The Safety Factor q
The number of toroidal transits a field line makes per poloidal transit, central to tokamak stability.
Field-line pitch
In a tokamak the magnetic field lines wind helically around the torus. The safety factor q counts how many times a field line goes the long way around (toroidally) for each time it goes the short way around (poloidally):
q = (toroidal transits) / (poloidal transit) ~ r B_toroidal / (R B_poloidal)The name safety factor comes from its role in stability: higher q is generally safer against the most dangerous kink instabilities.
Rational surfaces
Where q equals a ratio of small integers m/n, field lines close on themselves after m toroidal and n poloidal transits. These rational surfaces are where tearing modes and magnetic islands form, because a resonant perturbation can reconnect field lines there. The q-profile therefore predicts where instabilities live.
Stability rules of thumb
- The Kruskal-Shafranov limit requires q > 1 at the edge to avoid the external kink
- q on axis near or below 1 drives the internal kink and sawtooth cycle
- Reversed or elevated central shear can access improved-confinement regimes
How it is computed
The q-profile falls directly out of the equilibrium solve: integrate the field-line pitch over each flux surface from the Grad-Shafranov solution. It evolves slowly on the resistive current-diffusion timescale as the current profile relaxes.
Design use
Choosing the current and field so the q-profile avoids dangerous low-order rationals in the wrong places is a core part of scenario design. For the Hyperion breeder, the q-profile is engineered together with the negative-triangularity shaping to balance stability against high beta and bootstrap current.