Rotational transform: Difference between revisions
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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 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> | |||
where ''ψ'' is the poloidal magnetic flux, and | :<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 [[Tokamak|tokamak]] research, the quantity ''q = 2π/ι'' is preferred (called the "safety factor"). | ||
In a circular | In a circular tokamak, | ||
the equations of a field line on the flux surface are, approximately: | 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> | <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 == | ||
<references /> | <references /> | ||
Latest revision as of 12:31, 26 January 2023
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]
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]
where and θ are the toroidal and poloidal angles, respectively. Thus can be approximated by
Where the poloidal magnetic field is mostly produced by a toroidal plasma current. The principal significance of the safety factor q is that if 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
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