ION TAIL FORMATION AND SATURATION OF THE ION-ACOUSTIC INSTABILITY

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ION TAIL FORMATION AND SATURATION OF THE ION-ACOUSTIC INSTABILITY

Tu. Khiet

To cite this version:

Tu. Khiet. ION TAIL FORMATION AND SATURATION OF THE ION-ACOUSTIC INSTABIL- ITY. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-605-C7-606. �10.1051/jphyscol:19797293�.

�jpa-00219282�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZQment au n07, Tome 40, JuiZZet 1979, page C7- 605

ION

TAU, FORMATION

AND

SATURAflON OF THE

ION-ACOUSTIC

INSTABILITY

Tu. Khiet.

Laboratoire de Physique des PZasms, Universite' de Paris XI, centre drUrsay, Ba^t 212, 91405 Orsay Cddex, Frame.

I. INTRODUCTION

I n a turbulent plasma

,

e s p e c i a l l y when an ion- acoustic i n s t a b i l i t y i s p r e s e n t , i t has been shown experimentally (''and numerically ( 2 ) t h a t t h e ion d i s t r i b u t i o n function shows two d i f f e r e n t populati- ons constituted by a bulk having a lower temperatu- r e and a t a i l corresponding t o a much higher one.

This phenomenon i s very s i g n i f i c a n t f o r t h e fusion since t h e most e f f e c t i v e contribution t o it comes from t h e high energy p a r t of t h e ion population.

I n t h i s communication we s h a l l f i r s t propose a simple method allowing t o follow t h e time evolution of t h e average d i s t r i b u t i o n function of a p a r t i c l e species.0ur formalism r e v e a l s t o be more powerful and more accurate than t h e usual quasilinear one, b e s i d e s , i t presents an advantage with respect t o t h e l a t t e r i n t h e sense t h a t we have not t o solve a diffusion eauation with a diffusion c o e f f i c i e n t depending on ve1ocity.A~ an application,we subse-

ceeding i n magnitude t h e ion-acoustic speed c s 9

where ;c ( \ / M ) ' / ~ , M (m) being t h e ion(e1ectron).

mass.In t h e s e conditions,ion waves a r e generated C t h frequecies and growth-rates given by

2 2 1 / 2 .

+ r .

@ = kcs/(l+k

h

)

,

_{e 1}

where t h e notations a r e standard and Fi i s t h e ave rage d i s t r i b u t i o n function of t h e ion species. In t h e e a r l y s t a g e of t h e i n s t a b i l i t y , i o n damping i s negligiljle but when t h e time grows,the waves grow i n amplitude and then drive t h e ionsto higher velo c i t y and. simultaneously heat them which i n t u r n s enhance t h e absortion of t h e waves.In order t o t a - ke i n t o account of t h i s retro-effect,we do not t r y here t o reduce t h e equation f o r Fi t o a diffusion- l i k e equation.In t h i s respect

,

we make b e n e f i t of t h e r e l a t i o n F(?,t) = ( ~ ( ? ( t ) ) ) where

<. ..>

^denotes

ensemble average, H($) i s t h e value tagen by F a t quently consider t h e problem of s a t u r a t i o n of t h e

t = O ,?(t) i s t h e v e l o c i t y of a p a r t i c l e moving un- ion-acoustic i n s t a b i l i t y driven by an electron

der t h e influence of t h e turbulent f i e l d from t h e d r i f t current i n a plasma having a l a r g e electron-

point

(?,?)

a t t=O.Using a Fourier analysis of ~ ( v ) to-ion temperature r a t i o . We then show t h a t t h e

two-temperature behaviour of t h e ion d i s t r i b u t i o n which y i e l d s ~ ( ? , t ) i n t h e form function r e s u l t s from t h e exchange of energy and

momentum between waves and partic1es;The hot com- ponent then serves t o quench t h e i n s t a b i l i t y , t h e spectrum and t h e turbulent energy of t h e satura- t e d s t a t e can then be calculated within t h e b a s i c assumption t h a t enhanced ion-damping i s t h e most e f f i c i e n t mechanism of absorption.

II. BASIC CONSIDERATIONS.

We consider a plasma with T /T. )) 1 where

e 1 Te

(Ti) i s t h e electron (ion)temperature.The electron d i s t r i b u t i o n function i s assumed t o be a s h i f t e d Maxwellian with a drifted v e l o c i t y

5

l a r g e l y ex-

t h e problem i s now redaced t o evaluate an o r b i t - dependent function i n a s i m i l a r way a s i n DURREE- WEINSTOCK theoryeusing a well-known technique ( 3 ) devoted t o t h i s formalism, t h e two relevant cumu- l a n t s can be calculated i n t h e lowest order s s

where = ?(t )-? i s t h e v e l o c i t y increment .In order t o evaluate e x p l i c i t l y t h e above q u a n t i t i e s we assume t h a t t h e wave spectrum,i .e

. ^,

^Ik

^,

^{i s}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797293

nearly isotropic around the direction of vd

.

^Within

this basic assumption,and converting the sum appear- ing in (3) into integral,integration over angles can be carried out to yield the results in the form

in which the suffices refer to the direction of vd and the explicit expressions of various quantities appearing in

(4)

are quite long to be written down here .Putting

(4)

into (2) and performing integra- tion over p-space we obtain -$ F in the form

2 2

where v2 = 2Ti/M + 4at ; weff= 2Ti/M + haL are eff

still function of v by virtue of (3). Inspections of the analytic expressions of the various veloci- ties appearing in

(4)

show that U is always negli- gible and the two other quantities behave as

v2 _{ef f} S-w2ff z 2 T i ( l + d ~ e / ~ i ) / ~ for v

<

^cs

a

v 2

eff = (cz/v

?)/

^ak

E

^(k) for v> c s / n

dl

^{_ * .}

WEff= ( c ~ / ~ u d l ) ldkil-=)E(k) for v>cgfi k* k2V2

(6)

where the lower limit of integration is the non trivial solution of the equation

a

= k v ;

d

is the characteristic turbulent-to-kinetic energy ratio

,

^and E ( k ) =k2~(k)/4~2i

7

^=(Xm 1 8 ~ ) ~ ' ~ Choosing Te/Ti = 100 and d

=

as typical va- lues in an ion-heating experiment we deduce from

(6)

that the ion-body is only slighly heated and an ion-tail is formed in the region v) cs/

fi

which corresponds to the resonantinteraction be- tween waves and ions.

Balancing the electron growth rate with the ion damping given by (l),using

(5)

and

(6),

we obtain at saturation

-1 k2v2 =

[

^Log

---off

w 2

7')

as a first approximation,whe..-e v is to be cal- eff

culated by(6) with k * ~ k.

From (7) the wave spectrum results in the form

the total turbulent energy is then approximately

= 0.25

7

^{/Log(l/? )}

111. CONCLUSION.

We have developed a simple method using a tech- nique borrowed in strongly turbulent plasma theory, this technique is quite genera1,it can be applied also to explain the existence of a non-thermal elec tron distribution function which plays a crucial role in a laser-produced plasma

.

IY

.

REFERENCES.

1) D.S. Prono and C.B.Wharton,Plasma Phys. 15,253, (1973 )

.

2) C.T. Dum,R.Chodura,D.Biskamp,Phys.Rev.Lett.

32,

1231 (1974).

3) J. Wei~stock and B.Bezzerides,Phys.Fluids L6, 2287, (1973)

.