Fundamentals of Plasma Physics
150 Chapter 5. Streaming instabilities and the Landau problem and ∇^2 φ 1 =− 1 ε 0 ∑ σ qσnσ 1. (5.5) As before, all first-order ...
5.1 Streaming instabilities 151 The maximum growth rate is found by maximizing the right hand side of Eq.(5.10) with the minus s ...
152 Chapter 5. Streaming instabilities and the Landau problem Now consider the more realistic situation where electrons stream w ...
5.2 The Landau problem 153 are worked out in the assignments showing that the maximum growth rate is maxωi≃ √ 3 2 ( me 2 mi ) 1 ...
154 Chapter 5. Streaming instabilities and the Landau problem the relevantfluid equations were linearized, (ii) a perturbation∼e ...
5.2 The Landau problem 155 a relationship betweenn 1 andφ 1 .Another relationship betweenn 1 andφ 1 is Poisson’s equation ∂^2 φ ...
156 Chapter 5. Streaming instabilities and the Landau problem and can be considered as a “half of a Fourier transform” since the ...
5.2 The Landau problem 157 (a) (b) (c) Im p Im p Im p Rep Rep Rep complex pplane complex pplane p defined thisregion only ...
158 Chapter 5. Streaming instabilities and the Landau problem Before returning to physics, recall another peculiarity of Laplace ...
5.2 The Landau problem 159 This is similar to Eq.(5.24), except that now the Laplace variablepoccurs instead of the Fourier vari ...
160 Chapter 5. Streaming instabilities and the Landau problem 5.2.3 The relationship between poles, exponential functions, and a ...
5.2 The Landau problem 161 is simply fˆ(p)=^1 p−q , definedforallpprovidedfˆ(p)remainsanalytic. (5.58) The Bromwich contour can ...
162 Chapter 5. Streaming instabilities and the Landau problem 5.2.4 Asymptotic long time behavior of the potential oscillation W ...
5.2 The Landau problem 163 dropping pole, v∥ ip k (a) (b) (c) integration contour integration contour integration contour pole ...
164 Chapter 5. Streaming instabilities and the Landau problem contour may be deformed so that the vertical portion is pushed to ...
5.2 The Landau problem 165 as χσ = − 1 2 k^2 λ^2 Dσ 1 π^1 /^2 ∫∞ −∞ dξ 1 (ξ−ip/kvTσ) ∂ ∂ξ exp(−ξ^2 ) = 1 k^2 λ^2 Dσ ^1 π^1 /^ ...
166 Chapter 5. Streaming instabilities and the Landau problem Z(α)= 1 π^1 /^2 P ∫∞ −∞ dξ exp(−ξ^2 ) (ξ−α) +iπ^1 /^2 exp(−α ...
5.2 The Landau problem 167 The ‘P’ has been dropped from the right hand side of Eq.(5.69) because there is no longer any problem ...
168 Chapter 5. Streaming instabilities and the Landau problem 5.2.6 Landau damping of electron plasma waves The plasma susceptib ...
5.2 The Landau problem 169 Since the oscillation has been assumed to be weakly damped,ωi<<ωrand so Eq.(5.79) can be Taylor ...
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