Electrons have the normalised Lorentzian equilibrium distribution f_ 0e (v) = a (v^2+a^2)

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

Electrons have the normalised Lorentzian equilibrium distribution f^0e(v)=aπ(v2+a2),\hat f_{0e}(v) = \frac{a}{\pi(v^2+a^2)}, where a>0a>0 is a constant speed. This function is analytic in the complex vv-plane except for simple poles at v=±iav=\pm ia. Ions are fixed. Work with the linearised Vlasov-Poisson dispersion relation ϵ(k,ω)=1−ωpe2k2∫CLf^0e′(v)v−ω/k dv,\epsilon(k,\omega) = 1-\frac{\omega_{pe}^2}{k^2}\int_{C_L}\frac{\hat f_{0e}'(v)}{v-\omega/k}\,\mathrm{d}v, defined first for Im ω>0\mathrm{Im}\,\omega>0 on the real vv-axis and then continued by Landau's rule. ((a)) For Im ω>0\mathrm{Im}\,\omega>0, close the velocity contour in the lower half of the complex vv-plane (where the integrand vanishes fast enough at infinity) and pick up the pole of f^0e′\hat f_{0e}' at v=−iav=-ia. Show that this gives the closed form ϵ(k,ω)=1−ωpe2(ω+ika)2\epsilon(k,\omega) = 1-\dfrac{\omega_{pe}^2}{(\omega+ika)^2}, valid for every complex ω\omega (no further continuation is needed because this expression is already analytic). ((b)) Find the roots of ϵ=0\epsilon=0 and identify which one decays in time. ((c)) Say, in one sentence, why this result is exact rather than an approximation, unlike the Maxwellian case.

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