Langmuir waves are excited in a Maxwellian plasma at two wavenumbers

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Kinetic theory, collisions and transport · Nonlinear, shock and reconnection physics

Plasma Physics, Ch The Vlasov equation and Landau damping

Problem

Langmuir waves are excited in a Maxwellian plasma at two wavenumbers, kλDe=0.3k\lambda_{De}=0.3 and kλDe=0.2k\lambda_{De}=0.2, where the exact roots give γ=−1.26×10−2 ωpe\gamma=-1.26\times10^{-2}\,\omega_{pe} and γ=−5.51×10−5 ωpe\gamma=-5.51\times10^{-5}\,\omega_{pe}. A trapped electron bounces at ωB=k(eϕ0/me)1/2\omega_B=k(e\phi_0/m_e)^{1/2}, where ϕ0\phi_0 is the wave's potential amplitude. ((a)) For each wavenumber, find the amplitude eϕ0/Tee\phi_0/T_e at which ωB\omega_B equals ∣γ∣|\gamma|. ((b)) Both waves are launched with eϕ0/Te=1.0×10−4e\phi_0/T_e=1.0\times10^{-4}. Decide for each whether linear Landau theory describes its decay. ((c)) Explain why the wave that damps more slowly is the one for which linear theory fails.

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