Figure-8 stellarators are hard to make, especially since the coils have to be accurate enough to keep the field lines from wandering out to the walls.8 It was soon real­ized, however, that the necessary twist of the field lines can be produced without twisting the entire torus. We mentioned that the field lines in a toroidal magnetic bottle are twisted like the stripes on a candy cane. The way to produce such helical field lines can be visualized more easily if we decompose them into toroidal lines, as in Fig. 4.13a, and poloidal lines, as in Fig. 4.13b. Adding these two types of fields together will result in a field with helical field lines. To produce the toroidal part of the field, we can use coils like those in Fig. 4.14. Now we want to add coils that will produce the poloidal field. Figure 4.19 shows how this is done. Let there be a number of toroidal hoops placed all around the torus; two of these are shown in Fig. 4.19. If each hoop carries a current in the toroidal direction, as shown by the horizontal arrows, it will produce a magnetic field around itself in the direction shown by the arrows on the small circles around each hoop. The part of this field that extends into the plasma will be mostly in the poloidal direction. Imagine that there are an infinite number of these hoops covering the surface of the torus. Their fields inside the plasma will add up to give a purely poloidal field, as shown by the dashed arrows.

You have no doubt noticed the complementarity here: poloidal windings create toroidal magnetic fields (Fig. 4.14), and toroidal windings create poloidal fields (Fig. 4.19). In the same way that the toroidal and poloidal fields add up inside the torus to make helical field lines, the poloidal and toroidal windings can be combined into a helical winding! One turn of such a winding is shown in Fig. 4.20. The dotted line is a helical field line. Because it contains both toroidal and poloidal components, it may start near the top and then go to the bottom in another cross section. Now look at what an ion does.9 On the right, an ion starts drifting

Fig. 4.19 Generation of poloidal fields with coils

upwards — not downwards, as in Fig. 4.17 — because here I have drawn the mag­netic field going into the page instead of out of the page. When the ion reaches the left side, it is still drifting upwards — not downwards as in a figure-8 stellarator — but this is fine, because the ion is now near the bottom, and an upward drift will bring it back away from the wall. So there are two ways to skin the cat. Either a figure-8 stellarator or a stellarator with helical field lines made by helical windings can cancel the dreaded vertical drift of ions and electrons caused by bending a cylinder into a torus.

We started with the concept that field lines have to end on themselves so that particles moving along them will never leave the magnetic trap. Of course, the field lines do not have to meet their own tails exactly. All that is required is that the line never hits the wall. In general, field lines do not close on themselves. Rather, they come back to the same cross section in a different position after going around the torus the long way. This is illustrated in Fig. 4.21. An imaginary glass sheet has been cut through the torus so that we can see where the field lines strike this cross section. Let’s assume that a field line intersects this cross section at position 1. After going around the torus once, it might intersect at position 2. On successive passes, its position might be 3, 4, 5, 6, etc. On the seventh pass, the field line almost comes back to position 1, but it does not have to. One can define a mapping function such that for every position on that plane, there is a definite position for the next pass. Thus, whenever the line goes through position 2, it will pass near position 3 the next time. The line does not ever have to come back to itself. It can cover the entire cross section randomly, and the plasma will still be confined as long as the line never hits the wall.

At this point, we should define a quantity that will be very useful for under­standing twisted magnetic fields: the rotational transform. This is the average number of times a field line goes the short way around a cross section for each time

Fig. 4.21 Mapping of a field line

it goes the long way around the whole torus. In Fig. 4.21, suppose pass No. 7 fell exactly on pass No. 1, then it took six trips around the torus for the field line to make one trip around the cross section. The rotational transform is then about one-sixth. The field line does not have to trace a perfect circle in the cross section, and the crossings do not have to be evenly spaced. The rotational transform is an average that more or less measures the amount of twist.

You have no doubt heard of fractals and chaos theory, topics that have been developed since the invention of fast computers. It was the mapping of field lines in magnetic bottles that gave impetus to the development of these concepts. Ideally, with well designed and fabricated windings for creating the magnetic field, the locus of intersection points in a stellarator can be perfect circles, with the field lines coming back to a different angle on the circle each time. With a finite number of turns on the helical coil instead of an infinite number, the circle can be distorted into, say, a triangle; but a field line will come back to the same triangle on each pass. In real life, magnet coils are not made perfectly, and there are small perturba­tions. These can cause wild behavior in the map, causing strange attractors, where the points tend to clump at a particular place; or magnetic islands, which we will discuss later; or complete chaos in the way the points are distributed. The name of the game in stellarators is to create nested magnetic surfaces, in which the magnetic lines always stay on the same surface and intersect each cross section on the same curve. An idealized case is shown in Fig. 4.22. Once created on a magnetic surface, an ion or electron stays on that surface as it goes around the torus thousands or millions of times. The surfaces do not have to be circles, but they never touch or overlap, so the plasma remains trapped by the magnetic field.

A stellarator requires such precision in its manufacture that in the early days they could not hold a plasma very long. In the next chapter, we shall introduce the toka — mak. This is a torus, of course, since it has to be doubly connected; but its poloidal field is not generated by external coils but by a current in the plasma itself. This allows it to have self-healing features which can overcome small imperfections in its construction.

Fig. 4.22 Nested magnetic surfaces. A particle stays on its surface as it goes around and around the torus