Walking past Harold Furth’s office one day, I saw this huge Chiquita Banana bal­loon hanging down from the ceiling. “What’s going on?” I asked. “Welcome to banana theory,” he replied, “the fruitful approach to fusion!” This was the begin­ning of a new understanding of how particles move in a torus. We knew that bend­ing a cylinder into a torus would induce vertical drifts, and we knew how to counteract those by twisting the field lines into helices. But there were more subtle toroidal effects that we did not know about for the first 15 years. To explain banana orbits, we first have to describe magnetic mirrors.

If a magnetic field is not uniform — that is, if its strength changes as you move along a field line — it can reflect a charged particle and cause it to go backwards. This is the same effect that makes two permanent magnets repel each other when you turn one around so that their polarities don’t match. There are toys that use this repulsion effect to suspend a magnetic object in midair. In Fig. 4.3b in Chap. 4, we showed that an electromagnet can create a magnetic field with coils of wire carrying a current. The ions and electrons gyrating in their circular orbits in a mag­netic field are like electromagnets, since they are like one-turn coils carrying a current, even if the current is lumped into one charged particle. Figure 6.4 shows the field of a gyrating ion immersed in the nonuniform field of a normal electro­magnet. The ion’s magnetic field is always in the opposite direction to that of the field it’s immersed in. Why? Because a physical system always tries to fall into the

lowest energy state. By canceling part of the background magnetic field, the ions can lower the total magnetic energy. Electrons will do the same even though they have negative charge. They rotate in the opposite direction, but being negative, they carry current in the same direction as the ion do.

In Fig. 6.4, an ion, carrying the magnetic field that it generates, moves to the right. The field lines on the right are of a background magnetic field generated by large coils outside the plasma. The field lines generated by the current of the gyrating ion are shown in red. The opposing fields push the ion backwards, like two perma­nent magnets with opposite polarity. The ion’s motion to the right is slowed up. The ion is moving into a stronger field, since the black lines are getting closer together. When the external field gets too strong, the ion cannot go any farther and is reflected back. How far the ion goes depends on how fast it was moving from left to right. However, not all ions will get reflected because the background field has a maximum strength. If the ion comes in with enough energy to go through the maximum, it gets slowed up there, but it is able to go through and regain its velocity on the other side. A converging magnetic field is a magnetic mirror that can reflect all but the fastest ions. This mechanism of magnetic mirroring was used by Enrico Fermi to explain the origin of cosmic rays. There, the interstellar magnetic fields are moving very rapidly, and they can push ions up to very high energies. Why can’t we use magnetic mirrors to trap and hold a plasma? Indeed, we can, but magnetic mirror systems have not worked out as well as tokamaks. Mirrors will be described in Chap. 10.

Now we can get to the bananas. Tokamaks also have magnetic mirrors, but they hinder rather than help the confinement. Recall from Fig. 4.14 in Chap. 4 that the magnetic field is always stronger on the inside of a torus, near the hole, than on the outside because the coils are closer together in the hole, and therefore the field near one coil also gets contributions from the neighboring coils. That means that there is a nonuniform magnetic field, and particles going from a weak field to a strong field might get reflected. Ideally, particles travel along helical field lines on a mag­netic surface and never leave it. However, magnetic mirroring prevents this, as shown in Fig. 6.5.

Fig. 6.5

In this figure, the dashed line is a helical field line. An ion does not actually follow this line exactly unless its Larmor radius is zero. When it gyrates in a finite-sized circle, it will drift slowly from one line to another, as shown in Fig. 4.10, if the magnetic field strength is not the same on every side of its orbit.1 The helical twisting cancels out the vertical drift on the average, but the averaging is disrupted by the mirror effect. The actual ion orbit is like the one shown by the solid line in Fig. 6.5. This ion starts out on the outside of the torus, where the field is weak, and it loops around toward the inside, where the field is strong. If it is not moving fast enough, it will be reflected by the magnetic mirror effect and come back on a slightly dif­ferent path. Only ions with enough energy parallel to the field line will make it around to the inside of the torus and sample all parts of a magnetic surface as we envisioned in our earlier naive picture of magnetic bottles. If we project the path of the ion in Fig. 6.5 onto the cross section of the torus shown there, it will look some­thing like Fig. 6.6.

These are the so-called banana orbits. In each case, the outside of the torus is on the right side of the cross section, and the strong field near the hole in the doughnut is on the left. The small banana in panel (a) is for a particle with small velocity parallel to the magnetic field; it gets reflected before it gets very far toward the inside. The dashed line is the path of a passing particle, one that gets through the mirror and can come all the way around. In panel (b), the particle has larger parallel velocity and goes farther to the left, describing a larger banana. The limiting case is shown in panel (c), where the particle nearly makes it through the mirror. Tom Stix whimsically dubbed this the WFB, the World’s Fattest Banana.

Banana orbits were discovered theoretically. They have never been seen in experiment because it would be very hard to track the path of a single ion or elec­tron in a plasma with more than a trillion particles per cubic centimeter. However, theory predicts the consequences of banana orbits, and these unfavorable effects are well established by experiment. It’s easy to understand why these bananas bear

bitter fruit. When an ion makes a collision, instead of jumping from one Larmor orbit to an adjacent one, it jumps from one banana orbit to the next; and banana orbits are much wider.2 Instead of the very slow rate of “classical” diffusion that we described in Chap. 5, the rate of plasma transport across the magnetic field is much faster in a torus than in a straight cylinder. The rate of banana diffusion can be calculated easily and is called neoclassical diffusion. It is a characteristic of toruses that was not initially foreseen. The good news is that it is still a classical effect; that is, it can be calculated using a known theory. Figure 6.7 shows how banana diffu­sion differs from classical diffusion. At the left-hand side, the collision rate between ions and electrons is very small, so small that an ion can traverse one or many banana orbits before making a collision. In the middle, flat part of the curve, the trapped ions (those making banana orbits) make collisions during a banana orbit, but the passing particles, being faster, do not. In the right-hand part, the collision rate is high enough that all particles make collisions in traversing the torus. Under fusion conditions, the plasma is so hot and so nearly collisionless that it is well into the banana regime, at the extreme left of the graph. Therefore, it is clear that the banana diffusion rate is much higher than the classical one, shown by the straight line at the bottom.

One might think that the closer a torus is to a cylinder, the smaller the banana effects will be. The aspect ratio A of a torus is the major radius R divided by the minor radius a, as shown in Fig. 6.8. A fat torus would have small A and a skinny one, large A. One would think that large A would have smaller banana diffusion, but this is not always true. It depends on many subtle effects which can cancel one another. The Kruskal-Shafranov limit states that q (the inverse rotational transform) has to be larger than 1. For a given value of q, banana dif­fusion is actually larger for large A. This is primarily because the ion has to go a long way around the torus before it turns around, and it is drifting vertically the whole time.

An even stranger, counter-intuitive effect has to do with the width of a banana orbit. It turns out that this width depends only on the strength of the poloidal field Bp and not on the toroidal field Bt. Remember that Bp is only the small field generated by the plasma current that gives the field lines a small twist. The banana width is approximately the Larmor radius of an ion calculated with Bp instead of Bt. This is much larger than the real Larmor radius, calculated with Bt. Since banana diffusion goes by steps of the size of a banana width, which depends only on the relatively weak Bp, does this mean that the much stronger toroidal field is useless? No! The toroidal field is needed to make the real Larmor radius small so that we can consider only the movement of the guiding centers, not the actual particles. If the toroidal field were eliminated,4 the gyration orbits would be so large that magnetic confinement would be no good at all, and furthermore there would be nothing to hold the plasma pressure.5