Since tokamaks depend on an internal plasma current to produce the required twist of the magnetic field lines, the current has to be produced even if it is not needed for ohmic heating. Fortunately, the plasma automatically generates such a current,

figuratively “pulling itself up by its own bootstraps.” This comes about as follows. Since the plasma is not perfectly confined but gradually diffuses to the wall, there will be a density gradient, with the density high in the center and low near the walls, where the plasma can leave quickly. Think of a packed football or soccer stadium where, at the end of a game, the crowds storm onto the field in spite of the guards. The density of people is high at the top, but the crowd disperses on the field, where the density is low, forming a density gradient. It is this density gradient in a tokamak that causes the bootstrap current. Technically, it is the pressure gradient, where the pressure is density times temperature. Consider a tokamak with its helical magnetic lines, as shown in Fig. 7.17. The twist in the lines of the magnetic field is created by a toroidal current J, which generates the poloidal component, Bp of the field. It is this poloidal part of the field which is important here.

Figure 7.18 is a closer look at the minor cross section of the plasma showing the same tokamak current J seen in Fig. 7.17. The black arrows show the force on the electrons exerted by the plasma pressure pushing outwards. We can neglect the ions because they move so slowly that they cannot carry much current. The electrons gyrate in small circles, so we need only to consider the drift of their guiding centers. In Chap. 5, we showed the gyroscopic effect on guiding centers, in which a force moves the guiding center in a direction perpendicular to that force. The relevant part of Fig. 5.7 is reproduced in Fig. 7.18a, showing that the B-field, pressure force, and electron velocity are mutually perpendicular. Note that the current is opposite to the electron velocity because of their negative charge. In Fig. 7.18b, the force is in the radial direction, pushing outwards, while the poloidal field Bp is in the azi­muthal direction, going around in the circular direction. The electrons therefore drift in the direction perpendicular to both, which is the toroidal direction, the same as that of J. This toroidal electron drift constitutes the bootstrap current. It turns out that this current is always in the same direction as J, so that it adds to the total cur­rent. Once a seed current is induced in the torus so that the field lines twist enough to confine the plasma, the bootstrap current can then take over most of the work. There is, of course, also a pressure-caused drift perpendicular to the main toroidal component of the B-field, but this drift is in circles in the poloidal direction and does not contribute to the main tokamak current J.

Mother Nature made it hard for us to confine a plasma in a torus by requiring that the field lines be helical, but she then provided the benefit of bootstrap current so that this helicity can be mostly self-generated. It does not matter which direction the toroidal field is in, or which direction the toroidal current is in; the bootstrap current will always add to the toroidal current. In present experiments, the bootstrap current has been observed to contribute more than half the total current. In planned experiments, the bootstrap fraction will be more than 70%, and in fusion reactors more than 90%.

Detailed calculations3 of bootstrap current can be made using the neoclassical banana theory described in Chap. 6. Although collisions between passing particles and those in banana orbits cause the major part of bootstrap current, the final answer does not depend on knowing the collision rate. Collisions cause the pressure gradient, but it is only the resulting pressure gradient that matters. Going back to the stadium full of fans, we see that a density gradient of fans will occur regardless of whether they bump and shove one another or whether they do not touch.

In designing tokamaks, the shape of the bootstrap current depends on the shape of the magnetic field, which itself depends on the bootstrap current, so a delicate optimization problem has to be solved. The so-called Advanced Tokamak designs with high bootstrap fraction have hollow current profiles with larger current at the edge than in the center.