Rotational transformation

This article is about plasma physics . For the meaning of words in statistics, see Linear Combination and Rotation (Mathematics)

The rotational transformation or twist (small iota) is a geometric parameter for the course of the field lines of a toroidal magnetic field configuration ( tokamak , stellarator ) in the magnetic confinement of plasmas . ${\displaystyle \iota }$

Sketch for the calculation of the rotation transformation and decomposition of a magnetic field vector into poloidal and toroidal components . ${\displaystyle ({\vec {B}}_{\theta })}$ ${\displaystyle ({\vec {B}}_{\phi })}$

It is defined as the ratio of the change in the poloidal magnetic flux to the change in the toroidal magnetic flux : ${\displaystyle \psi _{\theta }}$ ${\displaystyle \psi _{\phi }}$

${\displaystyle \iota ={\frac {d\psi _{\theta }}{d\psi _{\phi }}}}$

In the case of toroidal magnetic fields, it is usually assumed that so-called magnetic flux surfaces are present. These are nested toroidal surfaces in which field lines for the magnetic confinement run. The rotation transformation as a characteristic quantity of a river area is to be averaged over such a river.

The concept of flux areas is important for the magnetohydrodynamics of fusion experiments, since it reduces the generally necessary consideration in three-dimensional spatial space to areas. The movement of the plasma particles is shaped by the flow areas.

The integration leading to the magnetic fluxes

${\displaystyle \psi =\int \limits _{S'}{\vec {B}}\cdot \mathrm {d} {\vec {S}}}$

in the case of the poloidal / toroidal flow occurs through a surface that captures the poloidal / toroidal component of the field. For the poloidal flow , this area is typically a finite, equatorial ring from the center of the torus with the large radius to a small radius . For the toroidal flow , the integration surface is a poloidal section at the same radius. For the circular torus the integration surface for the toroidal flow is thus a circle. ${\displaystyle {\vec {B}}}$ ${\displaystyle \psi _{\theta }}$ ${\displaystyle R}$ ${\displaystyle R+r}$ ${\displaystyle \psi _{\phi }}$

The size consequently changes with the small torus radius. For a pure torus geometry, as it occurs with the circular tokamak , the rotation transformation is ${\displaystyle \iota }$

${\displaystyle {\frac {\iota (r)}{2\pi }}={\frac {RB_{\theta }(r)}{rB_{\phi }(r)}}}$

It can be seen here that the rotation transformation represents the poloidal ( ) offset of a field line after a toroidal ( ) revolution. ${\displaystyle \theta }$ ${\displaystyle \phi }$

The inverse rotation transformation is the safety factor :

${\displaystyle q={\frac {2\pi }{\iota }}}$

An important role for the stability of magnetic configurations is played by the shear , which represents the radial change of the rotational transformation: ${\displaystyle S}$

${\displaystyle S=-{\frac {r}{\iota }}{\frac {d\iota }{dr}}}$

Individual evidence

1. ↑ F. Wagner, H. Wobig: Magnetic confinement . In: A. Dinklage, T.Klinger, G.Marx, L. Schweikhard (Eds.) Plasma Physics: Confinement, Transport and Collective Effects (= Lecture Notes in Physics , Vol. 670) Springer, Berlin / Heidelberg 2005, ISBN 3 -540-25274-6 , pp. 137-172.