THE TIME DEPENDENT TRANSPORT COEFFICIENT FOR THE TWO-COMPONENT PLASMA

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Submitted on 1 Jan 1979

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THE TIME DEPENDENT TRANSPORT COEFFICIENT FOR THE TWO-COMPONENT

PLASMA

W. Rozmus

To cite this version:

W. Rozmus. THE TIME DEPENDENT TRANSPORT COEFFICIENT FOR THE TWO- COMPONENT PLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-525-C7-526.

�10.1051/jphyscol:19797254�. �jpa-00219239�

JOURWL

PHYSIQUE CoZZoque C7, suppZt5men.t au

Tome

JuiZZet

page C7-

W. Rozmus.

~ ~ t ; t ~ t ~ fop flucZem

00681

p0zand.

t;

The time dependent transport coef f i- /Z/ $phlt) + Q ($J

q)pph(t] - \dzh*(x)~h\t-z) = RII)

0

oients for a two-component plasma /TcP/ where P is Zhe projector on the PD- will be found by systematic application subspace. The time dependent transport

of the projection operators technique. Coeficienta are given by the matrix Similar problem was recently studied by elements & k t 3

-<rut

others authors jl], L21, t33. We consider /3/ \(gs(t)= Id$ e ( ~ \ ~ - S J ) Q ( - t w + ~ ( s - 3 - ~ ~ 1 ~ ~ *

F

here a weakly coupled TCP described by Q (s-ql

the Vlasov-Landau kinetic equation, line- where Q = ^I - ^P is the projector on arized around the one-temperature equili- the non-PD subspace, and

brium /without external fields/. We have (p f) ⁼

2. ^\A% Y&) f t 4

found the important differences i n the fn the above qMX is the Maxwellian

behavlour of the shear viscosity and tribution function, y; are Hermite poly- friction ooefficients as compared to ana- nomical tensors spanning the basis of the l o ~ e u s quantities from [

The kinetic equation, we use, can be written in the form

atands for a vector

~ o u r i e r transforms of the dimensionless deviations of the both species distribution functions from the .equilibrium value,

the linearized Vlasov operctor matrix,

I-nee qe:

7 is the unit matrix and 3= I" 3* 3'L I

s3ei] describe oollisions.

13 is the only term responsible for

velocity space.

explicit evaluation of Rye /3/ is done by perturbation method with the acouraay up to terms proportio- nal to the square of the wave vector, Moreover the natural extension of the Grad 13-moments method C43 is used by

the finite dimensional appro- ximation for Q. The whole velocity space is spanned now by 26 Hermite polynomial tensors /13 for each spacies/. The inver- sion of the operator

+ ^Q

is now straightforward and requires knowledge of the eigenfunotions and coupling between electrons and ions. eigenvectors Of this operator in the

By projecting eq./l/ on the ten-dimen- finite dimensional subspace. These were skonal plasmadyllstmical /PD/ subspace calcwlated with required acouraoy in k /cf [3] / we derive the generalized PD expansion. Note, that w e should keep equations elements of C-iw

QIS-3-33)Q 1'"

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797254

which are of order k and k

, ^because

there are quantities like (plb3IQ

in /3/ which cannot be treated as small.

/They are of the zero order i n the small mass ratio

These terms give rise to new form of the transport coeffi- cient disscused below.

Presenting the results we restrict ourselves to the viscosity and

riotion coefficients, because these quantities show essential differences with those published previously [I . For this pur- pose we write down explicit expression for the Laplace transform of the dissipa-

ol

tive part of the pressure tensor n+, ^{, *=e&}

the frequency dependent shear viscosity coefficients are

i A I A

7;w

nok$ [ -=; ^{A .-]}

^OCb2)

/ 5 /

\$*I =n0b~[-eA:~a-~)'+ M-&-&) O ~ I

$= y ? ~ ,

R Q ~ T [A

- ^-

0k)

~eret(?d,~=e,i; j=1,2,3, denote the matrix elements of the Landau collision opera- tor- Explicitly.

A:=-Q.y.h

AT=

A,'

- q z z n ^A;

A =

P a ^t., (x-ljqe A = A ? / ( A ~ , - A , ' )

where ?I is the plasma parameter >=~ik-/<+lif?

and

is the electron plasma f re-

quency. Expressions / 5 / differ from that in [l] by terms proportional to A and A

.

By inspection theme terms arise from the additional terms in expansion of & ^/3/

discussed earlier.

For the friction coefficient 1; ^{we ob-}

tain the following formula

/6/ t x $ / ~ L Q , % ~ ( ~ * ) - ( A ; ' I ' / ( - ~ ~ - ~ ; \

usually omitted, contains the factor

( f ^.

spite of this , ^{this term}

should be kept as it produces contribu- tions of order1 in the dispersion laws for the plasmadynamical modes 152 .

References

1 G. Vasu, J. Plasma Phys. ,G/1976/, 299, 2 H. Baus, Physica,88A/1977/319,336,591, 3 R. Balescu,I. Paiva-Veretennicoff,

J. Plasma Phys. ,20/1978/, 231, 4 H. Grad,Handbuch der Physik, 1958,vol.XII 5 I?.

tad in J. Plasma Phys.

The second term on the r. h. s. of /6/,