As discussed in Sec. 10.1, the confinement of a plasma in a toroidal magnetic field requires a rotational transform of field lines in order to prevent local charge concentrations, plasma polarization and drifts to the wall. Unlike the tokamak which carries an externally induced current in its plasma, stellarator devices do not. They also feature a toroidal geometry but render the confining magnetic field lines helical, i. e. compel a rotational transform, either by a deformation of the torus itself-such early concepts showed poor stability-or by helical or contorted coil currents external to the plasma. These currents, as shown in Fig. 10.13, pass through helical conductors winding around the torus and make the magnetic field lines take on the form of a spiral. A rotational transform produced in this way, as well as the closed magnetic flux surfaces thus rendered, exist in a vacuum field and do not rely on a plasma current induced in a pulsed manner. In stellarators, all magnetic fields providing confinement of the plasma are generated by means of currents flowing in external conductors. Hence, as an important advantage, stellarators allow for steady-state confinement and continuous fusion reactor operation.

In practical terms , a helical winding is a loosely wrapped solenoidal winding and generates, as desired, a toroidal and poloidal field. Furthermore, if viewed from above the torus, it represents also as a loosely wrapped vertical field coil and hence generates a vertical field as well. To eliminate this contribution, currents in adjacent helical windings of the same pitch flow in opposite

directions canceling out one another’s vertical fields and also their toroidal fields, on average. Thus, as seen in Fig. 10.13, a separate set of coils is needed to provide the essential toroidal magnetic field. Therefore, a stellarator still requires toroidal field coils as shown. The poloidal field produced from the helical windings together with the toroidal field from the separate toroidal field coils result in a flux which twists the magnetic field lines as they pass around the torus and thus generate magnetic surfaces of the shape shown in Fig. 10.14. Evidently, the geometrical simplicity of axisymmetry is lost. It is noted that the establishment of closed magnetic surfaces is possible only in a restricted region of the minor cross section of the toroidal tube. For a stellarator with l — 3 pairs of helical coils of opposite currents, we illustrate in Fig. 10.15 the shape of the generated magnetic surfaces. Closed magnetic surfaces are observed within the cross-sectional area embraced by the dashed separatrix line which may be identified also as the last closed magnetic surface. Outside this separatrix the field lines wrap around the individual conductors.

Due to the absence of a current in a stellarator, Ampere’s law (compare with Eq. (9.48)) yields here

— — ds = 0,

К

surface. For this to be true, the poloidal field must change sign and magnitude along s. Hence, unlike in a tokamak, the magnetic field lines in a stellarator do not wrap monotonically around the toroidal tube. Rather, they appear to oscillate periodically according to the qualitative structure given in Fig. 10.16a. Each such so-called fundamental field period incrementally rotates the field lines in the poloidal direction. Yet another inhomogeneity of the magnetic field is to be considered, which is due to the curvature associated with torus geometry. The resulting variation of the magnetic field along the toroidal direction is illustrated in Fig. 10.16b where the deep and more frequent oscillations of В are caused by the helical windings alternately carrying currents of different direction, and where the slow modulation of В corresponds to the toroidal curvature. It is obvious that in addition to the magnetic mirrors in the toroidal field which lead to particle trapping as in Sec. 10.3, there are also local mirrors of the helical field.

Analyzing the particle motion in such magnetic field configurations yields three distinctive types of orbits: (i) circulating particles which pass entirely around the torus without encountering a reflection, (ii) so-called ‘helically trapped’ particles reflected in the local mirrors of the helical field, and (iii) ‘toroidally trapped’ particles tracing banana orbits as they are reflected in the toroidal magnetic mirrors known from Sec. 10.3. It is possible that a helically trapped particle appears to be toroidally trapped as well. Such a particle is then called a ‘superbanana particle’.

Another approach to establish a rotational transform for toroidal plasma confinement is to partially rotate non-circular toroidal field coils, one with respect to the other. Further, specific designs-with their general principle demonstrated in Fig. 10.17-allow for practical modular composition. For reasons of simplification, in Fig. 10.17 the helical windings of a stellarator are reduced to consist of only two conductors with currents of opposite sign. Such an t — 1 configuration permits replacement by modules which combine parts of the helical coils with additional meridian-ring conductors. Advanced modular designs utilize non-planar twisted coils, that are sophisticated spatial elements as illustrated in Fig. 10.18. Modular coils are favourable from an engineering standpoint of view because they allow for assembly and disassembly of the coils without having to unwind or disconnect helical conductors encircling the major torus axis.

Closely related to the stellarator configuration is the so-called ‘torsatron’ in which the rotational transform is produced again by helical windings, however with like current directions. In this design there is no need for toroidal field coils, but, instead, equatorial ring conductors may be applied to generate a transverse field which can compensate the vertical magnetic field produced by the torsatron as a result of like helical current directions. This necessity of a compensating vertical field may be avoided by specifically winding the helical conductors according to

тв = ф + a sin ф + P sin(20) (10.52)

with m being an integer and a, (3 denoting chosen constants. Such a device is

Much theoretical and experimental investigation is still devoted to the determination of the maximum plasma-beta in a stellarator/torsatron for which stable equilibrium can be sustained. It is thought that (3’s of several percent, perhaps up to 10%, can be stably achieved if the configuration exhibits a helical magnetic axis.

Non-axisymmetric configurations such as the stellarator may lead-in comparison with axisymmetric devices-to more complex transport processes due to the greater variety of particle orbits. The quality of particle and energy confinement is dominantly determined by the trapping of particles in the various

ripples of the magnetic field and by the frequency of collisions occurring in the plasma. Collisions may scatter particles from one region of trapping to an adjacent region and thereby alter the type of trapping. Interestingly, in non — axisymmetric toroidal systems, the electron and ion components of the confined plasma diffuse independently of each other.

Fig. 10.17: Depiction of replacement of helical conductors by modular elements: (a)
helical stellarator windings; (b) modular coils generating a stellarator magnetic
configuration much the same as established by (a).

For very high collision frequencies Vc= 1/XC (see Eq. (6.17)) in a stellarator or a torsatron, the particles do not travel sufficiently long distances without a scattering encounter to be reflected by either the helical or the toroidal mirrors. As a consequence, particle and energy transport in a stellarator operated in this collisional regime is similar to that in collisional tokamaks. However, differences between these two configurations arise for lower collision frequencies, that is when the mean collision time xc appears to be in the order of the average time needed by a particle to bounce between the mirrors. If Vc is such that particles can be trapped helically but do not precess entirely around the minor torus axis before undergoing a collision, the helically trapped particle will drift from one magnetic surface to another. This spreading of particles obviously enhances particle diffusion. For vc lower than the frequency of precession around the magnetic axis, i. e. the single line around which the magnetic surfaces appear to be nested, these particle spreadings tend to cancel out, and the diffusion coefficient in this regime is expected to decline with decreasing collision frequency. If, however, superbananas are present, as featured by stellarators, the diffusion coefficient will not decline immediately with a reduced vc, but rather it remains constant at its high value over a limited collision frequency interval and thus exhibits the so-called superbanana plateau. Finally, as Vc becomes smaller than the superbanana bounce frequency, the diffusion coefficient is observed to decrease in stellarators as well.

Thermal energy diffusion appears to follow a dependence on vc similar to that seen for particle diffusion. The plasma-energy-confinement time of stellarator/torsatron devices is thought to scale similarly to that found for tokamaks, except for the weak-collision regime where the spreading of helically trapped particles enhances the diffusion. Suppressing this contribution is the objective of advanced stellarator designs.

In conclusion, we summarize some potential advantages of the stellarator/torsatron concept: Steady-state magnetic fields simplify the magnet design. Unlike in tokamak reactors, there is no need for pulsed superconducting coils and corresponding energy storage to drive these pulsed coils. Since a toroidal plasma current is not needed, a potential source of instabilities is eliminated. Early predictions of enhanced transport losses and increased instability have not materialized. Stellarator and torsatrons appear to be operating as effective plasma confinement machines, with dimension and performance parameters comparable to those of similar toroidal magnetic devices. The high aspect ratio, the absence of transformer coils and, particularily, modular construction make stellarator/torsatron devices well accessible. Further, steady — state operation of an ignited plasma would allow for a simplified blanket design due to reduced material durability requirements.