A calculation proceeds as follows for a Maxwellian electron plasma with (where the…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Plasma Physics · Kinetic theory, collisions and transport

Plasma Physics, Ch The Vlasov equation and Landau damping

Problem

A calculation proceeds as follows for a Maxwellian electron plasma with kλDe=0.3k\lambda_{De}=0.3 (where the exact Maxwellian root is known to give γ=−1.26×10−2 ωpe\gamma=-1.26\times10^{-2}\,\omega_{pe}, damping). 'The dielectric function contains the integral ∫f^0e′(v)/(v−ω/k) dv\int \hat f_{0e}'(v)/(v-\omega/k)\,\mathrm{d}v, singular at v=ω/kv=\omega/k once ω\omega is real. To continue it from the region where it is well defined down to the real axis, indent the velocity contour so that it passes above the pole. The original Laplace contour in ω\omega was drawn above every singularity, so the velocity path should stay on the corresponding side too. This is equivalent to using the Plemelj form 1/(v−x+i0)=P(1/(v−x))−iπδ(v−x)1/(v-x+i0)=P(1/(v-x))-i\pi\delta(v-x) at x=ωr/kx=\omega_r/k. Substituting into ϵi=−π(ωpe2/k2)f^0e′(ωr/k)\epsilon_i=-\pi(\omega_{pe}^2/k^2)\hat f_{0e}'(\omega_r/k) then flips its sign, and carrying that sign through γ=−ϵi/(∂ϵr/∂ω)\gamma=-\epsilon_i/(\partial\epsilon_r/\partial\omega) gives γ=+1.26×10−2 ωpe\gamma=+1.26\times10^{-2}\,\omega_{pe}. Conclusion: the Langmuir wave grows exponentially in this plasma.' ((a)) Identify the error in this argument. ((b)) State the correct rule and the correct sign and value of γ\gamma.

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