# Model B: REGCOIL-like current potential

This model is the closest conceptual match to REGCOIL:

> Instead of driving currents by injecting charge at points (electrodes), we represent the surface current as a divergence-free field generated by a **current potential** $\Phi$ on the surface.

## Divergence-free surface currents

On a smooth surface, a divergence-free surface current density can be written as:

$$
\mathbf K = \hat{\mathbf n} \times \nabla_s \Phi,
$$

where $\Phi(\theta,\phi)$ has units of **Amperes**. This representation automatically enforces:

$$
\nabla_s\cdot \mathbf K = 0.
$$

This is the “current potential” framework used by REGCOIL and related coil-design formulations.

## Net poloidal / toroidal currents and multi-valued $\Phi$

The current potential can contain secular (multi-valued) terms that correspond to net currents linking the torus:

$$
\Phi(\theta,\phi) =
\Phi_{\mathrm{sv}}(\theta,\phi)
 + \frac{I_{\mathrm{tor}}}{2\pi}\,\theta
 + \frac{I_{\mathrm{pol}}}{2\pi}\,\phi,
$$

where:

- $I_{\mathrm{pol}}$ is the **net poloidal current** [A] (sets a dominant toroidal $1/R$ field scale via Ampère’s law)
- $I_{\mathrm{tor}}$ is the **net toroidal current** [A] (sets a dominant poloidal field component)

The derivatives are:

$$
\Phi_\theta = \partial_\theta \Phi_{\mathrm{sv}} + \frac{I_{\mathrm{tor}}}{2\pi},\qquad
\Phi_\phi = \partial_\phi \Phi_{\mathrm{sv}} + \frac{I_{\mathrm{pol}}}{2\pi}.
$$

:::{important}
The secular terms $\theta$ and $\phi$ are discontinuous when wrapped to $[0,2\pi)$ (a sawtooth). **Do not** differentiate them with FFTs.

In `torus-solver`, the secular contributions are handled analytically by adding the constant derivatives shown above.
:::

## Implementation on the circular torus

For the circular torus, one convenient formula is:

$$
\mathbf K = \frac{\Phi_\theta\, \mathbf r_\phi - \Phi_\phi\, \mathbf r_\theta}{\sqrt{g}},
$$

which is equivalent to $\hat{\mathbf n}\times \nabla_s \Phi$.

In the code:

- `src/torus_solver/current_potential.py`
  - `surface_current_from_current_potential(...)` (single-valued $\Phi$ only)
  - `surface_current_from_current_potential_with_net_currents(...)` (recommended)

## Relation to REGCOIL

REGCOIL solves a regularized least-squares problem for $\Phi$ on a coil surface, typically minimizing some norm of $B_n$ on a plasma boundary (plus regularization in $|K|^2$ or related measures). In this repository:

- `examples/inverse_design/optimize_vmec_surface_Bn.py` implements an analogous idea on a **circular torus coil surface** using:
  - Fourier coefficients for $\Phi_{\mathrm{sv}}$
  - an objective based on **normalized** $B_n/|B|$
  - L-BFGS (from `jaxopt`) and JAX autodiff

This makes it easy to experiment with differentiable extensions (new objectives, constraints, and parameterizations) while remaining close to the REGCOIL viewpoint.