Flux-Surface Coordinates
Flux-surface and field-aligned coordinates align the computational grid with magnetic surfaces, matching the strong anisotropy of magnetized transport.
Grids that follow the field
In a magnetically confined plasma, transport is enormously anisotropic: parallel to the magnetic field it is fast and along smooth field lines, while across the field it is slow and structured on nested magnetic surfaces. A grid aligned with Cartesian axes would need extreme resolution to capture both. Flux-surface coordinates instead use the magnetic geometry itself: one coordinate labels nested flux surfaces (a radial-like variable), and angle-like coordinates run poloidally and toroidally around them.
Because the field lines lie within flux surfaces, aligning the grid with those surfaces means the numerically demanding parallel dynamics vary slowly along a grid direction, so far fewer points are needed than an unaligned grid would require.
Coordinate constructions
Straight-field-line coordinates choose the angular variables so that field lines appear as straight lines in the coordinate plane, simplifying the parallel derivative. Ballooning and flux-tube coordinates follow a single field line or bundle over many transits, ideal for local turbulence studies. The Clebsch (field-aligned) representation writes the field in terms of coordinate gradients so that the parallel derivative involves only one coordinate.
Complications
Flux coordinates become singular or ill-defined where the geometry breaks down: at the magnetic axis, at the separatrix and X-point of a diverted plasma, and in stochastic field regions. Handling the edge and scrape-off layer, where surfaces are open, requires specialized meshes that transition from flux-aligned in the core to field-aligned or unstructured near material surfaces.
- Radial coordinate labels nested flux surfaces
- Angular coordinates run poloidally and toroidally
- Straight-field-line and flux-tube constructions
- Singularities at axis, separatrix, and X-point need care
Flux-surface and field-aligned coordinates are the natural setting for equilibrium, stability, and turbulence codes, letting them resolve anisotropic magnetized transport at a fraction of the cost of an unaligned grid.