Computer simulation of collective phenomena in a turbulent plasma

Computer simulation of collective phenomena in a turbulent plasma

Physica 2D (1981) 165-169 © North-Holland Publishing Company COMPUq'ER SIMULA'I'ION OP COLLECq'IVE IN A " P U R B U L E N ' I ' P L A S M A Yurii S...

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Physica 2D (1981) 165-169 © North-Holland Publishing Company

COMPUq'ER

SIMULA'I'ION OP COLLECq'IVE IN A " P U R B U L E N ' I ' P L A S M A Yurii

S.

PHENOMENA

Sigov

K e l d y s h I n s t i t u t e of A p p l i e d Mathematics USSR Academy of S c i e n c e s Moscow, 125047 USSR

A short r e v i e w of the results of o n e - a n d t w o - d i m e n s i o n a l c o m p u t e r investigations of strong L a n g m u i r t u r b u l e n c e i s given, T h e m a i n topics of interest are the following: formation of high energetic electron t a i l s ; ion S o u n d t u r b u l e n c e a c c o m p a n y i n g collapse a n d p l a s m o n s ' c o n v e r s i o n l modulation i n s t a b i l i h 2" (MI) in X , X - g e o m e t F y for Te/q'i:~ I a n d T /n/.._~. I| M I in t r a n s v e r s e m a g n e t i c field. "l~he c a s e of the "forced" i collapse ( q u a s i c o U a p s e ) i n p r e s e n c e of a n external p u m p E ext,~ E Cos6~) t i s consi•-ox o dered.

rl~he s t u d y

of Langmuir

turbulence

is

important

from

several

points

of view.

5'or fast methods of plasma heating (by an electron beam or laser radiation) the energy is initially transfered to Langmuir waves, and in most cases the physics of adopting of this energy is entirely determined by developing Langmuir turbulence / 1 - 3 / . + It d e t e r m i n e s such fundamental quantities as relaxation Iength of electron beam, laser radiation absorption and reflection coefficients and the form of the particles' distribution functions of h e a t e d plasma. F r o m the other h a n d , L a n g m u i r tdrbulence is o n e o-f the m o s t simple i y p e s of p l a s m a t u r b u l ence a n d in g e n e r a l - o n e of the m o s t investigated e x a m p l e s of a %urbulence, w h i c h doesn't r e s e m b l e a turbulenc e in a n i n c o m p r e s s i b l e fluid. q-~he data of c o m p u t e r e x p e r i m e n t s u n d e r consideration w e r e obtained b y C I C m e t h o d /4,5/ for d o u b l e periodic bounda*uy- conditions. B e c a u s e of the limited size of the p a p e r w e give only brief c o m m e n t s of the m a i n results, p a y i n g m o r e attention to the results that w e r e p r e s e n t e d in the p r e v i o u s w o r k s a n d r e f e r e n c e s m o r e scantily. "Phe physic, s of high energetic electron tail .~eneration is in a close c o n n e c t i o n w i t h d y n a m i c s of turbulent spectra. A l r e a d y the first c o m p u t e r e x p e r i m e n t s o n parametric p l a s m a heating d e m o n s t r a t e d the strong correlation b e t w e e n this p h e n o m e n o n a n d the formation of c o n t i n u o u s p o w e r s p e c tra of L a n g m u i r oscillations. A n u m b e r of n u m e r i c a l investigations carried out permits to construct a s e m i - p h e n o m e n o l o ~ i c a l theor~y- w h i c h is b a s e d o n hAro facts established: the quasi-linear m e c h a n i s m of the tails' origin a n d the relatively simple form of electron distribution function for large velocities v : f.... , ( v ) O ~ e x p [-~v~ "l~hus for description of a n o m a l o u s h e a • all ring it Is ~ppilopriate tO reta,n in the full s y s t e m of Q L "I~ e q u a t i o n s only the diffusion equation for the s p a c e a v e r a g e d distribution function fe(V,t) ~

.

.

~~ -fe

and the relation

for t h e n ( v )

.

"

= ~9- ~ n ( v )

~~ef

coefficient

(1)

nCv't)

k I

-* l

q'l~e d e p e n d e n c e experiments and

of f (v,t) on v,t is is t~ken in a form

assumed

165

to

be

known

from

computer

166

Yu.S. Sigov / Computer simulation

ft(v,t)

= fM(v

) e x p [ - (:~(t)

( lvl

- v~)l,

fM = ( ~ I ~ - ~ ) Here

we

use

for dimensionless

Ivl

~ v

>

0

(3)

e~p ( - v2/2 )

scaling of time, s p a c e ,

field strength the units as fo.ows [t } =0a -1

[El -(4Knome)ll2.

velocity a n d

[x]= A D, [~1 :

electric

(~'elm)~-i7

Pel

T h e aim of the theory is to construct a n d explain an analytical solution (which h a s to be in a g o o d a ~ r e e m e n t with kinetic simulations) starting from the original points (1)-(3). Let's note that strictly s p e a k i n g the propos e d theory won't to b~quasilinear o n e in a c o m m o n s e n s e b e c a u s e w e do not a s s u m e W / n T <:< :I_ arid the l i n e a r i n d e p e n d e n c e of E k - m o d e s . Su}3~tituting (3)

into (I)

D (v,t)=o.(G,t) e ~ (v-~)_ The

coefficient D(v,I)

a n d integrating w e

get

~I~3_&I,W, 2 ( v _ v ) + ~ e °C(v-v~)/~3

represented

(4)

in the form of the e x p a n s i o n

D(v,t):

,~. Bn(t) (v-~) n (~) n=O s a t i s f i e s the eq. (4) for a r b i t r a r y fbl4nctions D (t) _= D ( v it) ( b o u n d a r y condiUon at v - v ) and B ( t ) ( B _ ( t ) = ~ , < B ( t ) / 2 ) ~ i f the othg; expansion coef• i ~ O ficients are ~eterm,ned b y re4at,. ons B I - O~ D I B n : B OL n /n.(n~3):Bo=,< . Hereby the parameters ~ (t) and B ( t ) are connected by eq. ~ I o ~ 3 = B - D t or b y its i n t e g r a l ~ ( t ) = [ ~ o 2 + , l . l 2 ~t (D - B ) dt e~) o

%_ Xto)

~hus

the

self-consistent

D(Vit)=:B

solution

e(~(v-v~: ) + [ 1 +

for

d.f,

from

~/~ ( v - v : ) ]

class

(D

(3)

has

the

- B)

form

(7)

T h e steady-state condition]im(Df')=gonst is realized for B(t)--~D (t), het4eby( see(6.J )o~-~ ~ _ _ % " ~ (Pig.li2).~f'he connection b e t w e e n parameters v = k ~ "~ anc] the J--nitialvalue ~(to)..: ~ o c a n b e obtained from (3)

and

the condition of sign c h a n g e

In a c c o r d a n c e

with computer

of f M

experiments

Eo~e (v

(v~, to)

_f~/f~

= Oi/O :

for a v/ide range of E2=,O1 .:. "1 o

value a p p e a r s to b e in a v e r y w e a k d e p e n d e n c e on v ~ ~.5 )i that is natural c o n s e q u e n c e of the v e r y e n c e .fM~on v . T h e n according to (8) ~o:2"& .

stron~ d e p e n d -

S u m m a r i z i n g the results given a b o v e w e m a y c o n c l u d e that the electron tail evolution is determined b y the one.-parametrical system of formulae including (3),(5)-(8) a n d the relation for s p e c t r u m density of w a v e e n e r g y in o n e of ~ o equivalent representations:

l~kl~=~/~ k.~(t) e~ (t)(~/"-~/k=) / ~ k + ( ~ k

o

-~) [*+ ~(t)(~/.-~/k k

1/~. C~)

k

n=3 Here k the p u m ~

is the w a v e nl~mber of the m o s t unstable m o d e region of turbulent spectrum.

characterizing

167

Yu.S. Sigov / Computer simulation

"".,%

O~ (t)

D

(t)

\\ /

~eff ( t")k

P i g . 2. t

:

\

.~.~..__ _ t

F'i~.3. T i m e - b e h a v i o r

T

i

v

i

i v I

v

2

=

of W =E2/2, and % = x,y e,x,y i/2,y

t >y,x ] [ <- (~'n.)2

"(E2= o "1'60o ~6~, pe

t

v 3

q~eo/q)io= 2 5 )

v o

15"ig01. S c h e m e of the high energetic tail formation, /5/.

/7/~-

Pip_~4. q'~y-pical p i c t u r e of In f ( v ) at different t in the c a e e }90 -- O. Ma.x-wellian form of fe('~• "~)

during the stochastic heating/t0/.

F'ig.3.

.1

Wy(t)

.01

3

T

~

(t)

X

2

.

.01~

"

,

.

,'

,

,

I

I

.

200

,

,

,

,,



,

,

,

i

-,

600

i

,

,

,



.

.

,

I

2

V

Yu.S. sigov / Computer simulation

168

The

Fi~.5.

of f e s %e~dy-sfa%e

4

profiles a% f h e

(1n cgse )

sfage

1 - E

=.05 o 2 - " ".O1 + the i~ifial ,ion s o u n d noise of h i g h level/3,5/

3 -

E2=.1

4: _

,,

" 1.

~

1,2,3,4

O

e<

=

=1/2; 1/1.6; 1/3~ 1/6.

I

I

i

I

~t

.1

to

'e

w(t~ X'

"

~i~.6.

q~he s a m e a s o n E'ig.6 for f h e c a s e

bur

q~eo/q~io = I

.......... i .........

i

. ......

i .........

I .........

I

....

/7/.

No

sound

the

kinetic

of

af stage

quasicollapse.

4 Compare

~x,y wifh

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

! . . . . . . . . .

1000

I

. . . . . . . . .

I

. . . . . . . . .

i

2000

t

fig.3!