The Tokamak Approach

A tokamak is a device that confines plasma with magnetic fields inside a doughnut-shaped (toroidal) vacuum vessel, and it is the most extensively studied approach to magnetic confinement fusion today. The name comes from a Russian phrase meaning “toroidal chamber with magnetic coils.” This page starts from the intuition of why the doughnut shape is used, then works through how the magnetic fields are created, the conditions for stable operation, the operational limits that govern performance, and finally frontier research topics such as high-field tokamaks and AI-based control, so that you can understand each layer in turn.

Start with Intuition (High School)

Plasma is a super-hot gas in which atomic nuclei and electrons have come apart. To make fusion happen, you have to keep this gas heated to about 100 million degrees while confining it without letting it touch anything. If it touches a wall, it cools down in an instant.

This is where magnets come in. The particles in a plasma carry electric charge, so they move by winding around magnetic field lines. Think of a field line as an “invisible rail.” A particle spirals along the rail as it travels, and it is bad at crossing the rail to escape outward. So if you arrange the rails cleverly, you can keep the plasma floating without letting it touch a wall.

So how should you arrange the rails? If you run field lines through a straight tube, the particles escape from both ends of the tube. To get rid of the ends, you bend the tube around into a ring. This is the reason for choosing the doughnut shape. With no ends, the particles can keep circling along the rail forever.

However, simply bending it into a ring does not make confinement work well. Because the magnetic field is stronger on the inside of the doughnut than on the outside, the plasma is gradually pushed outward. To prevent this, a tokamak twists the field lines into a “helix.” These are twisted field lines that go around the large ring of the doughnut while at the same time circling around the small ring of the cross section. This twist makes the inward and outward pushes cancel each other out.

The clever part of the tokamak is that to create this twist, it drives an electric current through the plasma itself. Because plasma conducts electricity, running a large current through it creates a magnetic field around it, and that produces exactly the helical twist. The mechanism that generates the current works on the same principle as a familiar transformer. You place a coil at the center of the doughnut, and when you change the current in that coil, the plasma acts as the transformer’s secondary winding and a current is induced in it. In other words, the plasma itself is used as part of an electric circuit.

Understand the Physics (Undergraduate)

A tokamak’s magnetic field is made by superimposing two components. One is the toroidal field, produced by D-shaped TF coils (toroidal field coils) arranged so as to surround the doughnut, and it is a field that runs along the direction of the large ring. The other is the poloidal field, produced by the current flowing in the plasma itself, and it is a field that circles around the small ring of the cross section.

The device that generates this plasma current is the central solenoid (CS coil). When you vary the current in a coil placed on the central axis of the doughnut over time, a toroidal current $I_p$ is induced in the plasma by the same electromagnetic induction as in a transformer. The induced electromotive force is determined by the time variation of the magnetic flux, $-\,d\Phi/dt$, so in principle you can only drive a current while you can keep increasing the flux. This is why a tokamak tends to be a pulsed device by nature, which leads to the steady-state operation challenge discussed later.

When the two fields are combined, the field lines circle helically over the toroidal surface. The quantity that expresses how much this helix “twists” is the safety factor $q$. It is defined as how many times a field line goes around in the toroidal direction while it makes one poloidal loop, and for a simplified circular cross section it can be written as follows.

$$q = \frac{r B_\phi}{R B_\theta}$$

Here $r$ is the minor radius (the radius of the cross section), $R$ is the major radius (the distance from the center of the doughnut), $B_\phi$ is the toroidal field, and $B_\theta$ is the poloidal field. Reading the formula, the stronger the toroidal field and the weaker the poloidal field, the larger $q$ becomes (the looser the twist); conversely, the more you increase the plasma current to strengthen the poloidal field, the smaller $q$ becomes (the tighter the twist).

$q$ matters because it determines the stability of the plasma. If $q$ is too small, the energy of the magnetic field created by the plasma current is released, and an instability arises in which the plasma column twists or kinks (the kink instability). Empirically, for the whole plasma to be stable, the rule of thumb is to keep the edge (outermost) safety factor $q_a$ at 3 or above, and it is considered dangerous once $q$ drops below 2. The number of TF coils and the magnitude of the plasma current are designed so that this $q$ takes a safe value.

As for heating, ohmic heating (Joule heating) from the plasma current alone cannot reach fusion temperatures. This is because as the temperature rises, the plasma’s electrical resistance drops and the heating becomes ineffective. So auxiliary heating is used. Representative methods are neutral beam injection (NBI), which shoots in high-energy neutral atoms, and ion cyclotron resonance heating (ICRH) and electron cyclotron resonance heating (ECRH), which inject electromagnetic waves tuned to the resonant frequencies of the particles.

Deepen the Theory (Graduate)

The equilibrium and stability of a tokamak are described in the framework of magnetohydrodynamics (MHD). The balance between the plasma’s pressure gradient and the magnetic force is captured by the force-balance equation $\nabla p = \mathbf{j} \times \mathbf{B}$ and by the Grad-Shafranov equation, which describes axisymmetric equilibrium. For details, see MHD equilibrium and stability.

A measure of performance is the beta value $\beta$, the ratio of the plasma pressure to the magnetic pressure.

$$\beta = \frac{p}{B^2/2\mu_0}$$

$\beta$ expresses “how efficiently the applied magnetic field is being converted into plasma pressure,” and the more economical a fusion reactor is, the higher the $\beta$ that is desired. However, raising $\beta$ too much triggers pressure-driven instabilities. This limit is known as the Troyon beta limit, and using the normalized beta $\beta_N$ defined as

$$\beta_N = \beta \frac{a B_\phi}{I_p}$$

it is summarized in the empirical rule that the plasma becomes unstable once $\beta_N$ exceeds roughly 3 to 4 ($a$ is the minor radius, and care must be taken with the system of units).

There is also an upper limit on density, called the Greenwald density limit. When the line-averaged density $\bar{n}$ exceeds $n_G$, given by

$$n_G = \frac{I_p}{\pi a^2}$$

the plasma becomes prone to reaching a disruption due to effects such as increased radiation losses. Reading the formula, it says that the more you increase the plasma current and the smaller you make the cross section, the higher the allowed upper limit on density becomes. These three limits, the kink limit, the beta limit, and the density limit, each define the boundaries of the region in which the plasma can be operated (the operating window).

H-mode (high-confinement mode) is an operating state discovered in 1982 on the ASDEX device. When the auxiliary heating power exceeds a certain threshold, a transport barrier (the pedestal) with a steep pressure gradient forms spontaneously at the plasma edge, and the confinement performance jumps to about twice that of L-mode (low-confinement mode). Next-generation devices, starting with ITER, are designed with this H-mode as their standard operating scenario. However, in H-mode, ELMs (edge-localized modes), in which the edge pressure gradient collapses periodically, occur, and because large ELMs deposit instantaneous heat loads on the divertor, control by methods such as resonant magnetic perturbations (RMP) is being studied. The classification of the various instabilities is summarized in Plasma instabilities.

The most serious phenomenon that occurs when operational limits are exceeded is a disruption. It is a phenomenon in which the plasma current is lost over a short time: first, in the thermal quench, the stored thermal energy is released to the wall over roughly a millisecond, and then in the following current quench, large electromagnetic forces act on the structures. Even more troublesome are runaway electrons. Under the strong induced electric field that arises during the current quench, for some electrons the acceleration wins out over the friction from collisions, and they are accelerated without limit up to energies close to the speed of light. Because this high-energy electron beam causes serious damage if it strikes the wall locally, mitigation techniques such as massive injection of impurities via shattered pellet injection (SPI) are being studied.

Research Frontier (PhD)

Achieving steady-state operation is a central challenge in making the tokamak a practical reactor. Because inductive current drive by the central solenoid ends the pulse once the magnetic flux is used up, non-inductive current drive becomes necessary. Methods for driving current externally include lower hybrid current drive (LHCD), electron cyclotron current drive (ECCD), and drive by NBI. In addition, advanced tokamak scenarios, which make maximum use of the bootstrap current that the pressure gradient itself generates and reduce the externally injected power, are being studied vigorously.

The spherical tokamak is a configuration in which the aspect ratio (the ratio of the major radius to the minor radius) is reduced to roughly 1.2 to 2, making the plasma close to a sphere. It has the advantages of being able to achieve high $\beta$ more easily at the same magnetic field, and of allowing the device to be made compact. Programs such as the UK’s STEP aim at demonstrating power generation, but inherent engineering challenges are subjects of research, such as the limited space in the central column that makes the arrangement of the TF coils and central solenoid difficult, and resistance to neutron irradiation.

The high-field tokamak is a route that miniaturizes the device by strengthening the toroidal field. Because fusion output depends on a high power of the magnetic field, raising the field lets you obtain the same performance with a smaller device. The key here is high-temperature superconductors (HTS), such as rare-earth barium copper oxide (REBCO), which can generate higher magnetic fields than conventional low-temperature superconductors. SPARC, developed by MIT and Commonwealth Fusion Systems (CFS), is a leading example that aims to achieve a high magnetic field with these HTS magnets and to demonstrate a burning plasma with $Q$ well above 1 (for details, see SPARC). Reading it together with ITER brings out the contrast between the physics that the world’s largest ITER opens up and the miniaturization that the high-field route aims for. Also, technology for the long-pulse operation of superconducting tokamaks is being accumulated in devices such as JT-60SA.

In the field of control, AI-based plasma control is developing rapidly. Plasma is a nonlinear system involving many interacting variables, and adjusting its shape and instabilities by hand is difficult. Studies have been reported in which reinforcement learning automatically controls the currents in the magnetic-field coils to maintain a plasma of the intended shape, and in which machine learning predicts disruptions in advance to avoid them. Control methods that estimate the plasma state in real time and prevent destructive events before they happen are important research themes for the era of burning plasmas.

Check Your Understanding

Q1. Why does a tokamak adopt a doughnut shape rather than a straight tube?

A Because a doughnut shape makes the vacuum vessel cheaper to build B Because in a straight tube particles escape from both ends, and by bending it into a ring to remove the ends, particles can keep circling along the field lines C Because a doughnut shape makes magnets unnecessary D Because fusion reactions cannot happen in a straight tube

Q2. What creates a tokamak's poloidal field, and how is that current generated?

A External TF coils create it, and it is maintained by permanent magnets B The current flowing in the plasma itself creates it, and that current is induced in the plasma by varying the flux of the central solenoid over time, using the same electromagnetic induction as a transformer C The Earth's magnetic field creates it, and it is maintained naturally D Neutrons within the plasma create it, and it is maintained by the fusion reaction

Q3. If the safety factor q is too small, what instability becomes a problem, and roughly what value is the edge q kept at or above?

A A disruption from exceeding the density limit occurs, and q is kept at 10 or above B A kink instability in which the plasma column twists occurs, and the edge q is kept at roughly 3 or above C A transition to H-mode occurs, and q is kept below 1 D Runaway electrons are generated, and q is kept at zero

Q4. What kind of operating state is H-mode, and what phenomenon occurs in exchange?

A A state in which, once the heating power exceeds a threshold, a transport barrier (pedestal) forms at the edge and confinement becomes about twice that of L-mode, and in exchange ELMs occur in which the edge collapses periodically B A state in which strengthening the magnetic field makes the plasma hotter, and in exchange runaway electrons are generated C A state in which lowering the density improves confinement, and in exchange a kink instability occurs D A state that stabilizes when the plasma current is stopped, and in exchange a disruption occurs

Q5. Why is steady-state operation difficult with inductive current drive alone, and what kind of current is expected instead?

A Because the inductive current cools the plasma, an ohmic heating current is expected instead B Because induction by the central solenoid ends the pulse once the flux is used up, non-inductive current drive and the use of the bootstrap current spontaneously generated by the pressure gradient are expected C Because the inductive current exceeds the density limit, a Greenwald current is expected instead D Because the inductive current produces runaway electrons, a TF coil current is expected instead

Related Topics

• MHD equilibrium and stability - The theoretical framework that describes tokamak equilibrium
• Plasma instabilities - Classification of various instabilities such as kinks and ELMs
• ITER - The world’s largest tokamak and demonstration of a burning plasma
• JT-60SA - Long-pulse operation research with a superconducting tokamak
• SPARC - A high-field compact tokamak using high-temperature superconductors