Coordinate maps, ∇Ψ_toroidal, and B × ∇Ψ_toroidal vectors for 5/6, 5/5 and 5/4 island domains in W7-X

The edge island domain in Wendelstein 7-X consists of divertor islands that coincide with the location of rational values of the rotational transform ι≈(5/6, 5/5, 5/4) and surround the main confinement volume (the ‘main plasma’). The "5/5" edge is 5 individual islands that are unconnected. In contrast, a single island connects onto itself after 6 or 4 toroidal transits in the the “5/6” and “5/4” edge, respectively. Many interesting phenomena are related to these islands and diagnostic analyses require a mapping from `laboratory' or real space coordinates to the island coordinate system. Two procedures are described here to calculate several scalar and vector quantities for closed island structures which can be utilized in fast interpolation schemes for inverse maps. For the "5/5" edge, a fixed-boundary vacuum (zero beta) magneto-hydrodynamic solutions of the 5/5 island is found with VMEC.  The solution is compatible with already existing routines which determine ∇Ψ_toroidal (and other quantities) of VMEC solutions at arbitrary laboratory coordinates via stellarator symmetry. VMEC does not support solutions for the 5/4 and 5/6 island cases, so a field-line following technique for calculating the ∇Ψ_toroidal and B × ∇Ψ_toroidal vectors along the field line is utilized.