Coil fields, optimization, and constructible stellarator geometry

This note supplies the geometric and electromagnetic bridge from mathematical fields to hardware. Material/thermal constraints are in 07; continuous checker methods are in 05.

Contents

• Filament fields
• Finite conductors and plasma coupling
• Inverse coil design
• Geometry and manufacturing
• Robust design specifications
• A complete candidate workflow
• Theorem backlog

Filament fields

Let c:[0,2π]→ℝ³ be a regular closed parametrized curve and I its oriented current. At points away from the curve define

Define multi-coil fields by finite sums. Prove:

• linearity in current and superposition;
• reparametrization invariance;
• reversal of curve orientation/current and their sign effects;
• smoothness away from sources;
• div B=0 and curl B=0 in a current-free region;
• the chosen Ampère circulation/flux properties with their topological hypotheses.

For dist(x,c)≥d>0 and curve length L, direct kernel estimates give

The derivative bound uses the operator norm of D(r/|r|³), whose largest absolute eigenvalue is 2/|r|³. Derive higher derivative constants when needed. These are useful for interval quadrature, orbit error, and tolerance propagation.

Prove the circular-loop field on its axis as a sanity example:

Keep direction determined by the actual current orientation.

Finite conductors and plasma coupling

A filament field is singular on its source and cannot supply its own peak conductor field or self-force. Introduce a finite conductor Ω_c with vector current density J_cond:

under adequate integrability/current closure conditions. Handle points in or near the conductor through the appropriate regularity/PDE theory and quadrature, not by evaluating a singular filament formula there.

Prove a far-field/finite-width comparison between a specified winding-pack distribution and its centreline approximation, with distance and current-moment bounds. Use the finite-conductor field for critical-current and load certificates.

At finite plasma pressure the equilibrium field is not merely the vacuum field generated by coils:

with interface/boundary conditions and self-consistent plasma currents. A virtual-casing calculation can propose the external field to reproduce a fixed-boundary equilibrium, but needs a sound reconstruction/error theorem before certification.

For a conductor in an external field the force density is J_cond×B. Include self-consistent finite-width electromagnetic stress/self-force appropriately. Distinguish the magnetic pressure scale B²/(2μ₀) from a universal support-stress formula. The structural model must transmit the actual distributed loads.

Inverse coil design

Use SIMSOPT, DESC, REGCOIL, and coil-design papers as candidate generators. Record software version and conventions, particularly field periods, stellarator symmetry, Fourier signs, currents, and units.

For a winding surface S_w a surface-current potential has the form

with periodic and prescribed multivalued/circulation components as required by the chosen topology. Prove surface-current conservation and the field representation. Single-valued Φ alone omits possible net current periods.

The inverse problem may minimize

or optimize explicit curves. The achieved objective is evidence about that objective. It is not by itself a proof of a nearby exact equilibrium, invariant tori, robust confinement, or coil feasibility.

For practical bounds, prove how a field or normal-field residual in a declared norm controls the next physical quantity. An L² boundary residual does not imply a pointwise residual without extra regularity/inverse estimates.

For quasisymmetry define the intended residual precisely, including radial weights, angle measure, normalization, and omitted modes. Prove its zero characterization within the chosen function class. Compare near-axis, global-boundary, and simultaneous coil/plasma design formulations without identifying their objectives.

Geometry and manufacturing

Represent each physical coil by its centreline, a material frame, cross-section, casing, insulation, supports, and assembly state. Use an oriented ribbon or winding-pack model for tape; curvature of the centreline alone does not determine hard-way bending or torsional strain.

Required continuous properties include:

Prove continuous bounds via 05. Avoid self-distance checks that reject adjacent points on the same coil simply because parameter values differ.

The distances 0.52 m, 0.83 m, and 1.4 m in the starting note are not established universal limits. Select a machine/material source and an engineering budget. Scaling a geometry by s scales length and clearances by s and curvature by 1/s; required current and component forces have additional scaling laws to derive, so geometric scale invariance does not imply equal feasibility.

Intentional edge islands can serve an island divertor. Distinguish the confinement core from the exhaust region instead of forbidding every magnetic island. See W7-X divertor studies.

Robust design specifications

Let a be nominal design coefficients and θ∈U manufacturing/operating variations. Include coil displacements, current calibration, pressure/current-profile changes, thermal expansion, and material degradation.

Prove field perturbation bounds in the norms needed downstream, for example

with a common positive conductor-distance margin. A symmetry-breaking or normal-field scalar objective does not automatically yield this bound.

Translate downstream margins into tolerance budgets. Clearance margins use finite-solid displacement bounds. Stability margins require form perturbation bounds. Orbit margins require field derivatives and a time-dependent comparison. Magnet margins require field orientation, temperature, and strain.

Prove robust feasibility for a useful candidate before attempting global optimality. Keep objectives and hard certified constraints separate; a penalty term does not turn a violation into a theorem.

A complete candidate workflow

1. Specify the machine class and operating/material envelope.
2. Propose an axis/surface/coil/current parameterization from theory.
3. Search candidate geometries and choose a useful finite representation.
4. Certify embeddings, regularity, swept-solid clearances, and fields.
5. Validate a model-consistent equilibrium and its boundary conditions.
6. Certify the chosen MHD stability property on its full admissible space.
7. Certify particle loss/transport/exhaust quantities at the required level.
8. Bound heat, mechanical, magnet, irradiation, and lifetime constraints.
9. Propagate common uncertainties and compose the contracts for one candidate.
10. Record remaining model-validity evidence and the next physical validation step.

Separate proofs for different equilibria or operating profiles cannot be combined into one design certificate without a common-witness theorem.

Theorem backlog