By A. G. Sitenko

ISBN-10: 0080250513

ISBN-13: 9780080250519

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**Extra info for Fluctuations and Non-Linear Wave Interactions in Plasmas**

**Example text**

69) V ! 70) where ω = ωι+ω 2 , k = ki+k 2 ; «|f«(«i, ki; ω2, k 2 ; ω3, k3) = ki;<*>2, k 2 ; ω3, k3) + */$/(ω2, k 2 ; ωι, k x ; ω3, k3) + x%(co3,k3; co2,k2; ω ΐ5 kx)}, ^{^(«I. ]Ä)W'>· where ω = ωι4·ω24-ω3, k = ki-f-k2+k3, ω' = ω2+ω3, k' = k 2 +k 3 . , ki; ω2, k 2 ; . . ; ω„, k„) = £ — ^ ^~h;\ X 1 ΓΛ Α ω ι + ω , + . . +io„-({ki+k 2 + . . +k„}-v)+/0L\ vi ^ \ d y v > (ki'V)\g , . W A 1 9 o>i j M l _ r ω ι J Svfcl r/i (k«-*)\a . W A I I \ω 2 +ω 3 + . . -l-ωn-({k2+k3-l- . . 69) is sufficient for a complete description of the linear electrodynamic plasma properties.

2) [ " l ~fa + + " 2 äto» ΛιΛω'k) ,, / /7(ω ^ν ,χ η δ ' ^ 0? s ^ " ( ω ' k ) ~ ' " δ^7 Λ ( ω ' k ) ä ? 7 ^ 1 ) ( ω ' k ) · (2 93) · NON-LINEAR ELECTRODYNAMIC EQUATIONS 39 Since the times t, t\ /", . . , are supposed to be independent variables, this set of equa tions is in fact indefinite. To remove this uncertainty we require the equations to contain no secular terms. The conditions that the secular terms of each equation vanish make a supplementary set of equations. 90) on the times /, /', t'\ .

4) and the following interrelation occurs: *>ij == ~2GiklGjmnAmjc/Lnii (4·^) where eikl is the completely antisymmetric unit tensor. f An account of the linear theory of electromagnetic waves in a plasma is given by Stix (1962), Shafranov (1963), Allis, Buchsbaum, and Bers (1963), Ginzburg (1970), Ginzburg and Rukhadze (1970), and Akhiezer, Akhiezer, Polovin, Sitenko, and Stepanov (1975). Waves in bounded plasmas were studied in detail by Vandenplas (1968). 3) reduces to two requirements that the real and the imaginary parts of A(co9 k) be zero: Re Λ(ω, k) = 0 and Im Λ(ω, k) = 0.

### Fluctuations and Non-Linear Wave Interactions in Plasmas by A. G. Sitenko

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