Figure 6.1 of the last chapter showed how a plasma current circulating around a toka — mak generates a poloidal magnetic field to give a twist to the field lines. This twist, or helicity, is necessary to average out the vertical drifts that the particles have in a torus. These drifts arise when a straight cylinder is bent into a circle to form a torus. We then defined a quantity q, the quality factor, which tells how much twist there is; actually, how little twist there is. Large q means the twist is gentle, and small q means that the twist is tight. It is called the quality factor because the plasma is stable if q is larger than 1 and unstable if q is smaller than 1, so larger q gives better stability. You may recall that the culprit was the kink instability, and the boundary at q=1 was called the Kruskal-Shafranov limit. If q = 1, a field line goes around the torus the short ‘Numbers in superscripts indicate Notes and square brackets [] indicate References at the end of this chapter.

F. F. Chen, An Indispensable Truth: How Fusion Power Can Save the Planet,

DOI 10.1007/978-1-4419-7820-2_7, © Springer Science+Business Media, LLC 2011 way (the poloidal direction) exactly once after it goes once around the long way (the toroidal direction). It then joins on to its own tail. If q=2, the twist is smaller, so it takes two trips the long way before the field line joins onto itself, and so on.

In general, q is not a rational number like 1, 2, 3, 3/2, 4/3, and so forth. Except in such cases, a field line never comes back to itself; rather, after numerous turns, it traces out a magnetic surface. The field lines of neighboring surfaces cannot be parallel to one another either, because the magnetic field has to be sheared. Shear has a stabilizing effect on almost all instabilities. That means that q has to vary with radius within a cross section of the torus, so that the amount of twist is different on each magnetic surface. Scientifically, we say that q is a function of minor radius r, written as q(r). By now you may have guessed that something special happens when q is a rational number, like 2. At the radius where q(r) = 2, a field line joins onto itself after traversing the torus twice the long way. Remember that the tokamak current (the one that creates the helicity) is driven by an electric field (E-field). How this is done is shown later in this chapter. It is easier for the E-field to drive a current if the field lines are closed, since the electrons can then run around and around on the same field line. The current can break up into filaments. Each filament acts like its own little tokamak with its own magnetic surfaces, and the magnetic surface at q=2 breaks up into two magnetic islands. Other chains of islands could form at the q=3 surface, and so on. Between rational surfaces, the filamentation does not occur, and there are no islands. Figure 7.1 shows a computed picture of islands at the q=3/2 surface.1 Since the rotational transform is 1/q, it has the value of 2/3 here. That means that a field line inside the top island, after going around the whole torus once, will end up in the island at the lower right, say, two-thirds of the way around the cross section. After the next revolution, it will go another two-thirds of the way around, ending up in the island at the lower left. After the third traversal, it will be back in the top island, but not exactly where it started. It will be on the same small

magnetic surface inside the island, but at a different point. It is only after many, many traversals that the island is traced out. Our previous naive picture of nested magnetic surfaces has taken on a fantastic character!

Ions and electrons can cross an island in between collisions; and since the island width is much larger than a Larmor diameter, the escape rate is faster than classical just as in banana diffusion. Fortunately, not all island chains are large, and higher fractions like 5/6 would not yield noticeable islands at all.

Exactly where these island chains lie depends on how much current there is at each radius. The amount of current depends not only on the strength of the E-field, but also on the temperature of the electrons. The higher the temperature, the lower the resistivity, and the higher the current. Since the plasma tends to be hotter at the center, the plasma current generally has a peak at the center. Figure 7.2 gives an example of where island chains can occur, in principle. The curve shows how q typically varies with distance from the center of the plasma’s cross section. In this case, the rational surfaces q = 1, 2, and 3 occur at radii of about 3, 7, ad 9 cm, respectively, and there are no places where q is 4 or higher. There is a special region where q is less than 1.

The shape of the curve q(r) is determined by the distribution of the plasma cur­rent. Figure 7.3 gives examples of different current profiles J(r) and the q(r) curves that they produce. The uppermost curve, corresponding to the most peaked current, would have more rational q surfaces. Tokamak operators have some control over J(r), since there are various ways to heat the plasma. If the electron temperature changes, however, J(r) will change, and so will the magnetic topology. Where the q curves cross the line q = 1 is of utmost importance, as will be explained next.

Islands were first observed experimentally by Sauthoff et al. [1] in the famous “sombrero hat” experiment. Electrons emit a small X-ray signal when they collide with ions. By collecting these signals with detectors surrounding the plasma, the plasma density distribution can be reconstructed by computer the same way as in a medical CAT scan. Figure 7.4 shows a typical result at one instant of time.

The contours of constant density in Fig. 7.4a shows a q=2 island structure. A 3D plot of this in Fig. 7.4b resembles a sombrero.