While alternate magnetic concepts may differ from tokamaks in geometry, size, time scales, input power requirements and technology, the principal objectives remain, that is the heating of a D-T plasma to fusion ignition and confining it sufficiently long to yield a net energy gain.

Out of many different designs proposed and discussed in the literature, one such concept is the so-called ‘bumpy torus’, which links a number of mirror sections end to end into a high-aspect-ratio torus, depicted in Fig. 10.19. As illustrated, several mirror coils are equally spaced in a toroidal array. The plasma contained by this magnetic configuration threads the bores of these axisymmetric coils and thus takes on the shape of a bumpy toroidal ‘sausage’. The magnetic field lines close on themselves and the plasma particles are confined in two ways: trapped particles reflect back and forth in individual mirror sections, and passing particles circulate around the major circumference of the bumpy torus plasma. To stabilize such a configuration, electron cyclotron resonance heating (ECRH) is applied to form an annular high-energy electron plasma in the central part of each mirror section. When the currents generated by these hot-electron rings are sufficient to provide a minimum-B configuration, they thus stabilize the toroidal core plasma. In such a reactor, which is called the Elmo (electrons with large magnetic orbits) Bumpy Torus, the hot-electron annuli would typically possess densities of about 1018 m"3 and temperatures Te>100 keV, while the core plasma exhibits a density in the order of 1020nT3 and a temperature of aboutl5 keV. It is thus possible that the beta-value of the confined bumpy torus plasma can become comparable to that of the annuli. The toroidal plasma appears to be macroscopically stable as long as its beta-value is smaller than, or at most, approximately equal to the beta of the annuli. To produce the stabilizing minimum-B property, the latter beta has to exceed a threshold value in the range 5-15% depending on the annular shape. It was experimentally shown that such hot-electron rings can produce beta’s up to 50% in steady-state operation. Hence, the core plasma-^ of an Elmo Bumpy Torus may also be established at substantially increased values in comparison to the tokamak. As a consequence, the fusion power density, as limited by the magnetic pressure, Eq. (4.14), is elevated or, for a given power density, the magnetic field requirements are significantly reduced. Another advantage over the tokamak is the large aspect ratio (~ 5-10 times greater) allowing for simpler engineering design and construction. Further, there is no need for power interruption as associated with pulsed operation; an Elmo Bumpy Torus can thus be operated in a steady-state mode.

Avoiding the production of hot-electron annuli, which is relatively inefficient by means of ECRH and leads to increased radiation losses, a toroidal minimum-B configuration can also be generated by toroidally linking modular coils of specific shape such that each already represents a minimum-B magnetic mirror. To introduce a rotational transform, the coils, which do not exhibit poloidal

symmetry, are rotated about the magnetic axis with respect to each adjacent coil.

A device combining the effect of a z-pinch with that of a 0-pinch-recall Fig. 9.10-will contain a plasma with currents in the axial as well as in the poloidal direction and thus generate a confining magnetic field consisting of helical field lines. Due to the form of the field lines, this configuration is called a screw — pinch. Though similar to the tokamak, it is operated at relatively high [3-20%, but features only very short periods of sufficient plasma confinement.

Another toroidal confinement concept, which has received great attention due to its improved stability against MHD modes, is the so-called Reversed Field Pinch (RFP). It is much like the tokamak: the plasma is confined by a combination of toroidal and poloidal magnetic fields with the latter generated by a toroidal plasma current induced by transformer action. The toroidal field Вф is established primarily by external coils. The essential difference, however, is that in the RFP the plasma currents parallel to the toroidal minor axis do not only produce the poloidal field, Be, but also diamagnetically alter the toroidal field such that Вф can change sign near the plasma boundary (field reversal). Further, the plasma current and Be in RFP’s are much stronger than in comparable tokamaks, whereas Вф is modest. This gives rise to strongly sheared magnetic field lines with their pitch increasing rapidly with greater radial distance. The Reversed Field Pinch configuration is produced by the high magnetic shear near the edge of the plasma which suppresses local MHD instabilities. The field reversal is suggested to emerge from a turbulent state as a self-organization mechanism.

In contrast to a tokamak, where the safety factor q has to meet the Kruskal — Shafranov stability criterion, q(r)> 1 everywhere and q(r=a) > 2.5 (see Sec. 10.1), an RFP features q(r)<l with a negative q(r—>a) consistent with the reversal of the toroidal field component in this edge region. The evolution from a tokamak plasma to an RFP requires the presence of an electrically-conducting shell just outside the toroidal plasma or of closely fitting external conductors for assisting the tokamak plasma to remain confined while reducing q and turning to the RFP configuration. MHD stability theory for RFP indicates the plasma-(3 limitation at the high value of —30%. Due to the high [3, the deployment of advanced fusion fuel cycles in these devices is conceivable. A fusion plasma system not constrained by the Kruskal-Shafranov criterion provides the profit that it can be heated ohmically to ignition, if the energy confinement is good. Experiments, however, have shown so far a TH lower than that for tokamaks of similar size. An obvious advantage over the tokamak is the elimination of the requirement of minimizing the aspect ratio, such as previously demanded by Eq. (10.49). Hence, simplified designs with good maintenance access are possible. A handicap common with tokamaks is that the RFP is a pulsed device as well.

The reactor concepts discussed so far are physically large, they employ complex technology, represent expensive designs, and possess only a relatively
low power density. Evidently, a high power density would be a desirable feature of a fusion reactor. This may be accomplished in compact power reactors which achieve the same total power as a conventional magnetic fusion device in a significantly smaller geometry. Among various designs proposed in this context, e. g. a compact RFP, we choose here to describe the spheromak reactor as a distinctive representative.

A spheromak is an advanced toroidal plasma containment device in which the confining magnetic configuration, as displayed in Fig. 10.20, is characterized by an extremely low aspect ratio and by the absence of external toroidal fields. With the minor plasma radius a ~ R0, this configuration appears almost like a sphere and, obviously, this has inspired the given name. An axial current flows through a field-reversed 0-pinch plasma to internally produce the toroidal field. Thus, both the poloidal and the toroidal magnetic field are self-generated. Only the steady magnetic-bottle field is provided externally by coils. Topologically, spheromaks are open confinement systems with the plasma, however, contained within a closed separatrix surface. The region inside this separatrix is-similar to the RFP — associated with q<l, while, at the plasma boundary, the safety factor is zero. The great advantage of this design is its geometric simplicity and compactness. Experiments with spheromak configurations to date could be operated for only short pulse lengths.

Most pulsed toroidal confinement systems suffer from extremely low periods during which the plasma can be stably contained, and from relatively large radiation power losses due to the impurities released by intense plasma wall interaction. Though there exists a number of confinement concepts which are not discussed here but nevertheless are interesting in their specific design and/or the physics to be applied, we conclude this chapter by recalling that-among the closed magnetic systems-tokamak and stellarator concepts are the most promising candidates to be utilized for future fusion reactor operation.

Problems

10.1 Find the drift velocity, as given in Eq. (10.5) for the case of a purely toroidal magnetic field, by combining the effects of the grad-B drift and the curvature drift, which are simultaneously present in a curved В-field, for the

Вг=0,Вв =-Вв(а) , Вф = R° Вф(RB) a R„ +r cost?

may be applied to a tokamak reactor, estimate the size of such a reactor having a

circular plasma cross-section if it is to achieve ignition and contains a 50:50% d-t fusion plasma with an average ion density Ni = 1020 m’3 at average temperature Ti=Te=30 keV producing 1000 MW of fusion power. Use the ignition criterion of Sec. 8.4.

10.3 Using j = VxB, show that for a tokamak the magnetic field is proportional to 1/R.

10.4 What perspectives do high-beta devices offer relative to low-beta tokamaks? How can a high-beta configuration be realized?

10.5 For each of the following open magnetic confinement fusion reactor concepts, draw a sketch of the concept, label the major components, and describe their purpose. Describe how fusion fuel ions are confined, and list what the energy and particle losses are in the system, and where they occur. Describe all the different electrical currents, magnetic fields, and their purpose.

(a) Tokamaks

(b) Stellarators

(c) Spheromaks

(d) Reversed Field Pinch (it is a toroidal system)

10.6 List the favourable characteristics desirable for a future fusion power reactor.

10.7 Design an axisymmetric d-t tokamak reactor with circular cross-section capable of fusion plasma ignition demonstration assuming the empirical energy confinement time scaling

[‘] = 0 00338(/p [ma])° 85 (o[m])° 3 (r0 [m])1 2 (jV, [m’3 ]) (B, [tesla])0 2 (/с, л [mw])

where denotes the total fusion power in the plasma volume, and taking the

following fixed parameters:

ratio of first wall to plasma radius rw/a = 1.25 . 50:50% deuterium-tritium fuel mixture

. electron density Ne = 0.8 X 1020 m’3 . equal ion and electron temperature, T; = Te = 20 keV. plasma current Ip = 18 MA

toroidal magnetic field Bt = 6 Tesla. cyclotron radiation loss parameter |/ = 0.001 Specify the relevant plasma and reactor parameters allowing for ignition (i. e. so that Eq. (8.30) is met) and consistent with the constraints and requirements discussed in Sec. 10.4, in particular Eqs. (10.40b), (10.42) and (10.46), as well as with the engineering constraint of limiting the thermal power flux through the first wall by Pw < 1.5 MW m’2.