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(7.131).This asymptotic expression, accurate for large θ, shows an exponential decreasein damping with increasing θ. When θ falls below 10, Eq. (7.133) becomesinaccurate, and the damping must be computed from Eq. (7.128), which employsthe Z-function. For the experimentally interesting region 1 < θ < 10, the followingsimple formula is an analytic fit to the exact solution:Im ω=Re ω ¼ 1:1θ7=4 exp θ2ð7:134ÞThese approximations are compared with the exact result in Fig. 7.31.What happens when collisions are added to ion Landau damping? Surprisinglylittle.
Ion-electron collisions are weak because the ion and electron fluids moveFig. 7.31 Ion Landau damping of acoustic waves. (A) is the exact solution of Eq. (7.128); (B) isthe asymptotic formula, Eq. (7.133); and (C) is the empirical fit, Eq. (7.134), good for 1 < θ < 10.7.9 Ion Landau Damping255almost in unison, creating little friction between them.
Ion–ion collisions (ionviscosity) can damp ion acoustic waves, but we know that sound waves in air canpropagate well in spite of the dominance of collisions. Actually, collisions spoil theparticle resonances that cause Landau damping, and one finds that the total dampingis less than the Landau damping unless the collision rate is extremely large. Insummary, ion Landau damping is almost always the dominant process with ionwaves and it decreases exponentially with the ratio ZTe/Ti.Problems7.7. Ion acoustic waves of 1-cm wavelength are excited in a singly ionized xenon(A ¼ 131) plasma with Te ¼ 1 eV and Ti ¼ 0.1 eV. If the exciter is turned off,how long does it take for the waves to Landau damp to 1/e of their initialamplitude?7.8.
Ion waves with λ ¼ 5 cm are excited in a singly ionized argon plasma withne ¼ 1016 m3, Te ¼ 2 eV, Ti ¼ 0.2 eV; and the Landau damping rate ismeasured. A hydrogen impurity of density nH ¼ αne is then introduced.Calculate the value of α that will double the damping rate.7.9.
In laser fusion experiments one often encounters a hot electron distributionwith density nh and temperature Th in addition to the usual population with ne,Te. The hot electrons can change the damping of ion waves and hence affectsuch processes as stimulated Brillouin scattering. Assume Z ¼ 1 ions with niand Ti and define θe ¼ Te/Ti, θh ¼ Th/Ti, α ¼ nh/ni, 1 α ¼ ne/ni, ε ¼ m/M andk2Di ¼ ni e2 =ε0 KT i .(a) Write the ion wave dispersion relation for this three-component plasma,expanding the electron Z-functions.(b) Show that electron Landau damping is not appreciably increased by nh ifTh Te.(c) Show that ion Landau damping is decreased by nh, and that the effectcan be expressed as an increase in the effective temperature ratio Te/Ti.7.10. The dispersion relation for electron plasma waves propagating along B0z^ canbe obtained from the dielectric tensor ε (Appendix B) and Poisson’s equation,∇ · (ε · E) ¼ 0, where E ¼ ∇ϕ.
We then have, for a uniform plasma,∂∂ϕεzz¼ εzz k2z ϕ ¼ 0∂z∂zor Ezz ¼ 0. For a cold plasma, Problem 4.4 and Eq. (B.18) in Appendix B giveεzz ¼ 1 ω2pω2For a hot plasma, Eq. (7.124) givesor ω2 ¼ ω2p2567 Kinetic Theoryεzz ¼ 1 ω2pk2 v2thZ0ωkvth¼0By expanding the Z-function in the proper limits, show that this equationyields the Bohm–Gross wave frequency [Eq. (4.30)] and the Landau dampingrate [Eq. (7.70)].7.10Kinetic Effects in a Magnetic FieldWhen either the dc magnetic field B0 or the oscillating magnetic field B1 is finite,the v B term in the Vlasov equation (7.23) for a collisionless plasma must beincluded.
The linearized equation (7.45) is then replaced by∂f1q∂fq∂fþ v ∇ f 1 þ ðv B0 Þ 1 ¼ ðE1 þ v B1 Þ 0mm∂t∂v∂vð7:135ÞResonant particles moving along B0 still cause Landau damping if ω/k ’ vth, buttwo new kinetic effects now appear which are connected with the velocity component v⊥ perpendicular to B0.
One of these is cyclotron damping, which will bediscussed later; the other is the generation of cyclotron harmonics, leading to thepossibility of the oscillations commonly called Bernstein waves.Harmonics of the cyclotron frequency are generated when the particles’ circularLarmor orbits are distorted by the wave fields E1 and B1.
These finite-rL effects areneglected in ordinary fluid theory but can be taken into account to order k2r2L byincluding the viscosity π. A kinetic treatment can be accurate even for k2 r 2L ¼ Oð1Þ.To understand how harmonics arise, consider the motion of a particle in an electricfield:E¼Ex eiðkxωtÞ x^ð7:136ÞThe equation of motion [cf. Eq. (2.10)] isq€x þ ω2c x ¼ Ex eiðkxωtÞmð7:137ÞIf krL is not small, the exponent varies from one side of the orbit to the other.We can approximate kx by substituting the undisturbed orbit x ¼ rL sin ωctfrom Eq.
(2.7):€x þ ω2c x ¼qEx eiðkrL sin ωc tωtÞmð7:138Þ7.10Kinetic Effects in a Magnetic Field257The generating function for the Bessel functions Jn(z) isezðt1=tÞ=2 ¼1Xn¼1tn J n ðzÞð7:139ÞLetting z ¼ krL and t ¼ exp(iωct), we obtaineikrL sin ωc t ¼1X1J n ðkr L Þeinωc t1q X€x þ ω2c x ¼ ExJ n ðkr L Þeiðωnωc Þtm 1ð7:140Þð7:141ÞThe following solution can be verified by direct substitution:1q XJ n ðkr L Þeiðωnωc Þtx ¼ Exm 1 ω2c ðω nωc Þ2ð7:142ÞThis shows that the motion has frequency components differing from the drivingfrequency by multiples hof ωc, and that thei amplitudes of these components areproportional to J n ðkr L Þ= ω2c ðω nωc Þ2 . When the denominator vanishes, theamplitude becomes large.
This happens when ω nωc ¼ ωc, or ω ¼ (n 1)ωc,n ¼ 0, 1, 2, . . .; that is, when the field E(x, t) resonates with any harmonic of ωc.In the fluid limit krL ! 0, Jn(krL) can be approximated by (krL/2)n/n!, whichapproaches 0 for all n except n ¼ 0. For n ¼ 0, the coefficient in Eq. (7.142)1becomes ω2c ω2 , which is the fluid result [cf. Eq. (4.57)] containing onlythe fundamental cyclotron frequency.7.10.1 The Hot Plasma Dielectric TensorAfter Fourier analysis of f1(r, v, t) in space and time, Eq. (7.135) can be solved for aMaxwellian distribution f0(v), and the resulting expressions f1(k, v, ω) can be usedto calculate the density and current of each species.
The result is usually expressedin the form of an equivalent dielectric tensor e, such that the displacement vectorD ¼ e · E can be used in the Maxwell’s equations ∇ · D ¼ 0 and ∇ B ¼ μ0D tocalculate dispersion relations for various waves (see Appendix B). The algebra ishorrendous and therefore omitted. We quote only a restricted result valid fornonrelativistic plasmas with isotropic pressure (T⊥ ¼ T||) and no zero-order driftsv0j; these restrictions are easily removed, but the general formulas are too clutteredfor our purposes. We further assume k¼kx x^ þkz z^ , with z^ being the direction of B0;2587 Kinetic Theoryno generality is lost by setting ky equal to zero, since the plasma is isotropic in theplane perpendicular to B0.
The elements of eR ¼ e/e0 are thenexx ¼ 1 þe yy ¼ 1 þX ω2p ebsω2bζ01Xn2 I n ðbÞZðζ n Þ11 X ω2p eb Xζn2 I n ðbÞ02 bωs1exy ¼ e yx ¼ iXs 0þ 2b2 I n ðbÞ I n ðbÞ Z ζ n1 hiω2p b X0e ζ 0 n I n ðbÞ I n ðbÞ Zðζ n Þ2ω11XX ω2p eb0ζexz ¼ ezx ¼nI n ðbÞZ ðζ n Þ021=2ω ð2bÞs11 hXX ω2p b1=2 0 0eb ζ 0I n ð bÞ I n ð bÞ Z ζ n 2e yz ¼ ezy ¼ iω 21sezz ¼ 1 X ω2psωeb ζ 021X1ð7:143Þ0I n ðbÞζ n Z ðζ n Þwhere Z(ζ) is the plasma dispersion function of Eq.
(7.118), In(b) is the nth orderBessel function of imaginary argument, and the other symbols are defined byω2ps ¼ n0s Z 2s e2 =e0 msζ ns ¼ ðω þ nωcs Þ=kz vthsωcs ¼ Zs eB0 =ms v2ths ¼ 2KT s =msζ 0s ¼ ω=kz vthsð7:144Þ1bs ¼ k2⊥ r Ls ¼ k2x KT s =ms ω2cs2The first sum is over species s, with the understanding that ωp, b, ζ 0 and ζ n alldepend on s, and that the stands for the sign of the charge. The second sum is overthe harmonic number n.
The primes indicate differentiation with respect to theargument.As foreseen, there appear Bessel functions of the finite-rL parameter b. [Thechange from Jn(b) to In(b) occurs in the integration over velocities.] The elements ofe involving motion along Z^ contain Z0 (ζ n), which gives rise to Landau dampingwhen n ¼ 0 and ω/kz ’ vth. The n 6¼ 0 terms now make possible another collisionlessdamping mechanism, cyclotron damping, which occurs when (ω nωc)/kz ’ vth.Problem7.11. In the limit of zero temperature, show that the elements of e in Eq.
(7.143)reduce to the cold-plasma dielectric tensor given in Appendix B.7.10Kinetic Effects in a Magnetic Field2597.10.2 Cyclotron DampingWhen a particle moving along B0 in a wave with finite kz has the right velocity, itsees a Doppler-shifted frequency ω kzvz equal to nωc and is therefore subject tocontinuous acceleration by the electric field E⊥ of the wave. Those particles withthe “right” phase relative to E⊥ will gain energy; those with the “wrong” phase willlose energy. Since the energy change is the force times the distance, the fasteraccelerated particles gain more energy per unit time than what the slower decelerated particles lose.
There is, therefore, a net gain of energy by the particles, on theaverage, at the expense of the wave energy; and the wave is damped. Thismechanism differs from Landau damping because the energy gained is in thedirection perpendicular to B0, and hence perpendicular to the velocity componentthat brings the particle into resonance. The resonance is not easily destroyed byphenomena such as trapping. Furthermore, the mere existence of resonant particles0suffices to cause damping; one does not need a negative slope f0 (vz), as in Landaudamping.To clarify the physical mechanism of cyclotron damping, consider a wave withk¼kx x^ þkz z^ with kz positive.
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