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 $\tan (\alpha )=B_{\theta }/B_{\varphi }$. Assuming $B_{\varphi }\gg B_{\theta }$, one has $\sin (\alpha )\simeq \tan (\alpha )$, so that

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

where q is the safety factor, approximated by $q=r{B}_{\varphi }/R{B}_{\theta }$.

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.