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

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

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:

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

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

$q\simeq \frac{r{B}_{\phi }}{R{B}_{\theta }}$

See also

• Magnetic island
• Magnetic shear