Glossary and conventions

This page is a quick reference for the symbols and conventions used throughout the docs and code.

Geometry

  • Winding surface: The surface on which currents live. In this project it is always a circular torus.

  • Target surface: A surface inside the winding surface on which we evaluate objectives such as \(B_n/|B|\) (e.g. a VMEC plasma boundary scaled inward).

  • \(R_0\): Major radius of the circular torus [m].

  • \(a\): Minor radius of the circular torus [m].

  • \((\theta,\phi)\): Poloidal and toroidal angles on \([0,2\pi)\).

  • \(\mathbf r(\theta,\phi)\): Surface embedding in \(\mathbb{R}^3\).

  • \(\hat{\mathbf n}\): Outward unit normal of a surface.

Surface operators

  • \(\nabla_s\): Surface gradient (tangential gradient).

  • \(\nabla_s\cdot\): Surface divergence.

  • \(\nabla_s^2\): Laplace–Beltrami operator.

  • \(\sqrt{g}\): Surface Jacobian factor (area element density) so that \(dA=\sqrt{g}\,d\theta\,d\phi\).

Currents, potentials, and sources

  • \(\mathbf K(\theta,\phi)\): Surface current density [A/m] on the winding surface.

  • \(V(\theta,\phi)\): Electrostatic potential on the surface [V] used in the electrode-driven model.

  • \(\sigma_s\): Surface conductivity [S] (a model parameter; can be absorbed into scalings).

  • \(s(\theta,\phi)\): Injected/extracted surface source density [A/m\(^2\)] representing discrete electrodes or distributed sources/sinks.

Model A (electrodes)

  • Constitutive law: \(\mathbf K = -\sigma_s \nabla_s V\).

  • Continuity: \(-\nabla_s^2 V = s/\sigma_s\).

Model B (current potential / REGCOIL-like)

  • \(\Phi(\theta,\phi)\): Current potential [A] such that \(\mathbf K = \hat{\mathbf n}\times\nabla_s \Phi\).

  • \(\Phi_{\mathrm{sv}}\): Single-valued (periodic) part of \(\Phi\).

  • \(I_{\mathrm{pol}}\): Net poloidal current [A] (dominant control of the toroidal \(1/R\) field scale).

  • \(I_{\mathrm{tor}}\): Net toroidal current [A] (dominant control of the poloidal field scale).

Magnetic fields and objectives

  • \(\mathbf B(\mathbf x)\): Magnetic field [T] in vacuum from Biot–Savart.

  • \(B_n = \mathbf B\cdot\hat{\mathbf n}\): Normal component on a surface.

  • \(|B| = \|\mathbf B\|\): Magnitude of the magnetic field.

  • Normalized normal field: \(B_n/|B|\) (dimensionless). This is the quantity reported and optimized throughout the repository.

VMEC conventions (stellarator symmetry case)

VMEC input files represent the boundary by Fourier series:

\[ R(\theta,\phi)=\sum_{m,n} \mathrm{RBC}_{n,m}\cos(m\theta-nN_{\mathrm{fp}}\phi)+\mathrm{RBS}_{n,m}\sin(m\theta-nN_{\mathrm{fp}}\phi), \]
\[ Z(\theta,\phi)=\sum_{m,n} \mathrm{ZBC}_{n,m}\cos(m\theta-nN_{\mathrm{fp}}\phi)+\mathrm{ZBS}_{n,m}\sin(m\theta-nN_{\mathrm{fp}}\phi), \]

where:

  • \(N_{\mathrm{fp}}\) is the number of field periods (NFP in the VMEC input).