Application of Wiener-Hermite Expansion to Strong Plasma Turbulence

Notizen 583

Application of Wiener-Hermite Expansion to Strong Plasma Turbulence

G o o d a r z A h m a d i

D e p a r t m e n t of Mechanical and Industrial Engineering, Clarkson College of Technology,

Potsdam, N e w V o r k 13676, U S A Z. Naturforsch. 3 8 a , 5 8 3 - 5 8 4 (1983);

received March 8, 1983 *

A W i e n e r - H e r m i t e set of statistically orthogonal r a n d o m functions in phase space is introduced for investigation of electrostatic plasma turbulence. Expansions of the r a n d o m distribution functions in terms of the r a n d o m base are considered and the e q u a t i o n s governing the dynamics of the deterministic W i e n e r - H e r m i t e Kernels are derived.

Introduction

Expansion of a r a n d o m function in terms of an orthogonal r a n d o m base was introduced by C a m e - ron and Martin [1] and Wiener [2], M e e c h a m and Siegel [3] and M e e c h a m and Jeng [4] applied this technique to the p r o b l e m of h y d r o d y n a m i c turbu- lence. Recently, Jahedi and A h m a d i [5] used it in their study of nonlinear structures subjected to random loads. T h e t e c h n i q u e is now well known as the Wiener-Hermite expansion method.

T h e possible utility of the Wiener-Hermite ex- pansion in closure of strong plasma turbulence was pointed out by A h m a d i [6].

In the present work the W i e n e r - H e r m i t e method is applied to the problem of strong electrostatic plasma turbulence. Statistically orthogonal r a n d o m base functions in phase space are introduced. The r a n d o m distribution functions of ions and electrons are expanded in terms of the W i e n e r - H e r m i t e set and the equations for the deterministic kernels are derived. " C l o s u r e " is achieved by discarding the forth and higher order terms in the Wiener-Hermite series. Deterministic evolution equations for the Wiener-Hermite kernel functions are derived and discussed.

Basic Equations

The equations governing the evolution of the distribution functions of electrons and ions in an

* The first version was received September 19, 1983.

Reprint requests to Prof. G. A h m a d i , D e p a r t m e n t of Mechanical and Industrial Engineering, Clarkson College of Technology, Potsdam, N e w York 13676, USA.

electrostatic collisionless plasma are the well known Vlasov-Poisson equations. These are given by

61 oxjl my M x' (1)

t)MX,t)] = 0, a = 1 , 2 , x-x1 Or,

where X stands for the pair x, v. Here, x is the posi- tion vector, v is the velocity vector, ea and m , are charges and masses of particles, a = 1, 2 standing for electrons and ions, respectively. S u m m a t i o n conven- tion is employed on Latin indices. W h e n the plasma is in a turbulent state, the distribution functions become random functions of space and time.

To construct an a p p r o p r i a t e W i e n e r - H e r m i t e random base, we introduce a two c o m p o n e n t white noise process oca(X) d e f i n e d in the p h a s e space.

aa(X) has the following statistical properties:

<MA)> = 0, <MAV/?(A

)> = <M(*'-*

)' (2)

where an angular bracket stands for the expected value. T h e elements of the W i e n e r - H e r m i t e set are defined by

H^(X)= 1, H^{X) = a,{X), (3)

H$\X\ X2) = fla (A1) aß( X2) - öxßö(X* - X2) , . . . .

This set is a statistically orthogonal complete base.

T h e expansions of the distribution functions in terms of the W i e n e r - H e r m i t e set are given by

M X ) = ^ ( X ) + X f j Fty H<j\X>) d6*1

ß X1

+ X Z JJ f f F &

A , * u

)

ß y X' X<

x (A1, A2) d6Xl d6A2 + . . . ,

(4)

where F t y , . . . are deterministic kernel func- tions whose t i m e d e p e n d e n c e is implicitly under- stood. T h e first term in the series is the m e a n value of the corresponding distribution function. T h e second term is the G a u s s i a n part and the third and higher order terms correspond to the n o n - G a u s s i a n part of the distribution functions. Substituting the series given by (4) into (1), multiplying by various elements of the W i e n e r - H e r m i t e set taking expected values and m a k i n g use of orthogonality properties,

0340-4811 / 83 / 0500-0583 $ 01.3 0 / 0 . - Please o r d e r a reprint rather t h a n m a k i n g y o u r own copy.

584 Notizen we find Equations ( 6 - 7 ) are the basic equations governing

, , the evolutions of the deterministic unknown kernels 1 + , , J L fi0) ( A 0 + —S ^ J J d6* ' ;- T T F™ a n d F o r §i v e n i n i t i a l c o n di t i o n s , i s

dt dxjl ma M x1 x — x possible to find the time development of these func- r r ( 0 ) / v \ piO)( v h /cv t i o n s- Equation (4) then provides explicit expres- di'i * K J sions for the distribution functions of ions and elec- + X II F*/} (X. X2) Fj/J (A'1, X2) d6^2] = 0 , trons. Various order statistics of the distribution

ß xi functions could be generated by algebraic manipula- tion of the series expansion (4) and the known prop-

^ + r. d j ^ x ' } erties of the elements of the Wiener-Hermite set.

6t ' dxj

e* v _ ff j 6 v i -V/~A'/ Further Remarks

+ — X ^ J f d

^

X' Here, a formal derivation for the Wiener-Hermite

a

x _ (Fji°)(Arl) F\ljl(X, X') + F^^X) l*lUX\X') series expansions of the distribution functions of ions and electrons of an electrostatic plasma in + 2 X II d6!2! ^ ^ ! !2) F$y(X{ X' X2) turbulent state is presented. The random base con-

y x- ' ' sidered is time independent. However, generaliza- + F]}.}(X\X2) F{2fJ..(X X' X2)]', = 0 (6) tions to time dependent base could be carried out

with out difficulty. Finally, it should be noted that 6 0 \ ... , the positive definitness of the distribution functions

_ + r . _ f(2) ( X X' X") v u w , , , l l u w w lllv »««www»«

5t J dx j x ' ' imposes further restrictions on the kernel functions n v. — r1. a which must be addressed before a numerical solu-

^

\

r r J 6 v i

+ -L en JJ d X , 3 tion could be attempted.

MFI /I X' X X VI J

X [ j Fty (X, X') Fipy (X\ X") Acknowledgements

+ 4 F[\)(X,X") F$(X\X') The author would like to thank Dr. A. Hirose of +- F*0) X) F{2] X1 X' X" University of Saskatchewan for many helpful dis-

+ " ' ' ^ cussions. Thanks is also given to Ms. R. Price for + Fj,0^*1) F%{X, X\ X")} = 0 . (7) typing of the manuscript.

[1] R. H. C a m e r o n and W. T. M a r t i n . Ann. M a t h . 48, 885 (1947).

[2] N. Wiener, N o n l i n e a r Problem in R a n d o m Theory, John Wilev & Sons, Inc., N e w York 1958.

[3] W. C. M e e c h a m and A. Siegel, Phys. F l u i d s 7, 1178 (1969).

[4] W. C. Meecham and D. T. Jeng, J. Fluid Mech. 32, 225 (1968).

[5] A. Jahedi and G. A h m a d i , J. Appl. Mech. Trans.

ASME (in press).

[6] G. A h m a d i . Bull. Amer. Phys. Soc. 27, 1203 (1983).