If the plasma in a torus always thrashes around violently, there must be an energy source that drives the thrashing. An obvious source is the electric field applied to drive the current in ohmic heating. In the 1960s, a new method was devised for heating without a large DC current in the plasma. This was Ion Cyclotron Resonance Heating or ICRH. A radiofrequency (RF) power generator was hooked up to an antenna around the plasma, the way an FM station is hooked up to its antenna on a tower. The frequency was tuned to the gyration frequency of the ions in their cyclo­tron orbits. As the ions moved in circles, the RF field would change its direction so as to be pushing the ions all the time, just as in a real cyclotron. This could heat up the plasma without having to drive a DC current in it.8 Would this kill the turbu­lence and make the plasma nice and quiet, without Bohm diffusion? A case of champagne was bet on it. It didn’t work. The thrashing was as bad as ever. The darts in Bohm’s picture stayed there.

The problem was a failure of magnetohydrodynamics, MHD for short. MHD theory treats a plasma as a pure superconductor, with zero resistivity, and neglects the cyclotron orbits of the particles, treating them as points moving at the speed of their guiding centers. Though this simplified theory served us well in the design of toroidal confinement devices and in the suppression of the gravitational and kink instabilities, it did not treat a plasma in sufficient detail. First of all, there have to be some collisions in a fusion plasma or else there wouldn’t be any fusion at all! These infrequent collisions cause the plasma’s resistivity to be not exactly zero, and that has dire consequences on stability. The fact that the Larmor orbits of the ions are not mathematical points gives rise to the finite Larmor radius (FLR) effect. In some cases, even the very small inertia of the electrons has to be taken into account. Finally, instabilities could even be caused by distortions of the particles’ velocity distributions away from a pure Maxwellian, an effect called Landau Damping. These small deviations from ideal MHD turned out to be important, making the theorists’ task much more difficult.

The first inkling of what can happen was presented by Furth, Killeen, and Rosenbluth in their classic paper on the tearing mode [2]. If a current is driven along the field lines in a plasma with nonzero resistivity, the current will break up into filaments; and the initial smooth plasma will tear itself up into pieces! So “tearing” rhymes with “bearing,” and not “fearing,” though the latter interpretation may have been more appropriate. The tearing mode is too complicated to explain here, but we describe other instabilities which caused even more tears.

One of the tenets of ideal MHD is that plasma particles are “frozen” to the field lines, as shown in Fig. 4.10. Without collisions or one of the other microeffects named above, ions and electrons would always gyrate around the same field line, even if the field line moved. Bill Newcomb once proved a neat theorem about this [3], saying that plasma cannot move from one field line to another as long as E (E-parallel) is equal to zero. Here, E is the electric field along a magnetic field line, and it has to be zero in a superconductor, since in the absence of resistance even an infinitesimal voltage can drive an infinite current. But if there are collisions, the resistivity is not zero, E can exist, and plasma is freed from one of its constraints.

So it was back to the drawing board. While the theorists enjoyed a new challenge and a new reason for their employment, the experimentalists pondered what to do. In previous chapters, we showed that (1) a magnetic bottle had to be shaped like a torus, (2) bending a cylinder into a torus caused vertical drifts of ions and electrons, (3) these drifts could be canceled by twisting the field lines into helices, (4) this twist could be produced by driving a current in the plasma, and (5) this current could cause other instabilities, even in ideal MHD, but that those could be con­trolled by obeying the Kruskal-Shafranov limit. In spite of these precautions, the plasma is always turbulent, even when the current is removed by using a stellarator rather than a tokamak. How can we get a plasma so smooth and quiet that we can see a wave grow bigger and bigger until it breaks into turbulence, as in Fig. 6.9? Obviously, if one could straighten the torus back into a cylinder, much of the origi­nal cause of all the trouble would be removed. But how can one hold the plasma long enough just to do an experiment? The plasma will simply flow along the straight magnetic field into the endplates that seal off the cylinder so that it can hold a vacuum. The solution came with the invention of the Q-machine (Q for Quiescent). Developed by Nathan Rynn [4] and Motley [5], this is a plasma created in a straight cylinder with a straight magnetic field. Inside each end of the vacuum chamber is a circular tungsten plate heated to a red-hot temperature. A beam of cesium, potassium, or lithium atoms is aimed at each plate. It turns out that the outermost electron in these atoms is so loosely bound that it gets sucked into the tungsten plate upon contact. The electron is then lost, and the atom comes off as a positively charged ion.

Of course, a plasma has to be quasineutral, so the tungsten has to be hot enough to emit electrons thermionically, the way the filament in light bulb does. So both ions and electrons are emitted from the tungsten plates to form a neutral plasma. No electric field has to be applied! Only tungsten or molybdenum, in combination with the three elements above, can perform this kind of thermal ionization. In this clever device, all sources of energy to drive an instability have been removed or so we thought. Figure 6.11 shows a typical Q-machine, covered with the coils that create the steady, straight, and uniform magnetic field.

The plasma in a Q-machine has to be quiescent, right? To everyone’s surprise, it was still turbulent! The trace shown in Fig. 6.10 actually came from a Q-machine. Fortunately, it was possible to stabilize the plasma by applying shear, as shown in Fig. 5.9, or by applying a small voltage to the radial boundary of the plasma. A quiescent plasma in a magnetic field was finally achieved. Then, by adjusting the voltage, one could see a small, sinusoidal wave start to grow in the plasma; and, with further adjustment, one could see the wave get bigger and bigger until it broke into the turbulence seen in Fig. 6.10. With a regular, repetitive wave like a wave in open water, one could measure its frequency, its velocity, its direction, and how it changed with magnetic field strength. These were enough clues to figure out what kind of wave it was, what caused it to be unstable, and, eventually, to give it a name: a resistive drift wave.

As its name implies, the wave depends on the finite resistivity of the plasma. It also depends on microeffects: the finite size of the ion Larmor orbits. Before showing how a drift instability grows, let’s find the source of energy that drives it. In a Q-machine, we have eliminated all toroidal effects and all electric fields normally needed to ionize and heat the plasma. In fact, the plasma is quite cold, as plasmas go. It is the same temperature as the hot tungsten plates, about 2,300 K, so that the plasma temperature is only about 0.2 eV. You can heat a kiln up to that temperature, and it would stay perfectly quiescent. A magnetically confined plasma, however, has one subtle source of energy: its pressure gradient. When everything is at the same temperature and there are no energy sources such as currents, voltages, or drifts, there is still one source of energy when the plasma is confined. And confinement is the name of the game. Since ions and electrons recombine into neutral atoms when they strike the wall, plasma is lost at the walls. The plasma will be denser at the center than at the outside, and this causes a pressure that pushes against the mag­netic field. By Newcomb’s theorem, the plasma would remain attached to the field lines, and nothing can happen; but once there is resistivity, all bets are off. The plasma is then able to set up electric fields that allow it to move across the magnetic field in the direction that the pressure pushes it. Even if there are no collisions, other microeffects like electron inertia or Landau damping can cause the drift instability to grow. For this reason, the resistive drift instability and others in the same family are called universal instabilities. They are fortunately weak instabilities because the energy source is weak, and they can be stabilized with the proper precautions.