A picture of David Bohm was taped to the wall of Bob Motley’s office, and our group of experimentalists at Princeton’s Plasma Physics Laboratory took turns throwing darts at it. The frustration came from an unexplained phenomenon called “Bohm diffusion,” which caused plasmas in toruses to escape much faster than any classical or neoclassical theory would predict.6 In spite of all efforts to suppress the known instabilities, the plasma was always unstable, vibrating, rippling, and spit­ting itself out, like the foam on violently breaking surf. In Chap. 5, classical diffu­sion was described. This is a process in which collisions between ions and electrons cause them to jump from one field line to another one about one Larmor radius away. The classical confinement time is long, of the order of minutes. In this chap­ter, we described neoclassical diffusion, in which particles jump from one banana orbit to the next. The neoclassical confinement time is still of the order of seconds, longer than needed. Bohm diffusion caused the plasma to be lost in milliseconds. Major instabilities like the Rayleigh-Taylor or kink were no longer there, else the confinement time would have been microseconds. There were obviously some other instabilities that the theorists had not foreseen.

Bohm diffusion was first reported by physicist David Bohm when he was work­ing on the Manhattan project and, in particular, on a plasma device for separating uranium isotopes. From measurements of the plasma’s escape rate, he formulated a scaling law for this new kind of diffusion. It reads as follows. The diffusion rate across the magnetic field, given by the coefficient D± (pronounced D-perp), is pro­portional to 1/16 of the electron temperature divided by the magnetic field:

1 Te D, <x e

D 16 B

The 1/16 makes no sense here because I have not said what units T and B are in,

e

but that number has a historical significance. Whenever Bohm diffusion is observed, there are always randomly fluctuating electric fields in the plasma. Regardless of what is causing these fluctuations, the plasma particles will respond by drifting with their E x B drifts (Chap. 5). Since the size of the noise is related to Te, which supplies the energy for it, and the drift speed is inversely proportional to B, it is not hard to show that the TJB part is to be expected [1]. But how did Bohm come up with the number 1/16? Bohm had disappeared from sight after he was exiled to Brazil for un-American activities. In the 1960s, Lyman Spitzer tracked him down and asked him where the 1/16 came from. He didn’t remember! So we’ll never know. It turns out that the Bohm coefficient depends on the size and type of turbu­lence and can have different values, but always in the same ballpark.

Plasma turbulence is the operative term here. Any time there was unexplained noise, it was called “turbulence.” Doctors do the same thing with “syndrome” or “dermatitis.” Figure 6.9 is an example of turbulence; it is simply a wave breaking on a beach. As the wave approaches the beach, it has a regular, predictable up and down motion. But as the water gets shallower, the wave breaks and even foams. The

motion of the water is no longer predictable, and every case is different. That’s the turbulent part. The regular part is called the linear regime; this is a scientific term that has to do with the equations that govern a physical system’s behavior. Linear equations can be solved, so the linear behavior is predictable. The turbulent part, in the nonlinear regime, can be treated only in a statistical sense, since each case is different. Nonlinear generally means that the output is not proportional to the input. For instance, taxes are not proportional to income, since the rate changes with income. Compound interest is not proportional to the initial investment, even if the interest rate does not change, so the value increases nonlinearly. Population growth is nonlinear even with constant birth rate, in exact analogy with compound interest. Waves, when they are small and linear, will have sizes proportional to the force that drives them. But they cannot grow indefinitely; they will saturate and take on different forms when the driving force is too large. What a wave will look like after it reaches saturation can be predicted with computers, but the detailed shape will be different each time because of small differences in the conditions. Then you have turbulence. The smoke rising from a cigarette in still air will always start the same way, but after a few feet each case will look different.

The turbulence in every fusion device in early experiments was always fully devel­oped; we could never see the linear part, so we could not tell what caused the fluctua­tions to start in the first place. An example of plasma turbulence in a stellarator is shown in Fig. 6.10. This is what “foam” looks like in a plasma. These are fluctua­tions in electric field inside the plasma. These noisy fields make the particles do a random walk, reaching the wall faster than classical diffusion would take them.

Turbulence is well understood in hydrodynamics. If you try to push water through a pipe too fast, the flow breaks up into swirling eddies, slowing down the flow. Hydrodynamicist A. N. Kolmogoroff once gave an elegant proof, using only dimensional analysis, that the sizes of eddies generally follows a certain law; namely, that the number of eddies of a given size is proportional to the power 5/3 of the size.7 Attempts to do this for plasmas yielded a power of 5 rather than 5/3 [1],

and this has been observed in several experiments. However, plasmas are so com­plex (because they are charged) that no such simple relation holds in all cases.

The importance of turbulence and Bohm diffusion is not only that it is much faster than classical diffusion, but also that it depends on 1/B instead of 1/B2. In classical diffusion, doubling the magnetic field B would slow the diffusion down by a factor of 22 or 4. In Bohm diffusion, it would take four times larger B to get the same reduction in loss rate. It was this unforeseen problem of “anomalous diffusion” that held up progress in fusion for at least two decades. Only through the persistence of the community of dedicated plasma physicists, was the understanding and control of anomalous diffusion achieved. Modern tokamaks have confinement times approaching those required for a D-T reactor.