Sawteeth are an essential feature of tokamak discharges and are important because they show that a tokamak is self-healing. Toruses such as stellarators do not have such a feature because the magnetic structure is fixed by magnetic coils outside the plasma. A stellarator plasma cannot adjust its own magnetic topology by sawtooth-shaped hiccups. This brings up the general subject of self-organization. Many physical systems have been found which are self-organized. It may seem inconceivable that an insentient object can organize itself, but there are many examples in real life. Snowflakes are self-organized. No one had to program a computer to make these beautiful, symmetric art pieces (Fig. 7.7).

Our own bodies are self-organized. Complicated organs such as the eye, with its cornea, iris, lens, retina, and macula; and the ear, with its ossicles, cochlea, hair cells, and stereocilia, are self-organized, though some programming had to be done with the DNA. In the new field of nanotechnology, the objects are so small that they are difficult to make; and people are hoping that self-organization will help. In magnetic fusion, tokamaks have taken the lead partly because of their ability to heal themselves. There are magnetic bottles other than the standard tokamak empha­sized here that depend even more on self-organization (Chap. 10).

Magnetic Wells and Shapely Curves

Up to now, we have suppressed plasma instabilities by applying magnetic shear, creating a mesh of field lines that plasma cannot easily penetrate. There is another good way to eliminate instabilities, and that is to create a magnetic well. This is a magnetic bottle that surrounds the plasma with a stronger field on every side. The plasma then does not have enough energy to climb out of the hole that it is in. It is not possible to make such a container without a leak, which is why tokamaks do not depend on this effect as much as some other confinement concepts do. However, understanding the magnetic-well effect will help in the design of better tokamak shapes.

A simple magnetic well can be made with four infinitely long rods with opposite current in neighboring rods, as shown in Fig. 7.8. The magnetic field lines are the circles, and their spacing shows that the field gets stronger as one approaches each rod. A plasma trapped in the center would see the field increasing in every direction and would be held stably. However, there are leaks at each of the four cusps, where the field lines meet. An ion or electron following a field line toward one of the four cusps, where the field is strongest, would be reflected by the magnetic mirror effect described in Chap. 6. Unfortunately, that effect depends on the transverse momen­tum of the particles — the momentum that makes the particles gyrate in Larmor orbits. Those particles that have their velocities almost parallel to the field lines would not be reflected and would go right out at the cusps. There are enough of those particles to bring the confinement time of the plasma well below the many

seconds required in a fusion reactor.2 In the early days of fusion, one of the fanciful magnetic buckets that were proposed was the Picket Fence, a veritable Great Wall of China, as shown in Fig. 7.9. But if one had done his homework, he would have found that the leak at even one of the many cusps would have been insufferable.

Why does the magnetic field in cusp geometry look so different from the toka — mak fields we have seen so far? It is because the field bulges out towards the plasma instead of away from it. In a magnetic well, the field lines are convex as seen by the plasma, not concave. This generally means that the field is stronger on the outside than on the inside, and such field lines are said to have good curvature. Conversely, field lines that bulge outwards have bad curvature. This concept is much more general than its use in magnetic confinement. In Fig. 7.10, we see that a board which is bent upwards will support more weight than one which is level or sags downwards. Roman arches and those highly arched wooden bridges in Japanese gardens have good curvature.

Tokamaks have mostly bad curvature, but they can be designed, as we shall see, to minimize that effect. A true magnetic well is called a minimum-B device, where the plasma is in a magnetic field minimum. The twisting field lines in a torus can go through regions of both good and bad curvature. In that case, what matters is how much there is of each kind. If an electron sampling all regions of a magnetic surface sees mostly good curvature, it would be in an average-minimum-B device. It is hard to do this in a tokamak, but other toroidal systems which cannot be described here can be designed to be average-minimum-B. The idea is to minimize the time a particle spends in a region where the field is sharply bent in the bad direction. When an instability is concentrated in a region of bad curvature, it is

called a ballooning mode. The plasma escaping is such a region pulls the field lines with it, further weakening the field. This could be called a plasma hernia, but ballooning is a more dignified term!