t i c l e of s p e c i e s a.

where h i s t h e p a r t i c l e shape . Also i n Eq. 1 , !22 = (q/m)&? i c t h e

c y c l o t r o n Frequency ( v e c t o r ) . Although c, may depend on 5 and t i n t h e

g e n e r a l c a s e , 02 i s t a k e n as a s t a t i c c o n s t a n t i n t h e p r e s e n t paper.

'l'he s e l f - c o n s i s t e n t plasma model i s completed by t h e a p p r o p r i a t e

Maxwell e q u a t i o n s . For t h e e l e c t r o s t a t i c approx imat ion t a k e n h e r e ,

t h e s e a r e

To s o l v e t h e f i e l d Eq. 3 , a g r i d i s i n t r o d u c e d i n t h e u s u a l way

a n d p a r t i c l e q u a n t i t i e s a r e i n t e r p o l a t e d t o and from t h i s g r i d . Thus,

f o r a s e t o f g r i d p o i n t s x t h e d i f f e r e n c e o p e r a t o r s i n Eq. 3 a r e -g

evalua ted by f i n i t e d i f f e r e n c e ' s o r f i n i t e F o u r i e r t r a n s f o r m s ( d e t a i l s

a r e g i v e n below). The p a r t i c l e shape t l i s i n t e r p o l a t e d a s

tl(x -x.) - Z , h ( x -x r ) i ( x .-zj) , f o r some i n t e r p o l a t i o n f u n i t i o n i. -g -J g -g -g -&

T h i s d e t e r m i n e s t h e "raw" c h a r g e d e n s i r y

t h e sum t a k e n o v e r a l l p a r t i c l e s o f s p e c i e s a . I n terms of n,, p is

. . g i v e n by a c o n v o l u t i o n w i t h h , , . .

The p a r t i c l e f o r c e is i . n t e r p o l a t e d by t h e same r u l e s

ti v

s o t h a t p a r t i c l e s e l f f o r c e i s avo ided .

The interpolation function i may be chosen in many WAYS. For

c?xarnple, i may be chosen to give a subtracted dipole [ 1 3 ] expansion of

' h. However, the implicit time differencing algorithm adopted below

reqtlires derivatives of h to be computed. Thus, i is typically chosen

to be at least once differentiable. Linear or quadratic splines [ 1 4 ]

have been used for i to obtain the numerical results of Sectton 4.

To obtain the appropriate equations for low-frequency simuli~tion,

Eqs. 1-3 are normalized and differenced in time., Normalized variables - (denoted by carats) are defined in the conventional way as: x j = E~/A;

a A 8.

t = oeo t; At = oeo At; = Q2At; where A is the size of the N, by N~

cells' and At the time step of' the simulation, and where no is the mean

electron density of the plasma with associated plasma frequency'weo.

Denoting time levels by superscripts, the normalized difference

equations corresponding to Eq. 1 are written as a niodi.Eied leap-frog

sclleme. Dropping the carats henceforth for notational conven,ience,

where the exact definitions of and v?* are yet to be given. -J

The field Eqs. 3-5 become

where H ~ * h a s been w r i t t e n f o r N*H*.

b. Decen te red Equnt.ions f o r Q,>>l

. C o n s i d e r f i r s t t h e l a s t te rm I n Eq. 7b. Let vn* be d e f i n e d by - l i n e a r i n t e r p o l a t i o n ,

For y = 0 (and = g), Eq. 7b becomes t h e u s u a l t ime centerer i 1 . o r e n t ~

f o r c e ' e q u a t i o n f o r advanc ing p a r t i c l e d a t a [ 1 5 ] ? Thi:; d i f f e r e n c e

scheme i.s a b s o l u t e l y s t a b l e and second-order a c c u r a t e f o r Ra 1.

For R, >> 1 , Eq. 7 no l o n g e r c o r r e c t l y r e p r e s e n t s gyrolnotion.

R a t h e r , t h e c y c l o t r o n f requency a l i a s e s t o t h e Nyquis t f r e q u e n c y ,

u o = n / A t , and t h e o r b i t d e g e n e r a t e s t o .two p o s i t i o n s a l t e r n a t e l y ., .

assumed on a l t e r n a t e t ime s t e p s . I n t h i s c a s e , y > 0 i s chosen t o damp

t11 i's n l i a s e d motion by d e c e n t e r i n g t h e d . i f f e r e n c e ' equat i .ons . A'L;!.'

e f f e c t s of g y r a t i o n a r e t h e n e l i m i n a t e d from t h e ~ n o d e l , and , as is

shown below, Eq. 7 d e s c r i b e s the motion of t h e ~ e r n ' ~ ~ r ~ r a d i . t l s g u i d i n g

c e n t e r . It i s , s t r a i g h t f o r w a r d t o i n c l u d e f i n i t e g y r o r a d l u s e f f e c t s by

add ing a d d i t i o n a l f o r c e s t o Eq. 7 and s i m u l t i n e o u s l y modifying t h e

c h a r g e - c u r r e n t s o u r c e e q u a t i o n s [9., 12,161. Such a n e x t e n s i o n of t i l e

p r e s e n t work w i l l be d e s c r i b e d i n a f u t u r e p u b l i c a t i o n .

For row-f requency phenomena ( , t ) s l o w l y v a r y i n g ) , ~ q . 7

becomes 3 d i f f e r e n c e approximat i o n to t h e L o r e n t z f o r c e e q u a t i o n .

Northri tp [17] . h a s shown t h a t t h i s d i f f e r e n t i a l e q u a t i o n . d e s c r i b e s t h e

mot ion of a g u i d i n g c e n t e r when a p p r o p r i a t e gyroaveraged f o r c e s a r e

i n c l u d e d on t h e r i g h t . ' The d r i f t mot ions r e t a i n e d when ' t h e s e

a d d i t i o n a l f o r c e s a r e o m i t t e d a r e t h o s e which p e r s i s t a t z e r o

g y r o r a d i u s . These a r e t h e e l e c t r i c d r i f t and t h e i n e r t i a l d r i f t s ; o f

which the two most s i g n i f i c a n t a r e the. c e n t r i f u g a l and p o l a r i z a t i o n

d r i f t s . . ~.

The e f f e c t . o f i n t r o d u c i n g damping f o r y > O is t o . s e l e c t t h a t

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

c o n d i t i o n s a p p r o p r i a t e f o r t h e d r i f t mot ion . s o l u t i o n . [16,17]. .

Low-frequency motion i s l i t t l e a f f e c t e d by t h i s damping and, w h i l e ' t h e

r e s u l t i n g d i f f e r e n c e approx imat ion i s f o r m a l l y o n l y f i r s t - o r d e r

A i i t i c a L e Cur Y)O, 'y may be chosen s o s m a l l t h a t seconrl-nrdPr ' = c u r a c y . .

i s e f f e c t i v e l y a t t a i n ~ d . . .

Indeed , expanding i n a Tay lor s e r i e s i n t , Eqs. 7 , 9 g i v e a

d i f f e r c n t i a l e q u a t i o n whose s o l u t i o n f o r Q,>>l i s

Thus, second o r d e r a c c u r a c y i s a t t a i n e d f o r modes such t h a t

. . . .

As Llie r e s u l t s of S e c t i o n 3 conf i rm, y ,may be chosen. ve,ry - ,small . -

( t y p i c a l l y y-10-l) s o t h a t t h e c o n d i t i o n of. Eq. 11 i s . e a s i l y s a t i s f l e d .

c. Predictor-Corrector.FIethod f o r @,>>I

Cons ide r n e x t t h e e l e c t r i c a c c e l e r a t i o ~ l te'rm on t h e r i g h t of

Eq. 7b. T h i s term r e p r e s e n t s t h e s e l f - c o n s i s t e n t c o l l e c t i v e

i n t e r a c t i o n of a l l s i m u l a t i o n p a r t i c l e s t h r o u g l ~ t h e Maxwell Eqs. 3. It

i s ~ r i s e f u l t o r e p l a c e t h i s c o m p l i c a t e d i n t e r a c t i o n by a s i l n p l f f i e d model

s o t h a t t h e i m p l i c i t t ime d i f f e r e n c i n g scheme i s developed i n a

more t r a n s p a r e n t manner. Accord ing ly , Eq-. 7 i s r e p l a c e d by t h e

harmonic o s c i l l a t o r sys tem

where w , i s t h e ( t r u e ) mode f r e q u e n c y ( t i m e s A t ) and the ' s u p e r b a r

r e p r e s e n t s a n , a s y e t u n s p e c i f i e d , combina t ion o f v a r i o u s time l e v e l s

n o f x . For low-frequency plasma s i m u l a t i o n , ';;" shou.Sd be chosen s o t h a t

2 - xn f o r small f r e q u e n c i e s b u t s o t h a t tlie u s u a l s t a b i l i t y . c o n d f t i o n

lw,l < 2 i s avo ided . I n t h i s way, d e s i r a b l e p r o p e r t i e s a£ t h e u s u a l

e x p l i c i t l e a p f r o g scheme (yn = xn) a r e r e t a i n e d f o r t l ~ e modes of

i n t e r e s t . On t h e o t h e r hand, inodes f o r which urn -t (which a r e no t

r e p r e s e n t a b l e on t h e ' time g r i d cl loscn) st lould be s t r o n g l y damped a i d

- removed from t h e system. The optimum f i l t e r i n g of xn t o xn would t h u s ,

. . 'he s u c h t h a t ;"'= xn f o r loml < om, and s u c h t h a t t h e sys tem response

would he z e r o f o r lw,l > w,,.

T h i s may be approx imated by a n i m p l i c i t scheme i n whlch ;;". depends

on t ime advanced i n f o r m a t i o n . Thus, 2 s h o u l d depend on xn+l ( 1 t i s

shown below t h a t t h i s , i s n e c e s s a r y . f o r s t a b i l i t y a s U r n -+ -.). The

d e s i r e d f i . l t e r i n g of ,xn i s -accomplished by a 1 i n e n r . f i l t e r w i t h

f i n i t e number o f p o l e s . T h u s , . .

'L'o proceed Eqs. 7 ' , 1 1 are a n a l y z e d by the, Z transfarm. T t i s

assumed t h a t ( x " , v " + ~ / ~ ) = z n ( x , v ) , where Z = exp(- iwbt) and w i s the

normal mode f r e q u e n c y ' of t h e d i f f e r e n c e sys tem. The f o l l o w i n g

d i s p e r s i o n . r e l a t i o n > . r e s u l t s : . . . . .

. . .. .

(2-112 + 0; Z f ( Z ) = 0 (12) .

- . . . . . , . . . . -

In Eq . 1 2 , f ( Z ) i s t h e t r a n s f e r C11nctio.n .E,rom x t.o x. Suppose, .wl_t.l~out . >

. . . . . . . . _ . I . . .

l o s s of g e n e r a l i t y , t h a t . I . = J.. . E,q . 12 becomes the. ~po lynomina l , , .

- .

equa t i o n . . . .

.

I n s p e c t ' i o n of Eq. 1 3 g i v e s t h e r e q u i r e m e n t s f o r t h e undetermined ' : ' .. . . . . ' . .

c o e E f i c i . e n t s . Fnr 111.. = 0, there ape 1..1-2 L.OUL;S alld clle a ' s must , be m

chosen s o t h e z e r o s of t h e f i r s t ' f a c t o r i n Eq. 13 l i e i n s i d e t h e u n i t

c i r c l e .

As W; + " , t h e r e a re I+2 r o o t s a t t i h i t s Z provldcri a 8 0. If c

vanishes , a t l e a s t one of t h e t o o t l o c i ex tends t o and produces an

uns table roo t f o r w m s u f f i c i e n t l y l a r g e . ~ h u s , c + O i s necessary f o r

s t a b i l i t y a t l a r g e om.

For a m l a r g e , t h e r o o t s a r e Z = 0 and the ze ros of t he l a s t f a c t o r

i n Eq . 13. Choosing a l l 6 ' s = 0 causes a l l r oo t l o c i t o approach. z e r o

as urn + - and produces t he maximum d e s i r e d damping o f high-f requency

modes.

F i n a l l y , f o r second-order accuracy a t w, -- 0 , f -- 1 + 0(Z-l)2 a s

Z + 1. These cond i t i ons imply t h a t E must be of t h e form

f o r 1 . ) 2 , where. t h e a; a r e such t h a t a l l ze ros of t h e denominator l i e

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

The choice I = 2 ' g i v e s t he s i ~ n p l e s t . z e ro parametir scheme [ 9 ] i n

whicll t he bar opera ' t ion on a q u a n t i t y Q i s ' g i v e n by

For I=3 t h e r e is. one f r e e parameter ai. So lu t ion of t h e

d i spers ior i r e l a t i o n f o r f i n i t e am shows t h a t a i 6 -0.5 i s requi red f o r

s t a b i l i t y . The marginal ca se ininimizes damping of low-frequency modes

and g ives t he scheme

- The a m p l i t u d e respbnsk of Bq. 7 ' w i t h x d e f i n e d by scllernes 'r and 7

a r e shown i n F i g . 1. For compar ison, t h e a m p l i t u d e response of t h e

s i m p l e s t f i r s t o r d e r ( f u l l y i m p l i c i t ) scheme (2 = xn+l) i s a l s o , shown.

A s c a n he s e e n , t h e damping a t small w m i s much reduced f o r e i t h e r . .

second-order scheme compared t o t h e s i m p l e s t f i r s t o r d e r scheme. Note

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

£ , ' m o s t n e a r l y approx imates t h e i d e a l s t e p r e s p o n s e d e s c r i b e d above. ,

When 1 o r is, a p p l i e d t o t h e p l a s m a - f i e l d s i m u l a t i o n sys tem g i v e n

by Eqs. 7-8, a v e r y large sys tem of coup led d i f f e r e ~ l c e e q u a t i o n s

. . r e s u l t . It. i s i m p r a c t i c a l t o s o l v e : s u c h a . l a r g e i m p l i c i t sys tem. . .

F o r t u u a t e l y , i t is o n l y n e c e s s a r y t o s o l v e a much s m a l l e r sys tem f o r

t h e t ime . f i l t e r e d p o t e n t i a l p. Then t h e p a r t i c l e s a r e advanced i n . ,

. , . . . . . . . ,

t ime i n a p r e d i c t o r - c o r r e c t o r i t e r a t i o n . .

As d i s c u s s e d i n t h e i n t r o d u c t i o n , f i e l d c o r r e c t o r equations may be . >

. .

o b t a i n e d e i t h e r by t h e i m p l i c i t moment e q u a t i o n method [ 4 , 5 , 8 ] o r by . . . . . . L .

che d i r e c t method [ 6 , 7 1 . T h e . , . d e r i v a t i o n h e r e €ol.lows t h e l a t t e r

a p p r o a c h ' i n which t h e mecllanism f o r i n c l u d i n g f i n i t e s i q e d p a r t i c l e s

becomes t r a n s p a r e n t . S i m i l a r d e r i v a t i o n s [9,10] based on t h e mbment

e q u a t i o n s l e a v e s some i n d e t e r m i n a n c y i n t h e i n c l u s i o n of f i n i t e s i z e d

p a r r i c l e e f f e c t s . A s mentio*ed i n s e c t i o n 4 , numer ica l r e s u l t s c o n i i r m . . . . . .

t h e i m p o r t a n c e . o f c o n s i s t e n t l y t rcn tin$ t h e s e e f f e c t s as t ier ived Prom . .

. . . . . t h e d i r e c t method. .

I n t'he p r e d i c t o r - c o r r e c t o r method, t h e p a r t i c l e ' e q u a t i o n s a r e -

f . i r s t s o l v e d w i t h an e s t i m a t e d $n. Then t h e e r r o r in. t h e f i e l d Eq. 8

is c o r r e c t e d by a d j u s t i n g p, t a k i n g account of t h e change i n t h e r i g h t -

hand s i d e induced by a change i n 4".

Denote a p r e d i c t o r - c o r r e c t o r i t e r a t i o n l e v e l by a s u p e r s c r i p t R

w r i t t e n b e f o r e . Then l e t '3 be t h e raw d e l ~ s i t i e s r e s u l t i n g from

pushing t h e p a r t i c l e s w i t h 'gn. That i s ,

- - w i t h '3 g i v e n from 1 o r 2 with n,"+' r e p l a c e d by t h e above e s t i m a t e .

A s s o c i a t e d w i t h t h e Rth i t e r a t e t h e r e is a n e r r o r i n Eq. 8

A c o r r e c t i o n t o 'Tn i s sought such t h a t - 0,.

Let 6 d e n o t e f ' i r s t o r d e r changes a s s o c i a t e d w i t h R .+ ti-1, s o t h a t

- "Cn + 6 $ , e t c e . The change 6, i s computed t o g i v e s l i n e a r

- c o r r e c t o r e q u a t i o n from which 6$ i s determined. For t h i s 6 % i s

For y > 0, t h e s o l i t t o n of t h e d i f f e r e n c e Eqs. 7b,9 may be w r i t t e n

,The m a t r i c e s R and S a r e g i v e n ( i n d i s d i c r lotat io?) by

where

4 = { 1 + ( 1 / 2 * y ) 2 R ; . } - ~ , .

and I i s t h e u n i t d i a d i c .

P r u n ~ E q s . 7a,17 t h e estimate

i s o b t a i n e d . Thus, f rom Eq. 15 , . .

. . , . : .

The . f i e l d c o r r e c t o r i s o b t a i n e d f rom '+'r.- 'r+ 6~ = 0. Thus, 641

s h o u l d s a t i s f y

- Ln t h e a b o v e f,=1/2 f o r 1 and 215 f o r 2 , and t h e matrix

S = f n At2 N N /No 1 'n:+'sa. . A l s o in E q . 1 9 t h e last t e r n h a s been * Y a

a p p r o x i m a t e d as f o l l o w s

where t e rms of o r d e r kv (k wavenumber) have been ignored and wherk t h e

i n t e r p o l a t i o n h a s been p laced on t h e produ'ct of h2.: H*h and V 6 4 .

The n e g l e c t of t h e O(kv) terms i n Eq. 19 i n t r o d u c e s a convergence

c o n d i t i o n f o r t h e p r e d i c t o r - c o r r e c t o r i t e r a t i o n wh'ich i s found t o ' be

whe're k,,,ax Is t h e maximum wavenumber of t h e sys tem, u i s t h e mean

( f l u i d ) v e l o c i t y , and vT t h e the rmal v e l o c i t y of t h e e l e c t r o n s .

The convergence c o n d i t i o n of Eq. 20 i s s i m i l a r t o t h a t o b t a i n e d by

. o t h e r a u t h o r s [5 ,7] . T h i s t ime s t e p c o n s t r a i n t might be avo ided by a

more comp1.e~ p r e d i c t o r - c o r r e c t o r i t e r a t i o n . However, even i f

convergence were o b t a i n e d f o r l a r g e r A t , a ccuracy ,would be l o s t s i n c e

many p a r t i c l e s would t r a v e l a l a r g e f r a c t i o n of 3 wavelength i n ' a , . .

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

be t o combine i m p l i c i t f i e l d methods of t h e t y p e d e s c r i b e d h e r e w i t h

o r b i t a v e r a g i n g t e c h n i q u e s (181.

Many of t h e r e s u l t s of t h i s s u b s e c t i o n h a v e . been p r e v i o u s l y

o u t l i n e d o r d e s c r i b e d by t h e Livermore group [6,71. For Bxample,

- - acllemp 1 and 2 o.f the p r e s e n t work a re ~ 1 1 e s.?rne it:; t h e 1:)l ant1 112

scllemes ' o f Kef. 7. The d e r i v a t i o n i s r e p e a t e d h e r e t o rnalts t h i s paper

more s e l f c o n t a i n e d and t o s i n g l e o u t t h e s p e c i f i c a l g o r i t l ~ m used t o

o b t a i n t h e r e s u l t s of S e c t i o n 3:The m o t i v a t i o n f o r t h e d e r i v a t i o n of 1 -

and 2 g i v e n h e r e i n t e r m s of che l i n e a r f i l t e r of Eq:ll may a l s o be

more s a t i s f a c t o r y t h a n t h a t p r e v i o u s l y g iven .

Two i m p o r t a n l d l l i e r e n c e s bec i j een ' t h e p r e s e n t work and t h e

:! ~ ) ~ . e v l u u s work s h o u l d be emphasized. F i r s t , a p a r t i c u l a r s i ' q p l i f i e d

d i f f e r e n c i n g scheme is a d o p t e d f o r t h e f l e l d c o r r e c t o r ' e q u a t i o n , so

t h a t n u m e r i c a l solution. of E q . 19 i s 'convenient ; The d e t a i l s of t h i s 8

d i f f e r e n c i n g a r e g i v e n i n t h e nex t s u b s e c t i o n . Second, t h e r e c u r s i v e

f i l t e r i o r 7 i s a p p l i e d h e r e t o t h e mesh q u a n t i t y + r a t h e r than t o t h e

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

, ;IS ; e l l a s t h e push ing t ime s i n c e t h e p a r t i c l e p r e d i c t o r and

c o r r e c t o r pushes a r e done w i t h e x p l i c i t d i f f e r e n c e e q u a t i o n s . . . >

Recause t h e f i l t e r i s a p p l i e d on the mesh, t h e d i f f e r e n c e scheme , s . .

ciescr ibed above is no t s t r i c t l y momentum c o n s e r v a t i v e . Thus, moving . ,

p a r t i c l e s s u f f e r a ci,';,$g s e l f f o r c e which is p r o p o r t i o n a l t o : t h e s q u a r e , , , . . . 'I

of t h e i r v e l o c i t p a ~ * ~ ~ Numerical ly i.t i s found t h a t the c f f e c t of t h i s . ,

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

reduced by c o l l e c t i v e s h i e l d i n g and s i n c e t h e c o n s t r a i n t of E q . 20 . .

; a s s u r e s t h a t t h e v e l o c i - t y of a l l p a r t i c l e s i s s m a l l . . . . .

. . . . . . .

I m p l i c i t t ime d i f f e r e n c i n g methods o f t h e type d e s c r i b e d h'ere a l s o

i n t r o d u c e a s m a l l amount of damping of t h e c o l l e c t i v e plasma modes, as

s e e n above f o r t h e harmonic o s c i l l a t o r system. T h i s damping i s

obse rved t o l e a d t o ( p r i m a r i l y e l e c t r o n ) c o o l i n g i n imnplici t

s i ~ n u l a t i o n s [ 5 , 7 , 1 2 ] . Because t h e d i f f e r e n c e e q u a t i o n s d e s c r i b e d h e r e '

. a r e second-order a c c u r a t e a t low f r e q u e n c i e s and because t h e damping .

a s s o c i a t e d w i t h e i t h e r . i o r 2 i s e x t r e m e l y s m a l l e x c e p t a t t h e h i g h e s t

r e a l i z a b l e f r e q u e n c i e s , . t h i s c o o l i n g r a t e is v e r y low f o r r e a s o n a b l e

s i ~ n u l a t i o n pa ramete r s and i s no t a significant l i m i t a t i o n . on t h e

a p p l i c a b i l i t y of t h e method. T h i s i s conf i rmed by t h e numer ica l

r e s u l t s of S e c t i o n 3 below.

A more worrisome a s p e c t of t h e n u m e r i c a l l y i n t r o d u c e d d i s s i p a t i o n

i s t h e , p o s s i b i l i t y of d e s t a b i l z a t i o n .of . nega t ive -energy modes. .

Langdon [19] h a s p o i n t e d o u t t h a t d i s s i p a t i o n combined w i t h t h e l a c k of

G a l i l e a n i n v a r i a n c e i n t h e non-momentum c o n s e r v a t i v e scheme d e s c r i b e d

h e r e c a n , f o r i n s t a n c e , d e s t a b i l i z e t h e s low s p a c e charge ' w a v e f o r . .a

c o l d , d r i f t i n g e l e c t r o n beam. A s shown i n Ref. 7 , t h i s i n s t a b i l i t y is

avo ided i n t h e Dl scheme, s i n c e , w i t h o u t d i f f e r e r l t i a l motion of t h e

p a r t i c l e s , t h e r e is no p o s s i b i l i t y f o r d i s s i p a t i o n i n a G a l i l e a n

i n v a r i a n t scheme.

ljowever, a s - shown i n t h e append ix , t h i s i n s t a b i l i t y i s s t a b i l i z e d

aiai1.I i n t1;e p r e s e n t scheme i n t h e regime of most i n t e r e s t , i . e . f o r

l a r g e enough t ime s t e p s . The lower bound on A t depends on . t h e d r i f t

v e l o c i t y , b u t a l l modes become s t a b l e i n a c o l d , d r i f t i n g e l e c t r o n

plasma f o r A t > 2. I n f a c t , t h e s t a b i l i t y c o n d i t i o n A 1 0 i s e a s i l y

s a t i s f i e d i n t h e a p p l i c a t i o n of low-frequency methods of t h e t y p e

d e s c r i b e d h e r e . I n d i m e n s i o n l e s s v a r i a b l e s Eq. A 1 0 becomes;

-20-

(kXD) ( u / v T ) 6 1, where A D i s . t h e e l e c t r o n Debye l e n g t h , and where

E q . 20 h a s been used t o expand Eor s m a l l ku. S i n c e low-frequency

phc~lo~nerla a r e c h a r a c t e r i z e d by long w a v e l e n g t l ~ tl~ld s m a l l d r i f t

v e l o c i t i e s , b o t h f a c t o r s ' a b o v e a r e s m a l l and s t a b i l i t y i s e a s i l y

m a i n t a i n e d .

d. "Kenormalized" I t e r a t i o n Method

To comple te t h e p r e d i c t o r - c o r r e c t o r i t e r a t i o n scheme, Eq.. 19 must

be a o l v e d f u c S $ . For a aon-cons tan t d e n s i t y plasma, t h i s e q u a t i o n i s

a d l f f e r e n e r a l - i n t e g r a l e l l i p t i c e q u a t i o n with v a r i a h l ~ r n ~ f f i s i e n f c .

The most a p p r o p r i a t e s o l u t i o n schemes f o r such a . mu1t:t-dimensforla1 '

problem a r e i t e r a t i v e r e f i n e m e n t methods i n which a n approx imate

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

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

An e f f e c t i v e method of t h i s t y p e i s o b t a i n e d by w r i t i n g Eq. 19 i n

k - s p a c e and " renormal iz ing" [ l o ] . The F o u r i e r t r a n s f o r m of 19 i s

kZs@ (k) + Ih(k_) 121 k* S(k-k')a.&'6$(k8) = 'E, (k ) - . .

k ' e

The c o n v o l u t i o n sum o v e r k' r e p r e s e n t s t h e i n t e r a c t i o n of a l l 5'

modes w i t h t h e 5 mode t h r o u g h t h e d e n s i t y f l u c t u a t + o n s contai 'ned i n S.

T h i s . sum. may be decomposed i n t o a " s e l f - j n r c r a c t i o n " p a r t r ep reseuCic~g

t h e c o u p l i n g of a k' mode t o i t s e l f (k = k'), and a 'l~nodc-coupling" -,

p a r t (k P k'). - -

-21-

In t h e s p i r i t of weak t u r b u l e n c e t h e o r y , i t i s n a t u r a l t o

r e n o r m a l i z e t h e f r e e . s p a c e d i e l e c t r i c ( f i r s t rerm of Eq. 21) by

i n c l u d i n g t h e s e l f - i n t e r a c t i o n term i n a n approx imate i n v e r s e . and

c o n s i d e r i n g t h e mode c o u p l i n g Eerrns as s m a l l e r . An i n n e r L t e r a t i o n

( i n d i c a t e d by a s u p e r s c r i p t m below) t o s o l v e Eq. 19 i s o b t a i n e d by

tliis approach as

k '# m+'a$ (k) - = - kz + Ih (k ) ,., 12k- - S(0). k ..,

It i s e a s i l y v e r i f i e d t h a t t h e above approach amounts t o

c o n s t r u c t i n g a n approx imate i n v e r s e by r e p l a c i n g tlie v a r i a b l e

s u s c e p t i b i l i t y r e p r e s e n t e d by S by i t s a v e r a g e v a l u e o v e r t h e e n t i r e

c o m p ~ l t a t i o n a l domain. Accord ing ly , t h e convergence of Eq. 22 with m

might be e x p e c t e d t o be ve ry r a p i d f o r a n e a r l y uni.forrn plasma. A s

s e e n from t . 1 ~ numer ica l r e s u l t s o f S e c t i o n 3 , t h i s is indeed t h e c a s e .

I n f i l c t , even f o r r a d i c a l l y rion-uniform plasmas, t h e i t e r a t i o n of

Eq. 22 i s found t o converge i n a few s t e p s t o t h e n e c e s s a r y accuracy .

I n implement ing t h e i t e r a t i o n . of 22 , a combina t ion of r and k ,.,

s p a c e o p e r a t i o n s a r e used t o c o n v e n i e n t l y e v a l l ~ a t e t h e c o n v o l u t i o n

sums. Le t = m+16$ - m6$. Eq. 22 may be r e w r i t t e n a s

where t h e r e s i d u a l ms is t h e d i f f e r e n c e of t h e r i g h t . hand and l e f t hand

s i d e s of Eq. 21 w i t h used f o r 6 4 .

It i s s t r a i g h t f o r w a r d t o v e r i f y t h e fo l lowi r lg p rocedure which is

used t o i t e r a t e Eq. 22. F q r rn = 1, I s = .'e; 6+ = 0. For m + m+l,

o b t a i n "'5 from Eq. 23, t h e n '"+'6+ = m6$ + '"5. M u l t i p l y 9 by ik - and

i n v e r s e F o u r i e r t r a n s f o r i n t o - r space . M u l t i p l y t h e r e s u l t i n g v e c t o r by

S(r) - and t r a n s f o r m back t o k_ space . M u l t i p l y t h i s r e s u l t by i l h ( k ) 1 2 p - and add t o m s ( k ) . - F i n a l l y , s u b t r a c t k2? from t h i s r e s u l t t o o b t a i n

m+ls(?). Repeat u n t i l maxi l m s ( k ) 1 } s a t i s f i e s n s p e c i f i e d convergenee

c r i t e r i o n .

111. S i m u l a t i o n R e s u l t s

I n t h i s s e c t i o n , t h e g e n e r a 1 p r o p e r t i e s of t h e model a r e

i s l v e s t i g a t e d i n . two s i m p l e p h y s i c a l sys tems u s i n g a s i m u l a t i o n code

based Jn t h e low-f requency modc:L o f S e c t i on 2 . I n t h e f i t s t case, tile

i o n - a c o u s t i c f l u c t u a t i o n s of a un i fo rm, t h e r m a l , . unmagnet ized,

two- temperature , one-dimensional plasma a r e examined. S i n c e t h e

i o n - a c o u s t i c f l u c t u a t i o n s r e p r e s e n t a n e x t r e m e l y s m a l l p a r t of t h e

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

by e l e c t r o n Landau damping i n t h e pa ramete r r ange s t u d i e d , t h e s e

r e s u l t s r e p r e s e n t a s e v e r e t e s t of t h e a p p l i c a b i l i t y of t'he model.

I n t h e second c a s e , a v e r y s t r o n g l y non-uniform plasma i s

examined. A two-dimensional plasma i s suspended a .ga ins t a

g r a v i t a t i o n a l f o r c e by a magne t i c f i e l d normal t o t h e p l a n e of

s imul .a t ion. The i n i t i a l d e n s i t y p r o f i l e i s such t h a t t h e plasma

d e n s i t y is n e a r l y uniform i n t h e l e f t h a l f of t h e s i m u l a t i o n domain,

and n e a r l y ' z e r o i n t h e r i g h t h a l f . An u n s t a b l e g r a v i t a t i o n a l

i n t e r c h a n g e i s obse rved f o r p a r a m e t e r s s u c h t h a t ' t h e growth r a t e is

f i v e o r d e r s of magni tude lower t h a n ' t h e e l e c t r o n plasma o r c y c l o t r o n

f rec1uerlcies.

I n bo th c a s e s , t h e second-order a c c u r a t e scheme h a s been used . .

and q u a d r a t i c s p l i n e s have been used. t o i n t e r p o l a t e t h e c h a r g e d e n s i t y

f r o m and t h e f o r c e t o t h e p a r t i c l e s . The f l u c t u a t i o n spec t rum was

obse rved using a one-dimensional , unmagnetized v e r s i o n of t h e a l g o r i t h m

d e s c r i b e d above. S i m u l a t i o n p a r a m e t e r s were; ( s y s t e m i e n g t h ) Nx = 128,

(number of p a r t i c l e s ) . No = 9216, ( p a r t i c l e s i z e ) a = .3 , where the

Gauss ian p a r t i c l e shape i s g i v e n by h . = exp{-112 a262 }.

The t h e r m a l v e l o c i t y of t h e e l e c t r o n s i s such t h a t vT = 0.05, s o

t h a t t h e e l e c t r o r l Debye l e n g t h A D = 0.05. The i o n - t o - e l e c t r o n mass

r a t l o is mi/me = 100, w h i l e t h e e l e c t r o n - t o - i o n ' t e m p e r a t u r e r a t i o i s

Te/;ri = 20. E l e c t r o n s i.i11J i o n s a r c Londeci un i fo rmly on t h e s p a t i a l

mesh a t t' = 0 and g i v e n Maxwel l ian v e l o c i t y d i s t r i b u t i o n s f o r tl;is

t h e r m a l r u n . The t i m e s t e p is f i x e d a t A t = 10, a f a c t o r of 50-100

i n c r e a s e o v e r t h a t a l l o w e d f o r a n e x p l i c i t code i n which w e h a s t o be

r e s o l v e d . . The c a l c u l a t i o n compr i ses 16,384 t ime s t e p s o r 163,840 a;'

s o t h a t many i o n - a c o u s t i c wave p e r i o d s a r e r e s o l v e d . S i n c e t h e plasma

i s n e a r l y un i fo rm, t h ? i t e r a t i v a s n l i l t i n n of El., 22 r e q u i r c o o n l y twa

i t e r a t i o n s t o converge t o a r e l a t i v e e r r o r of 1 0 ' ~ .of t h e e q u i v a l e n t

mean d e n s i t y . No i t e r a t i o n of t h e p a r t i c l e pushing beyond t h e f i r s t

c o r r e c t i o n h a s been deemed n e c e s s a r y .

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

c - o b s e r v a t i o n s which l e d t o t h e c h o i c e of t h e s e p a r a m e t e r s and which b e a r

on t h e a p p l i c a b i l i t y of i m p l i c i t methods i n g e n e r a l . I n a d d i t i o n t o

t l ~ e l a r g e t i m e s t e p employed, two o t h e r s i ~ n u l a t i o n pa ramete r s c o n t r a s t

s h a r p l y t o t h o s e a p p r o p r i a t e f o r e x p l i c i t s i m u l a t i o n . F i r s t , t h e

p a r t i c l e s i z e i s much l a r g e r t h a n t h a t which i s optimum f o r e x p l i c i t

c a l c ~ l l a t i o n s ( t y p i c a l l y , a = 1 t h e r e ) . Other tests have shown t h a t "i. .

s t a b i l i t y . i s impraxed and c o o l i n g i s reduced as t h e p a r t i c l e s ize i s

i n c r e a s e d f rom a = 1, w i t h a t h r e s h o l d of a = 2 f o r t o l e r a b l e c o o l i n g .

F o r s m a l l e r v a l u e s of a , performance i s somewhat improved when

quadrarfc s p l i n e s are used compared t o , t h a t o b t a i n e d u s i n g l i n e a r

s p l i n e s o r m u l t i p o l e expans ion . However, a c c e p t a b l e performance s t i l l

r e q u i r e s a 2 2. F o r a = 3 , a s i r t t h e si.rn111ations d e s c r i b e d h e r e , t h e r e

is v e r y l i t t l e dependence o f t h e r e s u l t s on whether SUDS, l i n e a r

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

p a r t i c l e s . The c o o l i n g r a t e i s a l s o found t o be reduced t o a much

s m a l l e r e x t e n t when t h e n u m b e r ~ o f . p a r t i c l e s p e r c e l l , t h e t ime s t e p , o r

t h e sys tem l e n g t h is i n c r e a s e d i n s t e a d of t h e p a r t i c l e s i z e .

A second c o n t r a s t o c c u r s i n t h e c h o i c e of VT alld a s s o c i a t e d

A l l << 1. T h i s is r e q u i r e d by c o n d i t i o n 20, which i s e s s e n t i a l l y a

Courant-Friedrich-Lewy c o n d i t i o n on t h e t ime s t e p . I n c o n t r a s t t o

e x p l i c i t methods, however, A D << 1 does n o t l e a d t o s e v e r e f i n i t e g r i d

i n s t a b i l i t i e s s i n c e t h e plasma o s c i l l a t i o n branch is removed f o r

A t >> 1. Thus, i m p l i c i t methods are a p p r o p r i a t e f o r s t u d y i n g

long-wavelength modes i n a l a r g e sys tem ( Q u a s i n e u t r a l a l g o r i t h m s [ 2 1 ]

a l s o s h a r e t h i s p r o p e r t y b u t omit a l l e l e c t r o n k i n e t i c e f f e c t s ) .

S e v e r a l a d d i t i o n a l i n t e r e s t i n g p o i n t s can be s e e n from tests c a s e s

o t h e r t h a n t h e ones r e p o r t e d h e r e i n d e t a i l . F i r s t , t h e form of t h e

f i e l d c o r r e c t o r , Eq. 19 i s found t o be t h e o n l y one c o n s i s t e n t w i t h

F i n i t e s i z e d p a r t i c l e s . I f , f o r example, t h e smoothing o p e r a t o r , H ~ * ,

i s a p p l i e d t o e i t h e r o r bo th o f S and V 6$ b e f o r e t h e i r , p roduc t i s

t a k e n , s e v e r e e l e c t r o n c o o l i n g , h e a t i n g , o r i n s t a b i l i t y of t h e

; ~ l g o r i t h m is observed. The d i r e c t method g i v e s t h e c o r . r e c t form [ 7 ]

(modulo s i m p l i f y i n g t h e d i f f e r e n c i n g as i n d i c a t e d i n S e c t i o n 2c.) . The

momenf method l e a v e s i n d e t e r m i n a t e t h e form of t h e cor responding terms.

I f t h e s e terms are i n t e r p r e t e d as i n Eq. 19, t h e r e s u l t s of t h e moment

raethod and t h e d i r s c t method a r e e s s e n t i a l l y i d e n t i c a l .

Second ly , t h e recl1.1ction i n c o o l i n g o b t a i n e d f u r t h e second-order

a c c u r a t e methods d e s c r i b e d h e r e compared t o a f u l l y i m p l i c i t

f i r s t - o r d e r a c c u r a t e inethod i s no t a s d r a m a t i c a s t l ie r e d u c t i o n i n low

f r e q u e n c y mode damping ( s e e F i g . I ) . For tile pa ramete r s d e s c r i b e d

a b o v e , t h e c o o l i n g r a t e is o n l y reduced by a f a c t o r of two f o r t h e

second-order a c c u r a t e method ( i n one-dimensional geometry) . It a p p e a r s

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

v e l o c i t i e s v e x c i t e b a l l i s t i c modes w i t h Frequency w = k v . much l a r g e r j J

t h a n t h e c o l i e c t i v e ' m o d e f r e q u e n c y kcs. These h i g h e r Frequency modes

s u f f e r largpr nnmariaad damping tlSlall Ll~e collective modes and t h e i r

daulping seems c6 be t h e major c a u s e of t h e c o o l i n g . The h i g h e r o r d e r

a c c u r a t e method i s used h e r e and i n o t h e r a p p l i c a t i o n s t o improve t h e

f i d e l i r y of low-frequency r e s p o n s e ( r e d u c e d damping) a t no i n c r e a s e i n

c o m p u t a t i o n a l complex i ty .

R e t u r n i n g t o t h e i o n - a c o u s t i c s i m u l a t i o n d e s c r i b e d p r e v i o u s l y , the '

t ime e v o l i l t i o n of e l e c t r o n and i n n t m n p a r n t u r c ~ , ~ i u i ~ t l a l L ~ t ? d t 6 t h e i r

i n i t i a l v a l u e s , a r e shown i n Fig . 2a. The temperature is measured a s

C V ~ ) -.. < v j 7 , L l ~ r average b e i n g t a k e n o v e r a l l No p a r t i c l e s of e i t h e r

e l e c t r o n o r i o n s p e c i e s . The t o t a l momentum, p r o p o r t i o n a l t o <v>

rstnains l e s s than o f t h e v e l o c i t y a s s o c i a t e d w i t h t h e above

ternpera t l l re ; t h u s , t h e t e m p e r a t u r e s a l s o Inenstlres t h e t o t a l k i n e t i c

e n e r g i e s . The t ime e v o l u t i o n of t h e t o t a l e n e r g y , normal ized t o i t s

i n i t i a l v a l u e , i s shown i n F i g . 2b. The ion t e m p e r a t u r e i n c r c n s ~ s Sty

55X, wlli le t h e e l e c t r o n s c o o l by 20%. T o t a l e n e r g y d e c r e a s e s by o n l y .

12% o v e r 163,840 u z l . The e l e c t r o s t a t i c e n e r g y i s e s s e n t i a l l y c o n s t a n t

t h r o u g h o u t t h e c a l c u l a t i o n . I n t h i s c a s e , t h e plasma paramete r ( r a t i o

of a v e r a g e Couloumb e n e r g y t o i n i t i a l k i n e t i c e n e r g y ) i s g = 2. l r 3 .

E l e c t r o n c o o l i n g , which c a u s e s t h e t o t a l e n e r g y d e c r e a s e i s due t o t h e

numer ica l damping d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . L o c a l v e l o c i t y

s p a c e d i a g n o s t i c s show t h a t t h e f a s t o r t a i . 1 e l e c t r o n s a r e t h e ones

t h a t a r e s lowing down most r a p i d l y . The i o n h e a t i n g 'is p h y s i c a l and

due t o i o n Landau damping of t h e i o n - a c o u s t i c waves.

The c o l l e c t i v e b e h a v i o r of t h e plasma a t f r e q u e n c i e s w << w e i s

d i s p l a y e d i n F i g s 3a ,3b. The t ime averaged e l e c t r o s t a t i c ene rgy p e r

wavenumber, < E ~ / B T ~ > , normal ized t o t h e t h e r m a l e n e r g y p e r d e g r e e of

freedom, kgTe/2 (kB i s Boltzman's c o n s t a n t ) , o r f l ~ ~ c t u a t i o n spec t rum i s

sllown i n Fig . 3a. The i o n - a c o u s t i c f r e q u e n c y spec t rum, normal ized

f r e q u e n c y , @ / w e , v e r s u s normal ized wavenumber, kAD, i s shown i n

F i g . 3b.

For T >> Ti, and w << w e , t h e . f l u c t u a t i o n spec t rum of a

two- temperature Maxwell ian plasma call be w r i t t e n as [ 2 0 j

where

The f i r s t te rm on t h e r i g h t hand s i d e of Eq. 24 i s t h e t e rm one o b t a i n s

i f the ' e l e c t r o n s a r e t r e a t e d a d i a b a t i c a l l y a s i n q u a s i n e u t r a l

s i m u l a t i o n s . I ts m u l t i p l i e r r e p r e s e n t s t h e c o n t r i b u t i o n of i o n and

-28-

e l e c t r o n Landau damping t o t h e f l u c t u a t i o n e n e r g y from r e s o n a n t i o n s

and e l e c t r o n s . For T, >> Ti che second term i n t h e m u l t i p l i e r

d o ~ n i n a t e s o v e r O and t h e normal ized f l u c t l l a t i o n e n e r g y approaches

k21$1h(k) 1 2 / ( 1 + k21$lh(k) l 2 1.

A s t r i n g e n t t e s t o f t h e e l e c t r o n response a t 'low f r e q u e n c i e s i s

a f f o r d e d by compar ison of t h e s i m u l a t i o n r e s u l t s w i t h t h e p r e d i c t i o n of

Eq . 2'4. I f r e s o ~ i a n t e l e c t r o n r e s p o n s e i s r e t a i n e d by t h e i m p l i c i t

method, t h e f l u c t u a t i o n spec t rum w i l l obey Eq. 2 4 . I f t h e

low-f r equency e l e c t r o n r e s p o n s e i s e s s e n t i a l l y a d i a b a t i c , t h e

f l u c t u a t i o n s p e c t r u m w i l l c l o s e l y f o l l o w o n l y tlre f i r s t term on t h e

r i g h t of E q . 2 4 . A ~ i m u l a t i o n u ~ i n g t h e q l i a s i n e u t r a l 4 a i m u l a t i o ~ ~

model 1211 i n d e e d v e r i f i e s t h e l a t t e r s c a l i n g .

I n Fig . 3a t h e f l u c t u a t i o n s p e c t r a p r e d i c t e d by Eq. 24 and C l ~ a t

p r e d i c t e d f o r a d i a b a t i c e l e c t r o n s a r e p l o t t e d i n t h e upper and lower

s o l i d c u r v e s , r e s p e c t i v e l y . The r a t i o of t h e fo rmer t o t h e l a t t e r i s

a p p r o x i m a t e l y Te/TI. The s i r n u l n t i o n f l u c t ~ l a t i o n :;pectr~im I n d i c a t e d by

d o t s c l o s e l y f o l l o w s t h e p r e d i c t i o n of Eq. 24. Note a l s o t h a t t h e

wavenumbers w i t h maximum e n e r g y obse rved i n t h e s i m u l a t i o n and

p r e d i c t e d hy t h e o r y a g r e e e x a c t l y . A t l a r g e r wavenumhers, t h e t h e o r y

i s q u e s t i o n a b l e hecause of a p p r o x i m a t i o n s made i n t h e d e r i v a t i o n of

E q . 24. The e f f e c t of t h e p r e v i o u s l y mentioned b a l l i s t i c e l e c t r o n

modes on t h e f l u c t u a t i o n spec t rum i s t o i n c r e a s e i t s i n t e n s i t y above

t h e t h e d r e t l c a l l e v e l Eq. 24 . When most of t h e n o i s e due t o t h e s e

b a l l i s t i c modes i s s u p p r e s s e d by f r e q u e n c y f i l t e r i n g , as i n F i g . 3a

where a f i l t e r of w i d t h Aw = 0 . 0 2 ~ ~ a b o u t w = kcs i s u s e d , t h e

intensity of t h e s p e c t r u m d e c r e a s e s , bu t i t s shape is p r e s e r v e d , and

c l o s e r agreement w i t h t h e o r y r e s u l t s . S i n c e t h i s e x t r a b a l l i s t i c n o i s e

i s due t o p a r t i c l e d i s c r e t e n e s s , i n c r e a s i n g t h e number of p a r t i c l e s

f u r t h e r r educes t h e d i s c r e p a n c y .

I n summary, t h e f l u c t u a t i o n spec t rum i n d i c a t e s t h a t t h e r e s o n a n t

e l e c t r o n response a t low f r e q u e n c i e s i s d e s c r i b e d a c c u r a t e l y by t h e

i m p l i c i t method d e s c r i b e d h e r e . It i s a l s o c l e a r from t h e s e r e s u l t s

t h a t f l u c t u a t i q n s a t mode f r e q u e n c i e s h i g h e r t h a n t h e i o n - a c o u s t i c

range have . been s u p p r e s s e d ; o t h e r w i s e t h e spec t rum would have obeyed

t h e u s t ~ a l s c a l i n g 1 / ( 1 + k 2 ~ b l h ( k ) l 2 ).

The t h e o r e t i c a l i o n - a c o u s t i c d i s p e r s i o n r e l a t i o n , w /we v e r s u s I d D , .

o b t a i n e d from a numer ica l s o l u t i o n i n c l u d i n g f i n i t e g r i d e f f e c t s is

shown a s a s o l i d c u r v e i n F i g 3b. The measured s i m u l a t i o n f r e q u e n c i e s ,

shown by d o t s i n t h a t f i g u r e , a r e i n e x c e l l e n t agreement w i t h t h e

theory. No h i g h e r ,mode f r e q u e n c i e s were obse rved i n t h e spect rum. A

v e r y low i n t e n s i t y t a i l due t o t h e b a l l i s t i c e l e c t r o n modes is s e e n

e x t e n d i n g beyond t h e i o n - a c o u s t i c f r equency . These one-dimensional

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

d e s c r i b e d h e r e f o r ' s t u d y i n g low-frequency phenomena i n a thermal.

plasma. Next , a two-dimensional , magnet ized plasma i s examined usi.11g

t h e s e mctllods . An u n s t a b l e g r a v i t a t i o n a l i n t e r c h a n g e i s s t u d i e d i n two d i m e ~ l s i o n s

f o r a n inhomogeneous plasma. T h i s c a l c u l a t i o n i s c a r r i e d o u t f o r

( sys tem s i z e ) Nx = N = 32 , (number of e l e c t r o n s o r i o n s ) No - 4608, Y

(mass r a t i o ) mi/me = 100, ( e l e c t r o n c y c l o t r o n f requency) R e = 1, and

in article s i z e ) a = 3. The magnet ic f i e l d i s normal t o t h e p l a n e of

s i m u l a t i o n .

Elec t rons and ions a r e loaded i n i t i a l l y with t h e i r guiding c e n t e r

v e l o c i t y (9 = 0) i n such a way t h a t t h e d i s t r i b u t i o n of p a r t i c l e s is

r l l~iform i n t h e l e f t ha l f of t he s imu l s t ion domain. No part.lc1e.s are

loaded i n t he r i g h t h a l f of t he domain. A g r a v i t a t i o n a l a c c e l e r a t i o n

t o t he r i g h t d r i v e s an uns t ab le in te rchange l o c a l i z e d near t h e . . i n t e r f a c e a t the middle of t he domain.

T h e I ~ e r t u r b e d p o t e n t i a l J, f o r such an in te rchange s a t i s f i e s t he

. d i f f e r e n t i a l equat ion [ 2 2 ]

where g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n . For t h e sharp dens i ty

g r a d i e n t ca se cons idered h e r e , t h e above reduces t o t h e a l g e b r a i c

c o n d i t i o n

In Eqs. 26,27, t he e f f e c t of t he p o l a r i z a t i o n motion of t he ions

i s very important f o r h igh dens ' i ty plasmas ( a >> 1). I f t h e

p o l a r i ' z a t i o n motion is neglec ted , t h e f i r s t term i n the denominator of

Eq. 27 i s omit ted and the growth r a t e becomes unphys ica l ly l . ar -g~ f o r

t h e h igh d e n s i t y case . For t h e s imula t ion parameters considered he re ,

w i / R i = 10, so the phys i ca l growth r a t e y = 4 lk lg , while t h e growth Y

r a t e neg lec t ing p o l a r i z a t i o n i s - 7 t imes l a r g e r .

-31-

I n t h e s i m u l a t i o n , rerlorrnalizetl i . t e r a t i o n of E q . 22 . i s used t o

o b t a i n t h e f i e l d c o r r e o r w i t h e x t r e m e l y l a r g e A t = ' . lo3 . Convergence

t o lo-' of t h e e q u i v a l e n t mean d e n s i t y r e q u i r e s betwec.n 1 and' 20

i t e r a t i o n s , t h e l a t t e r number b e i n g r e q u i r e d o n l y b r i e f l y d u r i n g t h e

s t r o n g l y n o n l i n e a r phase of t h e s i m u l a t i o n .

The s i m u l n t i o n r e s u l t s f o r g = 2. a r e summarized i n F i g s 4,5.

I n t h i s c a s e , t h e plasma i s i n i t i a l l y p e r t u r b e d wi th t h e l o n g e s t y

wave leng th , k = 2m/N s o t h a t t h e growth r a t e of a s i n g l e mode may be Y Y '

more a c c u r a t e l y measured. With o n l y n o i s e a s t h e i n i t i a l p e r t u r b a t i o n ,

s e v e r a l l o n g y wavelength modes grow s i m u l t a n e o u s l y w i t h n e a r l y e q u a l

growth r a t e s . A v a l u e of 0.1 was s e l e c t e d f o r t h e d e c e n t e r i n g

pa ramete r y of Eqs. 7 , 9 t o o p t i m i z e code performance. I n Fig . 4 , t h e

e l e c t r o s t a t i c f i e l d energy i s shown i n a semi-l.og p l o t a s a f u n c t i o n of

t ime . The growth r a t e of y = 2.4. lo-' ' de te rmined from

W E = WE(0) e x p { 2 y t } i s i n good agreement w i t h t h e t h e o r e t i c a l l y

p r e d l c t ~ d v a l u e of Y = 2.7-lo-', v e r i f y i n g t h e c o r r e c t mode l l ing of i o n

p o l a r i z a t i o n motion.

Fig . 5 shows t h r e e d i f f e r e n t s n a p s h o t s of t h e f i n i t e p a r t i c l e s i z e

i o n d e n s i t y c o n t o u r s . The p o s i t i o n s of rough ly lo3 of t h e i o n s a r e

a l s o shown a s p o i n t s i n t h e f i g u r e . A s can be s e e n , t h e u n s t a b l e

i n t e r f a c e n e a r t h e middle of t h e s i ~ n u l a t i o n domain e v o l v e s th rough a

l i n e a r growth s t a g e , toward a n o n l i n e a r " s p i k e and bubble" s t a g e . The

o n l y s a t u r a t i o n mechanism i s p rov ided by t h e f i n i t e s i z e of t h e

p e r i o d i c s i m u l a t i o n boundary. Thus, plasma l e a v i n g on t h e r i g h t ( l e f t )

r e e n t e r s on t h e l e f t ( r i g h t ) . I n t h i s way t h e c o n f i g u r a t i o n e v o l v e s

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

s t a t i o n a r y v o r t i c e s .

-32-

The stmulation results presented in this section demonstrate the

accuracy and efEicacy of the methods described here when applied to the

s tlldy of low-f requency, magnetized plasma plienomena.

-33-

I V . Conc lus ions

The second-order a c c u r a t e s i i n u l ; ~ t . i o n method ' d e s c r i b e d h e r e i s

a p p r o p r i a t e . f o r - t h e s t u d y of low-frequency pher~oln&lla i n a magne t i zed

plasma. A f i e l d c o r r e c t o r d e r i v e d by t h e .' d i r e c t method ( w i t h

d i f f e r e n c i n g s i m p l i f i e d ) c o r r e c t l y t r e a t s f i n i t e s i z e d p a r t i c l e s . . T h e '

g u i d i n g c e n t e r mot ion of bo th i o n s and e l e c t r o n s is ' a c c u r a t e l y fo l lowed

f o r Q, >> 1 by . a s i m p l e . d e c e n t e r e d . d i f f e r e n c i n g of t h e L o r e n t z f o r c e

p a r t i c l e p u s l ~ i n g ~ e q u a t i o n s . A s t r a i g h t f o r w a r d i t e r a t i o n o f ' t h e f i e l d

c o r r e c t o r i s . d e v e l o p e d based on the , , r enormal . i zed plasma s i m u l a t i o n

met hod.

Numerical r e s u l t s c o n f i r m reduced e l e c t r o n c o o l i n g f o r t h e

second-order . a c c u r a t e method. The ~ l u m e r i c a l exper iments a l s o show t h a t

f i n i t e p a r t i c l e s i z e may o n l y be i n c o r p o r a t e d i n a n i m p l i c i t . .

c a l c u l a t i o n a s i n d i c a t e d by t h e d i r e c t method d e r i v a t i o n . The

- . e f f i c i e n c y of t h e i t e r a t i v e method f o r s o l v i n g . t h e f i e l d c o r r e c t o r i s

demons t ra ted f o r bo th n e a r l y un i fo rm and s t r o n g l y inhomegeneous

plasmas.

' .Accurate numer ica l r e s u l t s a r e o b t a i n e d i n two s t r i n g e n t t e s t

c a s e s . F i r s t , t h e i o n - a c o u s t i c f l u c t u a t i o n s of a t h e r m a l plasma

d e m o n s t r a t e t h e accuracy w i t h which k i n e t i c e l e c t r o n e f f e c t s on

low-frequency o s c i l l a t i o n s a r e r e p r e s e n t e d . Second, t h e s i m u l a t i o n of

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

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

phenomena a t e x t r e m c l y low f r e q u e n c i e s (growth raLo y = loW5we).

The method d e s c r i b e d h e r e may be e a s i l y extended i n t h r e e

d i r e c t i o n s . F i r s t , gyroaveraged f o r c e s may be a d d e d t o t h e Loren tz

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

f i n i t e i o n g y r o r a d i u s e f f e c t s . Second, t h e si lnuliat ion may be made

f u l l y e l e c t r o m a g n e t i c by t h e a d d i t i o n of a n e a r l y e x p l i c i t t ime advance

nf t h e v e c t o r p o t e n t i a l . T h i r d , t h e s i l n u l a t i o n geometry may be r e a d i l y

cllanged t o r e p r e s e n t mbre r e a l i s t i c c o n f i g u r a t i o n s w i t h more r e a l i s t i c

boundary c o n d i t i o n s . Many of t h e s e e x t e n s i o n s have been o u t l i n e d i n

t h e r e f e r e n c e s .and d e t a i l s w i l l be p r e s e n t e d i n f u t u r e pub l ica t i .ons .

Acknowledements:

T h i s work was s u p p o r t e d by the U. S. Department of Energy, under ,

c o n t r a c t number DE-FG05-80ET-,53088, tl?e N a t i o n a l Sc ience .Found 'a t ion

Grant NSF-ATM81-10539, and by t h e P l i n i s t r y of .Educat ion of JAPAN under

t h e U, S./JAPAN J o i n t I n s t i t u t e f o r Fus ion Theory ( J IFT) . One of u s

(D.C.B.) would l i k e t o acknowledge t h e k i n d h o s p i t a l i t y of t h e

JTPP-Nagoya s t a f f , d u r i n g h i s s t a y . t l ~ e r e . w h e n much.of t h i s work was . .

begun and t h e v i s i t of P r o f e s s o r T. Kamimura t o IFS-Texas d u r i n g which

t h i s work was completed. The a u t h o r s have a l s o b e n e f i t e d enormously

froin d i s c u s s i o n s w i t h H. Berk, J. U. B r a c k b i l l , B. I. Cohen,

A . R. Langdon, and M.' N. Rosenbluth . ,

Ref e r e ~ l c e s -.A ".-.----

1. .J. M. Jlawson, i n "Methods of Computationil l Phys.ic:sU ( H . A l d e r , . .

e t . al. .Eds . ) , Vol.. 9 , p. 1 , Academic t ' ress, New ~ o ' r k , ' 1970;

C. K. B i r d s a l l , A . R. Langdon and H. Okuda, i b i d . , Vol. 9 ,

p. 241.

2 . K. V . R o b e r t s and D. E. P o t t e r , in."Metl'rods oE C o m p ~ l t a t i o n a l

P h y s i c s " (B. A l d e r , e t . a l . E d s . ) , Vol. 9 , p. 333, ilcadelnic

P r e s s , New York, 1970; J. V. Hrac lch i l l , i n " ~ e t h o d s of

Computa t iona l Phys ics" ( b . A l d e r , e t . a l e Kds.), Vol. 16,

p. 1 , Academic P r e s s , New York, 1976.

3. A . R. Langdon, J . Comp. Phys. 30, 202 (1979).

4 . R. J. Mason, J. Comp. Phys. 41, 233 (1981) .

5. J. Denav i t , J. Comp. Phys. 42 , 337 (1981) . -

6. A. Friedman, A. R. Langdon, and H. I. Cohen, Comments on

Plasma Phys. and Cont r . Fus ion 6 , 225 (1981) . -

7. R. T. Cohen, A. R. Tangdon, and A. Friedman,

J. Coeip. Pilys. 4 6 , 15 (1982) ; A . B. Langdon, B. I. Cohcn, arld - A. Friedman, Lawrence Livermore L a b o r a t o r y , Univ. of C a L . ,

Repor t No. UCKL 86350, 1982.

8. J . U . K r a c k b i l l and D. W. P o r s l u n d , J. Comp. Phys. 46, 271

( '1 982).

9; 0. C. Barnes and T. Kamimura, I n s t i t u t e of Plasma P h y s i c s ,

Nagoya U n i v e r s i t y , Kesearch Kept.' No. LPYJ-b/U, 1982;

D. C. Barnes and T. Kamimura, B u l l . Amer. Phys. Soc. 26, 986 - (1981) .

10. T. Tajima a r ~ d J. N. Leboeuf , Bu l l . Ame,r. Phys. Soc. 2, 986

(1981).

-37-

11. T. Kamimura, T. Ta j ima , J. N. Leboeuf , and D. C. R a r n e s ,

B u l l . Am. Phys. Soc. (1982) .

12. J . N. Leboeuf and T. T a j i m a , i n " P r o c e e d i n g s o f the Annual.

Sherwood C o n t r o l l e d F u s i o n Theory c o n f e r e n c e " , p. 1E9., S a n t a

F e , New Mexico, A p r i l 2.5-28, 1982.

13. W. L. K r u e r , J. M. 'Dawson, a n d B. Rosen, J. Comp. Phys. 13,

14. H. Okuda, A. T. in, 'C. C. L i n , and J. M. Dawson, . . .

Comp. Phys. ~ o m m . 11, 227"(1979).

15. J. P. B o r i s , i n "P roceed ings ' of t h e F o u r t h Confe rence on

Numer i ca l s i m u l a t i o n of Plasma" si or is and Shanny Eds . ) , p. 4 ,

U. S. Govt. P r i n t i n g O f f i c e , Washington D. C. ( S t k . No.

16. A . Y. Aydemir, D. C. B a r n e s and T. Karnimura, i n " P r o c e e d i n g s .. ,

of t h e Annual Sherwood ~ o r i t r o l l e i F u s i o n Theory Confe rence" ,

p. 1E17, S a n t a Fe , New ~ e x i c o , ' . ~ ~ r i l 25-28, 1982.

17. T. G. N o r t h r u p , Anna l s of Phys. 15 , 1 9 (1961) . - 18. R. I. Cohen, T. A. R r e n g l e , D. 8. Conley , and R. P. F r i e s ,

J. Comp. Phys 38, 45 (1980) ; 3. I. Cohen, R. P. F r i e s , and

V. Thomas, J . Comp. Phys. 45, 345 (1982) .

19:A. R. 'Langdon, P r i v a t e Communication (1982) . . .

20. N. A. K r a l l ' and A. W. ~ r i v e l ~ i e c e , " P r i n c i p l e s o f P lasma , ,

P h y S i c s , McGrair-Ilill, New York, 1973. . .

21. H. Okuda, J. M. ~ a b s o n , A f T. L i n , and C. C. L i n , Phys. F l u i d s

21 , 476 (1978) . - 22. A. R. M i k h a i l o v s k i i , "Theory o f P lasma I n s t a b i l i t i e s " , V o l e 2 ,

p. 114, C o n s u l t a n t s Bureau , New York-London, 1974.

F i g u r e Capti 'ons -.-A

F i g u r e 1 - A r n p l i t u d e r e s p o n s e f o r t h e ,harmonic o s c i l l a t o r problem

wi th t h r e e d i f f e r e n t i m p l i c i t d i f f e r e n c e schemes: 1) f u l l y

i m p l i c i t - b r o k e n c u r v e ; 2 ) i - d o t t e d c u r v e ; 3 ) 3 - s o l i d

c u r v e . The a b s c i s s a i s t h e norlualized f requency of t h e

sys tem. The o r d i n a t e are c l o s e l y r e l a t e d t o t h e damping

d e c r e ~ n e n t . Also slluwn a re ehe t o o t l o c i f o r 2 r e l a t i v e t o t h e

rtnit c I P c L B . 'Lwo r o o t s move from 7; = 1 (marked by x crn

rtrr1,e) K O o r i g i n (0) a s 0 ( o 6 QJ. The o t l ~ e r two a r e

s t r o n g l y damped.

F i g u r e 2 - a ) E l e c t r o n and i o n . t e m p e r a t u r e s normal ized. t o t h e i r . .

t = 0 v a l u e s a s a f u n c i t o n of t i m e .

b) Total . e n e r g y normalizeci t o i t s t .= 0 v a l u e .as a

f u n c t i o n of t ime.

F i g u r e 3 - ?#J

a) Fluc&d&ion spec t rum f o r t h e r m a l plasma. Normalized

e l e c t r o s t a t i c f i e l d e n e r g y as a f u n c t i o n of wavenumber i s

shown. Upper c u r v e i s t h e o r y i n c l u d i n g e l e c t r o n Landau

damping; lower t h e o r y n e g l e c t i n g e l e c t r o n Landau damping;,

p o i n t s a r e s i m u l a t i o n r e s u l t s .

b) Frequency spec t rum . f o r t h e r m a l plasma. 'Normalized

f r e q u e ~ l c y v e r s u s normal ized wavenumber i s p l o t t e d . ' Curve i s

t h e o r y ; p o i n t s a r e s i ~ n u l a t i o n r e s u 1 . t ~ .

. . . F i g u r e 4 - E l e c t r o s t a t i c f i e l d , e n e r g y (normal ized ' u n i t s ) a s a

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

. , ' . F i g u r e 5, - . .

. ,

Snapsho t s 0.f f i n i t e s i z e i o n d e n s i t y c o n t o u r s a r e shown

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

s a t u r a t i o n of g r a v i t a t i o n a l i .nterc6ange. , .

a ) oeAt =' 3.10%;

b ) w e A t = 4 . 1 0 ~ ;

Appendix: D i s p e r s i o n R e l a t i o n f o r Cold d r i f t i n g E l e c t r o n Plasma --. - ------ In t h i s a p p e n d i x t h e d i s p e r s i q n r e l a t i o n ' f o r a c o l d ,

, . d r i f t i n g e l e c t r o n beam i s d e r i v e d and' c o n d i t i o n s f o r stability

are d i s c u s s e d . A small p e r t u r b a t i o n a b o u t a u n i f o r m ,

f i e l d - f r e e , one -d imens iona l e q u i l i b r i u m i s assumed. F i r s t ,

t h e p e r t u r b e d p a r t i c l e o r b i t s a r e computed f rom t h e d i f f e r e n c e

' Eqs. 7. Nex t , t h e p e r t u r b a t i o n i n t h e part-lc1.e p o s i t i o n s are

u s e d t o compute . t h e d e n s i t y p e r t u r b a t i o n t31. F i n a l l y , t h e

d e n s i t y p e r t u r b ~ k i n n s nre used t o o b t a i n t h e d i s p e r s i o n

relat ion f rom the f i e l d Eq, 8 . The d e r i v a t i o n f o l l u w s ' .

s t a n d a r d methods a n d d e t a i l s a r e o m i t t e d . , . .

. L e t 1. be as i n S e c t i o n 2 c , and . d e n o t e by f ( Z ) t h e - -

t r a n s f e r f u n c t i o r l a s s o c i a t e d w i t h f i l t e r 1 o r 2. Thus ,

L e t u be t h e ( n o r m a l i z e d ) d r i f t v e l o c i t y o f t h e c o l d

e l e c t r o n s . The d i s p e r s i o n r e l a t i o n , f o r tl!e scheme. -1 o r 2 w i t h

t h e f i l t e r i n g a p p l i e d o n t h e mesh as d i s c u s s e d i n Section 2c

where

-41- .

i k u W = Z e . ( A 4 ) ,

and t h e f i n i t e s i z e p a r t i c l e e f f e c t s have been n e g l e c t e d .

For A t s m a l l t h i s d i s p e r s i o n r e l a t i o n p r e d i c t s

i n s t a b i l i t y of t h e s low (w - ku - we) s p a c e c h a r g e wave [19] .

For s u f f i c i e n t l y l a r g e A t , however, a l l r o o t s of E q . ' A 3 become

s t a b l e . To show t h i s , examine Eq. A3 f o r A t such t h a t Z - 1.

P u t t i n g Z = 1 g i v e s t h e m a r g i n a l t ime ' s t e p

Expanding A t = A t o + r , Z = l + l Z j r j f o r r / A t O < < l

j d e t e r m i n e s Z j o r d e r by o r d e r . A f t e r some t e d i o u s a l g e b r a

t h e r e r e s u l t s , , . . . . . ,

- f o r 1, and

The c o n d i t i o n f o r s t a b i l i t y f o r A t - A to is found from

1 > 1z21. From A6-7 f o l l o w s

. .

I%* 1 1 - 253.1' T 3 o ( T ~ ) ( A R ) ,

. .

f o r 1, and

- f o r 2. Accurd ing ly , b o t h uchcmco g i v e as the s t a b i l i t y

c o n d i t i o n , T > 0 , o r

A t > 2 s i n k u / 2 (A10).

C o n d i t i o n A10 h a s been d e r i v e d f o r A t - A t o . However, a

numer ica l s o l u t i o n . o f t h e f u l l d i s p e r s i o n r e l a t i o n , A3,

c o n f i r m s t h a t a l l r o o t s remain s t r i b l e .as A t + -. Thus,

. c o n d i t i o n A10 i s n e c e s s a r y and s u f f i c i e n t f o r s t a h i l i t y .