A Special Kind of Torus

The name tokamak comes from the Russian words toroidalnaya kamera magnitnaya katushka meaning “toroidal chamber magnetic coils,” though it might have been appropriate to name it after the Russian word tok, meaning current. As mentioned in Chap. 4, this device was unveiled at the 1958 Geneva Conference. In those days, the Russians had the lead in space satellites, but their fusion research was done with poor equipment and considered primitive. The Americans and Britons, by contrast, had shiny, expensive, and well-engineered machines which they proudly displayed. The tokamak, however, turned out to be the one that worked the best and is the leading type of magnetic plasma container today. It was developed by a team led by Academician Lev Artsimovich on an idea of Andrei Sakharov and Igor Tamm and has been adopted by all nations working on magnetic fusion energy.

In Chap. 5, we showed that a magnetic bottle had to be a topological torus and that it had to have helically twisting magnetic field lines in order to compensate for the vertical particle drifts caused by the toroidal shape. The field lines also had to be sheared to stabilize the Rayleigh-Taylor interchange instability. In a stellarator, the proper magnetic field shape can be created with external helical windings car­rying current. In a tokamak, this is simplified by driving a large amount of current through the plasma itself. The current flows in the toroidal direction (the long way around the torus), and it generates a poloidal magnetic field (the short way around the cross section). When this poloidal field is added to the main toroidal field from the large outside coils, the magnetic field inside the plasma is twisted into helices. Moreover, since the poloidal field is not the same on every magnetic surface, the helical field also has shear. This is illustrated in Fig. 6.1. A strong field in the tor­oidal direction is created by external coils, of which only three are shown for clarity. Inside the plasma, one magnetic surface is shown. A toroidal current is driven through the plasma inside this surface, and this creates a poloidal field, which adds to the toroidal field to form a twisted helical field. Depending on how much current

there is inside each magnetic surface, the amount of twist differs from one surface to the next, and so the field is also sheared to prevent instabilities.

Using the plasma itself as a current-carrying coil to generate the twisting field would seem to be a great simplification, but we have not yet shown the hardware needed to drive this current. The advantage of the tokamak is more subtle. The cur­rent path for the poloidal field is not fixed by an external coil but can be varied by the plasma; and, fortuitously, the plasma has a self-curing property that distributes the current in a beneficial way. We explain this more fully later on.

The light at the end of the tunnel may be located at the spot marked A in southern France on the map of Fig. 8.22. It is here, in a town called Cadarache near Aix-en — Provence that ITER is being built. Magnetic confinement of plasma gets better with

size, and it has long been clear that a much larger machine has to be built to achieve ignition, a machine so large that no single country can bear the whole cost. Thus was born the international thermonuclear experimental reactor, now known only by its initials, ITER. Coincidentally, ITER in Latin means a path, a journey. It may indeed be the best way to get there.

The reason for the large size is that the amount of power generated is propor­tional to the volume of the plasma, which increases with the cube of its radius, while the losses are proportional to the surface area of the plasma, which increases only as the square of its radius. To take the next step beyond the four machines shown above, therefore, requires a much larger machine, one so large that its cost has to be shared among many countries. The idea of an international project to achieve fusion energy was born in the 1985 Geneva Superpower Summit, where President Mikhail Gorbachev of the USSR and President Ronald Reagan of the USA, with advice from President Frangois Mitterand of France, agreed to initiate a project involving the USSR, the USA, the European Union, and Japan. (It probably helped that Gorbachev’s advisers were Evgeniy Velikov and Roald Sagdeev, both plasma physicists.) More on what ensued afterwards will come later, but first let’s see what kind of machine ITER is.

Figure 8.23 is the diagram of the machine being built. Its size is indicated by the small figure at the bottom, representing a standard 2-meter person. The plasma chamber has the standard D-shape, 1.7 times as high as it is wide. The width is 4 m at its widest part, and the major radius (the distance between the center of the chamber and the axis of the whole machine) is 6.2 m. The D-shaped coils that produce the main magnetic field can be seen, but all the other equipment is shown simplified; otherwise, the vacuum chamber would not be visible at all!

That includes all the other coils for shaping the plasma, the neutral-beam injectors for heating, the neutron-absorbing blanket, the divertors for catching the plasma, pellet injectors for fueling, and a host of measurement devices. How much bigger ITER is compared with the current champion, JET, is shown in Fig. 8.24. The clutter surrounding a real machine can be seen in the pictures of existing large tokamaks in Figs. 8.3—8.6.

What is ITER designed to do? The primary goal is to produce, for the first time, a “burning” plasma. That is, a plasma that will keep itself hot once it has been heated to several hundred million degrees. Remember that 80% of the fusion energy from DT fuel is in the form of neutrons, and only 20% is in alpha particles (helium ions) which can give energy to the plasma because they are magnetically confined. Therefore, a Q value of at least 5 is needed for burning or ignition. To get a safety margin, ITER is designed to produce a Q of 10, where Q is the ratio of energy out of the plasma to the energy put into the plasma from external sources. Q = 1 is scientific breakeven (energy in equals total energy out), but most of that energy is in the form of neutrons, which produce the power plant energy but cannot heat the plasma. The best that JET could do was Q=0.65, below scientific breakeven. The large step from Q = 0.65 to Q = 10 is the reason that ITER has to be so big. The step is not trivial also from a physics point of view. The 3.5-MeV alphas may cause an instability that drives them out of the plasma. Although the stability conditions have been calculated, they have never been tested. The experiment will be consid­ered a success if enough self-heating occurs for these conditions to be established, even if Q = 10 is not achieved. The self-heating mechanism which powers the sun has never been seen on earth outside of a bomb, and plasma experts are eagerly

anticipating this critical test. The term “ignition” may invoke fear that the reaction will run away and cause an explosion. This cannot happen in a fusion reactor because if the density or temperature gets too high, the plasma will disrupt and fizzle out. This may cause melting of parts of the tokamak, but it would be no worse than leaving a pot on a stove after the water has boiled out. The “pot” here would be an expensive one, though!

There are other objectives for ITER besides achieving Q = 10. It will produce 500 MW of power, about one-sixth that of a full-size reactor. Many large key compo­nents of a fusion reactor have to be designed, manufactured, and tested in operation. This includes superconducting magnet coils, wall materials and divertors that can withstand the heat and neutron bombardment, tritium handling, and remote control and maintenance after the walls become radioactive and cannot be approached by personnel. Instability control has to keep the plasma confined steadily for as long as 8 min, using a large amount of bootstrap current and generating 500 MW of power. There will be a first test of a neutron-absorbing “blanket” that can breed tritium. Tritium does not occur naturally. Most of the time, ITER will use tritium coming from fission reactors, of which it is a byproduct; but in a fusion power plant the tritium has to be made internally. This is done in a blanket that captures the 14-MeV neutrons from the reaction, slows them down, and generates heat to run a steam plant. A part of this blanket can be used to breed tritium from lithium, which is an abundant element on earth.

ITER is the logical next step toward fusion power, but it is still primarily a physics experiment. It will lead to DEMO, a demonstration power plant that will run without breakdown and produce a usable amount of power. However, many believe that an

intermediate step between ITER and DEMO is necessary to develop engineering concepts that will work in a real reactor. Some of the difficult problems are, for instance, (1) the material to be used in the plasma-facing components (the “first wall”), (2) the handing and breeding of tritium, (3) continuous operation for long periods, (4) maintenance procedures, and (5) plasma exhaust and waste treatment. ITER can provide only a first try on such topics. Engineering will be the topic of the next chapter; this is only an introduction. As an example, the first-wall material has to take the heat of facing a 100,000,000-degree plasma, and it has to allow a large flux of neutrons to pass through without causing such damage that it has to be replaced often. It also cannot contaminate the plasma with impurities of high atomic number, which would cool the plasma. Tests of suitable materials can be done without a tokamak; a fission source of neutrons would do. In fact, most of these engineering tests can be done on a much smaller, cheaper machine than ITER, and such a machine can be built and operated simultaneously with ITER to save time. Most large laboratories have proposed such a machine. For instance, the Fusion Development Facility proposed by General Atomics is a tokamak using normal-conducting coils and producing only 100-250 MW of power at Q less than 5. But it is designed to run continuously for weeks at a time over 30% of a year and breed up to 1.3 kg of tritium per year. Such machines and DEMO are still in the talking stage, but the ITER project is up and running.

As can be imagined, a cooperative project among seven nations is an administra­tive nightmare. It took over 20 years to get to the present stage. After the initial Gorbachev-Reagan agreement, the four partner nations managed to agree to start Conceptual Design Activities in 1988, and the design was finished in 1990. The resulting tokamak was much larger than the present design. In 1992, an agreement was made to start more serious Engineering Design Activities. Each country had its own home team, and a Joint Central Team was stationed in La Jolla, California. The directors of ITER for this study was at first Paul-Henri Rebut and later Robert Aymar, both of France. After six years of work, it was decided that the tokamak was too large and too expensive, and the activity was extended to 2001. The final design, finished in 2001, is half the price but achieves almost the same objectives. The physics basis for ITER, which we discussed in Chap. 7, was worked out in this period and contrib­uted to the efficiency of the new design. Some $650M was expended to design ITER, with the original agreement that the European Union and Japan would each bear one — third of the cost, while the USSR and the USA shared the other third. To everyone’s chagrin, the USA withdrew from the project in 1999, not to return until 2003. The project continued without funding from the US Congress.

Meanwhile, in 1991, the USSR collapsed and was replaced by the Russian Federation. In 2003, the Peoples’ Republic of China and South Korea joined ITER. India joined in 2005, raising the number of partners to seven. Canada was temporarily involved but dropped out when its proposed site was turned down. With an area larger than that of Western Europe, Kazakhstan has been considering joining in spite of the fact that it has large fossil reserves. The seven current nations supporting ITER are shown in Fig. 8.25. These countries represent more than half the world’s population. Without public support, the USA has been a

lukewarm partner in this path-breaking enterprise, and again failed to contribute its financial share in 2008.

By 2003, ITER’s design had been agreed upon, and the project was ready to move ahead. The estimated cost was calculated to be five billion euros (about $7B) for ten years of construction, and another 5B euros for 20 years of operation.13 Then came a totally unexpected delay. There was a deadlock on the site for ITER. The site had to have sufficient power and accessibility for such a large machine. The final­ists were a site in Japan and a site in Europe, at first in Spain, but finally in France. The EU, China, and Russia voted for France; and Japan, Korea, and the USA voted for Japan. India had not yet joined. The impasse lasted for two years. Finally, in 2005, the deadlock was broken, and France was chosen. As compensation, Japan was to supply 20% of the staff and had the right to choose the Director. Furthermore, the EU was required to purchase 20% of its ITER material from Japan. As host, the EU has to bear 5/11ths of the cost of ITER, and the other six countries 1/11th each. Kaname Ikeda was chosen to be Director. The 45% contribution by the EU will stimulate its economy.

Once a Joint Implementation Agreement was signed in November 2006 by the seven parties, the ITER Organization sprang into action. Hundreds of scientists, engineers, and administrators began to migrate to Cadarache, settling into tempo­rary offices. Bulldozers began to move two million cubic meters of soil to prepare the flat site for ITER, shown in Fig. 8.26. This amount of dirt would fill the Cheops pyramid, and the area is that of 57 soccer fields. The roads had to be widened to accommodate nine-meter wide truck convoys which will carry the major compo­nents of the tokamak. Even traffic circles (roundabouts) like the one at the upper left of Fig. 8.26 had to be enlarged. Those parts manufactured outside Europe would be shipped to the Mediterranean port of Fos-sur-Mer and then barged and trucked to Cadarache. A three-story office building was built in 2008 to house 300

employees, but this was still temporary and off-site. To accommodate their families, a multilingual school was established in Manosque; by 2009 it had 212 students from 21 nations and 80 teachers. In 2010, the school will have its own building and include a nursery school and a junior high. The first ITER baby was delivered in 2008. A weekly bulletin14 covers not only technical and personnel news but also includes cultural events and introduces the entire international community to the his­tory and traditions of this region in southern France.

ITER is truly an international project. For instance, the vacuum vessel will be made by Europe and Korea, with other parts from Russia and India. The largest components, the magnet coils, will weigh 8,700 tons and will be made of Nb3Sn and NbTi superconductors. Many different types of magnet coils and their feed-ins are required, and the manufacture of the superconductor material and their forma­tion into coils are shared among most of the ITER partners. The USA will supply 40 tons of expensive Nb3Sn conductors for the toroidal field, and those for the poloidal field will be shared among China, Russia, and Europe. Superconductor wire is very complicated, wound in many strands and cooled with liquid helium. That these actually work in large coils has been tested in the LHD stellarator in Japan and will be further tested in the new superconducting tokamaks in China, Korea, and Japan.

Domestic Agencies have been established in each country to organize the manu­facture of its in-kind contributions to ITER by local industries. Through these agencies, Procurement Agreements have to be drawn up and signed by each member country. As of 2010, 28 PAs have been signed. The site in Fig. 8.25 has been completely leveled, and the construction of 38 buildings on it has begun. The first of these is a six-story 253-m long building for winding the poloidal field coils, which are too large to be shipped, and the superconductor cable is all in one piece.

New office buildings will replace the temporary ones. Off-site in Manosque, a new school, will be built for the community.

It is clear that the ITER project is in for the long pull. Figure 8.27 shows the originally agreed schedule for the construction and operation of ITER. The site preparation will not be finished until 2012, but meanwhile the components are being designed, fabricated, and tested in various countries. It will take four years to get all the parts delivered and the tokamak assembled. The first plasma is scheduled to be made near the end of 2016. At first, experiments will be done with hydrogen, which is not radioactive. Remote handling will then be implemented so that deuterium can be used; the D-D reaction creates some neutrons, but not as many as does DT. In 2020, operation with DT will start, first in pulsed (low-duty) operation, to achieve the designed Q value. In the later stages, emphasis will be on quasi-steady state operation (high-duty) to test whether bootstrap current and non­inductive (no transformer) drive can sustain the plasma. At the end of 2026, a decision will be made whether to decommission the machine or to continue it with modifications. De-activating, decommissioning, and disposing of the machine is expected to take another 11 years. The ITER machine will have 30,000 components in ten million pieces. To get these to be delivered on time and fit together requires numerous groups and oversight committees. Their acronyms are overwhelming, but that’s the price you pay for organizational efficiency.

At this time, the goal of achieving first plasma in 2016 seems a long way off, but the worldwide economic downturn in 2008-2009 has made it even worse. Both the budget and the schedule had to be revised in 2010. The project will be delayed two years or more by economic constraints. The new construction schedule will look something like Fig. 8.28. DT plasmas will not be attempted before 2027.

These estimates notwithstanding, the project is proceeding nicely under new Director Osamu Motojima. The digging and flattening of the ITER site has been finished and is shown in Fig. 8.29. Parts of the machine are coming in from different countries. Figure 8.30 shows the buildings planned for the site. These will be earth­quake-proof, and some will have containment for radioactivity. The long coil-winding

Fig. 8.28 The revised ITER timeline [32]

building mentioned above can be seen at the top for scale. It is exciting to see international teamwork functioning so well.

Contrary to popular perception, fusion is no longer in a guessing stage. The timeline for its development has been set. Each country has its own ITER organiza­tion and its own specialized manufacturing capabilities to contribute to the project.

At the current level of funding, it will take until 2026 to get the information from this experiment. Concurrently, materials testing facilities can be built and run to support DEMO. Design, construction, and operation of DEMO will take until 2050; and, if it is successful, commercial reactors can follow soon thereafter. The present plan is to achieve fusion power by 2050, in time for the present generation of children to enjoy. However, with increased international ambition, the time can be shortened.

There may be some confusion in the public’s mind between ITER and another large experiment, the Large Hadron Collider, or LHC, at CERN near Geneva. Geneva can be seen in Fig. 8.21 north of Cadarache. It is quite a coincidence that the two largest physics experiments in the world should be located only a few hundred kilometers from each other. The LHC is a particle accelerator 27 km (17 miles) in circumference, buried in a circular tunnel under France and Switzerland. It is similar to ITER in internationality, cost (6.3B euros), and the extensive use of superconductors; but it is entirely different in technology and purpose. The LHC is a basic physics experiment to explore the subatomic structure of matter and energy: quarks, Higgs bosons, dark matter, and so forth. Protons and antiprotons are accelerated to multi-TeV (trillions of eV) energies and hurled against one another to break them up, one particle at a time. ITER, on the other hand, deals with a gas of multi-billions of particles at KeV (thousands of eV) energies. In the LHC, large magnetic fields are used to bend the protons into circular orbits, their Larmor radius being measured in kilometers. In ITER, large magnetic fields are used to hold a plasma, which exerts a large pressure not because the particles are so energetic but because there are so many of them.

The LHC and its predecessors were inspired by man’s urge to understand his place in the universe, not by any practical need. ITER, on the other hand, is being built to develop an energy source that will save mankind, and, if done soon enough may also solve current problems in climate change and fossil fuel depletion. We are living in a golden age in which civilization has advanced to such a point that we can afford to reach for lofty goals. Let us hope that our reach does not exceed our grasp.

Methodology

In spite of the fact that we do not yet know how a fusion reactor will be constructed, or even if it is at all possible, detailed calculations have been made on the COE based on the reactor models described in the previous section. The work of Ward et al. [27], which we will summarize here, is based on the European PPCS designs. Their calculated costs for each component of a power plant compare well with those from the ARIES studies in the USA. Being a renewable power source, fusion shares with wind, solar, and hydro the benefit of essentially zero fuel cost. However, the capital cost is large. A breakdown is given by Ward [28] in Fig. 9.41. The capital cost of the tokamak power core is almost as large as that of the balance-of-plant, which is the power conversion system and electrical generators shown in Fig. 9.34. Compared with fossil fuel plants, the capital cost and replacement of blankets and divertors take the place of fuel costs. These fusion-specific costs depend on the reactor model. The models A, B, C, and D in Fig. 9.39 range from ITER-like primitive designs with steel chambers and water cooling to speculative advanced designs with Pb-Li liquid cooling and SiC/SiC first walls. Computer programs are used to calculate the costs of each component under different assumptions.

The story of Astron is more about a person than about a fusion concept [50]. Nick Christofilos was a self-made Greek physicist who independently co-invented the alternating-gradient focusing principle for accelerators and later the Astron machine. Since he was a Greek citizen working on US-classified material, he was not allowed to access his own work once it had been filed away. The Astron was a very large machine at Livermore which was to produce an FRC (Fig. 10.33) with a ring of relativistic electrons injected from an induction linac of his own design. Accumulating the electron layer from multiple pulses was not successful, and only 6% field reversal was attained. Meanwhile, Hans Fleischmann at Cornell achieved 100% field reversal using pulsed power. Without sufficient understanding, Christofilos also did not realize that the electrons would lose their energy by syn­chrotron radiation. But his persuasiveness finally gave way to reason, and the Atomic Energy Commission prepared to shut down the project. Before this hap­pened, however, Christofilos, a hard-driving, hard-drinking smoker, died of a heart attack at 55 in 1972.

We have already seen that multijunction solar cells use thin films made of III-V or II-VI materials. The problem with crystalline silicon is that it is what is called an indirect bandgap material. We need not go into the physics of this. What it means is that a palpable thickness of silicon (about 0.1 mm) is needed to absorb photons, and we saw in Box 3.5 how hard it is to make pure silicon. Thin-film materials, on the other hand, have direct bandgaps. The absorption is so good that thicknesses are measured in microns,39 typically 1 Mm, which is a thousandth of a millimeter. By comparison, the thickness of an ordinary piece of paper is about 100 Mm (0.1 mm or 0.004 in.), the same as a human hair. Thin films that can absorb 98% of sunlight are only 1% of that thickness. No wonder that even a thin layer of sunscreen spread on the skin can protect against sunburn. Since crystalline silicon in a solar cell has to be over 100 Mm thick, thin-film solar uses 100 times less semiconductor material than silicon.

However, the small amount of material required for thin-film solar cells is not the main reason for their success. It is because manufacturing techniques developed by First Solar, Inc. of the USA and other companies have reduced the cost so that solar power is commercially viable. Development advances much more rapidly when support moves from the government to private industry, where the monetary incentive is strong. First Solar became dominant in its field by optimizing the use of CdTe (cadmium telluride). This material, with a bandgap of 1.45 eV, combines the best combination of voltage and current for the higher power output from a single layer. First Solar started with a plant in Ohio with 90 MW/year of production capability, then added a 120-MW/year plant in Germany and a 240-MW/year plant in Malaysia. It has contracted with China to produce 30 MW in 2010, then 100 and 870 MW by 2014, and finally a total of 1,000 MW by 2019. The entire production process, from deposition of all the layers to assembly and to testing, takes only 2.5 h on their automated production line. Benefiting from economy of scale, First Solar has lowered the cell cost to below $1/W and the module cost to $110/m2. The goal is to bring this down to $0.50/W or $1.50/W including balance-of-system. The cost of electricity would be 6-8 0/kWh.40 Producing 1 GW/year in solar cells would give the company one-sixth of the world’s share.

The layers of a CdTe solar cell are shown in Fig. 3.41. The layers are deposited on a 60 cm x 120 cm glass superstrate 5 mm thick. This is about the size of a quarter-sheet of 4 x 8-feet plywood and will yield many cells. Below that is a thin SiO2 layer for insulation, followed by a transparent conducting layer of SnO2, which is the top electrical contact. A thin layer of CdS (cadmium sulfide) follows. Only about 0.1 Mm thick, it serves as the n-doped layer in Fig. 3.33. It must be thin to allow the light to reach the absorbing layer of CdTe. Sulfur is a Column VI element, which has been left out of Fig. 3.34 to avoid clutter, so CdS is a II-VI compound. It turns out that CdS is naturally slightly n-doped in production, and CdTe is slightly p-doped [8], so the other layers in Fig. 3.38 are not necessary to separate the electrons from the holes, greatly simplifying the device. The main CdTe layer

Incident sunlight

Glass superstrate

Silicon dioxide

Fluorine doped

tin oxide

Cadmium sulfide

Cadmium tellunde

Nickel

Aluminum

Ethyl vinyl acetate

Glass laminate

Fig. 3.41 Schematic of a CdS/CdTe solar cell (IEEE Spectrum, August 2008) is 1-5 мт thick; Gupta et al. [9] have shown that the performance does not improve much beyond 0.75 Mm. At the bottom is the other electrode, made of gold, nickel, or aluminum, followed by a plastic binder and a glass protector. Laser scribing is used between the deposition of the various layers to divide the cell into smaller cells and to connect them in a series to raise the voltage to 70 V. After all this, the whole sheet is annealed between 400 and 500°C in CdCl2 gas to improve the efficiency by as much as a factor of 2.41 The reason for this is not well understood. Such a cell puts out about an ampere of current and up to 75 W of power at 10.6% efficiency.41 Improvement to 12% may be possible.

The record efficiency achieved in the laboratory is 16.5%. To do this, the trans­parent conductor at the top, usually tin oxide, was replaced by cadmium stannate, which has higher conductivity and is more transparent. A buffer layer of zinc stan­nate was then added below it.42 As current flows through the cell, its internal resistance causes energy to be lost as heat. This loss is measured by the filling factor, which is the percentage of the ideal power that is actually usable. The best that can be achieved is 77%.42 Although the general production process is well known41 (see ref. [8]), the know-how details are closely guarded secrets. For instance, the bottom contact tends to be unstable, and adhesion is affected by the annealing step.

Thin-film materials competing with CdTe are amorphous silicon (a-Si:H) and copper indium gallium diselenide (CIGS). Amorphous silicon has a low efficiency of 6-7%, but it has had a head start because the manufacturing equipment had been developed in the semiconductor industry. This material loses the red part of the
solar spectrum, and there are attempts to add a 2-pm layer of microcrystalline silicon to add the blue part. The efficiency might then go up to 11% to compete with CdTe. CIGS has a laboratory efficiency of 19.5% vs. 16.5% for CdTe. In modules, the efficiencies are 13 and 11%, respectively; and in production they are 11.5 and 9% [8]. CIGS is harder to make, but it is being pursued because of the possibility of 25% efficiency. Currently, it has only a 1% market share, compared with 30% for CdTe and 60% for a Si.43

Geothermal energy comes from the hot rock deep down that makes geysers and warm pools for spas and mud baths. It mostly occurs at the junctions of tectonic plates. Worldwide, 10.7 GW of electricity is generated geothermally in 24 coun­tries, and another 28 GW is used for heating. The USA produces the most geother­mal power, 3 GW, in 77 plants mostly in California. The Philippines is second with 1.9 GW and gets more than a quarter of its energy from geothermal, as does Iceland. These numbers are very small on a world scale, and we need not say much about this energy source.

The capital expense of geothermal plants, used for exploration and drilling, is comparatively large. There is no fuel cost, but electricity is used to run the pumps. Once a bed of hot rock is found, a production well is drilled to extract the steam. If this is hot, above 180°C (360°F), it can be used directly to drive steam turbines to generate electricity. If it is cooler [below 150°C (300°F)], it is used for space or water heating. The used, cooled water is injected back into the rock in an injection well. With the steam, GHGs also came: CO2, methane, ammonia, and hydrogen sulfide, which smell. Whether these emissions are lower than from a comparable fossil-fuel plant depends on the location. The water also contains undesirable chemicals: mercury, arsenic, antimony, boron, and salt. All in all, geothermal energy is not going to be a solution to the world’s problems.

Sawteeth are an essential feature of tokamak discharges and are important because they show that a tokamak is self-healing. Toruses such as stellarators do not have such a feature because the magnetic structure is fixed by magnetic coils outside the plasma. A stellarator plasma cannot adjust its own magnetic topology by sawtooth-shaped hiccups. This brings up the general subject of self-organization. Many physical systems have been found which are self-organized. It may seem inconceivable that an insentient object can organize itself, but there are many examples in real life. Snowflakes are self-organized. No one had to program a computer to make these beautiful, symmetric art pieces (Fig. 7.7).

Our own bodies are self-organized. Complicated organs such as the eye, with its cornea, iris, lens, retina, and macula; and the ear, with its ossicles, cochlea, hair cells, and stereocilia, are self-organized, though some programming had to be done with the DNA. In the new field of nanotechnology, the objects are so small that they are difficult to make; and people are hoping that self-organization will help. In magnetic fusion, tokamaks have taken the lead partly because of their ability to heal themselves. There are magnetic bottles other than the standard tokamak empha­sized here that depend even more on self-organization (Chap. 10).

Magnetic Wells and Shapely Curves

Up to now, we have suppressed plasma instabilities by applying magnetic shear, creating a mesh of field lines that plasma cannot easily penetrate. There is another good way to eliminate instabilities, and that is to create a magnetic well. This is a magnetic bottle that surrounds the plasma with a stronger field on every side. The plasma then does not have enough energy to climb out of the hole that it is in. It is not possible to make such a container without a leak, which is why tokamaks do not depend on this effect as much as some other confinement concepts do. However, understanding the magnetic-well effect will help in the design of better tokamak shapes.

A simple magnetic well can be made with four infinitely long rods with opposite current in neighboring rods, as shown in Fig. 7.8. The magnetic field lines are the circles, and their spacing shows that the field gets stronger as one approaches each rod. A plasma trapped in the center would see the field increasing in every direction and would be held stably. However, there are leaks at each of the four cusps, where the field lines meet. An ion or electron following a field line toward one of the four cusps, where the field is strongest, would be reflected by the magnetic mirror effect described in Chap. 6. Unfortunately, that effect depends on the transverse momen­tum of the particles — the momentum that makes the particles gyrate in Larmor orbits. Those particles that have their velocities almost parallel to the field lines would not be reflected and would go right out at the cusps. There are enough of those particles to bring the confinement time of the plasma well below the many

seconds required in a fusion reactor.2 In the early days of fusion, one of the fanciful magnetic buckets that were proposed was the Picket Fence, a veritable Great Wall of China, as shown in Fig. 7.9. But if one had done his homework, he would have found that the leak at even one of the many cusps would have been insufferable.

Why does the magnetic field in cusp geometry look so different from the toka — mak fields we have seen so far? It is because the field bulges out towards the plasma instead of away from it. In a magnetic well, the field lines are convex as seen by the plasma, not concave. This generally means that the field is stronger on the outside than on the inside, and such field lines are said to have good curvature. Conversely, field lines that bulge outwards have bad curvature. This concept is much more general than its use in magnetic confinement. In Fig. 7.10, we see that a board which is bent upwards will support more weight than one which is level or sags downwards. Roman arches and those highly arched wooden bridges in Japanese gardens have good curvature.

Tokamaks have mostly bad curvature, but they can be designed, as we shall see, to minimize that effect. A true magnetic well is called a minimum-B device, where the plasma is in a magnetic field minimum. The twisting field lines in a torus can go through regions of both good and bad curvature. In that case, what matters is how much there is of each kind. If an electron sampling all regions of a magnetic surface sees mostly good curvature, it would be in an average-minimum-B device. It is hard to do this in a tokamak, but other toroidal systems which cannot be described here can be designed to be average-minimum-B. The idea is to minimize the time a particle spends in a region where the field is sharply bent in the bad direction. When an instability is concentrated in a region of bad curvature, it is

called a ballooning mode. The plasma escaping is such a region pulls the field lines with it, further weakening the field. This could be called a plasma hernia, but ballooning is a more dignified term!

The dominant features of a tokamak or any other magnetic bottle are the heavy coils that generate the large magnetic field used to confine the plasma. Until recently, all tokamaks had magnet coils made of copper, which conducts electricity better than any other metal except silver. Even so, it takes a lot of energy to drive megamperes of current through copper coils, and fusion reactors will have to use superconducting coils. Superconductors have zero resistivity, and once the current has been started in them, it will keep going almost forever. The hitch is that superconductors have to be cooled below 4.2 K with liquid helium. A cryogenic plant has to be built to supply the liquid helium, and the magnet coils (and hence the whole machine) have to be enclosed in a cryostat to insulate them from room temperature. The good news is that this technology is well developed and is not one of the serious obstacles to fusion power. In 1986, the world’s largest superconducting magnet, the MFTF (mirror fusion test facility), was completed at the Lawrence Livermore Laboratory in California. It was a different type of magnetic bottle that we will describe in Chap. 10. However, the program was almost immediately canceled by the Reagan administration in favor of the tokamak because the USA could not afford to follow two expensive paths to fusion. The MFTF was so large that for a while it became a museum that one could walk through. Currently, three superconducting tokamaks are in operation: the Tore Supra in France, the EAST (Experimental Advanced Superconducting Tokamak) in Hefei, China, and K-STAR, in Daejon, Korea. Soon to join them is an upgrade to Japan’s JT-60U (Fig. 8.6) called JT-60SA. In addition, the Large Helical Device, a superconducting stellarator-type machine, has been operating for two decades in Japan. ITER will, of course, have superconducting magnets.

Two superconducting materials are available on a large scale: niobium-titanium (NbTi) and niobium-tin (Nb3Sn). NbTi is cheaper and easier to make, but it loses its superconductivity above 8 T. A tesla is a large unit of magnetic field equal to 10,000 G, the old unit. Common magnets rarely go above 0.1 T, but some magnetic resonance imaging (MRI) machines in medicine can go up to 1.5 T. The earth’s magnetic field is only about 0.5 G or 0.00005 T. In ITER, fields up to 13.5 T are needed, so some coils are made of Nb3Sn, and others (for lower fields) are made of NbTi. The dividing line is around 5 T [14]. Superconducting cables are complicated to make because they have to be made of a thousand thin strands. This is because the current in superconductors flows only on the surface, and thin strands have large surface areas compared to their volumes. Also, the cables have to be bendable.

A spheromak (Fig. 10.17) is a toroidal plasma in a chamber with no hole in the middle. There can be toroidal coils to generate a poloidal B-field, but there cannot be any coils to generate a toroidal B-field since that would require a conductor going down the middle. Plasma with imbedded fields is injected into the chamber from external sources, and then the plasma self-organizes into a toroidal shape with both toroidal and poloidal fields (Fig. 10.18). Contrary to stellarators, which do away with the self-organization of tokamaks, spheromaks depend completely on self-organization. The classic method of injection with “plasma guns” is shown in Fig. 10.19.

The stable configuration that results from a period of instability and rearrangement of fields and currents is predicted by the classic theory of J. B. Taylor [9]. The main point is that the force exerted by a current in a B-field is perpendicular to B. These forces will move the plasma around until there are no more forces. That happens when the current J is parallel to B everywhere, so that there is no perpendicular force. Then the lines of B are the same as the lines of J. The B-field created by each element of J is just that which the neighboring elements of J follow. This means that the field is purely poloidal on the outside and purely toroidal on the minor axis, but the fields do not have to be neatly arranged as in Fig. 10.18. There can be a jumble of field lines that satisfy the minimum-force condition. The plasma will organize itself. Also needed is a conducting shell that keeps the whole plasma from expanding. When it tries to expand, image currents in the shell will push the plasma back.

These force-free configurations are interesting to physicists because they occur in many places, including outer space. However, spheromaks are unlikely candi­dates for fusion reactors. So far the confinement times have been short, and the plasmas have to be pulsed. Experiments have been aimed mostly at the problem of magnetic reconnection, which is important in space physics.

Hydrogen cars are electric cars whose energy is carried by pressurized hydrogen. The technology is in its infancy, especially on the manufacture of fuel cells at reasonable cost. Right now, hydrogen is made from natural gas, and the only gain, at great expense, is barely a doubling of the efficiency of burning the gas directly in reciprocating engine. Carbon dioxide is still emitted in the generation of hydrogen. Hydrogen is clean energy only when fission or fusion plants supply the energy to hydrolyze water to make it. Other nonpolluting sources such as hydroelectricity and solar and wind farms are not sufficient to replace the 383 million gallons of gasoline we consume per day in the USA [29]. The infrastructure for distributing hydrogen [4] will cost perhaps half a trillion dollars.