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.

It is certainly not a thin, soft cover to keep the plasma warm. It is a thick, massive, complex structure that serves three major purposes: (1) capture the neutrons gener­ated by fusion and convert their energy into heat, (2) produce the tritium to fuel the DT reaction, and (3) shield the superconducting magnets from the neutrons. The blanket is divided into modules for easier replacement. Figure 9.9 shows where the blanket is located inside a tokamak. In Fig. 9.9a, we see that the plasma first strikes the first wall (FW), which is also the front surface of the blanket. Then, the neutrons go into the blanket, where their energy is captured, and where the tritium breeding takes place. The heat is taken away by hot gas or liquid coolants to heat exchangers outside. Shielding material protects the vacuum walls and supercon­ducting magnets from the heat and the neutrons. Figure 9.9b gives an idea of how the blanket surrounds the plasma and lies inside the vacuum. Outside the vacuum vessel are the magnetic coils. The Central Solenoid coil is critical, since there is not much room in the hole of the torus to fit this coil into. The symmetry axis of the torus is at the left. The entire machine fits inside a cryostat which insulates the mag­net coils from the outside world, keeping them at superconducting temperatures.

In a reactor there could be hundreds of blanket modules, each weighing a ton. There are many ideas for blanket design, and ITER will have three ports available for test blanket modules (TBMs). There are six TBM proposals competing for these three spots [5].

Research on closed magnetic bottles started with stellarators such as the figure-8 stellarator shown in Fig. 4.18. In 1969, the Model C Stellarator in Princeton, the largest at the time, was converted to a tokamak because of the good results coming from Russia with their configuration. Soon almost all new machines were tokamaks. This was because of the self-healing properties of tokamaks, as described in Chap. 7. When the temperature profile in the plasma became too peaked, sawtooth oscillations would arise and smooth it out to maintain stability. All this is now changed, and stellarators have come back as a hope for the future.

The difference between these two toroidal devices, tokamaks and stellarators, is the way the poloidal magnetic field (the component that gives the field lines their helical twists) is generated. In tokamaks, a large current in the plasma generates that field. In stellarators, external coils generate that field, and no large plasma cur­rent is necessary. But these external poloidal-field coils are hard to make. Present — day Advanced Tokamaks no longer rely on the self-healing features that were initially useful. We have learned how to shape the plasma current with radiofre­quency and microwave power to keep the plasma stable. In fact, sawtooth oscilla­tions are now deliberately eliminated. Since self-organization is no longer necessary, we can reconsider stellarators. Stellarators are less subject to effects such as disruptions that are connected with the large plasma current. In effect, they elimi­nate a source of energy that allows a plasma to self-organize destructively in its efforts to escape from confinement. Furthermore, since transformer action to drive the plasma current is not necessary, stellarators are more suitable for steady-state, continuous operation.

• Developing fusion power will cost less than putting a man on the moon. The Manhattan and Apollo programs have shown that the scientific and engineering communities have the ingenuity to achieve almost unimaginable goals once it is driven by national priorities, a sense of urgency, personal challenge, and a sense of national pride. With seven nations having banded together to push forward on fusion, the USA has lost its chance to do this alone. However, we are still far

from the goal because the most difficult problems of materials engineering have yet to be solved. The USA can regain its former leadership in fusion research by building one or more large FDFs to solve these problems simultaneously with ITER to shorten the time to a working reactor.

• The development of wind and solar power in private industry has stimulated the economy. Fusion machines are big and must be funded by the government, but the economic stimulus can also be generated by the subcontracts awarded to small companies. For instance, such components as superconducting strands, silicon carbide tiles, blanket modules, RF antennas, and even 3D computations can all be parceled out to start-up companies. New jobs will be created, and new financing will be secured.

• A high-priority Apollo-like program to put fusion on a fast track will cost less than Apollo did and will solve the CO2 problem, the fossil-fuel shortage problem, and the oil dependence problem all at once.

We started with the fact that the sun gives the earth 1 kW/m2, enough energy in 1 hour to supply the earth for a whole year. Now we understand why it is so hard to capture that energy. The atmosphere absorbs part of the sunlight. The sun does not shine at night and does not rise high in the winter. There are cloudy and stormy days. There is little sunlight at high latitudes, where the power is most needed. Solar cells can capture only part of the solar spectrum and are not efficient at that. The peak efficiencies quoted apply only when the sun is directly overhead. The color of sunlight changes near sunset and no longer matches the color the solar cells are optimized for. Solar panels cannot economically be turned to follow the sun as it moves across the sky. We are lucky to capture a few percent of solar energy, but even that is a lot of energy that should not be wasted.

Local solar panels on rooftops and exterior walls should be popularized and widely accepted as standard for new structures. These can contribute a few percent to the grid’s power, but no more because solar power is intermittent and cannot economically be stored. Selling excess power back to the power station is just a gimmick; the utilities could care less about this small disturbance.47

The advances in thin-film technology have made photovoltaic solar power com­petitive with conventional power sources. The energy payback time will fall below one year, which is short enough, though not a short as for wind power. To use this technology for large solar farms to provide central-station power, however, is fraught with problems. The main problem is that the sun does not shine at night, the time when people turn on their lights. There is no cheap, proven method for storing that much energy. Alternatively, one can build high-tech transmission lines to carry the electricity across time zones from daylight to moonlight, but this will take many decades to implement.

Solar power is an important supplement to grid power, but it is not suitable as a primary central-station power source. Fifty years from now, only coal, fission, and fusion are capable of supplying the dependable, steady backbone power that the civilized world can count on.

That the binding energy curve peaks in the middle is the reason that both fission and fusion can produce energy, but the way to tap these resources leads to entirely different types of reactors. In a fission reactor, a chain reaction is sustained as neutrons created in one fission move on to split other atoms nearby. The material, uranium or

plutonium, is held in tubes which can be moved so that the number of “nearby” atoms can be controlled. If control is lost, the reaction runs away, and there is an accident. In fusion, the hydrogen fuel is heated into a gaseous, electrified state called a plasma. Since the plasma is hotter than the interior of the sun, it must be held in place by a magnetic field rather than a walled container. The problem is that this magnetic bottle leaks, and the problem is to keep the fire burning. There is certainly no possibility of a runaway reaction in this case. However, plugging these leaks has been a long and difficult journey for fusion researchers, whose story will soon unfold.

People used to confuse astronomy with astrology. With the great success of the Hubble telescope, the difference between science and fortune-telling is now clear in the public’s mind. If fusion should succeed, perhaps the difference between fission and fusion would be equally well recognized.

This is a beneficial but baffling effect that is still unexplained. In comparing the confinement times of tokamak discharges using hydrogen, deuterium, and helium, it has been carefully documented [5] that the confinement time increases with the mass of the ion, contrary to all neoclassical and instability theories. Heavier ions have larger Larmor radii, so their step size in diffusion across the magnetic field should be larger, leading to shorter rather than longer confinement times. If ions cross the field not by collisions but by instability, most theories predict one of two scalings with atomic number A. (Here A is not the aspect ratio that we used before but the more familiar A used in chemistry. A is 1 for hydrogen, 2 for deuterium, 3 for tritium, and 4 for helium.) The crudest estimate is Bohm diffusion, which we discussed in Chap. 6. That diffusion rate is independent of A. More refined theories predict gyro-Bohm scaling, which takes into account the Larmor radii of ions, which vary as the square root of A or A1/2. That leads to confinement times that vary as 1/A1/2, shorter for heavier ions. What is observed, however, is that confinement varies more like A1/2, improving by a factor between 1.4 and 2 between hydrogen and deuterium. This means that confinement will be even better with tritium, which is not normally used in small experiments because it is radioactive.

The isotope effect seems to be universal, occurring in many different types of tokamak discharges. At first it was proposed that it is caused by impurities in the gas, but very clean discharges also exhibit this effect. There have been several theo­ries on specific instabilities that could have nonlinear behavior that depends on A in this fashion, but so far these have not been confirmed in tokamak experiments. A factor of 1.5 or 2 may be trivial in an experiment but would have great commer­cial benefits in a power plant.

One of the aims of ITER is to reach ignition, when the alpha particles generated by the D-T reaction are able to keep the plasma hot. To get to this point, however, immense power has to be injected to raise the temperature to the order of 50 keV (500,000,000°). This is done mainly with NBI. ITER will have 33 MW of NBI. The injectors, three or four of them, are usually the largest appendages sticking out of the tokamak. In the first stage, deuterium atoms are given an extra electron to produce negative ions. Once charged, the ions can be accelerated electrostatically. Before entering the tokamak, the negative ions go through a little gas, which strips off the extra electron, restoring the atom to a neutral state. Being neutral, the atom is not affected by the magnetic field and can go well into the plasma until it is ionized by the electrons in the plasma. How far it goes depends on its energy. All large tokamaks use NBI, which is a well-established technology; but since ITER is so large, neutral beams of 1 MeV energy are needed to get to the center. NBI technology for 1 MeV has not yet been developed [15].

If these theoretical ideas prove to be feasible, the escaping plasma can be used to generate electricity directly, as shown in Fig. 10.29. The ions pass through elec­trodes, inducing electric currents in them. Note that this exhaust is a natural divertor spreading the heat over a large area. The product alpha particles are well contained in the main plasma and keep it hot; their release rate through the mirror can be controlled by design.

DECELERATION AND COLLECTION ELECTRODES

FUSION REACTOR

A battery stores a lot of energy in its chemicals, but chemical reactions are slow and cause a battery to charge and discharge slowly. A capacitor, on the other hand, can charge and discharge extremely fast. It stores energy with two electrodes and a separator the way a battery does, but it does not involve chemical reactions. It also can be recycled limitlessly and does not decay with time. Capacitors are used in almost all electronic circuits and come in many sizes. Millions of small ones can be made on a computer chip, and large ones the size of a waste basket (trash bin to Anglophiles) are used by power companies. Supercapacitors are capacitors that still use no chemicals but can hold much more energy than previously possible. Used in combination with batteries, they help overcome some of the drawbacks of batteries. Pseudocapacitors are supercapacitors with reacting chemicals, thus combining the virtues of capacitors and batteries. A few diagrams will show how interesting these new developments in transportable energy storage are.

Figure 3.56a shows a normal capacitor. The positive and negative electrodes are metal sheets separated by an insulator called a dielectric. When the capacitor is charged by applying voltage between the electrodes, the charges move to the inner surfaces of the dielectric, and they attract opposite charges onto the surfaces of the dielectric. There are then sheets of opposite charges on each interface, and they stay

there when the switch is opened. These charges cannot move together to annihilate one another because the dielectric is an insulator. The energy is stored in the dielec­tric. When the switch is closed to hook up a load, the opposite charges on the electrodes move through the load to combine with one another, thus applying the energy that was stored. The dielectric, which had zero total charge all along, then redistributes its charges to match the charges left on the metal sheets, if any. The energy storage capacity of a capacitor (hence its name) depends on three factors: the area of the sheets, the thinness of the dielectric, and “dielectric constant.” The latter is a number varying from 1 (for air or vacuum) to 3 (for plastic), to 5 (for glass), and as high as 80 for water. The higher the number, the more energy the dielectric can hold for a given voltage between the electrodes.

To get more energy into a capacitor, one can work with these three factors. Capacitors are already made as thin as possible and rolled up to get the largest area for their size. Supercapacitors, however, can have much thinner dielectrics and much larger areas by virtue of nanotechnology. This can be explained step-by-step. In Fig. 3.56b, we show two simple capacitors connected in series. The inner elec­trodes are not metal but conducting liquids (electrolytes). The gaps are filled not with a dielectric but with air. This lowers the dielectric constant to 1, but thickness of the gap is much, much smaller. Now if we connect the two capacitors not by a wire but by simply extending the electrolyte as in Fig. 3.56c, we have a capacitor whose capacitance depends on the thicknesses of the two gaps, and not by the thickness of the electrolyte layer. Next, we can increase the area by roughening up the inner surfaces, as shown in Fig. 3.57a. This is done by coating the electrodes with a layer of “activated” carbon, which consists of fine particles. Special process­ing techniques make the surfaces of these particles break up into channels nanometers in size, as shown in Fig. 3.57b. The electrolyte goes into these channels but does not actually touch the carbon because of a nanoscopic surface tension effect.

This forms an air gap of nanometers thick. The capacitance is increased to tens of thousands of times.

Capacitance is measured in farads (named after Michael Faraday). The energy a capacitor can hold is proportional to its capacitance and the square of the voltage it can take before arcing over. While usual capacitors have capacitances of picofarads to microfarads and a rare one may have a farad, supercapacitors (also called ultra­capacitors) can have 5,000 farads. They can hold 5% as much energy as a automo­tive Li-I cell in the same size package.61 They can supplement Li-I batteries in electric cars by storing and releasing braking energy more quickly than the batteries can. They can store enough energy to be used on short trips by buses, garbage trucks, and the like.

Pseudocapacitors add porous electrode structures like those of Fig. 3.57 to a Li-I battery using molybdenum trioxide (MoO3). The trick is to find a material that can make a chemical battery and yet can be processed in such a way as to have a large area, rough surface. This has been accomplished in the laboratory by Brezesinski et al [35]. Still in their infancy, pseudocapacitors have the potential to store enough energy fast enough to be useful in smoothing the output of intermittent energy sources such as wind and solar.62 The development of such electrochemical capacitors will fill the gap in Fig. 3.58 between batteries and capacitors in their abilities to store large amounts of energy and to cycle the storage fast. There are still other types of batteries which lurk in the future, such as metal-air batteries, especially zinc-air and lithium-air batteries. Since the cathodes are air, these could have very large storage per unit weight. They are the only batteries that could approach the energy density of gasoline. However, there are several performance defects, most seriously inability to be recharged completely. The physics of the reversible reaction is still unknown;62 but, with intensified research, there is hope for a paradigm­changing advance with these new types of batteries.