There are many microinstabilities, but they all share the same types of plasma motion. As an example, we shall try to explain how a resistive drift wave goes unstable. This instability has stood the test of time as other theoretical predictions have come and gone. In general, it is easier to derive an instability mathematically than to figure out exactly what the plasma is doing. If this part is difficult to follow, you can skip to the next section without losing essential information. Let’s start with a plasma in a straight cylinder with a straight magnetic field, as shown in Fig. 6.12. The plasma is necessarily denser at the center than on the outside. The white arrows show a density ripple, like a wave, propagating in the azimuthal direc­tion. We shall focus on the plasma’s behavior inside the small rectangle at the bottom. This rectangle is shown enlarged in Fig. 6.13. On the left, we see Larmor orbits of ions whose guiding centers may be outside the rectangle. If the magnetic field is

out of the page, as shown, the ions will be rotating clockwise. Remember that the plasma density is higher at the top than at the bottom because the top is closer to the center of the plasma. To show this, we have drawn two orbits at the top and only one at the bottom. There are obviously more ions going left than going right. The ion fluid in this small volume therefore has an average flow toward the left. This effect is called the ion diamagnetic drift, and the drift velocity is called vDi. Note that this drift is perpendicular both to the magnetic field and to the direction in which the density is changing. The diagram on the right is the same thing for elec­trons. With their negative charge, electrons gyrate counterclockwise. Their diamag­netic drift velocity, vDe, is therefore in the opposite direction, to the right. This motion of the ions and electrons, considered as fluids occupying the same space, is there even if the guiding centers are not moving. The existence of the diamagnetic drift depends on the gradient in density and would be zero if the density were uniform everywhere. If you have a problem with two fluids occupying the same space, just think of the vermouth and vodka in a martini.

Now we can proceed with the wave. Our little rectangle is shown three times in Fig. 6.14. At the bottom of the first diagram, (a), a density ripple is shown. A slice of the rectangle near the peak of the wave, where the density is high, is shown in a darker shade. The background density is high at the top and low at the bottom, as seen in Fig. 6.12. The diamagnetic drift of the ions in the background density gradi­ent is to the left for ions and to the right for electrons, as shown in Fig. 6.13. Because the wave density is high near its peak, the diamagnetic drifts bring an excess of positive charge to the left side of the small slice and an excess of negative charge to the right side. These electric charges create the electric field E shown in panel (b). Recall from Chap. 5, Fig. 5.4, that an electric field causes an E x B drift, vE, perpendicular to both E and B. In this case, the drift is downwards, as shown in panel (c). Since the background density is high at the top, vE brings more density into the slice, and the wave gets more density where the wave density was already high. Therefore, the wave grows; it is unstable. Figure 6.15 shows what happens at a wave trough. There, the density is less than average, so the diamagnetic drifts bring less density to the edges of the slice, causing the buildup of charges of the opposite sign. The resulting electric field, shown in panel (b), is in the opposite direction from before. This causes the E x B drift in panel (c) to be upwards instead

of downwards. But an upward motion brings lower background density into the slice where the wave density is already low. This adds to the growth of the wave. We can now give it its rightful name: a drift wave. If we average over the cycles of the drift wave, more density is moved downwards at the peaks of the wave than is lost at the troughs, and consequently the wave causes plasma to move outwards, away from the center, toward the wall. Another insidious, cunning way the plasma finds to escape from its magnetic trap.

However, we are not quite finished; there is a three-dimensional part, shown in Fig. 6.16. The rectangular slices at the peaks and troughs of the wave in the last two figures are shown together at two cross sections of the cylinder, now considered as part of a torus. There are four slices: peak, trough, peak, trough. Between the slices are the electric charges shown in Figs. 6.14 and 6.15. Recall that a toroidal confinement requires a poloidal field to twist the magnetic field. This twist causes the field line going through a positive charge to connect to a negative charge in a cross section further downstream. Electrons, being very light and mobile, almost instantaneously move along the field line to cancel the charges. The electric field of the drift wave is nullified, and the wave can never grow. Ah, but if there are collisions, the electrons are slowed down, and they cannot cancel the charges fast enough. This is another example of Newcomb’s theorem: if is not zero, all bets are off! The growth of the drift instability depends on the existence of resistivity, one of our microeffects. Even without collisions, electron inertia or Landau damping can slow down the electrons and allow the instability to grow. Hence, it is a universal instability which can occur whenever there is a density gradient in a magnetic confined plasma.

The obvious question is, “What if the plasma density is uniform all the way out to the wall?” That can’t happen, since the density has to be essentially zero at a cold wall. If the density gradient occurs in a thin layer near the wall, the sharp gradient there will make the instability grow even faster. It then eats away the plasma so that the thickness of the gradient layer gets larger and larger. Drift instabilities can be stabilized by shortening the connection length between the cross sections shown in Fig. 6.16, so that the electrons can move between them fast enough. This requires a larger helical twist of the magnetic field. Fortunately, this can be done without violating the Kruskal-Shafranov limit.

There are many other possible microinstabilities. The ion-temperature-gradient instability is another one that is worrisome. This example of the resistive drift insta­bility serves to give an idea of how complicated plasma behavior is and how Bohm diffusion was solved. What happens when an unstable wave breaks and becomes turbulent? It is no longer possible to identify which instability started the turbu­lence, but one can apply known stabilization methods to see if the fluctuations can be suppressed. There are turbulence theories that purport to predict how the turbu­lence will look and how much anomalous diffusion it will lead to. A powerful modern method is to do a computer simulation. A computer does not care whether an equation is nonlinear or not. It does not even need to solve an equation; it just follows the particles around to see where they go. There will be some examples later; it’s not as simple as it sounds. Or, one can use physical intuition to make a guess. Figure 6.17 shows a guess on what a resistive drift instability might become when it goes nonlinear. The waves break up into blobs of density which are drifted out to the boundary by their internal electric fields. Thus, plasma is lost in bunches. This guess was made in 1967, before diagnostic techniques were available to detect such blobs. However, in 2003, physicists at M. I.T. (Massachusetts Institute of Technology) developed a special technique which allowed them to take pictures of blobs as they carried plasma radially outward. One such picture of two simultane­ous blobs is shown in Fig. 6.18 [6]. This behavior is not accidental, since it was observed also in several other tokamak machines. However, this is just a example

of how an instability, starting as a simple wave, can grow and carry plasma outwards. Other instabilities have been found to develop into other shapes as they do their dirty work.

In Chap. 4, we showed why a torus was chosen as a possible shape of a magnetic bottle used to hold a plasma hot enough to produce fusion power. In Chap. 5, we discussed the general features that had to be built into toruses in order to hold plas­mas. In this chapter, we described the unexpected difficulties that were encountered in tokamaks and how these were overcome. These are the concepts which guided our work in the early days of fusion. In the four decades since that time, experiments on dozens, or even hundreds, of tokamaks, stellarators, and other magnetic devices throughout the world have led to improvements in design and advances in theoretical understanding. Tokamaks no longer look like simple circular toruses. The next chapter will tell why.