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 toroidal flux), q95.

See also

References

  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