The Safety Factor Formula

The number of toroidal transits a field line makes per poloidal transit, the master parameter of tokamak stability.

Definition

The safety factor q counts how many times a magnetic field line winds the long way around the torus for each time it winds the short way. In a large-aspect-ratio circular tokamak, q(r) = r B_t / (R B_p), where B_t is the toroidal field, B_p the poloidal field, r the minor radius, and R the major radius. It is dimensionless and varies across the profile, giving q(r).

Why It Is Called Safety

The name reflects its role in stability: low q means a strongly wound, current-heavy field that is prone to kink instabilities, while higher q is safer. The Kruskal-Shafranov condition requires q at the edge to exceed one to avoid the most dangerous external kink, and in practice edge q of two to four is typical. The overall level of q scales inversely with the plasma current.

Rational Surfaces

Where q equals a ratio of small integers m/n, field lines close on themselves and the surface is resonant with an m,n perturbation. These rational surfaces host tearing modes and are focal points for MHD activity. The q=1 surface governs sawtooth oscillations, and the q=2 surface is frequently associated with disruptive tearing modes.

Relevance

The q profile ties together current, field, stability, and confinement, and shaping it is central to scenario design. Spherical tokamaks like the Hyperion breeder concept, with low aspect ratio, carry high plasma current at modest field, giving distinctive q profiles that the design-stage stability analysis evaluates. The reported currents such as 9.86 MA are design targets for a machine in simulation.