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The parametric decay instability can enhance the heating efficiency by converting laser energy into plasma wave energy, which is thentransferred to electrons by Landau damping. If an iodine laser with 1.3-μmwavelength is used, at what plasma density does parametric decay take place?8.13 (a) Derive the following dispersion relation for an ion acoustic wave in thepresence of an externally applied ponderomotive force FNL: 2ω þ 2iΓω k2 v2s n1 ¼ ikFNL =M,3008 Nonlinear EffectsFig.
8.17 Oscillograms showing the frequency spectra of oscillations observed in the device ofFig. 8.16. When the driving power is just below threshold, only noise is seen in the low-frequencyspectrum and only the driver (pump) signal in the high-frequency spectrum. A slight increase inpower brings the system above threshold, and the frequencies of a plasma wave and an ion wavesimultaneously appear. [Courtesy of R. Stenzel, UCLA.]where Γ is the damping rate of the undriven wave (when FNL ¼ 0).
(Hint:introduce a “collision frequency” v in the ion equation of motion,evaluate Γ in terms of v, and eventually replace v by its Γ-equivalent.)(b) Evaluate FNL for the case of stimulated Brillouin scattering in terms ofthe amplitudes E0 and E2 of the pump and the backscattered wave,respectively, thus recovering the constant c1 of Problem (8.10). (Hint:cf. Eq. (8.64).)8.14 In Fig. 8.17 it is seen that the upper sideband at ω0 + ω1 is missing. Indeed, inmost parametric processes the upper sideband is observed to be smaller thanthe lower sideband. Using simple energy arguments, perhaps with a quantummechanical analogy, explain why this should be so.8.6 Plasma Echoes301Fig. 8.18 Geometry of an ionospheric modification experiment in which radiofrequency waveswere absorbed by parametric decay. [From W.
F. Utlaut and R. Cohen, Science 174, 245 (1971).]8.6Plasma EchoesSince Landau damping does not involve collisions or dissipation, it is a reversibleprocess. That this is true is vividly demonstrated by the remarkable phenomenon ofplasma echoes. Figure 8.19 shows a schematic of the experimental arrangement.A plasma wave with frequency ω1 and wavelength λ1 is generated at the first gridand propagated to the right. The wave is Landau-damped to below the threshold ofdetectability. A second wave of ω2 and λ2 is generated by a second grid a distancel from the first one. The second wave also damps away. If a third grid connected to areceiver tuned to ω ¼ ω2 ω1 is moved along the plasma column, it will find anecho at a distance l0 ¼ lω2/(ω2 ω1).
What happens is that the resonant particlescausing the first wave to damp out retains information about the wave in theirdistribution function. If the second grid is made to reverse the change in theresonant particle distribution, a wave can be made to reappear. Clearly, this processcan occur only in a very nearly collisionless plasma. In fact, the echo amplitude hasbeen used as a sensitive measure of the collision rate. Figure 8.20 gives a physicalpicture of why echoes occur. The same basic mechanism lies behind observations3028 Nonlinear EffectsFig. 8.19 Schematic of a plasma echo experiment.
[From A. Y. Wong and D. R. Baker, Phys. Rev.188, 326 (1969).]of echoes with electron plasma waves or cyclotron waves. Figure 8.20 is a plot ofdistance vs. time, so that the trajectory of a particle with a given velocity is astraight line. At x ¼ 0, a grid periodically allows bunches of particles with a spreadin velocity to pass through. Because of the velocity spread, the bunches mixtogether, and after a distance l, the density, shown at the right of the diagram,becomes constant in time. A second grid at x ¼ l alternately blocks and passesparticles at a higher frequency. This selection of particle trajectories in space–timethen causes a bunching of particles to reoccur at x ¼ l0 .The relation between l0 and l can be obtained from this simplified picture, whichneglects the influence of the wave electric field on the particle trajectories. If f1(v) isthe distribution function at the first grid and it is modulated by cos ω1t, thedistribution at x > 0 will be given byω1 f ðx; v; tÞ ¼ f 1 ðvÞ cos ω1 t xvð8:84ÞThe second grid at x ¼ l will further modulate this distribution by a factorcontaining ω2 and the distance x l:hiω1 ω2f ðx; v; tÞ ¼ f 12 ðvÞ cos ω1 t x cos ω2 t ðx lÞvvω 2 ð x l Þ þ ω1 x1¼ f 12 ðvÞ cos ðω2 þ ω1 Þt 2vω2 ðx lÞ ω1 xþ cos ðω2 ω1 Þt vð8:85Þð8:86Þ8.6 Plasma Echoes303Fig.
8.20 Space–time trajectories of gated particles showing the bunching that causes echoes. Thedensity at various distances is shown at the right. [From D. R. Baker, N. R. Ahern, and A. Y.Wong, Phys. Rev. Lett. 20, 318 (1968).]The echo comes from the second term, which oscillates at ω ¼ ω2 ω1 and has anargument independent of v ifω 2 ð x l Þ ¼ ω1 xor0x ¼ ω2 l=ðω2 ω1 Þ lð8:87Þ3048 Nonlinear EffectsFig.
8.21 Measurements ofecho amplitude profiles forvarious separationsl between the driver grids.The solid circles correspondto the case ω2 < ω1, forwhich no echo is expected.[From Baker, Ahern, andWong, loc. cit.]The spread in velocities, therefore, does not affect the second term at x ¼ l0 , and thephase mixing has been undone. When integrated over velocity, this term gives adensity fluctuation at ω ¼ ω2 ω1. The first term is undetectable because phasemixing has smoothed the density perturbations.
It is clear that l0 is positive only ifω2 > ω1. The physical reason is that the second grid has less distance in which tounravel the perturbations imparted by the first grid, and hence must operate at ahigher frequency.Figure 8.21 shows the measurements of Baker, Ahern, and Wong on ion waveechoes. The distance l0 varies with l in accord with Eq. (8.87). The solid dots,corresponding to the case ω2 < ω1, show the absence of an echo, as expected. Theecho amplitude decreases with distance because collisions destroy the coherence ofthe velocity modulations.8.7Nonlinear Landau DampingWhen the amplitude of an electron or ion wave excited, say, by a grid is followed inspace, it is often found that the decay is not exponential, as predicted by lineartheory, if the amplitude is large. Instead, one typically finds that the amplitude8.7 Nonlinear Landau Damping305Fig.
8.22 Measurement ofthe amplitude profile of anonlinear electron waveshowing nonmonotonicdecay. [From R. N.Franklin, S. M. Hamberger,H. Ikezi, G. Lampis, andG. J. Smith, Phys. Rev. Lett.28, 1114 (1972).]decays, grows again, and then oscillates before settling down to a steady value.Such behavior for an electron wave at 38 MHz is shown in Fig. 8.22. Althoughother effects may also be operative, these oscillations in amplitude are exactly whatwould be expected from the nonlinear effect of particle trapping discussed inSect. 7.5.
Trapping of a particle of velocity v occurs when its energy in the waveframe is smaller than the wave potential; that is, when21 jeϕj > m v vϕ2Small waves will trap only these particles moving at high speeds near vϕ. To trap alarge number of particles in the main part of the distribution (near v ¼ 0) wouldrequirejqϕj ¼1212mv2ϕ ¼ mðω=kÞ2ð8:88ÞWhen the wave is this large, its linear behavior can be expected to be greatlymodified.
Since jϕj ¼ jE/kj, the condition Eq. (8.88) is equivalent toω ffi ωB ,whereω2B jqkE=mjð8:89ÞThe quantity ωB is called the bounce frequency because it is the frequency ofoscillation of a particle trapped at the bottom of a sinusoidal potential well(Fig. 8.27). The potential is given byϕ ¼ ϕ0 ð1 cos kxÞ ¼ ϕ01 2 2k x2þ ð8:90Þ3068 Nonlinear EffectsFig. 8.23 A trappedparticle bouncing in thepotential well of a waveThe equation of motion ismd2 xdϕ¼ qkϕ0 sin kx¼ mω2 x ¼ qE ¼ qdt2dxð8:91ÞThe frequency ω is not constant unless x is small, sin kx kx, and ϕ is approximately parabolic.
Then ω takes the value ωB defined in Eq. (8.89). When theresonant particles are reflected by the potential, they give kinetic energy back tothe wave, and the amplitude increases. When the particles bounce again from theother side, the energy goes back into the particles, and the wave is damped. Thus,one would expect oscillations in amplitude at the frequency ωB in the wave frame.In the laboratory frame, the frequency would be ω0 ¼ ωB + kvϕ; and the amplitudeoscillations would have wave number k0 ¼ ω0 /vϕ ¼ k[1 + (ωB/ω)].
The condition ωB≳ ω turns out to define the breakdown of linear theory even when other processesbesides particle trapping are responsible.Another type of nonlinear Landau damping involves the beating of two waves.Suppose there are two high-frequency electron waves (ω1, k1) and (ω2, k2). Thesewould beat to form an amplitude envelope traveling at a velocity (ω2 ω1)/(k2 k1) dω/dk ¼ vg.
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