Rotational transform: Difference between revisions

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Rotational transform (view source)

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The rotational transform (or field line pitch) ''ι/2π'' is defined as the mean number of toroidal transits (''n'') divided by the mean number of poloidal transits (''m'') of a field line on a toroidal flux surface.
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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 ''&psi;'' is the poloidal magnetic flux, and &Phi; the toroidal magnetic flux.
 
== Safety factor ==
 
In [[Tokamak|tokamak]] research, the quantity ''q = 2&pi;/&iota;'' is preferred (called the "safety factor").
In a circular [[Tokamak|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 ''&theta;'' 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. Is is customary to use ''q'' at the 95% flux surface (''q<sub>95</sub>'').
 
== See also ==
 
* [[Magnetic island]]
* [[Magnetic shear]]
 
== References ==
<references />
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