Rotational transform

The rotational transform (or field line pitch) ι/2π is defined as the mean number of toroidal transits (n) divided by the mean number of poloidal transits (m) of a field line on a toroidal flux surface. The definition can be relaxed somewhat to include field lines moving in a spatial volume between two nested toroidal surfaces (e.g., a stochastic field region).

Assuming the existence of toroidally nested magnetic flux surfaces, the rotational transform on such a surface may also be defined as

${\frac {\iota }{2\pi }}={\frac {d\psi }{d\Phi }}$

where ψ is the poloidal magnetic flux, and Φ the toroidal magnetic flux.

Safety factor

In tokamak research, the quantity q = 2π/ι is preferred (called the "safety factor"). In a circular tokamak, the equations of a field line on the flux surface are, approximately:

${\frac {rd\theta }{B_{\theta }}}={\frac {Rd\varphi }{B_{\varphi }}}$

where $\phi$ and θ are the toroidal and poloidal angles, respectively. Thus $q=m/n=\left\langle d\varphi /d\theta \right\rangle$ can be approximated by

$q\simeq {\frac {rB_{\varphi }}{RB_{\theta }}}$

Where the poloidal magnetic field ${B_{\theta }}$ is mostly produced by a toroidal plasma current. The principal significance of the safety factor q is that if $q\leq 2$ at the last closed flux surface (the edge), the plasma is magnetohydrodynamically unstable.

In tokamaks with a divertor, q approaches infinity at the separatrix, so it is more useful to consider q just inside the separatrix. Is is customary to use q at the 95% flux surface (q₉₅).

See also

• Magnetic island
• Magnetic shear