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Figure 8.11 shows measurements of the density fluctuation in the shockwave as a function of time and probe position. It is seen that the wavefront steepensand then turns into a shock wave of the classic shape. The damping of theoscillations is due to collisions.Problem8.5 Calculate the maximum possible velocity of an ion acoustic shock wave in anexperiment such as that shown in Fig.
8.10, where Te ¼ 1.5 eV, Ti ¼ 0.2 eV, andthe gas is argon. What is the maximum possible shock wave amplitude in volts?8.3.5Double LayersA phenomenon related to sheaths and ion acoustic shocks is that of the double layer.This is a localized potential jump, believed to occur naturally in the ionosphere,Fig. 8.10 Schematic of a DP machine in which ion acoustic shock waves were produced anddetected. [Cf. R. J.
Taylor, D. R. Baker, and H. Ikezi, Phys. Rev. Lett. 24, 206 (1970).]8.3 Ion Acoustic Shock Waves283Fig. 8.11 Measurements of the density distribution in a shock wave at various times, showinghow the characteristic shape of Fig. 8.5 develops. [From Taylor et al., loc cit.]which neither propagates nor is attached to a boundary. The name comes from thesuccessive layers of net positive and net negative charge that are necessary to createa step in ϕ(x). Such a step can remain stationary in space only if there is a plasmaflow that Doppler-shifts a shock front down to zero velocity in the lab frame, or ifthe distribution functions of the transmitted and reflected electrons and ions on eachside of the discontinuity are specially tailored so as to make this possible. Doublelayers have been created in the laboratory in “triple-plasma” devices, which aresimilar to the DP machine shown in Fig.
8.10, but with a third experimentalchamber (without filaments) inserted between the two source chambers. Byadjusting the relative potentials of the three chambers, which are isolated bygrids, streams of ions or electrons can be spilled into the center chamber to forma double layer there. In natural situations double layers are likely to arise wherethere are gradients in the magnetic field B, not where B is zero or uniform as inlaboratory simulations. In that case, the μ∇B force (Eq.
(2.38)) can play a large rolein localizing a double layer away from all boundaries. Indeed, the thermal barrier intandem mirror reactors is an example of a double layer with strong magnetictrapping. In Sect. 8.11 we shall see that a double layer can arise in “mid-air”when a dense plasma is injected into a diverging magnetic field. Ions acceleratedby the potential drop in the double layer can be used to push a spacecraft.2848.48 Nonlinear EffectsThe Ponderomotive ForceLight waves exert a radiation pressure which is usually very weak and hard todetect. Even the esoteric example of comet tails, formed by the pressure of sunlight,is tainted by the added effect of particles streaming from the sun. When highpowered microwaves or laser beams are used to heat or confine plasmas, however,the radiation pressure can reach several hundred thousand atmospheres! Whenapplied to a plasma, this force is coupled to the particles in a somewhat subtleway and is called the ponderomotive force.
Many nonlinear phenomena have asimple explanation in terms of the ponderomotive force.The easiest way to derive this nonlinear force is to consider the motion of anelectron in the oscillating E and B fields of a wave. We neglect dc E0 and B0 fields.The electron equation of motion ismdv¼ e½EðrÞ þ v BðrÞdtð8:34ÞThis equation is exact if E and B are evaluated at the instantaneous position of theelectron. The nonlinearity comes partly from the v B term, which is second orderbecause both v and B vanish in the equilibrium, so that the term is no larger thanv1 B1, where v1 and B1 are the linear-theory values.
The other part of thenonlinearity, as we shall see, comes from evaluating E at the actual position ofthe particle rather than its initial position. Assume a wave electric field of the formE ¼ Es ðrÞ cos ωtð8:35Þwhere Es(r) contains the spatial dependence. In first order, we may neglect thev B term in Eq.
(8.34) and evaluate E at the initial position r0. We havemdv1 =dt ¼ eEðr0 Þð8:36Þv1 ¼ ðe=mωÞEs sin ωt ¼ dr1 =dtð8:37Þδr1 ¼ e=mω2 Es cos ωtð8:38ÞEðrÞ ¼ Eðr0 Þ þ ðδr1 ∇ ÞEjr¼r0 þ ð8:39ÞIt is important to note that in a nonlinear calculation, we cannot write eiωt and takeits real part later. Instead, we write its real part explicitly as cos ωt. This is becauseproducts of oscillating factors appear in nonlinear theory, and the operations ofmultiplying and taking the real part do not commute.Going to second order, we expand E(r) about r0:8.4 The Ponderomotive Force285We must now add the term v1 B1, where B1 is given by Maxwell’s equation:∇ E ¼ ∂B=∂tB1 ¼ ð1=ωÞ∇ Es jr¼r0 sin ωtð8:40ÞThe second-order part of Eq. (8.34) is thenmdv2 =dt ¼ e½ðδr1 ∇ ÞE þ v1 B1 ð8:41ÞInserting Eqs.
(8.37), (8.38), and (8.40) into (8.41) and averaging over time, wehave dv2e2 1m½ðEs ∇ ÞEs þ Es ð∇ Es Þ ¼ f NL¼dtmω2 2ð8:42ÞHere we used sin 2 ωt ¼ cos 2 ωt ¼ ½. The double cross product can bewritten as the sum of two terms, one of which cancels the (Es · ∇)Es term. Whatremains isf NL ¼ 1 e2∇E24 mω2 sð8:43ÞThis is the effective nonlinear force on a single electron. The force per m3 is fNLtimes the electron density n0, which can be written in terms of ω2p . Since E2s ¼ 2 E2 , we finally have for the ponderomotive force the formulaFNLω2p ε0 E2¼ 2∇ω2ð8:44ÞIf the wave is electromagnetic, the second term in Eq. (8.42) is dominant, and thephysical mechanism for FNL is as follows.
Electrons oscillate in the direction of E,but the wave magnetic field distorts their orbits. That is, the Lorentz force ev Bpushes the electrons in the direction of k (since v is in the direction of E, and E Bis in the direction of k). The phases of v and B are such that the motion does notaverage to zero over an oscillation, but there is a secular drift along k. If the wavehas uniform amplitude, no force is needed to maintain this drift; but if the waveamplitude varies, the electrons will pile up in regions of small amplitude, and aforce is needed to overcome the space charge.
This is why the effective force FNL isproportional to the gradient of hE2i. Since the drift for each electron is the same,FNL is proportional to the density—hence the factor ω2p /ω2 in Eq. (8.44).If the wave is electrostatic, the first term in Eq. (8.42) is dominant. Then thephysical mechanism is simply that an electron oscillating along k || E moves fartherin the half-cycle when it is moving from a strong-field region to a weak-field regionthan vice versa, so there is a net drift.2868 Nonlinear EffectsFig. 8.12 Self-focusing of a laser beam is caused by the ponderomotive forceAlthough FNL acts mainly on the electrons, the force is ultimately transmitted tothe ions, since it is a low-frequency or dc effect.
When electrons are bunched byFNL, a charge-separation field Ecs is created. The total force felt by the electrons isFe ¼ eEcs þ FNLð8:45ÞSince the ponderomotive force on the ions is smaller by Ω2p =ω2p ¼ m=M, the forceon the ion fluid is approximatelyFi ¼ eEcsð8:46ÞSumming the last two equations, we find that the force on the plasma is FNL.A direct effect of FNL is the self-focusing of laser light in a plasma.
In Fig. 8.12we see that a laser beam of finite diameter causes a radially directed ponderomotiveforce in a plasma. This force moves plasma out of the beam, so that ωp is lower andthe dielectric constant ε is higher inside the beam than outside. The plasma then actsas a convex lens, focusing the beam to a smaller diameter.Problems8.6 A 1-TW laser beam is focused to a spot 50 μm in diameter on a solid target. Aplasma is created and heated by the beam, and it tries to expand. Theponderomotive force of the beam, which acts mainly on the region of criticaldensity (n ¼ nc, or ω ¼ ωp), pushes the plasma back and causes “profile modification,” which is an abrupt change in density at the critical layer.(a) How much pressure (in N/m2 and in lbf/in.2) is exerted by theponderomotive force? (Hint: Note that FNL is in units of N/m3 and thatthe gradient length cancels out.
To calculate hE2i, assume conservativelythat it has the same value as in vacuum, and set the 1-TW Poynting fluxequal to the beam’s energy density times its group velocity in vacuum.)(b) What is the total force, in tonnes, exerted by the beam on the plasma?(c) If Ti ¼ Te ¼ 1 keV, how large a density jump can the light pressuresupport?8.7 Self-focusing occurs when a cylindrically symmetric laser beam of frequencyω is propagated through an underdense plasma; that is, one which hasn < nc ε0 mω2 =e28.5 Parametric Instabilities287In steady state, the beam’s intensity profile and the density depression causedby the beam (Fig.
8.12) are related by force balance. Neglecting plasma heating(KT KTe + KTi ¼ constant), prove the relation2n ¼ n0 eε0 hE i=2nc KT n0 eαðrÞThe quantity α(0) is a measure of the relative importance of ponderomotivepressure to plasma pressure.8.5Parametric InstabilitiesThe most thoroughly investigated of the nonlinear wave–wave interactions are the“parametric instabilities,” so called because of an analogy with parametric amplifiers, well-known devices in electrical engineering. A reason for the relativelyadvanced state of understanding of this subject is that the theory is basically alinear one, but linear about an oscillating equilibrium.8.5.1Coupled OscillatorsConsider the mechanical model of Fig.
8.13, in which two oscillators M1 and M2 arecoupled to a bar resting on a pivot. The pivot P is made to slide back and forth at afrequency ω0, while the natural frequencies of the oscillators are ω1 and ω2. It isclear that, in the absence of friction, the pivot encounters no resistance as long as M1and M2 are not moving. Furthermore, if P is not moving and M2 is put into motion,M1 will move; but as long as ω2 is not the natural frequency of M1, the amplitudewill be small.
Suppose now that both P and M2 are set into motion. The displacement of M1 is proportional to the product of the displacement of M2 and the lengthof the lever arm and, hence, will vary in time asFig. 8.13 A mechanicalanalog of a parametricinstability2888 Nonlinear Effectscos ω2 t cos ω0 t ¼1212cos ½ðω2 þ ω0 Þt þ cos ½ðω2 ω0 Þtð8:47ÞIf ω1 is equal to either ω2 + ω0 or ω2 ω0, M1 will be resonantly excited and willgrow to large amplitude. Once M1 starts oscillating, M2 will also gain energy,because one of the beat frequencies of ω1 with ω0 is just ω2. Thus, once eitheroscillator is started, each will be excited by the other, and the system is unstable.The energy, of course, comes from the “pump” P, which encounters resistance oncethe rod is slanted.
If the pump is strong enough, its oscillation amplitude isunaffected by M1 and M2; the instability can then be treated by a linear theory. Ina plasma, the oscillators P, M1, and M2 may be different types of waves.8.5.2Frequency MatchingThe equation of motion for a simple harmonic oscillator x1 isd 2 x1þ ω21 x1 ¼ 0dt2ð8:48Þwhere ω1 is its resonant frequency. If it is driven by a time-dependent force which isproportional to the product of the amplitude E0 of the driver, or pump, and theamplitude x2 of a second oscillator, the equation of motion becomesd2 x1þ ω21 x1 ¼ c1 x2 E0dt2ð8:49Þwhere c1 is a constant indicating the strength of the coupling.
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