Characterisation of X- and O-points in Wendelstein 7-X with respect to coil currents

TL;DR

This work develops a fast automated scheme to locate fixed points in Wendelstein 7-X vacuum magnetic fields and to compute the Jacobian trace $\mathrm{Tr}(\mathsf{M})$ of the field-line map, enabling characterization of X- and O-points and island branches. By performing both individual coil scans around standard, high iota, and low iota configurations and an extensive random-sampled scan across $\sim 2.8\times 10^5$ configurations, the study quantifies how non-planar coils NPC, planar coils PC, and the control coil CC shape island location, phase, and size. It establishes that planar coils mainly shift fixed-point positions and adjust the iota profile, while the CC strongly modulates $\mathrm{Tr}(\mathsf{M})$ and can drive X to O transitions, with the effect increasing for fixed points at larger minor radii. A key finding is that the proxy $|\mathrm{Tr}(\mathsf{M})-2|$ correlates with island size for internal islands, offering a practical metric to select experimental configurations with desired island widths. The fast scheme and large-configuration analysis provide actionable guidance for tailoring edge magnetic topology in W7-X and similar stellarators.

Abstract

This work analyses vacuum magnetic field topology in Wendelstein 7-X (W7-X) with respect to changes in the current in the superconducting coils. We develop a fast automated scheme to locate fixed points (such as X- and O-points) and calculate the trace of the Jacobian of the field line map for them (Tr(M)), which represents several important properties of the fixed point. We perform two sets of coil current scans: (1) scans where each coil current is varied individually, using the "standard", "high iota" and "low iota" configurations as starting points; (2) a scan of over 200,000 magnetic configurations in which the coil currents are randomly sampled. In both cases we constrain the coil currents to the normal range of W7-X. We verify the principal roles of the non-planar, planar and control coils: the non-planar coils establish island chains with a certain phase; the planar coils modify the location of the island chain by both controlling the iota profile and shifting the configuration "inward" and "outward"; the control coil affects the island size and phase. We also find that |Tr(M)-2| (a quantity closely related to the magnitude of the Greene's residue) tends to increase with the minor radius of the fixed points, and that Tr(M) for X- and O-points can be very differently affected by the control coil current. Finally, we show that |Tr(M)-2| serves as a proxy for island size for internal island chains, which may help identification of suitable experimental candidates.

Figures (13)

• Figure 1: Top-down view of W7-X for the "standard configuration". The plasma boundary is shown as a yellow surface and coils as black lines, with the coils of one field period shown bold and coloured (each unique coil geometry having its own colour). The magnetic axis is shown as a red line and the X- and O-points of the edge island chain are shown as dark and light green lines.
• Figure 2: A Poincaré section for the "standard" configuration at several toroidal locations, also showing the magnetic axis and edge $\iota=1$ X- and O-points. The X-point and O-point in the midplane ($Z=0$) at $\phi=36^\circ$ are shown by larger markers (blue and orange) to illustrate their motion.
• Figure 3: A Poincaré section for the "high iota" (upper row) and "low iota" (lower row) configurations at $\phi=36^\circ$ (left) and $\phi=0^\circ$ (right).
• Figure 4: Coil currents scans with the standard configuration as the starting point. a) Magnetic axis location at $\phi=36^\circ$ ($R_\text{MA}$) as planar coils (PC) are varied. b) $R_\text{MA}$ variation with non-planar coils (NPCs). c) On-axis rotational transform $\iota_\text{MA}$ against PC currents. The range of $\iota_\text{MA}$ when NPC currents are scanned is shown as shaded yellow region. d) Location of $\phi=36^\circ$ outboard midplane fixed point $R_\text{(fixed point)}$ against PC currents, with NPC range shown as shaded yellow region and control coil (CC) range shown as shaded grey region. e) Jacobian trace $\mathrm{Tr}(\mathsf{M})$ for the $\phi=36^\circ$ outboard midplane fixed point (solid lines) and the $\phi=36^\circ$ inboard midplane fixed point (dashed line) against PC currents with NPC range shown in shaded yellow. f) $\mathrm{Tr}(\mathsf{M})$ against CC current for both fixed points.
• Figure 5: a) Poincaré sections, showing edge island chain as control coil current $I_\text{CC}$ is scanned. b) Island area and Jacobian trace $\mathrm{Tr}(\mathsf{M})$ as a function of $I_\text{CC}$.