Non-axisymmetric MHD equilibria with an arbitrary number of field periods

Several closed-form non-axisymmetric magnetic fields with nested toroidal flux surfaces have recently been discovered as exact benchmarks for verifying three-dimensional magnetohydrodynamic equilibrium codes. However, all of these solutions have at most two field periods (N ≤ 2), i.e., identical segments that repeat toroidally. Here, we derive closed-form non-axisymmetric magnetohydrodynamic equilibria for an arbitrary integer number of field periods N ≥ 1. The equilibria have non-constant pressure and closed magnetic field lines with either zero rotational transform ($ι= 0$) for N ≥ 1 or integer rotational transform (ι= ± N) for N ≥ 3. For the reflection-symmetric $ι= 0$ equilibria, we also construct a matched exterior vacuum field with continuous magnetic field across the plasma boundary, providing a benchmark for free-boundary equilibrium codes.