Rotational transform: Difference between revisions
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<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> | <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 \ | :<math>\frac{\iota}{2 \pi} = \frac{d \psi}{d \Phi}</math> | ||
where ''ψ'' is the poloidal magnetic flux, and | where ''ψ'' is the poloidal magnetic flux, and Φ the toroidal magnetic flux. | ||
== Safety factor == | == Safety factor == | ||
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<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_\ | :<math>\frac{r d\theta}{B_\theta} = \frac{Rd\varphi}{B_\varphi}</math> | ||
where | where <math>\phi</math> and ''θ'' are the [[Toroidal coordinates|toroidal and poloidal angles]], respectively. | ||
Thus | 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> | :<math>q \simeq \frac{r B_\varphi}{R B_\theta}</math> | ||
Revision as of 09:30, 16 December 2010
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. 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
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