Tokamak: Bottling a Star with Magnetic Fields

Confining a star in a donut

Fusing hydrogen isotopes into helium releases enormous energy, but only if you can hold a plasma at 100-150 million kelvin — ten times the core of the Sun — long enough and dense enough for nuclei to collide and fuse faster than the plasma leaks its heat away. No solid material survives contact with that plasma, so it has to be held in place with magnetic fields instead. The tokamak (a Russian acronym for 'toroidal chamber with magnetic coils'), invented in the Soviet Union in the 1950s, does this by bending the confining field into a closed donut shape so field lines never hit a wall — a charged particle gyrating around a field line just keeps circling the torus forever, in principle.

Why a purely toroidal field isn't enough

A simple donut-shaped (toroidal) field alone doesn't confine a plasma, because the field is stronger on the inside of the donut (small major radius) than the outside — field lines are more tightly packed there. That gradient makes ions and electrons drift vertically in opposite directions, which builds up an electric field, which in turn produces an E×B drift that pushes the whole plasma outward, straight into the wall. The fix is to add a second field component, poloidal, that winds the short way around the torus's minor circumference. Combine toroidal and poloidal components and each field line spirals around the torus on a helical path, so a particle following it samples both the inside and outside of the donut in turn and the vertical drifts cancel out on average.

B_total = B_toroidal (long way around, the donut hole)
        + B_poloidal (short way around, the donut's cross-section)

resultant field lines trace helices on nested toroidal "flux surfaces"

The safety factor q

How tightly that helix twists is measured by the safety factor, q, roughly the number of times a field line goes around the torus the long way for each time it goes around the short way. Low q (a tightly wound helix) is prone to violent kink and disruption instabilities; the classic Kruskal-Shafranov limit requires q > 1 at the plasma edge for basic stability, and real devices typically run with edge q around 3 to 5. In a tokamak the poloidal field is generated mostly by driving a huge electric current directly through the plasma itself (up to millions of amps), which is also what ohmically heats the plasma during start-up.

Poincaré maps and flux surfaces

Because the field-line helix is deterministic but the torus is periodic, the cleanest way to see the structure is a Poincaré map: pierce a single poloidal cross-section every time a field line passes through it, going around the torus over and over. If the confinement is good, the punctures trace out closed nested curves — the flux surfaces — like the rings of an onion, and the plasma pressure and temperature are constant on each one. If field lines instead wander and fill in an area rather than tracing a closed loop, that is a sign of magnetic chaos — overlapping resonances that let heat and particles leak across surfaces far faster than intended, which is one of the central practical problems in fusion confinement.

From tokamak physics to ITER

The Lawson criterion sets the bar for a self-sustaining ('ignited') fusion reaction: the product of plasma density, confinement time and temperature must exceed a threshold, so a reactor needs to be simultaneously hot, dense, and well-insulated for long enough. Real tokamaks add extra field-shaping coils to control the plasma's cross-sectional shape (elongation and triangularity improve stability and confinement), auxiliary heating beyond ohmic current (neutral-beam injection, radio-frequency heating) to push past ohmic heating's practical temperature ceiling, and divertors to channel exhaust heat and impurities away from the core plasma. ITER, under construction in France, is designed to be the first tokamak to produce substantially more fusion power than the power put in to heat the plasma, a milestone called breakeven.

Frequently asked questions

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