We saw in Chap. 5 that the plasma in a fusion reactor has to have a temperature of at least 10 keV (about 100 million degrees), but most of our deliberations have been about the problem of keeping a plasma from leaking out of its magnetic container. Isn’t heating to 50 times the temperature of the sun a bigger problem? The problem is nontrivial, but there have been no unexpected effects comparable to, say, micro­instabilities. The simplest way to heat a plasma is to drive a current through it.

A current is needed in a tokamak anyway to produce the poloidal field. This is ohmic heating, which happens whenever there is resistance in a wire carrying a current, such as in a toaster. The plasma in a tokamak can be considered as a one — turn wire loop, even though it is a gaseous one. It has a resistivity due to electron — ion collisions. When a voltage is applied around the loop, the electrons carry the current; and when they collide with ions, their velocities get randomized into a bell­shaped distribution, raising the temperature. The usual way to apply an electric field to loops of wire is to use a transformer, a common household device. It is the heavy piece of iron found in fluorescent lights and in the power bricks of electronic devices like cell phone chargers. Very large transformers are used to convert the high voltage of the power line (as much as 10,000 V) down to the household 115 V AC that we use in the USA or the 230 V in Europe. We know about these because they sometimes blow up, causing a power outage.

The first tokamaks used transformers for ohmic heating, as illustrated in Fig. 7.14. A pulse of current in the primary winding (shown as the three turns on the outer legs) drives a larger current in the plasma, which forms a one-turn secondary winding. This method was OK for small research machines, but the transformer would be too large in a large machine. Instead, one can use an air-core

transformer without the iron, as shown in Fig. 7.15. What are shown are toroidal coils, known as OH (ohmic heating) coils, which go around the torus the long way. A pulsed current in the OH coils induces a current in the opposite direction in the plasma. This is inefficient compared to an iron transformer, but it is easier to drive a large current in the OH coils than to create the space for a large iron transformer. The “Equilibrium Field Coil” in that figure generates the vertical field described at the end of Chap. 6. Note that Fig. 7.15 is intended only to show the principle; actual “poloidal-field coils” are numerous toroidal coils located mostly on the outside of the torus and combine the currents necessary for equilibrium, ohmic heating, and shaping of the plasma.

At this point, the words poloidal and toroidal have been used so often that it may be well to review what these terms mean to avoid any further confusion. A toroidal line goes along a doughnut, or even a pretzel, the long way, tracing out a circle in the case of a doughnut and a figure-8 in the case of a pretzel. A poloidal line goes the short way around the cross section of a doughnut, encircling the dough but not the hole. What is confusing is that magnetic and electric fields are generated differently by currents flowing in coils. For magnetic fields, a toroidal field is generated by poloidal coils which pass through the hole and encircle the plasma. Thus, the main toroidal magnetic field of a tokamak is generated by poloidal coils called toroidal-field coils! These are the blue coils seen in Fig. 7.15. A toroidal coil generates a magnetic field passing through the coil. Thus, the largest red coils in Fig. 7.15 generate a more or less vertical magnetic field, which is poloidal even though it does not actually encircle the plasma the short way. For electric fields, the opposite is true: a toroidal coil will generate a toroidal current. Thus, the smaller toroidal red coils in Fig. 7.15 are used to induce toroidal currents in the plasma. These are the OH coils. It is not necessary to understand this. Creating the fields we need is straight electrical engineering, and there are no unexpected plasma instabilities!

Ohmic heating cannot be the primary heating method in a fusion reactor for two reasons. First, OH cannot raise the plasma temperature high enough for fusion because, as explained in Chap. 5, the plasma is almost a superconductor at those temperatures. Collisions are so rare that the plasma’s resistance is almost zero, and resistive heating becomes very slow. Second, transformers work only on AC, whereas a fusion reactor must be on all the time in a DC fashion. The current induced in the secondary depends on an increasing current in the primary, and that current cannot increase forever. That is why tokamaks up to now have been pulsed, though very long pulses, of the order of minutes, are now possible. Other heating methods are used which can operate in steady state. Remember, however, that aside from ohmic heating, a current is necessary in a tokamak for producing a rotational transform — the twisting of the field lines. Fortunately, there are other ways to gen­erate DC current for that purpose. One way is to launch a wave in the plasma that can push electrons along the magnetic field. Another is the “bootstrap current,” a naturally occurring phenomenon that we describe in the “Mother Nature Lends a Hand” section. Stellarators are toroidal machines that do not need a current, since the rotational transform is generated by twists in the external coils. Hence, stellara — tors avoid the problem of current drive. They may ultimately be the way fusion reactors are constructed, but up to now we have had much more experimental expe­rience with tokamaks.

Another way to heat a fusion plasma to the required millions of degrees is Neutral Beam Injection or NBI to those who like acronyms. This is now the preferred method, and it works as follows. Neutral atoms of deuterium with high energy (between 100 and 1,000 keV) are injected into the plasma. Being neutral, these atoms can cross the magnetic field. Once inside the plasma, the atoms are rapidly ionized into ions and electrons, producing beams of energetic deuterium ions. The velocity of the neutral atoms is adjusted so that they can go far into the plasma before they are ionized. Once ionized, the beam becomes a beam of fast deuterium ions, and these give their energy to electrons by “electron drag” and to the plasma ions by colliding with them, raising their temperature. Neutral atoms cannot be accelerated by an electric field because they have no charge, so to make a neutral beam one must start with charged particles. One can start with a positive ion, accelerate it, and then add an electron to make it neutral; or one can start with

a negative ion and then strip its extra electron to make it neutral. It is easier to do the latter. Hydrogen has an affinity for electrons, so negative deuterium (D-) ions are not hard to make. They are then accelerated in a relatively simple accelerator. The extra electron in D- is loosely bound, so it is easily stripped off when the beam passes through a little bit of gas; and a fast neutral is formed. Neutral beam injectors are very large and tend to take up more space than the tokamak itself. Figure 7.16 shows what a tokamak looks like when surrounded by neutral beam injectors. These beams can be aimed in different directions to give momentum to the plasma. Normally, it is best to use co-injection; that is, injection in the same direction as the tokamak current. This method of heating is powerful and necessarily changes the conditions of the plasma from what simple theory would predict. On the other hand, adjusting the beam affords another way to control the plasma.

There are three other major methods for heating worth noting: ion cyclotron resonance heating (ICRH), electron cyclotron resonance heating (ECRH), and lower-hybrid heating (LHH). In cyclotron heating, a high-frequency electric field is launched into the plasma, and its frequency is adjusted to match the gyration frequency of the particles as they rotate around in the magnetic field. These circular Larmor orbits were shown in Fig. 4.9. The electric field changes its direction at the cyclotron frequency, so that as the particle moves in a circle and changes its direction, the electric field follows it so that it is always pushing the particle. Those particles that start out out-of-phase are decelerated by the field but then get into phase and are pushed. They collide with one another to thermalize, thus raising the tempera­ture of the whole gas. This works for both ions and electrons, but the technology is entirely different.

ICRH requires power generators with frequency in the tens of MHz (million cycles per second). This is in the radiofrequency range, between the bands used by AM and FM radios. Therefore, the generators are like those used by radio stations, only more powerful. The antenna, however, is not mounted on tall towers. It is a series of coils inside the vacuum chamber of a tokamak but outside the plasma, so that it does not get damaged.

ECRH requires generators of the much higher cyclotron frequency of electrons, around 50 GHz (billion cycles per second). This is in the microwave range. Microwave ovens and some telephones operate at the standard frequency of 2.4 GHz, some 20 times lower. The magnetron that is used in microwave ovens puts out about a kilowatt of power. In fusion, special gyrotrons have been developed which can produce tens of megawatts continuously. As in a microwave oven, ECRH does not need an antenna; the waves go through a hole. A very useful feature of cyclotron heating is that it is localized. Cyclotrons work because the frequency does not change with particle energy (until it goes beyond an MeV), but the fre­quency does change with magnetic field. Since the magnetic field in a tokamak is not the same everywhere, this means that only the plasma located at the right mag­netic field gets heated, and this position can be changed by changing the frequency. We have seen how the profile of the tokamak current can change the magnetic topology and the q-value of the rotational transform. Localized heating can change all this, giving operators a way to control the stability of the tokamak.

Heating can also be accomplished by launching waves into the plasma using different frequencies and different types of antennas. These waves bear names like lower-hybrid wave or fast Alfven wave and belong to a large array of waves that can exist in a magnetized plasma. By contrast, the unmagnetized, un-ionized air that we breathe can support only two kinds of waves, light and sound. It remains to be seen whether wave heating will be practical in a real fusion reactor.