For an electron plasma with

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Kinetic theory, collisions and transport

Plasma Physics, Ch Hot plasmas and free energy: beam instabilities, cyclotron damping and Bernstein waves

Problem

For an electron plasma with ωpe/∣Ωe∣=3\omega_{pe}/|\Omega_e| = 3, consider waves propagating exactly perpendicular to B0\mathbf{B}_0 (k∥=0k_\parallel = 0). The hot magnetised electrostatic response built from integrating along helical orbits gives ϵ⊥(ω)=1−2ωpe2Ωe2b∑ℓ≥1ℓ2Γℓ(b)(ω/Ωe)2−ℓ2,b=k⊥2rLe2.\epsilon_\perp(\omega) = 1 - \frac{2\omega_{pe}^2}{\Omega_e^2 b}\sum_{\ell\ge1}\frac{\ell^2\Gamma_\ell(b)}{(\omega/\Omega_e)^2-\ell^2}, \qquad b = k_\perp^2r_{Le}^2. ((a)) State the cold-plasma function from Stix's dielectric tensor that ϵ⊥\epsilon_\perp must reduce to as b→0b\to0 (with Te→0T_e\to0), and give the frequency at which that cold function vanishes (the upper hybrid resonance) in terms of ωpe\omega_{pe} and ∣Ωe∣|\Omega_e|. ((b)) One Bernstein branch is known to start, as k⊥rLe→0k_\perp r_{Le}\to0, exactly at that upper hybrid frequency. Evaluate this frequency as a multiple of ∣Ωe∣|\Omega_e| for ωpe/∣Ωe∣=3\omega_{pe}/|\Omega_e|=3, and state which two consecutive cyclotron harmonics it lies between.

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