1629373397-425d4de58b7aea127ffc7c337418ea8d (846389), страница 46
Текст из файла (страница 46)
That backwardwaves actually exist in the laboratory has been verified not only by ωvs. k measurements of the type shown in Fig. 7.35, but also by wave interferometertraces which show the motion of phase fronts in the backward direction fromreceiver to transmitter.7.10Kinetic Effects in a Magnetic Field265Fig. 7.35 Pure ion Bernstein waves: agreement between theory and experiment in a Q-machineplasma [From J. P. M. Schmitt, Phys.
Rev. Lett. 31, 982 (1973)]Finally, we consider neutralized Bernstein waves, for which ζ 0e is small andZ0 (ζ 0e) ’ 2. The electron n ¼ 0 term in Eq. (7.156) becomes simply k2De . Assumingthat ζ 0i 1 still holds, the analysis leading to Eq. (7.157) is unchanged, andEq. (7.156) becomesk2z"#2k2D Ω p b1 þ 2 2 e I 0 ðbÞωkz"#1ω2pΩ2p 2 b XI n ð bÞ2eþ k⊥ 1 2¼0þω ω2c Ω2c b n¼1 ðω=nΩc Þ2 1ð7:159Þfor k2z k2⊥ , an approximate relation for neutralized ion Bernstein waves can bewritten1þk2 λ2D"#1ω2pΩ2p 2 b XI n ð bÞeþ1 2¼0ω ω2c Ω2c b n¼1 ðω=nΩc 1Þ2ð7:160ÞNote that electron temperature is now contained in λD, whereas pure ion Bernstein waves, Eq.
(7.157), are independent of KTe. If k2λ2D is small, the bracket in2667 Kinetic TheoryFig. 7.36 Neutralized ionBernstein modes:agreement between theoryand experiment in a Hemicrowave discharge [FromE. Ault and H. Ikezi, Phys.Fluids 13, 2874 (1970)]Eq. (7.160) must be large; and this can happen only near a resonance ω ’ nΩc.Thus the neutralized modes are not sensitive to the lower hybrid resonanceω ’ ωl. Indeed, as k⊥rLi ! 0 the envelope of the dispersion curves approachesthe electrostatic ion cyclotron wave relation (4.67), which is the fluid limit forneutralized waves.Neutralized Bernstein modes are not as well documented in experiment aspure Bernstein modes, but we show in Fig.
7.36 one case in which the formerhave been seen.Chapter 8Nonlinear Effects8.1IntroductionUp to this point, we have limited our attention almost exclusively to linearphenomena; that is, to phenomena describable by equations in which the dependentvariable occurs to no higher than the first power. The entire treatment of waves inChap.
4, for instance, depended on the process of linearization, in which higherorder terms were regarded as small and were neglected. This procedure enabled usto consider only one Fourier component at a time, with the secure feeling that anynonsinusoidal wave can be handled simply by adding up the appropriate distribution of Fourier components. This works as long as the wave amplitude is smallenough that the linear equations are valid.Unfortunately, in many experiments waves are no longer describable by thelinear theory by the time they are observed.
Consider, for instance, the case of driftwaves. Because they are unstable, drift waves would, according to linear theory,increase their amplitude exponentially. This period of growth is not normallyobserved—since one usually does not know when to start looking—but insteadone observes the waves only after they have grown to a large, steady amplitude. Thefact that the waves are no longer growing means that the linear theory is no longervalid, and some nonlinear effect is limiting the amplitude. Theoretical explanationof this elementary observation has proved to be a surprisingly difficult problem,since the observed amplitude at saturation is rather small.A wave can undergo a number of changes when its amplitude gets large. It canchange its shape—say, from a sine wave to a lopsided triangular waveform.
This isthe same as saying that Fourier components at other frequencies (or wave numbers)are generated. Ultimately, the wave can “break,” like ocean waves on a beach,converting the wave energy into thermal energy of the particles. A large wave canThe original version of this chapter was revised. An erratum to this chapter can be found athttps://doi.org/10.1007/978-3-319-22309-4_11© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_82672688 Nonlinear Effectstrap particles in its potential troughs, thus changing the properties of the medium inwhich it propagates.
We have already encountered this effect in discussingnonlinear Landau damping. If a plasma is so strongly excited that a continuousspectrum of frequencies is present, it is in a state of turbulence. This state must bedescribed statistically, as in the case of ordinary fluid hydrodynamics. An importantconsequence of plasma turbulence is anomalous resistivity, in which electrons areslowed down by collisions with random electric field fluctuations, rather than withions. This effect is used for ohmic heating of a plasma (Sect. 5.6.3) to temperaturesso high that ordinary resistivity is insufficient.Nonlinear phenomena can be grouped into three broad categories:1. Basically nonlinearizable problems.
Diffusion in a fully ionized gas, forinstance, is intrinsically a nonlinear problem (Sect. 5.8) because the diffusioncoefficient varies with density. In Sect. 6.1, we have seen that problems ofhydromagnetic equilibrium are nonlinear.
In Sect. 8.2, we shall give a furtherexample—the important subject of plasma sheaths.2. Wave–particle interactions. Particle trapping (Sect. 7.5) is an example of thisand can lead to nonlinear damping. A classic example is the quasilinear effect, inwhich the equilibrium of the plasma is changed by the waves.
Consider the caseof a plasma with an electron beam (Fig. 8.1). Since the distribution function has aregion where df0/dv is positive, the system has inverse Landau damping, andplasma oscillations with vϕ in the positive-slope region are unstable (Eq. (7.67)).The resonant electrons are the first to be affected by wave–particle interactions,and their distribution function will be changed by the wave electric field. Thewaves are stabilized when fe(v) is flattened by the waves, as shown by the dashedline in Fig. 8.1, so that the new equilibrium distribution no longer has a positiveslope.
This is a typical quasilinear effect. Another example of wave–particleinteractions, that of plasma wave echoes, will be given in Sect. 8.6.3. Wave–wave interactions. Waves can interact with each other even in the fluiddescription, in which individual particle effects are neglected. A single wave candecay by first generating harmonics of its fundamental frequency. These harmonics can then interact with each other and with the primary wave to formother waves at the beat frequencies.
The beat waves in turn can grow so largethat they can interact and form many more beat frequencies, until the spectrumFig. 8.1 A double-humped,unstable electrondistribution8.2 Sheaths269becomes continuous. It is interesting to discuss the direction of energy flow ina turbulent spectrum. In fluid dynamics, long-wavelength modes decay intoshort-wavelength modes, because the large eddies contain more energy andcan decay only by splitting into small eddies, which are each less energetic.The smallest eddies then convert their kinetic motion into heat by viscousdamping. In a plasma, usually the opposite occurs.
Short-wavelength modestend to coalesce into long-wavelength modes, which are less energetic. This isbecause the electric field energy E2/2 is of order k2ϕ2/2, so that if eϕ is fixed(usually by KTe), the small-k, long-λ modes have less energy. As a consequence,energy will be transferred to small k by instabilities at large k, and somemechanism must be found to dissipate the energy. No such problem exists atlarge k, where Landau damping can occur. For motions along B0, nonlinear“modulational” instabilities could cause the energy at small k to be coupled toions and to heat them. For motions perpendicular to B0, the largest eddies willhave wavelengths of the order of the plasma radius and could cause plasma lossto the walls by convection.Although problems still remain to be solved in the linear theory of waves andinstabilities, the mainstream of plasma research has turned to the much less wellunderstood area of nonlinear phenomena.
The examples in the following sections willgive an idea of some of the effects that have been studied in theory and in experiment.8.28.2.1SheathsThe Necessity for SheathsIn all practical plasma devices, the plasma is contained in a vacuum chamber offinite size. What happens to the plasma at the wall? For simplicity, let us confine ourattention to a one-dimensional model with no magnetic field (Fig. 8.2). Supposethere is no appreciable electric field inside the plasma; we can then let the potentialϕ be zero there.
When ions and electrons hit the wall, they recombine and are lost.Since electrons have much higher thermal velocities than ions, they are lost fasterFig. 8.2 The plasmapotential ϕ forms sheathsnear the walls so thatelectrons are reflected. TheCoulomb barrier eϕwadjusts itself so that equalnumbers of ions andelectrons reach the walls persecond2708 Nonlinear Effectsand leave the plasma with a net positive charge. The plasma must then have apotential positive with respect to the wall; i.e., the wall potential ϕw is negative.This potential cannot be distributed over the entire plasma, since Debye shielding(Sect. 1.4) will confine the potential variation to a layer of the order of severalDebye lengths in thickness. This layer, which must exist on all cold walls withwhich the plasma is in contact, is called a sheath.
The function of a sheath is to forma potential barrier so that the more mobile species, usually electrons, is confinedelectrostatically. The height of the barrier adjusts itself so that the flux of electronsthat have enough energy to go over the barrier to the wall is just equal to the flux ofions reaching the wall.8.2.2The Planar Sheath EquationIn Sect. 1.4, we linearized Poisson’s equation to derive the Debye length. Toexamine the exact behavior of ϕ (x) in the sheath, we must treat the nonlinearproblem; we shall find that there is not always a solution.
Figure 8.3 shows thesituation near one of the walls. At the plane x ¼ 0, ions are imagined to enter thesheath region from the main plasma with a drift velocity u0. This drift is needed toaccount for the loss of ions to the wall from the region in which they were createdby ionization. For simplicity, we assume Ti ¼ 0, so that all ions have the velocity u0at x ¼ 0. We consider the steady state problem in a collisionless sheath region. Thepotential ϕ is assumed to decrease monotonically with x. Actually, ϕ could havespatial oscillations, and then there would be trapped particles in the steady state.This does not happen in practice because dissipative processes tend to destroy anysuch highly organized state.Fig.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
