Components of polar basis vectors are not NFP periodic

[unalmis]

Issue

In the singular integral routines used for free boundary and in #1360 , the Fourier transforms of the components of the polar basis vectors of NFP symmetric vector fields are taken. Therefore, all vector quantities, in particular the virtual casing current K_vc and the boundary normal vector n_rho, are interpolated incorrectly for any finite resolution due to Gibbs phenomena.

Fix

The fix is to interpolate with Fourier transforms after converting the vectors to Cartesian form.

Math

Recall that a vector field $A$ has discrete integer field period symmetry, also known as NFP symmetry, if

$$\mathbf{A}(R, \phi + 2\pi/\text{NFP} , Z) = \mathbf{A}(R, \phi, Z)$$

Therefore, $\mathbf{A}$ may be represented as a vector of Fourier series.

$$ \mathbf{A} = A^x \hat{x} + A^y \hat{y} + A^z \hat{Z} = A^{R} \hat{R} + A^{\phi} \hat{\phi} + A^z \hat{Z} $$

However, only the components of $\mathbf{A}$ belonging to a coordinate system with constant basis vectors, such as the Cartesian components are NFP periodic. In particular for $\text{NFP} \neq 1$, the components of the polar basis vectors are not NFP periodic because $\hat{R}$ and $\hat{\phi}$ have toroidal variation. Instead a vector field $A$ with NFP symmetry satisfies

$$ \begin{bmatrix} A^{R}(\phi) \\ A^\phi(\phi) \end{bmatrix} = \begin{bmatrix} \cos(2\pi/\text{NFP}) & -\sin(2\pi/\text{NFP}) \\ \sin(2\pi/\text{NFP}) & \cos(2\pi/\text{NFP}) \end{bmatrix} \begin{bmatrix} A^{R}(\phi + 2\pi/\text{NFP}) \\ A^\phi(\phi + 2\pi/\text{NFP}) \end{bmatrix} $$

Therefore, interpolation of $A^{R}$ and $A^{\phi}$ with Fourier series over the domain

$$(\theta, \zeta) \in [0, 2\pi) \times [0, 2\pi / \text{NFP})$$

will suffer from Gibbs phenomena and not converge with finite resolution.

[unalmis]

The NFP definition in the reference I'm looking at isn't right; it should be cylindrical components are periodic.