# 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
^{[1]}

- $ \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]}

- $ \frac{r d\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{r B_\varphi}{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 \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 toroidal flux), *q _{95}*.

## See also

## References

- ↑ A.H. Boozer,
*Physics of magnetically confined plasmas*, Rev. Mod. Phys.**76**(2004) 1071 - ↑ K. Miyamoto,
*Plasma Physics and Controlled Nuclear Fusion*, Springer-Verlag (2005) ISBN 3540242171 - ↑ Wesson J 1997 Tokamaks 2nd edn (Oxford: Oxford University Press) p280 ISBN 0198509227