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Unformatted text preview: 16. Wave-particle interaction Reading: Shu, Vol.II, Ch.29 16.1 Landau damping We started our discussion of hydromagnetic waves with simple one-dimensional electrostatic fluctuations, the Langmuir waves. We derived their dispersion relation and their phase velocity as a function of wavenumber. What the MHD treatment did not tell us was the relationship between the waves and the plasma. Can the waves change plasma properties or, vice versa, can plasma excite or damp waves? This question is different from the finite amplitude reasoning that we did for acoustic waves, where we noted that non-linear behaviour occurs for waves of finite amplitude account of the perturbations showing up in the dispersion relation. Here we ask for interactions between individual particles and waves. We therefore need to embark on an analysis in kinetic theory. The one-dimensional Boltzmann equation for electrons reads ∂f ∂t + v ∂f ∂x- e m E ∂f ∂v = 0 (16 . 1) Suppose there is no large-scale electric field but only perturbations E 1 , and the distribution function is the sum of a steady-state term plus a small perturbation, f = f + f 1 . Then ∂f ∂t + v ∂f ∂x = 0 ⇒ ∂f 1 ∂t + v ∂f 1 ∂x = e m E 1 ∂f ∂v (16 . 2) With the usual wave ansatz for all perturbed quatities we obtain- ıω f 1 + vık f 1 = e m E 1 ∂f ∂v (16 . 3) and to close the system the Poisson equation ∂E 1 ∂x =- 4 π e n e, 1 ⇒ ık E 1 =- 4 π e Z ∞-∞ f 1 dv (16 . 4) Inserting f 1 from 16.3 gives 1 = ω 2 p n e k Z ∂f ∂v vk- ω dv (16 . 5) So if ∂f ∂v doesn’t vanish around the zero of the denumerator, then ω must be complex to satisfy Eq.16.5 and the Langmuir waves are unstable. We wish to know the sign of the imaginary part of the frequency, ω I , for E 1 ∝ exp( ı k x- ı ω R t ) exp( ω I t ) → ω I > wave growth < wave damping (16 . 6) 1 Setting ω = ω R + ı ω I we can rewrite Eq.16.5 as 1 = ω 2 p n e k Z ∂f ∂v ( vk- ω R + ıω I ) ( vk- ω...
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.
- Fall '08