Rotational transform

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

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

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

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

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 $q\le 2$ 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. It is customary to use q at the 95% flux surface (the flux surface that encloses 95% of the toroidal flux), q₉₅.

See also

• Magnetic island
• Magnetic shear