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For the ions, we haveMn0∂vi1¼ en0 ðE1 þ vi1 B0 Þ γ i KTi ∇n1∂tð4:129ÞWith our choice of E1 and k, this becomesvix ¼ie Ex þ viy B0 csMωð4:130Þ1324 Waves in PlasmasFig. 4.50 Geometryof a magnetosonicwave propagatingat right angles to B0viy ¼iek γ i KTi n1ðvix B0 Þ þMωω M n0ð4:131ÞThe equation of continuity yieldsn1 k¼ viyn0 ωð4:132Þso that Eq. (4.131) becomesviy ¼ iek2 γ KTivix B0 þ 2 iviyMωω Mð4:133ÞWith the abbreviationAk2 γ i KTiω2 Mthis becomesviy ð1 AÞ ¼ iΩcvixωð4:134ÞCombining this with Eq. (4.130), we haveieiΩciΩcEx þvix ¼ð1 AÞ1 vixMωωωΩ2c =ω2ieExvix 1 ¼Mω1Að4:135Þ4.19Magnetosonic Waves133This is the only component of vi1 we shall need, since the only nontrivial component of the wave equation (4.81) isε0 ω2 c2 k2 Ex ¼ iωn0 eðvix vex Þð4:136ÞTo obtain vex, we need only to make the appropriate changes in Eq.
(4.135) and takethe limit of small electron mass, so that ω2 ω2c and ω2 k2 v2the :ie ω2k2 γ e KT eik2 γ e KT evex ¼Ex :!1Exmω ω2cω2 mωB20 eð4:137ÞPutting the last three equations together we have2"ie1AExε0 ω2 c2 k Ex ¼ iωn0 eMω1 A Ω2c =ω2ik2 M γ e KT eEþxωB20 eM!ð4:138ÞWe shall again assume ω2 Ω2c ; so that 1 A can be neglected relative to Ω2c /ω2.With the help of the definitions of Ωp and vA, we have 2Ω2pk2 c2 γ KT eω c 2 k 2 ¼ 2 ω 2 ð 1 AÞ þ 2 eMvAΩcΩ2pγ e KT e22 γ i KT i22 2ω c k 1þþ 2 ω k¼0MMv2AΩcð4:139ÞSinceΩ2p =Ω2c ¼ c2 =v2Að4:140ÞEq.
(4.139) becomesc2γ e KT e þ γ i KT iv2s2 22 2ω 1þ 2 ¼c k 1þ¼c k 1þ 2vAMv2AvA2ð4:141Þwhere vs is the acoustic speed. Finally, we have22ω22 vs þ vA¼cc2 þ v2Ak2ð4:142Þ1344 Waves in PlasmasThis is the dispersion relation for the magnetosonic wave propagating perpendicularto B0. It is an acoustic wave in which the compressions and rarefactions areproduced not by motions along E, but by E B drifts across E. In the limitB0 ! 0, vA ! 0, the magnetosonic wave turns into an ordinary ion acoustic wave.In the limit KT ! 0, vs ! 0, the pressure gradient forces vanish, and the wavebecomes a modified Alfvén wave.
The phase velocity of the magnetosonic modeis almost always larger than vA; for this reason, it is often called simply the “fast”hydromagnetic wave.4.20Summary of Elementary Plasma WavesElectron waves (electrostatic)B0 ¼ 0 or k k B0 :32ω2 ¼ ω2p þ k2 v2th2ω ¼k⊥B0 :ω2pþω2c¼ω2hIon waves (electrostatic)B0 ¼ 0 or k k B0 :ðPlasma oscillationsÞUpper hybridoscillationsω2 ¼ k2 v2sγ e KT e þ γ i KTiðAcoustic wavesÞMElectrostatic ion2 222ω ¼ Ωc þ k vscyclotron waves¼ k2k⊥B0 :or2ω ¼ω2l¼ Ω c ωcLower hybridoscillationsð4:143Þð4:144Þð4:145Þð4:146Þð4:147ÞElectron waves (electromagnetic)B0 ¼ 0 :k⊥B0 , E1 k B0 :k⊥B0 , E1 ⊥B0 :ω2 ¼ ω2p þ k2 c2ðLight wavesÞω2pc2 k 2ðO waveÞ¼1ω2ω2ω2p ω2 ω2pc2 k2¼1ðX waveÞω2 ω2 ω2hω2ð4:148Þð4:149Þð4:150Þ4.21The CMA Diagramk k B0 :135ω2p =ω2c2 k 2¼1ω21 ðωc =ωÞðR waveÞðwhistler modeÞð4:151Þω2p =ω2c2 k 2¼121 þ ðωc =ωÞωðL waveÞð4:152ÞðAlfven waveÞð4:153ÞIon waves (electromagnetic)B0 ¼ 0 :Nonek k B0 :ω2 ¼ k2 v2Ak⊥B0 :22ω22 vs þ vA¼cc2 þ v2Ak2ðMagnetosonic waveÞð4:154ÞThis set of dispersion relations is a greatly simplified one covering only theprincipal directions of propagation.
Nonetheless, it is a very useful set of equationsto have in mind as a frame of reference for discussing more complicated wavemotions. It is often possible to understand a complex situation as a modification orsuperposition of these basic modes of oscillation.4.21The CMA DiagramWhen propagation occurs at an angle to the magnetic field, the phase velocitieschange with angle. Some of the modes listed above with kkB0 and k ⊥ B0 changecontinuously into each other; other modes simply disappear at a critical angle. Thiscomplicated state of affairs is greatly clarified by the Clemmow–Mullaly–Allis(CMA) diagram, so named for its co-inventors by T.
H. Stix. Such a diagram isshown in Fig. 4.51. The CMA diagram is valid, however, only for cold plasmas,with Ti ¼ Te ¼ 0. Extension to finite temperatures introduces so much complexitythat the diagram is no longer useful.Figure 4.51 is a plot of ωc/ω vs. ω2p /ω2 or, equivalently, a plot of magnetic fieldvs. density. For a given frequency ω, any experimental situation characterized by ωpand ωc is denoted by a point on the graph.
The total space is divided into sections bythe various cutoffs and resonances we have encountered. For instance, the extraordinary wave cutoff at ω2 ¼ ω2c þ ω2p is a quadratic relation between ωc/ω and ω2p /ω2; the resulting parabola can be recognized on Fig. 4.51 as the curve labeled“upper hybrid resonance.” These cutoff and resonance curves separate regions ofpropagation and nonpropagation for the various waves. The sets of waves that canexist in the different regions will therefore be different.1364 Waves in PlasmasFig. 4.51 A Clemmow–Mullaly–Allis diagram for classification of waves in a cold plasmaThe small diagram in each region indicates not only which waves are present butalso how the phase velocity varies qualitatively with angle.
The magnetic field isimagined to be vertical on the diagram. The distance from the center to any point onan ellipse or figure-eight at an angle θ to the vertical is proportional to the phasevelocity at that angle with respect to the magnetic field. For instance, in the4.21The CMA Diagram137triangular region marked with an * on Fig. 4.51, we see that the L wave becomes theX wave as θ varies from zero to π/2. The R wave has a velocity smaller thanthe L wave, and it disappears as θ varies from zero to π/2. It does not turn into theO wave, because ω2 < ω2p in that region, and the O wave does not exist.The upper regions of the CMA diagram correspond to ω ωc : Thelow-frequency ion waves are found here.
Since thermal velocities have beenneglected on this diagram, the electrostatic ion waves do not appear; they propagateonly in warm plasmas. One can regard the CMA diagram as a “plasma pond”: Apebble dropped in each region will send out ripples with shapes like the ones shown.Problems4.26. A hydrogen discharge in a 1-T field produces a density of 1016 m3.(a) What is the Alfvén speed vA?(b) Suppose vA had come out greater than c.
Does this mean that Alfvénwaves travel faster than the speed of light?4.27. Calculate the Alfvén speed in a region of the magnetosphere whereB ¼ 108 T, n ¼ 108 m3, and M ¼ MH ¼ 1.67 1027 kg.4.28. Suppose you have created a laboratory plasma with n ¼ 1015 m3 andB ¼ 102 T. You connect a 160-MHz signal generator to a probe insertedinto the plasma.(a) Draw a CMA diagram and indicate the region in which the experiment islocated.(b) What electromagnetic waves might be excited and propagated in theplasma?4.29. Suppose you wish to design an experiment in which standing torsional Alfvénwaves are generated in a cylindrical plasma column, so that the standingwave has maximum amplitude at the midplane and nodes at the ends. Tosatisfy the condition ω Ωc, you make ω ¼ 0.1Ωc.(a) If you could create a hydrogen plasma with n ¼ 1019 m3 and B ¼ 1 T,how long does the column have to be?(b) If you tried to do this with a 0.3 T Q-machine, in which the singlycharged Cs ions have an atomic weight 133 and a density n ¼ 1018 m3,how long would the plasma have to be? Hint: Figure out the scalingfactors and use the result of part (a).4.30.
A pulsar emits a broad spectrum of electromagnetic radiation, which isdetected with a receiver tuned to the neighborhood of f ¼ 80 MHz. Becauseof the dispersion in group velocity caused by the interstellar plasma,the observed frequency during each pulse drifts at a rate given by df/dt ¼ 5 MHz/s.(a) If the interstellar magnetic field is negligible and ω2 ω2p ; show that1384 Waves in Plasmasdfc f3 2dtxfpwhere fp is the plasma frequency and x is the distance of the pulsar.(b) If the average electron density in space is 2 105 m3, how far away isthe pulsar? (1 parsec ¼ 3 1016 m.)4.31.
A three-component plasma has a density n0 of electrons, (1 ε)n0 of ions ofmass M1, and ε n0 of ions of mass M2. Let Ti1 ¼ Ti2 ¼ 0, Te 6¼ 0.(a) Derive a dispersion relation for electrostatic ion cyclotron waves.(b) Find a simple expression for ω2 when ε is small.(c) Evaluate the wave frequencies for a case when ε is not small: a 50–50 %D–T mixture at KTe ¼ 10 keV, B0 ¼ 5 T, and k ¼ 1 cm1.4.32. For a Langmuir plasma oscillation, show that the time-averaged electronkinetic energy per m3 is equal to the electric field energy density 1/2 E0hE2i.4.33. For an Alfven wave, show that the time-averaged ion kinetic energy per m3 isequal to the magnetic wave energy hB21 i/2μ0.4.34. Figure P4.34 shows a far-infrared laser operating at λ ¼ 337 μm.
WhenB0 ¼ 0, this radiation easily penetrates the plasma whenever ωp is less thanω, or n < nc ¼ 1022 m3. However, because of the long path length, thedefocusing effect of the plasma (cf. Fig. 4.30) spoils the optical cavity, andthe density is limited by the conditions ω2p < εω2 ; where ε 1. In the interestof increasing the limiting density, and hence the laser output power, amagnetic field B0 is added.(a) If ε is unchanged, show that the limiting density can be increased if lefthand circularly polarized waves are propagated.(b) If n is to be doubled, how large does B0 have to be?Fig.
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