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Let ω ¼ ωs + iγ and assume γ 2 << ω2s and n nc. Show thatγ¼ vosc ω0 1=2Ωp2c ωswhere vosc is the peak oscillating velocity of the electrons.8.5.4Physical MechanismThe parametric excitation of waves can be understood very simply in terms of theponderomotive force (Sect. 8.4). As an illustration, consider the case of an electromagnetic wave (ω0, k0) driving an electron plasma wave (ω2, k2) and alow-frequency ion wave (ω1, k1) (Fig. 8.14b).
Since ω1 is small, ω0 must be closeto ωp. However, the behavior is quite different for ω0 < ωp than for ω0 > ωp. Theformer case gives rise to the “oscillating two-stream” instability (which will betreated in detail), and the latter to the “parametric decay” instability.Suppose there is a density perturbation in the plasma of the form n1 cos k1x; thisperturbation can occur spontaneously as one component of the thermal noise. Letthe pump wave have an electric field E0 cos ω0t in the x direction, as shown inFig. 8.15. In the absence of a dc field B0, the pump wave follows the relationω20 ¼ ω2p þ c2 k20 , so that k0 0 for ω0 ωp.
We may therefore regard E0 as spatiallyuniform. If ω0 is less than ωp, which is the resonant frequency of the cold electronfluid, the electrons will move in the direction opposite to E0, while the ions do notmove on the time scale of ω0. The density ripple then causes a charge separation, asshown in Fig. 8.15. The electrostatic charges create a field E1, which oscillates atthe frequency ω0.
The ponderomotive force due to the total field is given byEq. (8.44):2948 Nonlinear EffectsFig. 8.15 Physical mechanism of the oscillating two-stream instabilityFNL ¼ω2p 2ω0∇DEð E0 þ E1 Þ 22ε0ð8:63ÞSince E0 is uniform and much larger than E1, only the cross term is important:FNL ¼ ω2p ∂ h2E0 E1 iε02ω20 ∂xð8:64ÞThis force does not average to zero, since E1 changes sign with E0. As seen inFig. 8.15, FNL is zero at the peaks and troughs of n1 but is large where ∇n1 is large.This spatial distribution causes FNL to push electrons from regions of low density toregions of high density.
The resulting de electric field drags the ions along also, andthe density perturbation grows. The threshold value of FNL is the value justsufficient to overcome the pressure ∇ni1(KTi + KTe), which tends to smooth thedensity. The density ripple does not propagate, so that Re(ω1) ¼ 0. This is called theoscillating two-stream instability because the sloshing electrons have a timeaveraged distribution function which is double-peaked, as in the two-stream instability (Sect.
6.6).If ω0 is larger than ωp, this physical mechanism does not work, because anoscillator driven faster than its resonant frequency moves opposite to the directionof the applied force (this will be explained more clearly in the next section). Thedirections of ve, E1, and FNL are then reversed on Fig. 8.15, and the ponderomotiveforce moves ions from dense regions to less dense regions. If the density8.5 Parametric Instabilities295perturbation did not move, it would decay.
However, if it were a traveling ionacoustic wave, the inertial delay between the application of the force FNL and thechange of ion positions causes the density maxima to move into the regions intowhich FNL is pushing the ions. This can happen, of course, only if the phase velocityof the ion wave has just the right value. That this value is vs can be seen from thefact that the phase of the force FNL in Fig. 8.15 (with the arrows reversed now) isexactly the same as the phase of the electrostatic restoring force in an ion wave,where the potential is maximum at the density maximum and vice versa. Consequently, FNL adds to the restoring force and augments the ion wave.
The electrons,meanwhile, oscillate with large amplitude if the pump field is near the naturalfrequency of the electron fluid; namely, ω22 ¼ ω2p þ 32k2 v2th . The pump cannot haveexactly the frequency ω2 because the beat between ω0 and ω2 must be at the ionwave frequency ω1 ¼ kvs, so that the expression for FNL in Eq. (8.64) can have theright frequency to excite ion waves. If this frequency matching is satisfied, viz.,ω1 ¼ ω0 ω2, both an ion wave and an electron wave are excited at the expense ofthe pump wave. This is the mechanism of the parametric decay instability.8.5.5The Oscillating Two-Stream InstabilityWe shall now actually derive this simplest example of a parametric instability withthe help of the physical picture given in the last section. For simplicity, let thetemperatures Ti and Te and the collision rates vi and ve all vanish.
The ion fluid thenobeys the low-frequency equations∂vi1¼ en0 E ¼ FNL∂t∂ni1∂vi1¼ n0¼0∂t∂xMn0ð8:65Þð8:66ÞSince the equilibrium is assumed to be spatially homogeneous, we may Fourieranalyze in space and replace ∂/∂x by ik. The last two equations then give2∂ ni1 ikþ FNL ¼ 0M∂t2ð8:67Þwith FNL given by Eq. (8.64). To find E1, we must consider the motion of theelectrons, given bym∂ve∂þ ve ve ¼ eðE0 þ E1 Þ∂x∂twhere E1 is related to the density ne1 by Poisson’s equationð8:68Þ2968 Nonlinear Effectsikε0 E1 ¼ ene1ð8:69ÞWe must realize at this point that the quantities E1, ve, and ne1 each has two parts: ahigh-frequency part, in which the electrons move independently of the ions, and alow-frequency part, in which they move along with the ions in a quasineutralmanner.
To lowest order, the motion is a high-frequency one in response to thespatially uniform field E0:∂ve 0ee¼ E0 ¼ E^ 0 cos ω0 tmm∂tð8:70ÞLinearizing about this oscillating equilibrium, we have∂ve1eeþ ikve0 ve1 ¼ E1 ¼ ðE1h þ E1l Þmm∂tð8:71Þwhere the subscripts h and l denote the high- and low-frequency parts.
The first termconsists mostly of the high-frequency velocity veh, given by∂veheneh e2¼ E1h ¼m∂tikε0 mð8:72Þwhere we have used Eq. (8.69). The low-frequency part of Eq. (8.71) iseikve0 veh ¼ E1lmThe right-hand side is just the ponderomotive term used in Eq. (8.65) to drive theion waves. It results from the low-frequency beat between ve0 and veh. The left-handside can be recognized as related to the electrostatic part of the ponderomotive forceexpression in Eq. (8.42).The electron continuity equation is∂ne1þ ikve0 ne1 þ n0 ikve1 ¼ 0∂tð8:73ÞWe are interested in the high-frequency part of this equation. In the middle term,only the low-frequency density nel can beat with ve0 to give a high-frequency term,if we reject phenomena near 2ω0 and higher harmonics.
But nel ¼ ni1 byquasineutrality so we have∂nehþ ikn0 veh þ ikve0 ni1 ¼ 0∂tð8:74ÞTaking the time derivative, neglecting ∂ni1/∂t, and using Eqs. (8.70) and (8.72), weobtain8.5 Parametric Instabilities2972∂ nehikeni1 E0þ ω2p neh ¼m∂t2ð8:75ÞLet neh vary as exp (iωt):ikeni1 E0ω2p ω2 neh ¼mð8:76ÞEquations (8.69) and (8.76) then give the high-frequency field:E1h ¼ e2 ni1 E0e2 ni1 E0ε0 m ω2p ω2ε0 m ω2p ω20ð8:77ÞIn setting ω ω0 we have assumed that the growth rate of ni1 is very smallcompared with the frequency of E0.
The ponderomotive force follows fromEq. (8.64):FNL ω2p e2 ikni1 2 Eω20 m ω2p ω20 0ð8:78ÞNote that both E1h and FNL change sign with ω2p ω20 . This is the reason theoscillating two-stream instability mechanism does not work for ω20 > ω2p . Themaximum response will occur for ω20 ω2p , and we may neglect the factor (ω2p /ω20 ). Equation (8.67) can then be written2∂ ni1e2 k2 E^ 20 ni12Mm ω2p ω20∂t2ð8:79ÞSince the low-frequency perturbation does not propagate in this instability, we canlet ni1 ¼ ni1 expγt, where γ is the growth rate.
Thusγ2 E^ 20e2 k 22Mm ω2p ω20ð8:80Þand γ is real if ω20 < ω2p . The actual value of γ will depend on how small thedenominator in Eq. (8.77) can be made without the approximation ω2 ω20 . Ifdamping is finite, ω2p ω2 will have an imaginary part proportional to 2Γ2ωp,where Γ2 is the damping rate of the electron oscillations.
Then we have1=2γ / E^ 0 =Γ2ð8:81Þ2988 Nonlinear EffectsFar above threshold, the imaginary part of ω will be dominated by the growth rate γrather than by Γ2. One then hasγ2 /E^ 20γ 2=3γ / E^ 0ð8:82ÞThis behavior of γ with E0 is typical of all parametric instabilities. An exactcalculation of γ and of the threshold value of E0 requires a more careful treatmentof the frequency shift ωp ω0 than we can present here.To solve the problem exactly, one solves for ni1 in Eq. (8.76) and substitutes intoEq. (8.79):2∂ ni1ikeneh E0¼M∂t2ð8:83ÞEquations (8.75) and (8.83) then constitute a pair of equations of the form ofEqs.
(8.49) and (8.50), and the solution of Eq. (8.55) can be used. The frequencyω1 vanishes in that case because the ion wave has ω1 ¼ 0 in the zero-temperaturelimit.8.5.6The Parametric Decay InstabilityThe derivation for ω0 > ωp follows the same lines as above and leads to theexcitation of a plasma wave and an ion wave. We shall omit the algebra, which issomewhat lengthier than for the oscillating two-stream instability, but shall insteaddescribe some experimental observations. The parametric decay instability is welldocumented, having been observed both in the ionosphere and in the laboratory.The oscillating two-stream instability is not often seen, partly because Re(ω) ¼ 0and partly because ω0 < ωp means that the incident wave is evanescent. Figure 8.16shows the apparatus of Stenzel and Wong, consisting of a plasma source similar tothat of Fig. 8.10, a pair of grids between which the field E0 is generated by anoscillator, and a probe connected to two frequency spectrum analyzers.
Figure 8.17shows spectra of the signals detected in the plasma. Below threshold, the highfrequency spectrum shows only the pump wave at 400 MHz, while thelow-frequency spectrum shows only a small amount of noise. When the pumpwave amplitude is increased slightly, an ion wave at 300 kHz appears in thelow-frequency spectrum; and at the same time, a sideband at 399.7 MHz appearsin the high-frequency spectrum. The latter is an electron plasma wave at thedifference frequency.
The ion wave then can be observed to beat with the pumpwave to give a small signal at the sum frequency, 400.3 MHz.This instability has also been observed in ionospheric experiments. Figure 8.18shows the geometry of an ionospheric modification experiment performed with the8.5 Parametric Instabilities299Fig.
8.16 Schematic of an experiment in which the parametric decay instability was verified.[From A. Y. Wong et al., Plasma Physics and Controlled Nuclear Fusion Research, 1971, I,335 (International Atomic Energy Agency, Vienna, 1971).]large radio telescope at Platteville, Colorado. A 2-MW radiofrequency beam at7 MHz is launched from the antenna into the ionosphere. At the layer where ω0 ≳ωp, electron and ion waves are generated, and the ionospheric electrons are heated.In another experiment with the large dish antenna at Arecibo, Puerto Rico, the ωand k of the electron waves were measured by probing with a 430-MHz radar beamand observing the scattering from the grating formed by the electron densityperturbations.Problems8.12 In laser fusion, a pellet containing thermonuclear fuel is heated by intenselaser beams.
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