Rotational transform

2022-03-03T20:31:34Z

Odstrcilt: Wrong definition of q95, it is 95% of poloidal flux (J. Wesson, Tokamaks pg. 411)

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 surface|flux surfaces]], the rotational transform on such a surface may also be defined as
<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2004) 1071]</ref>

:<math>\frac{\iota}{2 \pi} = \frac{d \psi}{d \Phi}</math>

where ''ψ'' is the poloidal magnetic flux, and Φ the toroidal magnetic flux.

== Safety factor ==

In [[Tokamak|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:
<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>

:<math>\frac{r d\theta}{B_\theta} = \frac{Rd\varphi}{B_\varphi}</math>

where <math>\phi</math> and ''θ'' are the [[Toroidal coordinates|toroidal and poloidal angles]], respectively.
Thus <math>q = m/n = \left \langle d\varphi /d\theta \right \rangle </math> can be approximated by

:<math>q \simeq \frac{r B_\varphi}{R B_\theta}</math>

Where the poloidal magnetic field <math>{B_\theta}</math> is mostly produced by a toroidal plasma current. The principal significance of the safety factor ''q'' is that if <math>q \leq 2</math> at the last closed flux surface (the edge), the plasma is [[:Wikipedia:Magnetohydrodynamics|magnetohydrodynamically]] unstable.<ref>Wesson J 1997 Tokamaks 2nd edn (Oxford: Oxford University Press) p280 ISBN 0198509227</ref>

In [[Tokamak|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), ''q<sub>95</sub>''.

== See also ==

* [[Magnetic island]]
* [[Magnetic shear]]

== References ==
<references />

FusionWiki - User contributions [en]

Rotational transform