The number of times a magnetic field line wraps around the torus the long way for each short-way circuit -- a master parameter governing MHD stability in tokamaks.
The safety factor, universally denoted q, is one of the most important dimensionless parameters in magnetic confinement fusion. It measures the helical pitch of the magnetic field lines on a given flux surface: specifically, q is the number of toroidal transits a field line makes for each complete poloidal circuit. The name reflects the fact that higher values of q generally correspond to greater stability against magnetohydrodynamic (MHD) perturbations.
MHD stability theory places critical constraints on the q profile. The Kruskal-Shafranov limit requires q at the plasma edge, qa, to exceed unity; in practice, tokamaks operate with qa > 2–3 to avoid global kink modes. Internally, the appearance of rational surfaces where q = m/n (with m and n integers) can seed tearing modes and magnetic islands. The most dangerous is the q = 1 surface, associated with the sawtooth instability, and the q = 2 surface, where 2/1 neoclassical tearing modes (NTMs) frequently lock and trigger disruptions.2
The radial distribution q(r) is called the q profile or current profile, since q is determined by how the toroidal current is distributed across the plasma. In a standard H-mode tokamak, q is lowest on axis (q0 ≈ 0.8–1.0) and rises monotonically to the edge. Advanced tokamak scenarios aim for non-monotonic (reversed-shear) q profiles with an internal transport barrier at the minimum-q radius, enabling higher confinement and bootstrap-current fraction.3
The safety factor is the reciprocal of the rotational transform ι used in stellarator physics: q = 1/ι. The two notations reflect different historical conventions—the tokamak community counts toroidal transits per poloidal turn, while the stellarator community counts poloidal turns per toroidal transit—but describe the same geometric property of the field line.