Rotating-Ellipse FCI¶

The rotating-ellipse (l = 2) stellarator is the canonical minimal non-axisymmetric magnetic field: a torus whose elliptical cross-section rotates as it is followed around toroidally, so the metric depends on all three logical coordinates. It is the reference geometry for checking that the flux-coordinate-independent (FCI) parallel operators are correct on a genuinely three-dimensional field, and it is the flagship non-axisymmetric benchmark in drbx.

Geometry¶

The logical coordinates are (x, theta, zeta) with a minor-radius label x and periodic poloidal / toroidal angles. The physical embedding into Cartesian space is

p0  = (1 + delta) * x * cos(theta)
q0  = (1 - delta) * x * sin(theta)
lam = n_field_periods * zeta
p   = cos(lam) * p0 - sin(lam) * q0
q   = sin(lam) * p0 + cos(lam) * q0
R   = r0 + p
X, Y, Z = R cos(zeta), R sin(zeta), q

so a surface x = const is an ellipse of elongation (1 + delta) / (1 - delta) whose major axis rotates n_field_periods times per toroidal turn. The left panel of the figure shows the nested flux surfaces at four toroidal angles: the ellipse orientation genuinely turns with zeta.

The metric is obtained by automatic differentiation of this embedding, g_ij = d_i X . d_j X, rather than by hand. This keeps the construction exact (the contravariant and covariant metrics are a consistent inverse pair to machine precision) and differentiable with respect to the shape parameters themselves — the same autodiff that builds the metric gives the gradient of any geometric diagnostic with respect to the elongation, which is what makes stellarator-shape optimization through the FCI stack possible.

The magnetic field is helical on the flux surfaces, B^x = 0, B^theta = iota * c_phi / J, B^zeta = c_phi / J, so its field lines wind with rotational transform iota and never leave a surface.

The constructor is drbx.geometry.build_rotating_ellipse_geometry; the embedding is exposed separately as rotating_ellipse_position.

What is checked¶

The gate in tests/test_rotating_ellipse_fci.py pins:

• Non-axisymmetry and metric consistency — the covariant metric varies along the toroidal axis (an axisymmetric geometry would be constant in zeta), the contravariant/covariant pair inverts to the identity to better than 1e-10, and the Jacobian stays positive.
• Second-order parallel-gradient convergence — with a manufactured field, the FCI parallel gradient converges at order 2 on the rotating-ellipse metric, both for the direct b^i d_i f operator and for the traced-field-line operator grad_parallel_op_fci, which follows the field lines between toroidal planes (the FCI-specific path). The right panel of the figure shows both tracking the slope--2 reference line.
• Shape differentiability — jax.grad of a metric diagnostic with respect to the ellipse elongation matches a central finite difference.

The rotating ellipse is a standard FCI convergence test case (see Stegmeir et al., Comput. Phys. Commun. 198, 139 (2016), the GRILLIX field-line-map approach).

Seeded-filament dynamics¶

Beyond the linear operator gate, the four-field drift-reduced FCI model (density, vorticity, ion/electron parallel velocity) runs a seeded filament on this geometry. A localized density blob is initialised with no vorticity; the curvature drive then generates vorticity from the pressure blob -- the interchange mechanism that moves a filament -- and because the flux surfaces rotate with the toroidal angle, the filament evolves differently in each toroidal plane.

The figure shows, in physical (R, Z) cross-sections at three toroidal angles (each a differently oriented ellipse), the seeded density blob, the evolved density, and the evolved vorticity: a clear interchange dipole develops, oriented with the local rotated cross-section. The gate tests/test_rotating_ellipse_filament.py pins that the run stays finite, the density stays positive and bounded, and both vorticity and a parallel ion flow are generated from the seed. Regenerate the figure with

PYTHONPATH=src python examples/stellarator/rotating_ellipse_filament.py

Reproduce¶

PYTHONPATH=src python examples/stellarator/rotating_ellipse_fci.py

writes output/rotating_ellipse_fci/ with the two-panel figure (rotating flux surfaces + parallel-operator convergence) and a JSON summary. Run the gate with

pytest -q tests/test_rotating_ellipse_fci.py