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Generalizations such as those fromthe early days can no longer be made. With the advent of fast computers, turbulencein the three-dimensional geometry of a tokamak can be simulated. Figure 8.40shows such a computation. The magnetic field lines are no longer in the wellordered magnetic island structure but are completely scrambled.Basic experiments on plasma turbulence have been done on Walter Gekelman’sLAPD (Large Area Plasma Device) machine at the University of California, LosAngeles. Shown in Fig. 8.41 the machine produces a plasma 60 cm in diameter and17 m long, ionized by electrons emitted from a hot cathode and accelerated through8.10TurbulenceFig.
8.39 Turbulencespectrum in the PLTtokamak, taken byMazzucato at three radialpositions3271022< r > = 5 cm< r > = 16 cm< r > = 30 cmS (k) [cm-3]1021102010190515102025k [cm-1]Fig. 8.40 Computer simulation of turbulent field lines in a tokamak. [Courtesy of W.W. Lee,Princeton Plasma Physics Lab]3288 Nonlinear EffectsFig. 8.41 The LAPD machine of W. GekelmanFig. 8.42 Flux ropesobserved in the LAPDmachinea mesh anode.
Pulsed for 10–20 ms at 1 Hz, the plasma reaches n 4 1018 m3and KTe 8 eV in B 0.3 T (3 kG). The machine is ideally suited for studies ofAlfvén waves, which require long length and high B-field.Fig. 8.42 gives an example of nonlinear effects observed in LAPD which wouldbe hard to predict theoretically. By measuring the currents’ B-fields, a current in theplasma was found to filament into curving flux ropes which themselves are composed of smaller filaments.8.118.11Sheath Boundaries329Sheath BoundariesLaboratory plasmas are not infinitely long cylinders, as many theories assume.Plasmas of finite length in a B-field have sheaths at the endplates which control thedensity profiles.
As an example, consider the discharge in Fig. 8.43, which is shortenough that gradients in the z direction can be neglected but long enough that theend sheaths do not overlap. The magnetic field is strong enough to confine theelectrons to move rapidly only in the z direction, and to move in the r direction onlyvia collisions. But the B-field is weak enough that we can neglect the curvature ofthe ion orbits inside the radius a. We now describe the Simon short-circuit effect,which causes the electrons to behave as if they could cross the magnetic fieldrapidly although they actually do not.In Fig.
8.44a, the top of each figure is the radial boundary; the bottom is theinterior of the discharge. We assume that radiofrequency power is applied at r ¼ a,so that initially n is higher in tube ① than in tube ②. The sheath at the endplateadjusts itself to allow the electron loss to be equal to the ion loss at the Bohmvelocity cs at each r, thus keeping the plasma neutral.
The sheath drop ϕp(r) istherefore given byKT e ½ eϕ p =KT eKT e ½ne¼n2πmM∴eϕ pM 1=2¼ ln;2πmKT eð8:156Þwhere the bracket on the l.h.s. is the random electron velocity in one dimension, andthe one on the r.h.s. is the acoustic speed. Since ϕp is independent of n, the sheathdrop is normally independent of radius.
However, when ions are injected at r ¼ a,they diffuse inward from tube ① to tube ②. Electrons can’t follow them, but thesheath drop on tube ② can increase slightly to trap more electrons to keep tube ②neutral. The sheath on tube ② is becomes thicker than the one on tube ①, as shownin Fig. 8.44a. This mechanism allows electrons to fall into thermal equilibrium andfollow the Boltzmann relation for over all radii:Fig. 8.43 Model of a finite-length laboratory plasma3308 Nonlinear EffectsFig. 8.44 Behavior of sheaths (a) initially, (b) during approach to equilibrium, and (c) inequilibriumnðr Þ ¼ n0 eeϕðrÞ=KT e ðrÞ :ð3:73ÞDuring the approach to equilibrium, the higher density on the outside causes ϕ to behigher there, and this corresponds to a radial E-field pointing inwards, as shown inFig.
8.44b. The field drives the ions toward the center at a fast rate scaled to KTe.There cannot be an E-field along B because that would cause a large electron8.11Sheath Boundaries331current, and the sheaths have stopped that. Therefore, the ions cannot leave thecenter except at their low thermal velocities. The ion density builds up there until anoutward E-field following Eq. (3.73) drives outward any ions that are created in theinterior. If all ions are created at the boundary, n(r) would be flat.
The equilibriumsheaths and fields are shown in Fig. 8.44c. A 2011 theory by Curreli and Chenshowed that a discharge with the geometry of Fig. 8.43 with sheaths at the endplateswill fall into an equilibrium with a “universal” density profile which is independentof pressure and plasma radius a.Chapter 9Special Plasmas9.19.1.1Non-Neutral PlasmasPure Electron PlasmasThe concepts of plasma physics have so far been couched in terms of well-behaved,quasineutral plasmas, but there are other plasmas with special properties.
Particleaccelerators have a single species, but the kinematic effects are so large that thecollective effects are not important. It is possible to generate single-species plasmasat low densities such that the electric fields are manageable. Ronald Davidson,Malmberg and O’Neil, Dan Dubin, and others have developed this interesting topic.Consider an infinite cylinder of electrons of uniform density n0 in a uniform coaxialr will arise, (where Er ismagnetic field B (Fig.
9.1). A large electric field E ¼ Er ðr Þ^negative), and a typical fluid element of electrons will drift in a circular orbit, sinceeverything is azimuthally symmetric. Those with small, off-axis orbits will driftaround the axis in “diocotron” orbits, as shown at the right. We wish to calculate therotation frequency ωr:ωr ðr Þ vθ ðr Þ=r:ð9:1ÞIn equilibrium, the inward and outward forces will balance:mv2θ ðr Þ=r ¼mrω2r¼eEr ðr ÞeErevθ ðr ÞBevθ B ¼eBrωreErð9:2ÞThe radial E-field is found from Poisson’s equation:© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_93333349 Special PlasmasFig. 9.1 A non-neutral plasma of electrons.
(Adapted from R.C. Davidson, Physics of Nonneutralplasmas (Addison-Wesley, Redwood City, CA, 1990)).1 dðrEr Þ ¼r drene =ε0ðene r 0 0ene r 2r dr ¼rEr ¼2ε0ε0 0r enerm 2Er ¼ω¼ra2 ε02e pð9:3ÞEquation (9.2) then becomesω2r þ ωc ωr þeEr¼ ω2r þ ωc ωr ½ω2p ¼ 0mr1=2 22ωr ¼ ½ωc 1 1 2ω p =ωc:ð9:4ÞSince ωr is independent of radius, we see that such an pure electron plasma rotatesas a solid body.
When ω2p =ω2c > ½, there is no solution; otherwise, there are twosolutions. For 2ω2p =ω2c << 1, expanding the square root yields the frequenciesωr ωc , and the lower frequencyωr ω2p =2ωc ωD :ð9:5ÞCalled the diocotron frequency, ωD has an orbit like the one at the right in Fig. 9.1.With appropriate modifications, these equations can also describe pure ionplasmas. Both types of single-species plasmas have been produced and studied inthe laboratory.9.1.2ExperimentsA device called a Malmberg-Penning trap, shown in Fig.
9.2, is commonly used tostudy single-species plasmas. This useful device evolved from the work of9.2 Solid, Ultra-Cold Plasmas335Fig. 9.2 A Malmberg-Penning trap for electronsWolfgang Paul, Hans Dehmelt, and Norman Ramsey,1 who won the Nobel Prize in1989.
The trap consists of three coaxial, conducting cylinders, with the ends biasednegatively to trap electrons and positively to trap ions. Conservation of canonicalangular momentum can be used to show that the particles are trapped unless theymake a collision with a stray neutral atom or with a error field due to imperfectmachining of the container.
The physical reason for this can be seen from Fig. 5.16,which shows that like-particle collisions do not cause diffusion.As an example of what can be done with these plasmas, F. Anderegget al. studied diffusion due to long-range interactions. Recall that Spitzer diffusion(Eq.
(5.71)) had an arbitrary cutoff related to the Debye length. Ion-ion diffusionwas quantified in an ion trap with LIF (laser-induced fluorescence) diagnostics. Inthe data shown in Fig. 9.3, one sees that the measured diffusion coefficient is anorder of magnitude larger than “classical” but is closer when collisions in largeorbit E B drifts are included.9.2Solid, Ultra-Cold PlasmasBy freezing the plasma in an ion trap to sub-Kelvin temperatures, it is possible tocreate liquid and solid plasmas. As the thermal motions of the particles decrease,their thermal energies become comparable to the Coulomb energy Wc betweenparticles.
This condition can be quantified by defining a coupling parameter Γ, theratio between the average Wc and KT. Two singly charged particles separated by adistance a have a Coulomb energy e2/a. HenceΓ ¼ e2 =aKT:1The author’s thesis adviser.ð9:6Þ3369 Special PlasmasFig. 9.3 Ion-ion diffusion coefficient compared with theory. The double lines show the expectedrange. (Adapted from F. Anderegg et al., Phys. Rev. Lett.
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