[rogeriojorge]

Stacked on #146. It adds DESC and near-axis fields, and full orbits in toroidal fields, so every ESSOS model runs on coils, permanent magnets, VMEX/VMEC, MRX and DESC.

DescField(ToroidalField)

A DESC equilibrium, evaluated as DESC writes it:

• R, Z and the stream function lambda as Fourier-Zernike series, with the radial Zernike polynomials from their Jacobi recurrence;
• iota tabulated on 1025 radii and interpolated by cubic Hermite polynomials;
• in (s, theta, phi) = (rho^2, theta, zeta), B^theta = Psi (iota - d_phi lambda) / (2 pi sqrt(g)) and B^phi = Psi (1 + d_theta lambda) / (2 pi sqrt(g)).

The field is exact on the axis.

DescField.from_desc takes a DESC Equilibrium, a DESC output file, or the .npz that save writes. Reading the .npz needs no DESC install, which helps because DESC pins jax < 0.10.

Against DESC's own compute at 500 random points:

equilibrium	x	B	|B|	sqrt(g)	iota
precise QA (vacuum)	1.7e-15	4.4e-13	1.6e-13	2.2e-15	3.9e-12
QA at 2.5% beta, bootstrap current	3.9e-15	2.1e-9	3.8e-10	3.0e-15	7.8e-7

NearAxisField(MagneticField)

A first-order pyQSC_JAX near-axis field in its Boozer coordinates (r, theta, varphi). It uses the expansion's own covariant and contravariant components, |B| and Jacobian. to_xyz maps through the axis's Frenet frame and is first order like the field: |e_i B^i| / |B| - 1 is 1% at r = 0.01 and 7% at r = 0.1.

(r, theta) is singular on the axis, so orbits must stay off it. Full orbits are refused with a clear error, because the field has no inverse map to Cartesian points.

Full orbits in toroidal fields

Before this PR, FullOrbit* models with a VMEC field read (s, theta, phi) as (x, y, z). Now, for a ToroidalField:

• Tracing maps each guiding-center start through to_xyz;
• it traces in field.cartesian(), a Cartesian view built on B_xyz;
• by default it stops orbits at s = 1, where the inverse map is clamped.

Every model on every field

Run on Landreman-Paul QA where available. Settings:

• 8 orbits per row: 100 keV alphas for 2 us, field lines for 50 samples;
• the near-axis field at r = 0.01-0.05;
• the energy error is the largest over the orbits;
• the collisional model has no conserved energy.

field	field lines	guiding centers	full orbits (adaptive)	full orbits (Boris)	collisional GC
coils	3.0 s	18 s, 1e-8	1.2 s, 1e-15	3.3 s, 4e-16	64 s
coils + magnets (PointDipoleField)	1.9 s	2.4 s, 2e-8	1.0 s, 3e-9	2.3 s, 3e-14	78 s
VMEX	2.6 s	3.9 s, 1e-7	45 s, 1e-10	4.7 s, 2e-14	31 s
MRX	1.1 s	5.1 s, 5e-9	8.3 s, 4e-9	3.1 s, 3e-14	33 s
DESC	1.2 s	2.5 s, 8e-11	13 s, 4e-8	15 s, 1e-14	61 s
near-axis	0.7 s	2.7 s, see below	not available	not available	16 s

• Near-axis guiding centers. With pitch 0.5 they conserve energy to 2e-4, the size of the first-order expansion's own inconsistency: B_i B^i differs from |B|^2 by 1.4% at r = 0.04. With random pitch, one of the 8 orbits reached the axis and the error grew to 0.34.
• VMEX adaptive full orbits. Before the stop at s = 1, orbits that left the plasma ran 6912 s; they now take 45 s.

Tests

tests/test_fields.py:

• Fourier-Zernike against closed forms;
• DescField: covariant and contravariant consistency, zero divergence, Hermite iota, the .npz round trip;
• full orbits in a VMEC field (start and energy);
• near-axis first-order consistency (skipped without pyQSC_JAX).

The DESC comparisons against DESC itself ran in a DESC environment.

[rogeriojorge]

Split into focused PRs, each reviewable on its own:

• Vmec: C1 radial interpolation (guiding centers stalled at grid points) #150 Vmec C1 radial interpolation (guiding-center stall)
• Use every CPU core and GPU by default; cache compiled traces #151 CPU/GPU defaults and compile cache
• Magnetic field classes: Cartesian defaults, field algebra, ToroidalField #152 Magnetic field classes (base of the next five)
• PointDipoleField: point dipoles with the moments as dofs #153 PointDipoleField
• MRXField: trace MRX states (vacuum or relaxed, with islands) #154 MRXField, examples and islands movie
• DescField: trace DESC equilibria as DESC writes them #155 DescField
• NearAxisField: trace pyQSC_JAX near-axis fields in Boozer coordinates #156 NearAxisField
• Full orbits in toroidal fields #157 Full orbits in toroidal fields