The D layer of the ionosphere (roughly 60-90 km altitude) is weakly ionised and…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Plasma Physics, Ch Plasma oscillations and light in an unmagnetised plasma

Problem

The D layer of the ionosphere (roughly 60-90 km altitude) is weakly ionised and collisional. Take the electron density there as ne=1.0×109 m−3n_e=1.0\times10^{9}\,\mathrm{m^{-3}} and the electron-neutral collision frequency as ν=1.0×106 s−1\nu=1.0\times10^{6}\,\mathrm{s^{-1}}. ((a)) Starting from the electron momentum equation with a friction term, me ∂v/∂t=−eE−meνvm_e\,\partial\mathbf v/\partial t=-e\mathbf E-m_e\nu\mathbf v, derive the conductivity σ(ω)\sigma(\omega) and the dielectric function ϵ(ω)=1−ωpe2ω(ω+iν)\epsilon(\omega)=1-\dfrac{\omega_{pe}^2}{\omega(\omega+i\nu)} for a Fourier mode ∝ei(k⋅x−ωt)\propto e^{i(\mathbf k\cdot\mathbf x-\omega t)}. ((b)) Show that as ω→0\omega\to0, σ(ω)→nee2/meν\sigma(\omega)\to n_ee^2/m_e\nu, the Drude (steady-current) conductivity, and explain why this cold collisional plasma behaves as a Lorentz oscillator with no restoring force (ω0=0\omega_0=0) but a finite damping rate ν\nu. ((c)) Evaluate ϵ(ω)\epsilon(\omega) numerically at f=1.0 MHzf=1.0\,\mathrm{MHz} for the D-layer numbers above, giving its real and imaginary parts, and say whether the sign of the imaginary part is consistent with absorption of the wave.

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