A toroidal plasma current serves two purposes: it generates the necessary twist in the magnetic field, and it can also raise the plasma temperature by ohmic heating. However, there is a limit to how much current can be driven because of yet another instability: the kink instability. Figure 6.2 shows an initially straight current path in the plasma that has bent itself into a kink. The circles show the field lines of the poloidal field that the current generates (the toroidal field is from left to right). Note that the lines are closer together on the inside of the kink than on the outside, indicating that the field is stronger on the inside. The magnetic pressure, therefore, is stronger at the bottom of this picture than at the top, and the kink is pushed further out. The bigger the kink, the larger the pressure difference; and the instability grows rapidly and disrupts the current. Remember that the poloidal field shown here is not the main (toroidal) field that supports the plasma pressure; it is the relatively small field that provides the twist. The toroidal field has a stabilizing influence, since it resists being pushed around by the plasma current. The onset of instability, therefore, depends on
how strong the toroidal field is relative to the current. Conversely, onset of instability depends on how much current there is for a given toroidal field strength.

The limiting current for stable operation is called the Kruskal-Shafranov limit, and it is conveniently expressed in terms of the rotational transform, which is the number of times a field line goes around a torus the short way for each time it goes around the long way (Chap. 4). The critical rotational transform is exactly ONE! The critical current is that which creates a poloidal field large enough to twist the field lines just enough to give unity rotational transform, taking into account the strength of the main toroidal field. Transforms larger than 1 are unstable to kinks; transforms smaller than 1 are stable. The criterion for kink stability is actually quite complicated, since it depends on how the current varies across the plasma, but we can give a rough picture of why a rotational transform of 1 is a magic number.

The kink shown in Fig. 6.2 is in a straight plasma, but the current channel actu­ally flows around the torus and joins back on itself. Figure 6.3 shows the largest unstable kink, which is actually an off-center displacement of the plasma. The plasma has been made unrealistically thin in order to have room to show the effect. In the top view (a), the dashed lines indicate the cross sections viewed in panel (b). Let us assume that the rotational transform is exactly 1. On the right-hand side of either view, the plasma has been displaced toward the outer wall. On the left-hand side, half-way around the torus, the field lines have rotated half-way around the cross section, so the plasma is now close to the inside wall. If the transform is

Fig. 6.3 A large kink distortion of the plasma in a torus: (a) top view and (b) cross-sectional view

exactly 1, when the field lines come back to the right-hand side, they will be in the same place where they started, so the current can flow in a closed path. Remember that the plasma is almost a superconductor; so, without collisions, the electrons carrying the current must stay on the same field line. Now let us assume that the rotational transform is less than 1. Then, upon coming back to the right-hand cross section, the current channel is in the position shown by the cross-hatched circle, which does not match up with its initial position. Since current must flow in a con­tinuous path, this distortion of the current channel is not possible, and this kink cannot form. The plasma is stable for rotational transforms less than 1. In this simple picture, the plasma would also be stable if the transform is greater than 1, as long as it is not exactly 1. However, in that case, the current is strong enough to drive other shapes of kinks, and the plasma is kink-unstable in a way that is not easy to explain.

Since small rotational transform is good while large transform is bad, the recip­rocal of the transform is used in tokamak lore. This is the quality factor q (“little q”), which is high when the plasma is kink-stable and low when it is kink-unstable. If the rotational transform is larger than 1, q is less than 1, and the plasma iskink — unstable. If the rotational transform is smaller than 1, q is larger than 1, and the plasma is kink-stable. What if q is a rational fraction so that the current channel joins up to itself after several trips around the torus? Then very interesting things happen, which we will get to.