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Being an astronomer, Spitzer named it a stellarator. The idea of a figure-8 cameto Spitzer during a long ride on a ski lift at Garmisch-Partenkirchen. This was asolution to the problem of vertical drifts in a torus, as shown in Fig. 10.21. The37010Plasma ApplicationsFig. 10.21 Vertical driftsin a torusFig. 10.22 Principle of a figure-8 stellaratorproblem is this. In a torus B must be stronger on the inside (near the major axis, tothe left here) than on the outside because the field lines are crowded together there.This causes the Larmor radii to be larger on one side of each orbit than on the other,giving rise to the grad-B drift [Eq. (2.24)], which depends on the sign of the charge.The electrons drift upwards, and the ions downwards, creating a vertical E-field.The resulting E B drift is always outward, causing a loss of plasma.To cancel this drift, one can twist the torus into a figure-8 shape, as shown inFig.
10.22. As a particle follows a field line, its E B drift is towards the outside inone half, and back towards the inside in the other half. A demonstration model of afigure-8 stellarator is shown in Fig. 10.23. This was built for the Atoms for Peaceconference in Geneva in 1958, an event at which all nations revealed their secretwork on fusion. An electron gun could be inserted into the model to accelerateelectrons that traced out the field lines. An A-2 stellarator (fondly called the Etude)was operated by Kees Bol.
The B-1 stellarator was run by the author in 1954,showing that electrons (not plasma) could be confined for millions of traversesaround the 8, in spite of errors in the fabrication of the coils. A B-2 stellarator,shown in Fig. 10.24, was shipped to Geneva also, together with its power suppliesand controls so that it could be operated normally. Meanwhile, T. Stix andR. Palladino at Princeton constructed a figure-8 stellarator in the form of a square,which they named B-64, or 82.10.2Fusion Energy371Fig.
10.23 Exhibit model of a figure-8 stellaratorFig. 10.24 The B-2 stellarator at PrincetonIt was soon realized that the twist needed to cancel the vertical drifts could beproduced without a figure-8 machine. What was needed was another set of coilswound helically around the torus, as shown in Fig. 10.25. This method had theadditional advantage that the helical current could be varied. Th B-series ofmachines was replaced by the C-stellarator, a much larger machine with twoimportant innovations: ion cyclotron heating, and divertors for capturing the escaping plasma. Previously, the plasma was heated only by ohmic heating, in which acurrent was induced around the torus by a transformer of which the plasma was thesecondary winding.
Since the plasma has a resistance caused by collisions of the37210Plasma ApplicationsFig. 10.25 A stellaratorwith helical windings[Google Images, 2015]Fig. 10.26 The LHD stellarator in Japan [Google Images, 2015]current-carrying electrons with ions, it was heated by the I2R losses. The currentcreated a poloidal B-field so that the field lines went through a poloidal angle ι (iota)each time they went around the torus the long way.
A kink instability occurs wheniota exceeds 2π. This is called the Kruskal-Shafranov limit. These difficulties havebeen overcome in tokamaks, which differ from stellarators in that the poloidal fieldis generated by the plasma current itself, and not by external coils. Modernstellarators use electron-cyclotron, lower-hybrid, and neutral-beam heating, as intokamaks, and do not depend on ohmic heating.Stellarators have been evolving since 1951, and variations with names such astorsatron, heliotron, heliac, and helias have been built in different countries. Themost spectacular of these, the LHD (Large Helical Device) in Japan, is shown inFig. 10.26.
The entire vacuum chamber was shaped to follow the magnetic field. InGermany, D. Pfirsch and H. Schlüter combined the toroidal and helical coils into aseries of 20 planar and 50 non-planar coils, several of each shape, to form theWendelstein 7-X, shown in Fig.
10.27. This machine is under construction in10.2Fusion Energy373Fig. 10.27 Drawing of theWendelstein 7-X stellarator[T. Klinger, Max-PlanckInstitute for Plasma Physics,Greifswald, Germany]Greifswald, Germany. As with the LHD, the vacuum chamber is not round. Tomake room for all these coils, stellarators must have large aspect ratio.With the large advances in the development and understanding of tokamaks,stellarators are no longer preferred for plasma confinement in fusion experiments.Nonetheless, stellarators have advantages which may make them more suitable thantokamaks for reactors.
For instance, their poloidal field is created by helicalwindings and is not dependent on the plasma current, whose shape is not in directcontrol. Stellarators are also immune from “disruptions”, which terminate tokamakplasmas unexpectedly and are not yet well understood.10.2.1.5TokamaksInvented in Russia, TOKAMAK is a Russian acronym for toroidal chamber with anaxial magnetic field. Since tokamaks have become the favored form of toroidalfusion devices, so much is known about their behavior that only a few generalcharacteristics can be given here. In tokamak literature, the rotational transform ι(iota) of stellarators is replaced by its reciprocal q.
Thus, a field line that comes backto the same poloidal position after going around the torus twice the long way hasι ¼ ½ or q ¼ 2. A typical q profile is shown in Fig. 10.28. The rational-q surfaceshave field lines which close upon themselves and are special. The region q < 1corresponds to ι > 1 and is unstable. In tokamaks this instability has the form ofsawtooth oscillations, as shown in Fig. 10.29 for Ti and Te vs. time at q ¼ 1. As thetemperature rises at the center, the resistivity lowers, and the current densityincreases, thus driving the temperature higher until the configuration can no longerbe sustained. Then the sawtooth crashes, ejecting a hot plasma outwards.
Unlikestellarators, tokamaks have a self-organized plasma that generates its own behaviorto achieve an equilibrium.It is clear that the profile q(r) should depend on the current profile J(r). Anexample of this variation is shown in Fig. 10.30. It is seen that more peaked J(r)’sgive higher q(a)’s. The particles, both ions and electrons, travel in interesting orbitson these magnetic surfaces.374Fig. 10.28 A typical q profile in a tokamakFig. 10.29 Sawtooth oscillations in a tokamakFig.
10.30 The q profile for three different J profiles10Plasma Applications10.2Fusion Energy375Islands and BananasAt the radii where q is a rational number, the field lines form magnetic islands. Anexample of this is shown in Fig. 10.31. In the islands, a particle will hop from oneisland to the next, eventually returning to the initial island in a different position. Itsintersections with this cross section will trace out the islands. However, someparticles can’t go all the way around the torus. Consider the trajectory shown inFig. 10.32. Since the B-field is stronger on the inside of the torus (nearer the majoraxis) than on the outside, a particle with small v|| can be mirror-trapped and reflectedback.
Projected onto a cross-sectional plane, this orbit resembles that in Fig. 10.33and is appropriately called a banana orbit. These orbits can cause enhanced diffusionFig. 10.31 Magneticislands at the q ¼ 3/2surfaceFig. 10.32 Mirror-trappingof a particle in a torus37610Plasma ApplicationsFig. 10.33 A banana orbit.A particle with larger v||/v⊥would follow the dashedorbitFig. 10.34 Mechanismof the Ware Pinchof plasma. Normally, a collision can result in a random-walk step the size of a Larmordiameter (Fig. 5.17); but now a particle can jump from one banana to another. Theresult is that diffusion depends no longer on the toroidal field Bϕ, but on the weakerpoloidal field Bθ.
These effects contribute to “neoclassical diffusion” (see Fig. 5.22).An effect associated with bananas is the Ware Pinch, illustrated in Fig. 10.34.Particles in banana orbits have an Eϕ Bp drift which is always inward, just as in a10.2Fusion Energy377linear z-pinch. Thus, bananas tend to move inward, countering collisional diffusionoutward. The inward velocity of a Ware pinch is given byvWare ¼ ð2 0:5ÞA1=2 Eϕ =Bθ ;ð10:8Þwhere A is the aspect ratio, and the range covers details such as Zeff.Bootstrap CurrentAnother interesting effect in tokamaks is the bootstrap current, a current that arises,enhancing the toroidal current, as the plasma diffuses outward, as if the tokamak ispulling itself up “by its own bootstraps”.
This expression, originated around 1781,means to overcome an impediment without outside help. Figure 10.35a is areminder of Eq. (2.17) showing the velocity of its guiding center when a particleis pushed by a force perpendicular to B:vf ¼1F B:q B2ð10:9ÞIn Fig.
10.35b, the black arrows show the outward pressure force on the plasma in amonotonic density profile. This causes the azimuthal drift of electrons, which isinnocuous. The toroidal current generates an azimuthal Bp (blue arrows). Theelectrons drift in this field Bp, driven by ∇p, is the bootstrap current. It is alwaysin the same direction as the main current and hence adds it. The bootstrap currentcan be as large as 70–90 % of the total current, and the tokamak is really pullingitself up by its own bootstraps.Fig.
10.35 Origin of the bootstrap current [the small red arrows pointing out of the paper in (b)]37810Plasma ApplicationsThe H-ModeIn 1982, Friedrich (Fritz) Wagner was heating the ASDEX tokamak with neutralbeams when, at a certain threshold power, the plasma density suddenly doubled,and instabilities quieted down. This is now called the H (high)-mode. The densityand temperature of the plasma did not fall to near zero at the edge but stoppedfalling at a pedestal level (Fig. 10.36a) as if there were a transport barrier there(Fig. 10.36b). Later experiments showed that there is a highly sheared azimuthalflow at the barrier which prevented normal outward diffusion of plasma.
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