Connection length

The connection length is understood to be the distance between two points, measured along a magnetic field line passing through these points.

In the fusion context, a distinction is made between closed field lines (inside the Last Closed Flux Surface) and open field lines (outside).

Closed field lines

In a toroidally confined plasma, inside the Last Closed Flux Surface, the connection length is commonly defined as the length, measured along the magnetic field, to complete a poloidal turn.

In a circular tokamak, the poloidal circumference is 2πr. The connection length is $L=2\pi r/\sin(\alpha )$, where $\alpha$ is the pitch angle of the field line, namely $\tan(\alpha )=B_{\theta }/B_{\phi }$. Assuming $B_{\phi }\gg B_{\theta }$, one has $\sin(\alpha )\simeq \tan(\alpha )$, so that

$L=2\pi r{\frac {B_{\phi }}{B_{\theta }}}=2\pi Rq$

where q is the safety factor, approximated by $q=rB_{\phi }/RB_{\theta }$.

Open field lines

Outside Last Closed Flux Surface, the connection length associated with a given point is defined as the shortest distance from that point to any material surface measured along the field line through that point.