The continued kinetic dielectric function for electrostatic waves in an unmagnetised…

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

The continued kinetic dielectric function for electrostatic waves in an unmagnetised plasma can be split, via the Plemelj formula, into ϵ(k,ω)=ϵr(k,ω)+iϵi(k,ω)\epsilon(k,\omega)=\epsilon_r(k,\omega)+i\epsilon_i(k,\omega), with ϵi\epsilon_i coming from the residue at the pole on the Landau contour. Suppose a root of ϵ(k,ω)=0\epsilon(k,\omega)=0 has ω=ωr+iγ\omega=\omega_r+i\gamma with ∣γ∣≪ωr|\gamma|\ll\omega_r, and that near ωr\omega_r, ∣ϵi∣≪∣ϵr∣|\epsilon_i|\ll|\epsilon_r|. ((a)) By Taylor-expanding ϵ(k,ωr+iγ)=0\epsilon(k,\omega_r+i\gamma)=0 to first order in γ\gamma (treating ϵi\epsilon_i itself as already small, so a term γ ∂ϵi/∂ω\gamma\,\partial\epsilon_i/\partial\omega can be dropped), show that γ=−ϵi(k,ωr)∂ϵr/∂ω∣ωr.\gamma=-\frac{\epsilon_i(k,\omega_r)}{\partial\epsilon_r/\partial\omega\big|_{\omega_r}}. ((b)) For a single electron species with ϵi(k,ω)=−π(ωpe2/k2)f^0e′(ω/k)\epsilon_i(k,\omega)=-\pi(\omega_{pe}^2/k^2)\hat f_{0e}'(\omega/k) and, near ωr≈ωpe\omega_r\approx\omega_{pe}, the cold-fluid approximation ϵr≈1−ωpe2/ω2\epsilon_r\approx 1-\omega_{pe}^2/\omega^2, show that γ≈π2ωpe3k2f^0e′(ωpe/k).\gamma\approx \frac{\pi}{2}\frac{\omega_{pe}^3}{k^2}\hat f_{0e}'(\omega_{pe}/k).

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