Fundamentals of Plasma Physics
430 Chapter 15. Wave-wave nonlinearities hole in the plasma density. The pump wave can dig itself a channel in the plasma and th ...
15.2 Manley-Rowe relations 431 where it is assumed that A ̇j Aj <<ωjandδ ̇j<<ωj. (15.5) Each oscillator is considere ...
432 Chapter 15. Wave-wave nonlinearities Usingω 2 −ω 3 =−ω 1 ,δ 2 −δ 3 =θ−δ 1 ,ω 1 −ω 3 =−ω 2 ,δ 1 −δ 3 =θ−δ 2 and discarding no ...
15.2 Manley-Rowe relations 433 A corresponding set of relations for the action can be obtained by multiplying pairs of equations ...
434 Chapter 15. Wave-wave nonlinearities Figure 15.2: Photon at frequencyω 3 decaying into a photon at frequencyω 1 and a photon ...
15.3 Application to waves 435 it is seen that dA 2 dt = dτ dt dA 2 dτ =sinθ dA 2 dτ (15.27) and so d^2 A 2 dτ^2 = λ^2 A^23 16 m^ ...
436 Chapter 15. Wave-wave nonlinearities replacingδj.Hence, so long as the selection rulesω 3 =ω 1 +ω 2 andk 3 =k 1 +k 2 are sat ...
15.3 Application to waves 437 frequency modes, then the ponderomotive force contains a term at the difference (i.e., beat) frequ ...
438 Chapter 15. Wave-wave nonlinearities If three waves satisfy the selection rules ω 3 = ω 1 +ω 2 (15.44) k 3 = k 1 +k 2 , (15. ...
15.3 Application to waves 439 not be an electromagnetic wave because then all three waves would haveto be electromag- netic wave ...
440 Chapter 15. Wave-wave nonlinearities the low frequency beat component of ( ̃uhe ) 2 can be considered as modulating the effe ...
15.3 Application to waves 441 However, the high frequency linear electron equation of motion is ∂ ̃ue ∂t = qe me E ̃−^3 κTe men ...
442 Chapter 15. Wave-wave nonlinearities High frequency wave is an electromagnetic wave Here the wave is transverse so ∇·E ̃=0an ...
15.3 Application to waves 443 Although the two velocities are oscillating 90^0 out of phase relative to their respective elec- t ...
444 Chapter 15. Wave-wave nonlinearities where λ= ωpeqek 1 √ mimeω 2 ω 3 . (15.81) Defining the mode frequencies as ω^23 = ω^2 p ...
15.4 Non-linear dispersion formulation and instability threshold 445 Thus, if 1 /LC >> R/Lthe imaginary part of the freque ...
446 Chapter 15. Wave-wave nonlinearities where the sign of the exponent is chosen to be consistent with the convention that a si ...
15.4 Non-linear dispersion formulation and instability threshold 447 and also ω−ω 3 =x+iy−ω 3 ≃−ω 1. (15.105) These assumptions ...
448 Chapter 15. Wave-wave nonlinearities and substitution forx ̄andx ̄−∆in Eq.(15.113) results in the expression for the growth ...
15.5 Digging a hole in the plasma via ponderomotive force 449 and so Eq.(15.49) reduces to 1 n ∇ ̃n=− 1 2 me κTe ∇ ( ̃uhe ) 2 (1 ...
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