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Similarly, thefactor i(V2/6)t is just i(3/2)k0 2 t0 , which can be recognized from Eq. (8.149) asthe Bohm–Gross frequency for δn0 ¼ 0, the factor ½ coming from expansion ofthe square root. Since ω0 ’ ωp, the terms ω0 + (V2/6) represent the Bohm–Grossfrequency, and A is therefore the frequency shift (in units of ωp) due to the cavity inδn0 . The soliton amplitude and width are given in Eq. (8.154) in terms of the shift A,and the high-frequency electric field can be found from Eq. (8.138).Cavitons have been observed in devices similar to that of Fig. 8.16.
Figures 8.29and 8.30 show two experiments in which structures like the envelope solitonsdiscussed above have been generated by injecting high-power rf into a quiescentplasma. These experiments initiated the interpretation of laser-fusion data in termsof “profile modification,” or the change in density profile caused by theponderomotive force of laser radiation near the critical layer, where ω0 ’ ωp,ω0 being the laser frequency.8.8 Equations of Nonlinear Plasma Physics321Fig. 8.29 A density cavity,or “caviton,” dug by theponderomotive force of anrf field near the criticallayer. The high-frequencyoscillations (not shown)were probed with anelectron beam.
[From H. C.Kim, R. L. Stenzel, andA. Y. Wong, Phys. Rev.Lett. 33, 886 (1974).]Problems8.21 Check that the relation between the frequency shift A and the solitonamplitude in Eq. (8.154) is reasonable by calculating the average densitydepression in the soliton and the corresponding average change in ωp. (Hint:Use Eq. (8.146) and assume that the sech2 factor has an average value of ’ ½over the soliton width.)8.22 A Langmuir-wave soliton with an envelope amplitude of 3.2 V peak-to-peakis excited in a 2-eV plasma with n0 ¼ 1015 m3. If the electron waves havekλD ¼ 0.3, find (a) the full width at half maximum of the envelope (in mm),(b) the number of wavelengths within this width, and (c) the frequency shift(in MHz) away from the linear-theory Bohm–Gross frequency.8.23 A density cavity in the shape of a square well is created in a one-dimensionalplasma with KTe ¼ 3 eV.
The density outside the cavity is n0 ¼ 1016 m3, andthat inside is ni ¼ 0.4 1016 m3. If the cavity is long enough that boundaryresonances can be ignored, what is the wavelength of the shortest electronplasma wave that can be trapped in the cavity?3228.98 Nonlinear EffectsReconnectionIn a collisionless plasma, electrons and ions gyrate around magnetic field lines andare tied to them. If these field lines are brought close together, the plasma willbecome very dense, and hence highly collisional.
In that case, plasma can diffuseacross field lines and change the magnetic geometry. Due to kinetic effects, this canhappen even without collisions. Reconnection of magnetic field lines occurs in theearth’s field on the night side, as shown in Fig. 8.31. The plasma of the solar windpushes the earth’s dipole field away from the sun into a magnetotail, where fieldlines in opposite directions are brought close to one another. The magnetic configuration can change, as shown in Fig. 8.32. In the magnetotail, the left-hand partbecomes a loop connected to the earth’s poles, while the right-hand part breaks offto connect to the interplanetary field.
At the magnetopause, the lines break andconnect to the solar wind.What happens in the thin reconnecting layer shown in Fig. 8.33 is extremelycomplicated because both collisional and collisionless damping mechanisms, aswell as finite-Larmor-radius effects can occur there. There are many models for theconditions in the layer. In the Sweet-Parker model, B and entrained n diffuse intothe layer by resistive diffusion, building up a high density there which must escapeat a large velocity v0 of the order of the Alfvén speed.
H. Petschek showed thatMHD shocks can occur in fast reconnection, but these do not stay long enough toFig. 8.30 Coupled electron and ion wave solitons. In (a) the low-frequency density cavities areseen to propagate to the left. In (b) the high-frequency electric field, as measured by wire probes, isfound to be large at the local density minima.
[From H. Ikezi, K. Nishikawa, H. Hojo, andK. Mima, Plasma Physics and Controlled Nuclear Fusion Research, 1974, II, 609, InternationalAtomic Energy Agency, Vienna, 1975.]8.9 Reconnection323Fig. 8.31 Schematic diagram of the earth’s magnetic field, as it is blown by the solar wind into amagnetotailFig.
8.32 Reconnection offield linesFig. 8.33 An idealized, thin reconnection layerhave much effect. With the advent of fast computers, W. Daughton has studiedthree-dimensional reconnection and the generation of turbulence. Reconnectionoccurs on the surface of the sun itself. These regions manifest themselves assunspots.A controlled experiment on reconnection was built at Princeton by M. Yamadaand H. Ji, with theoretical support by R.
Kulsrud. A diagram of the magnetic field intheir machine is shown in Fig. 8.34. The two large rings shown vertically carrycurrent to generate the field lines shown. As the current in the rings is decreased, the3248 Nonlinear EffectsFig. 8.34 Schematic of thefield lines in Yamada’sreconnection machinefield lines move in the direction of the arrows, and a reconnection sheet ofhorizontal cross section forms. If the current is increased, the field lines moveopposite to the arrows, and the reconnection sheet has a vertical cross section.8.10TurbulenceIn Chap.
4 we treated waves in a plasma by linearizing the equations of motion, thuslimiting the discussion to waves of small amplitude. Sinusoidal waves cannot growindefinitely, however; nonlinear effects change their shapes, limit their amplitudes,and ultimately turning them into random turbulence.
Figure 8.35 shows a jet of airforming vortices and then breaking into turbulence. In pipes carrying a liquid, theflow is smooth at low velocities, when the velocity has a smooth profile from slowat the edge to fast at the center, as seen in Fig. 8.36 (top).
At high velocities, the flowbreaks up into turbulence, as in Fig. 8.36 (bottom). The flow slows down, andenergy goes into noise. Nonlinearity can happen in a plasma in other ways. Forinstance, in the plasma oscillation of Fig. 4.2, the amplitude can become so largethat the excursion of an electron sheet can overlap the next wavelength. In earlyexperiments in which a current was drawn between an anode and a cathode in amagnetized cylinder, probes inserted into the plasma always detected turbulentfluctuations like those shown in Fig.
8.37. The basic waves that grew into this8.10Turbulence325Fig. 8.35 Turbulence in a jet of air, made visible by smoke. [from M. Van Dyke, An Album ofFluid Motion, The Parabolic Press, Stanford, CA (1982)]Fig. 8.36 Turbulence in a water pipeFig. 8.37 Turbulent potential fluctuations observed in a fusion plasma (Data by the author)3268 Nonlinear Effectsab-4.6f0-10-10~2> (dB)-20 <n-20-30-30-40~2> (dB)<nETUDE SPECTRUM (BOL)L-2 SPECTRUM (CHEN)-40-5010f-5.10-50100f (kc)100010100f (kc)1000Fig. 8.38 Turbulence spectra in (a) the L-2, a reflex arc (2000 G, 1012 cm3); and (b) the Etude, astellarator (6700 G, 1013 cm3).
Here, frequency is measured rather than knonlinear state were not found for years. We now know of many instabilities thatcan cause this.A turbulent field can be decomposed into a spectrum of eddies of different sizes.After excitation, the distribution of sizes—the turbulent spectrum S(k)—will settledown to a form predictable by theory.
In two-dimensional systems, such as a sheetof graphene, small eddies will coalesce into larger eddies. In general, however,large eddies will break up into small eddies with a given size distribution. Inordinary hydrodynamics, Kolmogoroff found from dimensional analysis that thespectrum should vary as k5/3. In laboratory plasmas, the turbulence is driven byinstabilities such as the Rayleigh-Taylor or drift-wave instabilities, and electronmotions are dominated by their E B drifts.
Early experiments at Princetonshowed consistent spectra varying as k5 even in entirely different machines, asshown in Fig. 8.38. Because plasma moves very differently from normal fluids, it isnot surprising that the exponent of k is quite different. The boundary-dependentpeaks in Fig. 8.38 are from the low-frequency instabilities driving the turbulence.By contrast, the spectrum in an early tokamak (PLT) at Princeton, shown inFig. 8.39, varies as k2.8.In the modern era, electrostatic probes cannot be inserted into a tokamak, andnon-invasive diagnostics must be used, such as reflectometry. The turbulent spectrum will vary greatly with position because of sawtooth oscillations in the interiorand Edge Localized Modes near the boundary.
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