Fundamentals of Plasma Physics
390 Chapter 13. Fokker-Planck theory of collisions and so, using Eq.(13.44), hF(v)= nFmT μ ( mF 2 πκTF ) 3 / 2 ∫ exp ( −mFv′^2 / ...
13.2 Statistical argument for the development of the Fokker-Planck equation 391 which is the ratio of the beam velocity to the t ...
392 Chapter 13. Fokker-Planck theory of collisions Test particle is much faster than both ion and electron thermal velocities, ...
13.3 Electrical resistivity 393 and the slowing down time becomes τs= 3 √ ππε^20 nee^2 lnΛ m^2 T q^2 T 1 ( Z ( 1+ mT mi )( mi 2 ...
394 Chapter 13. Fokker-Planck theory of collisions The current density will be J=niqiui+neqeue. (13.75) It is convenient to tran ...
13.5 Assignments 395 13.4 Runaway electric field The evaluation of the error function in Eq.(13.78) involved the assumptionthatu ...
396 Chapter 13. Fokker-Planck theory of collisions (a) Show that the Rosenbluthhpotential can be approximated as hF(v) = mT μ ∫ ...
13.5 Assignments 397 (e) Show that a steady-state equilibrium can develop where, because of collisions with background electrons ...
14 Wave-particle nonlinearities 14.1 Introduction Linear models are straightforward and rich in descriptive power becausethey ar ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 399 interpreted in terms of conservation of wave energy and ...
400 Chapter 14. Wave-particle nonlinearities where it is implicit that the magnitude of terms with subscriptnis of orderǫnwhere ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 401 so that ∫ f ̄ 0 (v,t)dv=1. (14.5) This definition causes ...
402 Chapter 14. Wave-particle nonlinearities On applying this postulate, Eq.(14.10) reduces to thequasi-linear velocity-space di ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 403 This expression can be further evaluated by invoking the ...
404 Chapter 14. Wave-particle nonlinearities and ω(−k)+ω(k)=2iωi(k) (14.28) in which case Eq.(14.19) reduces to 〈E 1 f 1 〉= i 2 ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 405 where thequasi-linear velocity space diffusion coefficie ...
406 Chapter 14. Wave-particle nonlinearities 14.2.2Conservation properties of the quasilinear diffusion equation Conservation of ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 407 Using Eq.(14.36) gives ∂WP ∂t = − ∫ dvmv 2ie^2 ε 0 m^2 ∫ ...
408 Chapter 14. Wave-particle nonlinearities while the non-resonant portion can be written as a principle-part integral.Thus, Eq ...
14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 409 and the principle part integral in Eq.(14.58) can simila ...
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