Fundamentals of Plasma Physics
410 Chapter 14. Wave-particle nonlinearities resonant particle velocity range respectively map to the lower and upper bounds of ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 411 conserved, but the second moment is not. Change of the r ...
412 Chapter 14. Wave-particle nonlinearities In the range of velocities where∂^2 f 0 ,non−res/∂v^2 > 0 there will be a decrea ...
14.3 Echoes 413 damping mechanism and raises some interesting questions about how Landau dampingre- lates to entropy. This arise ...
414 Chapter 14. Wave-particle nonlinearities to this situation is the process of making a holographic image of an object. The ho ...
14.3 Echoes 415 ̃v 1 (ω,t)consists of a particular solution satisfying the inhomogeneous part of the equation (i.e., balances th ...
416 Chapter 14. Wave-particle nonlinearities so v ̃ 1 =− eE/m ̄ (p+iω)(p+ikv 0 ) . (14.95) The inverse Laplace transform gives ̃ ...
14.3 Echoes 417 In analogy to Eq.(14.88) the linearized density can be expressed as n 1 (t)= 1 2 π ∫ n ̃ 1 (ω,t)dω (14.101) wher ...
418 Chapter 14. Wave-particle nonlinearities 14.3.4The self-consistent non-linear Vlasov-Poisson problem These ideas carry over ...
14.3 Echoes 419 ping terms higher than second-order gives the second-order equation ∂f 2 ∂t +v ∂f 2 ∂x − eE 2 m ∂f 0 ∂v − e m E ...
420 Chapter 14. Wave-particle nonlinearities is the usual self-consistent linear dielectric response function. Thus, the driven ...
14.3 Echoes 421 a similar procedure for Fourier transforms gives F(g(x)h(x)) = ∫∞ −∞ g(x)h(x)eikxdx = ∫∞ −∞ [ 1 2 π ∫∞ −∞ ̃g(k′) ...
422 Chapter 14. Wave-particle nonlinearities Equation (14.129) may be solved forf ̃ 2 (p,k)to give f ̃ 2 (p,k)=−ie m k (p+ikv) ∂ ...
14.3 Echoes 423 from thebpulse in the second ̃φextfactor in Eq.(14.135) will be considered. We therefore substitute theacontribu ...
424 Chapter 14. Wave-particle nonlinearities andkb→−kb.The inverse Laplace transform gives φ ̃upper 2 (t,x) = φaφb 4(ka−kb)^2 e ...
14.3 Echoes 425 Theφ ̃ lower (t,x)term is obtained by lettingka−kb→−(ka−kb)and so ̃φ 2 (t,x)= ̃φupper 2 (t,x)+ ̃φlower(t,x). (14 ...
426 Chapter 14. Wave-particle nonlinearities 14.3.6Higher order echoes If one expands the Vlasov equation to higher than second- ...
14.4 Assignments 427 Show that the steady-state solution to this equation has the form (Fisch1978) FT∼exp − ∫w dw (2+Z)w ...
15 Wave-wave nonlinearities 15.1 Introduction Wave non-linearity is a vast subject that is not particularly specific to plasma p ...
15.1 Introduction 429 generator amplitude is increased further, the pump wave amplitude no longer increases in proportion to the ...
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