# Rotational transform

(Redirected from Safety factor)

The rotational transform (or field line pitch) ι/2π is defined as the number of poloidal transits per single toroidal transit 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 [1]

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

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: [2]

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

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

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

Where the poloidal magnetic field ${\displaystyle {B_{\theta }}}$ is mostly produced by a toroidal plasma current. The principal significance of the safety factor q is that if ${\displaystyle q\leq 2}$ at the last closed flux surface (the edge), the plasma is magnetohydrodynamically unstable.[3]

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