Applications and workflows

This page connects the models in torus-solver to common research workflows in toroidal confinement.

Tokamak-like vacuum fields (validation and intuition)

Even though this project is motivated by stellarators, a circular torus is also a convenient sandbox for tokamak-like vacuum fields.

A canonical reference profile is an ideal toroidal field:

\[ B_\phi(R) = B_0\frac{R_0}{R}, \]

which follows from Ampère’s law when there is a net poloidal current linking the torus.

In this repository:

  • torus_solver.fields.ideal_toroidal_field(...) provides the analytical \(1/R\) field.

  • torus_solver.current_potential.surface_current_from_current_potential_with_net_currents(...) lets you set a net poloidal current \(I_{\mathrm{pol}}\) and compute the resulting surface current density.

  • torus_solver.biot_savart.biot_savart_surface(...) evaluates the field from the surface current so you can compare numerically against the analytical \(1/R\) profile.

  • torus_solver.fieldline.trace_field_lines_batch(...) traces field lines to confirm that trajectories remain well-behaved inside the torus.

This “tokamak mode” is useful for:

  • unit/regression tests (known analytical scaling)

  • sign-convention checks (direction of \(B_\phi\) vs sign of \(I_{\mathrm{pol}}\))

  • performance profiling (Biot–Savart + tracing in a controlled geometry)

Stellarator-style design proxy: minimize normalized \(B_n/|B|\)

For stellarator coil design, a common ingredient is to choose currents on a winding surface so that the normal component of the magnetic field on a target surface is small:

\[ \mathrm{BnOverB}(\theta,\phi) = \frac{\mathbf B(\theta,\phi)\cdot\hat{\mathbf n}(\theta,\phi)}{|\mathbf B(\theta,\phi)|}. \]

The script examples/inverse_design/optimize_vmec_surface_Bn.py implements a REGCOIL-like optimization loop:

  1. Read a VMEC boundary surface (Fourier coefficients) from a input.* file.

  2. Scale that target surface to fit safely inside a circular torus (the winding surface).

  3. Parameterize the winding-surface current using either:

    • an electrode-driven potential model, or

    • a current potential \(\Phi\) (recommended for stringent targets).

  4. Minimize a smooth proxy for max-norm, typically a weighted \(p\)-norm with \(p\gtrsim 8\).

  5. Validate with field-line tracing and diagnostics such as max|Bn/B|.

Note

Driving \(B_n/|B|\to 0\) on a single interior surface is a powerful objective, but it does not by itself guarantee globally nested flux surfaces. Use field-line tracing and Poincaré plots as sanity checks.

Connecting to near-axis / quasisymmetry tools

Many near-axis and quasisymmetry workflows (e.g. Garren–Boozer constructions) output either:

  • an axis curve + surface shape, or

  • a VMEC input file describing a boundary by Fourier series.

This repository currently focuses on the second case: VMEC-style Fourier surfaces. If you generate a quasisymmetric configuration with a near-axis tool, you can export a VMEC boundary and use it as a target in torus-solver.

What this code is (and is not)

This code is designed to be:

  • differentiable end-to-end (JAX)

  • a research sandbox (easy to modify objectives/parameterizations)

  • focused on a circular torus winding surface (for clarity and analytic checks)

This code is not yet:

  • a full coil engineering tool (manufacturing constraints, coil thickness, discrete coil sets)

  • a plasma-response / equilibrium solver (vacuum field only)

  • a general winding-surface geometry solver (non-circular surfaces)

The Development → Roadmap page lists concrete extension directions.