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

Abstract

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 azimuthal Fourier modes $m=0,...,N_m,\quad N_m \ll N_θ$, reducing the costly three-dimensional field solve $\mathcal{O}(N_zN_rN_θ)$ to a family of decoupled independent two-dimensional problems $\mathcal{O}((N_m+1)N_zN_r)$ 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. For initial validation, the method is tested against two benchmark problems with restricted axial dimension: The diocotron instability of a hollow electron annulus across three geometrically distinct configurations, recovering linear growth rates and eigenpotential structures within 7% of closed-form analytic predictions, and reproducing the non-linear vortex dynamics characteristic of the instability saturation; and 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 represents a step toward addressing 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.