Tokamak Confinement

The tokamak is currently the most advanced magnetic confinement fusion device. The name derives from the Russian acronym for “toroidal chamber with magnetic coils” (toroidalnaya kamera s magnitnymi katushkami). Developed in the Soviet Union in the 1950s, tokamak research expanded worldwide following the success of the T-3 tokamak in 1968.

Magnetic Configuration

The tokamak combines two types of magnetic fields to form helical field lines that confine the plasma.

Toroidal Field

This magnetic field runs along the circumferential direction of the torus (doughnut shape). It is generated by toroidal field (TF) coils arranged around the torus.

The toroidal field strength is inversely proportional to the distance from the major axis:

$$B_\phi = \frac{B_0 R_0}{R}$$

where $B_0$ is the field strength at the magnetic axis, $R_0$ is the major radius, and $R$ is the distance from the major axis.

This field gradient causes drift motion of plasma particles. The drift is in opposite directions for ions and electrons, causing charge separation. A poloidal field is required to cancel this effect.

Poloidal Field

This magnetic field circles around the minor cross-section (poloidal cross-section) of the torus. In tokamaks, it is generated by the current flowing through the plasma itself (plasma current).

The plasma current is induced by the changing magnetic flux of the central solenoid (CS) coil:

$$V_{\text{loop}} = -\frac{d\Phi_{\text{CS}}}{dt}$$

When plasma current $I_p$ flows, it generates a poloidal magnetic field $B_\theta$ around it.

Helical Field Lines

The combination of toroidal and poloidal fields causes field lines to wind helically around the torus surface. These helical field lines effectively confine plasma particles.

Safety Factor q

The safety factor $q$ represents how many times a field line goes around toroidally while making one poloidal circuit:

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

The value of $q$ is directly related to plasma stability:

• $q < 1$: Kink instability occurs
• $q = 1$: Region of sawtooth oscillations
• $q > 2$: Stable region

The $q$ value at the plasma boundary (edge $q$) is typically maintained above 3. The $q$ profile, known as magnetic shear, has the effect of suppressing many instabilities.

Main Components

Toroidal Field Coils

Superconducting coils arranged around the torus that generate a strong toroidal field (5.3 T in ITER). ITER uses Nb$_3$Sn superconductors with 18 D-shaped coils.

Central Solenoid

A solenoid coil placed along the central axis of the torus. It induces plasma current through flux change and is also used for initial plasma startup. It is a limiting factor for pulsed operation.

Poloidal Field Coils

A set of coils for controlling plasma position and shape. They maintain vertical equilibrium and shape the plasma cross-section into a divertor configuration.

Vacuum Vessel

A stainless steel vessel that contains the plasma. It maintains high vacuum and supports shielding and cooling systems.

Divertor

A structure that receives heat and particles escaping from the plasma boundary. It handles impurity exhaust and helium ash removal. ITER uses a tungsten divertor designed to withstand heat fluxes exceeding 10 MW/m$^2$.

Plasma Current and Heating

Role of Plasma Current

The plasma current has three important roles:

1. Generation of poloidal field
2. Joule heating (ohmic heating)
3. Induction of bootstrap current

The ohmic heating power is proportional to the plasma resistivity $\eta$:

$$P_{\text{Ohmic}} = \eta j^2$$

However, in high-temperature plasma, the resistivity decreases proportional to temperature to the power of $-3/2$:

$$\eta \propto T^{-3/2}$$

Above several keV, auxiliary heating becomes necessary.

Auxiliary Heating

• Neutral Beam Injection (NBI): Injects fast neutral atoms into plasma
• Ion Cyclotron Resonance Heating (ICRH): Heats at the ion cyclotron frequency
• Electron Cyclotron Resonance Heating (ECRH): Heats at the electron cyclotron frequency
• Lower Hybrid Current Drive (LHCD): Also used for current drive

Disruption

A disruption is a dangerous phenomenon where the plasma current collapses rapidly.

Mechanism

1. Growth of MHD instabilities (especially tearing modes)
2. Destruction of magnetic surfaces and thermal quench
3. Rapid drop in plasma temperature and increase in resistivity
4. Current quench and generation of electromagnetic forces

Conditions for Occurrence

• Exceeding the density limit (Greenwald density)
• Exceeding the $\beta$ limit
• Excessive impurity contamination
• Vertical position instability

Effects and Countermeasures

Electromagnetic forces generated during current quench impose large loads on structures. Halo currents and runaway electrons are also serious concerns.

Research on countermeasures includes:

• Mitigation by pellet injection
• Early detection and control of instabilities
• Optimization of $q$ profile by current drive

Operating Modes

L-mode (Low Confinement Mode)

The normal operating mode with moderate confinement performance.

H-mode (High Confinement Mode)

A transport barrier (pedestal) forms at the plasma boundary, roughly doubling the confinement time compared to L-mode. It was discovered at ASDEX in 1982.

A threshold heating power must be exceeded to trigger the transition to H-mode. ITER operation is based on H-mode.

ELMs (Edge Localized Modes)

A periodic instability occurring in the pedestal region during H-mode. It causes pulsed heat loads on the divertor and requires control.

Representative Tokamak Devices

Device	Country	Major Radius	Plasma Current	Features
ITER	International	6.2 m	15 MA	Burning plasma demonstration
JT-60SA	Japan	2.96 m	5.5 MA	ITER support, advanced operation
KSTAR	Korea	1.8 m	2 MA	Superconducting, long-pulse
EAST	China	1.85 m	1 MA	Superconducting, steady-state
JET	Europe	2.96 m	4.8 MA	D-T experiments (ended 2021)

Challenges and Outlook

Main Challenges

1. Disruption avoidance and mitigation
2. Achieving steady-state operation (improving current drive efficiency)
3. Managing divertor heat loads
4. Material activation and lifetime

Future Outlook

New concepts such as spherical tokamaks (STEP, MAST-U) and compact tokamaks (SPARC, ARC) are being researched. The use of high-temperature superconductors is expected to enable stronger magnetic fields and more compact devices.

Related Topics

• Confinement Methods: Overview - Overview of confinement methods
• Stellarator/Helical Confinement - Alternative method without plasma current
• Glossary: Tokamak - Basic definition
• Glossary: Plasma - Plasma fundamentals
• ITER Project - The world’s largest tokamak