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By symmetry, derivatives inthe θ^ and z^ directions vanish. J is given by the time-independent fourth Maxwellequationμ0 J ¼ ∇B ¼∂Bθ Bθþz^ ;∂rrð10:2ÞSo thatμ0 JB ¼ ð∇BÞB ¼ ð∇BÞz z^ Bθ θ^dBθ Bθ1 dB2θ B2θþr^þrÞ ¼ ¼Bθ ð^2 drrdrrThe z-pinch equilibrium is thus given bydB2B2p þ θ þ θ ¼ 0:dr2μ0μ0 rð10:3ÞAnalogously, equilibrium of a θ-pinch is given bydB2p þ z ¼ 0;dr2μ0ð10:4Þwithout the extra azimuthal term. This shows that the plasma pressure is balancedby the magnetic-field pressure.Problem10.1. A z-pinch of radius a has a uniform current J ¼ J z z^ and a plasma pressure p(r) which is balanced by the J B force.
Derive the parabolic form of p(r).Theta pinch and reversed-field pinch. As shown in Fig. 10.9, a theta pinch has theplasma current going in the theta direction. After preionization, a pulse in the thetapinch coils drives an azimuthal current J, creating a plasma trapped in the B-field ofthe coils. By Newton’s third law, this action creates a reaction: the current J is in thedirection to produce a B-field on axis opposing the B-field of the coils. Withsufficient current, the internal B-field can become larger than that from the coils,and we have a reversed-field pinch.
The plasma can extend as far as the separatrix,which divides the internal field lines from those that extend beyond the coils.10.2Fusion Energy363Fig. 10.9 Schematic of a reversed-field pinch (RFP)The plasma is kept from sliding axially by the J θ Br force at its ends. At a criticalcurrent, the B-field on axis can be zero, and we have created a β ¼ 1 plasma!The plasma in Fig.
10.9 suffers from instabilities. If the coils are long or faraway, the plasma can tilt or slide in the horizontal direction. A hydromagneticinstability can occur at the ends because of the unfavorable curvature there. Thoughthe pinch is necessarily pulsed, it lasts much longer than the instability growth time.The effect of curvature is illustrated in Fig. 10.10, where the animals representplasma pressure from above. When the curvature is concave to the plasma, moreplasma pressure can be supported, and visa versa. The curvature drift is described inSect. 2.3.2 for a convex curvature.
In Fig. 10.9, the sharp bends at the ends of theplasma are highly unstable, but fortunately the unstable fields (see Fig. 6.11) areshort-circuited by the ions, which have large Larmor orbits and can cross theB-field. The finite-Larmor-radius effect was discussed in Sect. 2.4.The Taylor state. Consider now a toroidal z-pinch such as the Zeta machinementioned previously. The plasma is held in a toroidal B-field, and a current is driventhrough the plasma to produce a twist, turning the field lines into helices.
It was foundthat, after a period of violent shaking, the plasma settled into a quiescent state.J.B. Taylor of the U.K. found that, if the helicity of the B-field is conserved, thisrelaxed state is a force-free, minimum energy equilibrium following the equation∇ B ¼ λB;ð10:5Þwhere λ is a constant. The long derivation is omitted here, but Eq. (10.5) is anequation that occurs often in different fields of physics.10.2.1.2Magnetic MirrorsIt is possible to trap charged particles between magnetic mirrors, which reflectparticles with finite velocity v⊥ perpendicular B.
In Eq. (2.46) it was shown that the36410Plasma ApplicationsFig. 10.10 The effect ofcurvaturemagnetic moment μ ¼ ½mv2⊥ =B of a particle is conserved. The energy W of aparticle can then be written with constant μ:W ¼ ½mv2 ¼ ½mv2k þ ½mv2⊥ ¼ ½mv2k þ μB:Consider a barely trapped particle that is turned around at B ¼ Bmax. Let v|| ¼ v|| atB ¼ B0 and v|| ¼ 0 at B ¼ Bmax. Then we haveW ¼ ½mv2k þ μB0W ¼ 0 þ μBmaxatB¼0ð10:6ÞB ¼ BmaxatSince energy is conserved, this gives"#½mv2kv2kB0B01¼¼þ1þ 2 ,v⊥Bmax μBmax Bmax0v2k2BmaxvRm ¼1þ 2 ¼ 2 :B0v⊥ v⊥ð10:7Þ10.2Fusion Energy365pffiffiffiffiffiffiRm is called the mirror ratio, and particles starting with v=v⊥ > Rm are in the losscone (Fig.
2.9) and are not confined by the mirror (Fig. 10.11).The end losses from a simple mirror are so large that many modifications havebeen made to minimize these losses. Before we get to these, consider the stability ofthe plasma. At the center of the mirror, the field lines are convex to the plasma, andRayleigh–Taylor instabilities can occur. At the throats of the mirror, the field linesare concave to the plasma, providing a stabilizing effect. If the machine is longenough to hold a useful volume of plasma, however, the unstable region dominates.One way to stabilize that part is to add “Ioffe bars”, named after the inventor, whichare four conductors carrying current in the axial direction, as shown in Fig.
10.12.The azimuthal fields from the bars, added to the mirror field, form twisted magneticsurfaces which have the minimum-B property. That is, the field strength jBjincreases in every direction, forming a magnetic well for the plasma. This is avery stable arrangement, but the plasma still leaks out the ends of the mirror.
Inaddition, mirrors suffer from another instability, the cyclotron-ion instability.If one links the currents in the coils and the bars into a single conductor, a “baseballcoil” is obtained, as shown in Fig. 10.13. A very large baseball-type magnetic mirrorwas built at Livermore Laboratory in the U.S. This heavy device (shown inFig. 10.11 Particletrapping by a magneticmirror with Rm ~ 4Fig. 10.12 A magnetic mirror with Ioffe bars36610Plasma ApplicationsFig.
10.13 Diagram of abaseball coilFig. 10.14 The MFTF-B mirror machine being moved by the old Roman methodFig. 10.14, was being lifted into place when an earthquake struck, but what happenedis inconsequential because the funding for mirror research was cut off at that time.The MFTF-B was never used and was turned into a walk-in museum for visitors.Magnetic mirror research continued in Tsukuba, Japan, where a large axisymmetric mirror machine Gamma 10 was built (Fig. 10.15). This was a tandem mirror10.2Fusion Energy367Fig.
10.15 Magnetic field configuration in a tandem mirrorFig. 10.16 Formation of an ion beam outside a simple mirrorconsisting of several mirrors in series. For instance, one mirror could haveminimum-B stabilization, while the end mirror could be a short one with a largemirror ratio for confinement. The main interest in mirrors was the possibility ofdirect conversion of plasma energy into electricity without going through a heatcycle. As ions collide, some will enter the loss cone and escape.
These will beaccelerated in the z direction because as B decreases, the conservation of μ meansthat v⊥ will decrease, and hence v|| must increase. An ion beam will be ejected, asshown in Fig. 10.16, and it will be neutralized by electrons, which can escape easilyby their frequent collisions. The ion beam will be sorted by energy and collected bya series of bins, as seen in Fig.
10.17, with the more energetic ones going fartherbefore drifting sideways. Thus, the bins provide a DC current. Since the ions enterthe bins at low energy, little heat is lost; and this would be an efficient way toconvert fusion energy directly to electricity.Mirrors are fueled by tangential injection of ions at the midplane. Mirrors are ofcourse not immune to instability.
In addition to the flute interchange instability in thecentral cell, there are instabilities driven by the anisotropy of the distribution functions, which have an empty loss cone. For instance, there is an Alfvén ion-cyclotron36810Plasma ApplicationsFig. 10.17 Conceptual scheme for direct conversion of ion energy to electricityinstability. Compared with toroidal devices, mirrors do not confine plasmas as welland are not as suitable for fusion.
However, they are useful for industrial applicationswhere confinement is not as important as the ejected beams. They can also producehigh-β plasmas for experiments not possible in low-β devices.10.2.1.3Reversed-Field ConfigurationsThe possibility of creating a high-β plasma spawned a number of large experimentson reversed-field configurations (RFCs). An RFC is a high-β plasma requiring notoroidal field or conducting boundaries. A diagram of an RFC is shown inFig.
10.18. Though the plasma is pulsed, many instabilities can grow faster thanthe pulse length. At the ends of the plasma is a region of bad curvature wheregravitational instabilities are stabilized by finite ion Larmor radius rLi. This is notsimple, since rLi varies rapidly in the nonuniform field. The most dangerousinstability is thought to be the n ¼ 2 tilt mode, shown in Fig. 10.19, where n is theazimuthal mode number. There is also an n ¼ 1 rotational instability. Much of thework on FRCs is theoretical, but these instabilities have been observed. The plasmalasts microseconds, and the total temperature KTe + KTi can reach 800 eV.In the confined region of Fig. 10.18, plasma pressure is needed to balance themagnetic pressure.
It can be shown that the average β is given byhβi ¼ 1 ½ðr s =r w Þ2 , where rw is the wall radius. Since rs/rw 1, hβi must be>½, and this is an intrinsically high-β device.FRCs are translatable; that is, they can be pushed magnetically from onechamber to another. For fusion purposes, an FRC can, in principle, be translatedinto chamber with a DC magnetic field, a conducting wall for stabilization, andeven “blankets” for capturing the neutrons and converting their energy into heat.10.2.1.4StellaratorsConfinement of plasma in a torus eliminates endlosses but introduces new problems.
It is convenient to classify tori is by their aspect ratios. In a circular torus with10.2Fusion Energy369Fig. 10.18 Geometry of an RFC. The dashed line is the separatrix, with maximum radius rsFig. 10.19 Drawing of a tilt instability in an FRCFig.
10.20 A circular toruswith aspect ratio R/acircular cross section, we can define R as the major radius and a as the minor radius,as shown in Fig. 10.20. The aspect ratio is defined as R/a. Proceeding from large tosmall aspect ratio, we start with stellarators.Toroidal confinement began around 1951 when Lyman Spitzer, Jr., and MartinSchwarzschild built the figure-8 shaped Model A-1 machine at Princeton University.
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