A Reduced-Order Particle-in-Cell Method with Azimuthal Fourier-Decomposed Fields for Nominally Axisymmetric Plasmas

A reduced-order Particle-in-Cell method is introduced for kinetic simulation of otherwise axisymmetric cylindrical plasmas that exhibit azimuthal instabilities. The method spatially decomposes all field quantities into a small number of (mesh-less) azimuthal Fourier modes $m=0,\mathrm{.}\mathrm{.}\mathrm{.},N_m,N_m\ll N_{\theta }$, reducing the costly three-dimensional field solve $\mathcal{O}(N_z{N}_r{N}_{\theta })$ to a family of decoupled independent two-dimensional problems $\mathcal{O}((N_m+1)N_z{N}_r)$ on the meridional plane - one per mode - while particles continue to move in full three-dimensional space. Fields are reconstructed at particle positions by coherent superposition of these modal contributions, preserving complete azimuthal variation at a fraction of the cost of a conventional three-dimensional simulation. The method is validated against the diocotron instability of a hollow electron annulus across three geometrically distinct configurations, recovering linear growth rates and eigenmode structures within 7% of closed-form analytic predictions, and reproducing the non-linear vortex dynamics characteristic of the instability saturation. A further benchmark against the community-standard Landmark Penning discharge problem recovers the rotating-spoke frequency, radial plasma profiles, and modal energy hierarchy in quantitative agreement with long-time reference simulations, at approximately 640 CPU-hours - a factor of 46 speed-up compared to the median benchmark cost. The approach addresses the important gap between computationally prohibitive full three-dimensional kinetic simulation and the physically limited reduced-dimensionality models on which predictive modelling of anomalous transport in magnetised plasma devices currently relies.