1629373397-425d4de58b7aea127ffc7c337418ea8d (846389), страница 42
Текст из файла (страница 42)
Thus we taken1 ¼ nueE1 k1m ðω kuÞ2 ½ cos ðkx ωtÞ cos ðkx kutÞ ðω kuÞt sin ðkx kutÞð7:90ÞThis clearly vanishes at t ¼ 0, and one can easily verify that it satisfies Eq. (7.78).These expressions for v1 and n1 allow us now to calculate the work done by thewave on each beam. The force acting on a unit volume of each beam isFu ¼ eE1 sin ðkx ωtÞðnu þ n1 Þð7:91Þ2407 Kinetic Theoryand therefore its energy changes at the ratedW¼ Fu ðu þ v1 Þ ¼ eE1 sin ðkx ωtÞ nu u þ nu v1 þ n1 u þ n1 v1dt①④③②!ð7:92ÞWe now take the spatial average over a wavelength. The first term vanishes becausenuu is constant. The fourth term can be neglected because it is second order, but inany case it can be shown to have zero average.
The terms ② and ③ can be evaluatedusing Eqs. (7.87) and (7.90) and the identities1h sin ðkx ωtÞ cos ðkx kutÞi ¼ sin ðωt kutÞ21h sin ðkx ωtÞ sin ðkx kutÞi ¼ cos ðωt kutÞ2ð7:93ÞThe result is easily seen to be"# dWe2 E21sin ðωt kutÞsin ðωt kutÞ ðω kuÞt cos ðωt kutÞnuþ ku¼2mdt uω kuðω kuÞ2ð7:94ÞNote that the only terms that survive the averaging process come from the initialconditions.The total work done on the particles is found by summing over all the beams:X dW udtu¼ð ð^f 0 ðuÞ dWf 0 ðuÞ dWdu ¼ n0dununudt udt uð7:95ÞInserting Eq. (7.94) and using the definition of ωp, we then find for the rate ofchange of kinetic energyðdW kε0 E21 2sin ðωt kutÞ¼ω p ^f 0 ðuÞdudt2ω ku#ðsinðωtkutÞðωkuÞtcosðωtkutÞku duþ ^f 0 ðuÞðω kuÞ21¼ ε0 E21 ω2p2ð7:96Þð1^f 0 ðuÞdu sin ðωt kutÞ þ u d sin ðωt kutÞð7:97Þω kuduω kuð1^f 0 ðuÞdu d u sin ðωt kutÞduω ku1¼ ε0 E21 ω2p211ð7:98Þ7.6 A Physical Derivation of Landau Damping241This is to be set equal to the rate of loss of wave energy density Ww.
Thewave energy consists of two parts. The first part is the energy density ofthe electrostatic field: hW E i ¼ ε0 E2 =2 ¼ ε0 E21 =4ð7:99ÞThe second part is the kinetic energy of oscillation of the particles. If we againdivide the plasma up into beams, Eq. (7.84) gives the energy per beam:hΔW k iu ¼1 nu e2 E212ku1þ4 m ðω kuÞ2ðω kuÞð7:100ÞIn deriving this result, we did not use the correct initial conditions, which areimportant for the resonant particles; however, the latter contribute very little tothe total energy of the wave. Summing over the beams, we have1 e2 E21hΔW k i ¼4 mð11f 0 ð uÞðω kuÞ22ku1þduω kuð7:101ÞThe second term in the brackets can be neglected in the limit ω/k vth, which weshall take in order to compare with our previous results.
The dispersion relation isfound by Poisson’s equation:kε0 E1 cos ðkx ωtÞ ¼ e Σ n1uð7:102ÞUsing Eq. (7.79) for n1, we have1¼e2nue2Σ¼ε0 m u ðω kuÞ2 ε0 mð11f 0 ðuÞduðω kuÞ2ð7:103ÞComparing this with Eq. (7.101), we findhΔW k i ¼1 e2 E21 ε0 m ε0 E21¼ hW E i¼4 m e24ð7:104ÞThusW w ¼ ε0 E21 =2ð7:105ÞThe rate of change of this is given by the negative of Eq.
(7.98):dW w¼ W w ω2pdtð11^f 0 ðuÞ d u sin ðω kuÞt duduω kuð7:106Þ2427 Kinetic TheoryIntegration by parts givesdW w¼ W w ω2pdt(sin ðω kuÞtu ^f 0 ðuÞω ku1d ^f sin ðω kuÞtduu 0ω kudu11ð1)The integrated part vanishes for well-behaved functions ^f 0 ðuÞ, and we havedWwω¼ Ww ω2pkdtð11^f 0 ðuÞ sin ðω kuÞt du0ω kuð7:107Þwhere u has been set equal to ω/k (a constant), since only velocities very close tothis will contribute to the integral.
In fact, for sufficiently large t, the square bracketcan be approximated by a delta function:ω ksin ðω kuÞt¼ limδ ukπ t!1ω kuð7:108Þω2p 0 ωdWwπ ω ^ 0 ωf0¼ W w ω2p¼ W w πω 2 ^f 0kkkkdtkð7:109ÞThusSince Im(ω) is the growth rate of E1, and Ww is proportional to E21 we must havedWw =dt ¼ 2½ImðωÞWwð7:110Þπ ω2p 0 ωImðωÞ ¼ ω 2 ^f 02 kkð7:111ÞHencein agreement with the previous result, Eq. (7.67), for ω ¼ ωp. Contour integration isa shortcut to this result but gives no indication of the particle physics involved.7.6.1The Resonant ParticlesWe are now in a position to see precisely which are the resonant particles thatcontribute to linear Landau damping.
Figure 7.25 gives a plot of the factor multi0plying ^f 0 ðuÞ in the integrand of Eq. (7.107). We see that the largest contributioncomes from particles with jω kuj < π/t, or jv vϕjt < π/k ¼ λ/2; i.e., those particlesin the initial distribution that have not yet traveled a half-wavelength relative to thewave. The width of the central peak narrows with time, as expected. The subsidiary7.6 A Physical Derivation of Landau Damping243Fig. 7.25 A function which describes the relative contribution of various velocity groups toLandau dampingpeaks in the “diffraction pattern” of Fig. 7.25 come from particles that have traveledinto neighboring half-wavelengths of the wave potential. These particles rapidlybecome spread out in phase, so that they contribute little on the average; the initialdistribution is forgotten.
Note that the width of the central peak is independent of theinitial amplitude of the wave; hence, the resonant particles may include both trappedand untrapped particles. This phenomenon is unrelated to particle trapping.7.6.2Two Paradoxes ResolvedFigure 7.25 shows that the integrand in Eq. (7.107) is an even function of ω ku, sothat particles going both faster than the wave and slower than the wave add toLandau damping. This is the physical picture we found in Fig.
7.24. On the otherhand, the slope of the curve of Fig. 7.25, which represents the factor in the integrandof Eq. (7.106), is an odd function of ω ku; and one would infer from this thatparticles traveling faster than the wave give energy to it, while those travelingslower than the wave take energy from it. The two descriptions differ by anintegration by parts. Both descriptions are correct; which one is to be chosen0depends on whether one wishes to have ^f 0 ðuÞ or ^f 0 ðuÞ in the integrand.A second paradox concerns the question of Galilean invariance. If we take theview that damping requires there be fewer particles traveling faster than the wavethan slower, there is no problem as long as one is in the frame in which the plasma isat rest.
However, if one goes into another frame moving with a velocityV (Fig. 7.26), there would appear to be more particles faster than the wave thanslower, and one would expect the wave to grow instead of decay. This paradox is2447 Kinetic TheoryFig. 7.26 A Maxwellian distribution seen from a moving frame appears to have a region ofunstable sloperemoved by reinserting the second term in Eq. (7.100), which we neglected. Asshown in Sect.
7.5.1, this term can make hΔWki negative. Indeed, in the frameshown in Fig. 7.26, the second term in Eq. (7.100) is not negligible, hΔWki isnegative, and the wave appears to have negative energy (that is, there is moreenergy in the quiescent, drifting Maxwellian distribution than in the presence of anoscillation).
The wave “grows,” but adding energy to a negative energy wave makesits amplitude decrease.7.7BGK and Van Kampen ModesWe have seen that Landau damping is directly connected to the requirement thatf0(v) be initially uniform in space. On the other hand, one can generate undampedelectron waves if f(v, t ¼ 0) is made to be constant along the particle trajectoriesinitially. It is easy to see from Fig. 7.24 that the particles will neither gain nor loseenergy, on the average, if the plasma is initially prepared so that the density isconstant along each trajectory.
Such a wave is called a BGK mode, since it was I. B.Bernstein, J. M. Greene, and M. D. Kruskal who first showed that undamped wavesof arbitrary ω, k, amplitude, and waveform were possible. The crucial parameter toadjust in tailoring f(v, t ¼ 0) to form a BGK mode is the relative number of trappedand untrapped particles. If we take the small-amplitude limit of a BGK mode, weobtain what is called a Van Kampen mode. In this limit, only the particleswith v ¼ vϕ are trapped. We can change the number of trapped particles by addingto f(v, t ¼ 0) a term proportional to δ(v vϕ).
Examination of Fig. 7.24 will showthat adding particles along the line v ¼ vϕ will not cause damping—at a later time,there are just as many particles gaining energy as losing energy. In fact, by choosingdistributions with δ-functions at other values of vϕ one can generate undamped VanKampen modes of arbitrary vϕ. Such singular initial conditions are, however, not7.8 Experimental Verification245physical. To get a smoothly varying f(v, t ¼ 0), one must sum over Van Kampenmodes with a distribution of vϕ’s.
Although each mode is undamped, the totalperturbation will show Landau damping because the various modes get out of phasewith one another.7.8Experimental VerificationAlthough Landau’s derivation of collisionless damping was short and neat, it wasnot clear that it concerned a physically observable phenomenon until J.
M. Dawsongave the longer, intuitive derivation which was paraphrased in Sect. 7.6. Even then,there were doubts that the proper conditions could be established in the laboratory.These doubts were removed in 1965 by an experiment by Malmberg and Wharton.They used probes to excite and detect plasma waves along a collisionless plasmacolumn. The phase and amplitude of the waves as a function of distance wereobtained by interferometry. A tracing of the spatial variation of the damped wave isshown in Fig. 7.27. Since in the experiment ω was real but k was complex, the resultwe obtained in Eq. (7.70) cannot be compared with the data.
Instead, a calculationof Im(k)/Re(k) for real ω has to be made. This ratio also contains the factor expv2ϕ =v2th which is proportional to the number of resonant electrons in a Maxwel-lian distribution. Consequently, the logarithm of Im(k)/Re(k) should be proportionalto (vϕ/vth)2. Figure 7.28 shows the agreement obtained between the measurementsand the theoretical curve.A similar experiment by Derfler and Simonen was done in plane geometry, sothat the results for Re(ω) can be compared with Eq.
(7.64). Figure 7.29 shows theirmeasurements of Re(k) and Im(k) at different frequencies. The dashed curveFig. 7.27 Interferometer trace showing the perturbed density pattern in a damped plasma wave[From J. H. Malmberg and C. B. Wharton, Phys. Rev. Lett. 17, 175 (1966)]2467 Kinetic TheoryFig. 7.28 Verification ofLandau damping in theMalmberg–Whartonexperiment (loc. cit.)represents Eq. (7.64) and is the same as the one drawn in Fig. 4.5. The experimentalpoints deviate from the dashed curve because of the higher-order terms in theexpansion of Eq. (7.59).
The theoretical curve calculated from Eq. (7.54), however,fits the data well.Problems7.1 Plasma waves are generated in a plasma with n ¼ 1017 m3 and KTe ¼ 10 eV.If k ¼ 104 m1, calculate the approximate Landau damping rate jIm(ω/ωp)j.7.2 An electron plasma wave with 1-cm wavelength is excited in a 10-eV plasmawith n ¼ 1015 m3. The excitation is then removed, and the wave Landaudamps away. How long does it take for the amplitude to fall by a factor of e?7.3 An infinite, uniform plasma with fixed ions has an electron distribution functioncomposed of (1) a Maxwellian distribution of “plasma” electrons with densitynp and temperature Tp at rest in the laboratory, and (2) a Maxwellian distribution of “beam” electrons with density nb and temperature Tb centered at v¼V x^(Fig. P7.3). If nb is infinitesimally small, plasma oscillations traveling in thex direction are Landau-damped.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
