Rotational transform

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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 poloidal flux), q95.

See also


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