1629373397-425d4de58b7aea127ffc7c337418ea8d (846389), страница 68
Текст из файла (страница 68)
4 can easily be generalized to anarbitrary number of charged particle species and an arbitrary angle of propagation θrelative to the magnetic field. Waves that depend on finite T, such as ion acousticwaves, are not included in this treatment.First, we define the dielectric tensor of a plasma as follows. The fourth Maxwellequation is∇ B ¼ μ0 j þ E0 E_ðB:1Þwhere j is the plasma current due to the motion of the various charged particlespecies s, with density ns, charge qs, and velocity vs:Xns qs vsðB:2Þj¼sConsidering the plasma to be a dielectric with internal currents j, we may write Eq.(B.1) as∇ B ¼ μ0 D_ðB:3ÞwhereD ¼ E0 E þijωðB:4ÞHere we have assumed an exp (iωt) dependence for all plasma motions.
Let thecurrent j be proportional to E but not necessarily in the same direction (because ofthe magnetic field B0ẑ); we may then define a conductivity tensor σ by the relation© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4417418Appendix B: Theory of Waves in a Cold Uniform Plasmaj¼σEðB:5ÞiD ¼ E0 I þσ E¼eEE0 ωðB:6ÞEquation (B.4) becomesThus the effective dielectric constant of the plasma is the tensore ¼ E0 ðI þ iσ=E0 ωÞðB:7Þwhere I is the unit tensor.To evaluate σ, we use the linearized fluid equation of motion for species s,neglecting the collision and pressure terms:ms∂vs¼ qs ðE þ vs B0 Þ∂tðB:8ÞDefining the cyclotron and plasma frequencies for each species asq B0 ωcs s ms 2 n0 q ω2ps s ,E0 msðB:9Þwe can separate Eq.
(B.8) into x, y, and z components and solve for vs, obtainingiqs Ex iðωcs =ωÞE yvxs ¼ms ω 1 ðωcs =ωÞ2iqs E y iðωcs =ωÞExv ys ¼ms ω 1 ðωcs =ωÞ2vzs ¼iqsEzms ωðB:10aÞðB:10bÞðB:10cÞwhere stands for the sign of qs. The plasma current isXn0s qs vsj¼sðB:11Þso thatX in0s iq2 Ex iðωcs =ωÞE yisjx ¼E0 ω1 ðωcs =ωÞmωEωs0s¼Xsω2ps Ex iðωcs =ωÞE y 2ω1 ðωcs =ωÞðB:12ÞAppendix B: Theory of Waves in a Cold Uniform Plasma419Using the identities11 ðωcs =ωÞ¼21ωωþ2 ω ωcs ω ωcs1ωωþ¼,1 ðωcs =ωÞ2 2 ω ωcs ω ωcsωcs =ωðB:13Þwe can write Eq. (B.12) as follows:11X ω2psjx ¼ E0 ω2 s ω2ωωþExω ωcs ω ωcs ωωþþiE y :ω ωcs ω ωcsðB:14ÞSimilarly, the y and z components are11X ω2psjy ¼ E0 ω2 s ω2ωωþiExω ωcs ω ωcs ωωþþEyω ωcs ω ωcsX ω2psiEzjz ¼ ω2E0 ωsðB:15ÞðB:16ÞUse of Eq.
(B.14) in Eq. (B.4) gives" 11X ω2psωωD x ¼ Ex þExE02 s ω2 ω ωcs ω ωcs #ω2psωωþ 2iE y :ω ω ωcs ω ωcsðB:17Þ420Appendix B: Theory of Waves in a Cold Uniform PlasmaWe define the convenient abbreviationsR1L1X ω2ps ω2sX ω2ps ω2s1S ðR þ LÞ2P1ωω ωcsωω ωcs1D ðR LÞ*2ðB:18ÞX ω2psω2sUsing these in Eq. (B.17) and proceeding similarly with the y and z components, weobtainE10 Dx ¼ SEx iDE yE10 D y ¼ iDEx þ SE yE10 Dz¼ PEzðB:19ÞComparing with Eq. (B.6), we see that0Se ¼ E0 @ iD0iDS0100 A E 0 eRPðB:20ÞWe next derive the wave equation by taking the curl of the equation ∇ E ¼ B__ obtainingand substituting ∇ B ¼ μ0 e E,€€ ¼ 1 eR E∇ ∇ E¼μ0 E0 eR Ec2ðB:21ÞAssuming an exp (ik r) spatial dependence of E and defining a vector index ofrefractionμ¼*ck,ωNote that D here stands for “difference.” It is not the displacement vector D.ðB:22ÞAppendix B: Theory of Waves in a Cold Uniform Plasma421we can write Eq.
(B.21) asμ ðμ EÞ þ eR E ¼ 0:ðB:23ÞThe uniform plasma is isotropic in the x y plane, so we may choose the y axis sothat ky = 0, without loss of generality. If θ is the angle between k and B0, we thenhaveμx ¼ μ sin θμz ¼ μ cos θμy ¼ 0ðB:24ÞThe next step is to separate Eq.
(B.23) into components, using the elements of eRgiven in Eq. (B.20). This procedure readily yields0S μ2 cos 2 θ@REiDμ2 sin θ cos θiDS μ2010 1Exμ2 sin θ cos θA@ E y A ¼ 0:0EzP μ2 sin 2 θðB:25ÞFrom this it is clear that the Ex, Ey components are coupled to Ez only if one deviatesfrom the principal angles θ = 0, 90 .Equation (B.25) is a set of three simultaneous, homogeneous equations; thecondition for the existence of a solution is that the determinant of R vanish:jjRjj = 0.
Expanding in minors of the second column, we then obtainðiDÞ2 ðP μ2 sin 2 θÞ þ ðS μ2 Þ ½ðS μ2 cos 2 θÞðP μ2 sin 2 θÞ μ4 sin 2 θ cos 2 θ ¼ 0:ðB:26ÞBy replacing cos2 θ by 1 sin2 θ, we can solve for sin2 θ, obtainingsin 2 θ ¼Pðμ4 2Sμ2 þ RLÞ: PÞ þ μ2 ðPS RLÞμ4 ðSðB:27ÞWe have used the identity S2 D2 = RL. Similarly,cos 2 θ ¼Sμ4 ðPS þ RLÞμ2 þ PRL:μ4 ðS PÞ þ μ2 ðPS RLÞðB:28ÞDividing the last two equations, we obtaintan 2 θ ¼Pðμ4 2Sμ2 þ RLÞ:Sμ4 ðPS þ RLÞμ2 þ PRLSince 2S = R + L, the numerator and denominator can be factored to give the coldplasma dispersion relation422Appendix B: Theory of Waves in a Cold Uniform Plasmatan 2 θ ¼Pðμ2 RÞðμ2 LÞðSμ2 RLÞðμ2 PÞðB:29ÞThe principal modes of Chap.
4 can be recovered by setting θ = 0 and 90 . Whenθ = 0 , there are three roots: P = 0 (Langmuir wave), μ2 = R (R wave), and μ2 = L(L wave). When θ = 90 , there are two roots: μ2 = RL/S (extraordinary wave) andμ2 = P (ordinary wave). By inserting the definitions of Eq. (B.18), one can verifythat these are identical to the dispersion relations given in Chap. 4, with the additionof corrections due to ion motions.The resonances can be found by letting μ go to 1.
We then havetan 2 θres ¼ P=SðB:30ÞThis shows that the resonance frequencies depend on angle θ. If θ = 0 , the possiblesolutions are P = 0 and S = 1. The former is the plasma resonance ω = ωp, while thelatter occurs when either R = 1 (electron cyclotron resonance) or L = 1 (ioncyclotron resonance). If θ = 90 , the possible solutions are P = 1 or S = 0.
Theformer cannot occur for finite ωp and ω, and the latter yields the upper and lowerhybrid frequencies, as well as the two-ion hybrid frequency when there is more thanone ion species.The cutoffs can be found by setting μ = 0 in Eq. (B.26). Again usingS2 D2 = RL, we find that the condition for cutoff is independent of θ:PRL ¼ 0ðB:31ÞThe conditions R = 0 and L = 0 yield the ωR and ωL cutoff frequencies of Chap. 4,with the addition of ion corrections. The condition P = 0 is seen to correspond tocutoff as well as to resonance.
This degeneracy is due to our neglect of thermalmotions. Actually, P = 0 (or ω = ωp) is a resonance for longitudinal waves and acutoff for transverse waves.The information contained in Eq. (B.29) is summarized in the Clemmow–Mullaly–Allis diagram. One further result, not in the diagram, can be obtainedeasily from this formulation.
The middle line of Eq. (B.25) readsiDEx þ S μ2 E y ¼ 0ðB:32ÞiEx μ2 S:¼DEyðB:33ÞThus the polarization in the plane perpendicular to B0 is given byFrom this it is easily seen that waves are linearly polarized at resonance (μ2 = 1)and circularly polarized at cutoff (μ2 = 0, R = 0 or L = 0; thus S = D).Appendix C: Sample Three-Hour Final ExamPart A (One Hour, Closed Book)1. The number of electrons in a Debye sphere for n = 1017 m3, KTe = 10 eV isapproximately(A)(B)(C)(D)(E)1350.147.4 1031.7 1053.5 10102. The electron plasma frequency in a plasma of density n = 1020 m3 is(A)(B)(C)(D)(E)90 MHz900 MHz9 GHz90 GHzNone of the above to within 10 %3.
A doubly charged helium nucleus of energy 3.5 MeV in a magnetic field of 8 Thas a maximum Larmor radius of approximately(A)(B)(C)(D)(E)2 mm2 cm20 cm2m2 ft4. A laboratory plasma with n = 1016 m3, KTe = 2 eV, KTi = 0.1 eV, and B = 0.3 Thas a beta (plasma pressure/magnetic field pressure) of approximately(A) 107(B) 106© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4423424Appendix C: Sample Three-Hour Final Exam(C) 104(D) 102(E) 1015.
The grad-B drift v∇B is(A)(B)(C)(D)(E)always in the same direction as vEalways opposite to vEsometimes parallel to Balways opposite to the curvature drift vRsometimes parallel to the diamagnetic drift vD6. In the toroidal plasma shown, the diamagnetic current flows mainly in thedirection(A)(B)(C)(D)(E)^þϕ^ϕ^þθ^θþ^z7. In the torus shown above, torsional Alfvén waves can propagate in thedirections(A)(B)(C)(D)(E)^r^θ^ϕ^þθ only^ onlyθ8. Plasma A is ten times denser than plasma B but has the same temperature andcomposition.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
