Fundamentals of Plasma Physics
450 Chapter 15. Wave-wave nonlinearities By defining the normalized variables τ = ωpet/ 2 χ = A 2 √ κTe/me ξ = xωpe/c η = 2Γ/ωpe ...
15.5 Digging a hole in the plasma via ponderomotive force 451 Caviton instability The instability outlined above can be describe ...
452 Chapter 15. Wave-wave nonlinearities The maximumγis found by taking the derivative of both sides with respect tok^2 and sett ...
15.5 Digging a hole in the plasma via ponderomotive force 453 By letting g= √ 2Ω coshθ (15.153) it is seen that dg=− √ 2Ω cosh^2 ...
454 Chapter 15. Wave-wave nonlinearities 15.6 Ion acoustic wave soliton The ion acoustic wave dispersion relation is ω^2 = k^2 c ...
15.6 Ion acoustic wave soliton 455 We now introduce dimensionless variables by normalizing lengths to the Debye length, velociti ...
456 Chapter 15. Wave-wave nonlinearities where the boundary condition at infinity has been used again. This can be written as d( ...
15.7 Assignments 457 Drazin and Johns (1989)). This theory is called inverse scattering (Gardner et al. 1967) and involves mappi ...
458 Chapter 15. Wave-wave nonlinearities discussed in this problem can be expressed in terms of Jacobi elliptic functions (Sagde ...
15.7 Assignments 459 (c) Using the results from (a) express the instability threshold in a form where ε 0 E^2 /nκTeis a function ...
16 Non-neutral plasmas 16.1 Introduction Conventional plasmas, the main topic of this book, consist of a quasi-neutral collectio ...
16.2 Brillouinflow 461 negatively biased electrode negatively biased electrode cylindricalelectron cloud coils to make axialmagn ...
462 Chapter 16. Non-neutral plasmas is called the and it is seen that real rootsω 0 exist only if the density is sufficiently lo ...
16.3 Isomorphism to incompressible 2D hydrodynamics 463 16.3 Isomorphism to incompressible 2D hydrodynamics Consider now the equ ...
464 Chapter 16. Non-neutral plasmas There is thus an exact isomorphism between the non-neutral plasma equations and the equation ...
16.5 Diocotron modes 465 reduces to the simple relationship Pθ= ∑N i=1 Pθi≃ qeB 2 ∑N i=1 r^2 i=const. (16.25) Equation (16.25) c ...
466 Chapter 16. Non-neutral plasmas Thelthazimuthal mode is assumed to have a time dependenceexp(−iωlt)so that φ 1 (r,θ,t)= ∑ l ...
16.5 Diocotron modes 467 Equation (16.42) reduces to the special case of rigid body rotation when the density is uniform, but in ...
468 Chapter 16. Non-neutral plasmas whereSis a constant. Since Eq.(16.48) is a second-order ordinary differential equation, it m ...
16.5 Diocotron modes 469 erty and Levy 1970) w ̄ = ∫t −∞ dt 〈 E 1 · ( J 1 +ε 0 ∂E 1 ∂t )〉 = ε 0 2 E 12 + ∫t −∞ dt〈E 1 ·J 1 〉. (1 ...
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