# Field theory and structure-preserving geometric particle-in-cell algorithm for drift wave instability and turbulence

## TL;DR

This paper introduces a structure-preserving geometric Particle-In-Cell algorithm for simulating drift wave instability and turbulence in magnetized plasmas, ensuring long-term accuracy and conservation properties.

## Contribution

It develops a novel geometric PIC algorithm based on discrete exterior calculus and Hamiltonian splitting that preserves symplectic structure and gauge symmetry.

## Key findings

- The algorithm accurately simulates ion Bernstein and drift waves.
- Simulation results show turbulence energy diffusion scales between Bohm and gyro-Bohm.
- Density blobs form as prominent structures in fully developed turbulence.

## Abstract

A field theory and the associated structure-preserving geometric Particle-In-Cell (PIC) algorithm are developed to study low frequency electrostatic perturbations with fully kinetic ions and adiabatic electrons in magnetized plasmas. The algorithm is constructed by geometrically discretizing the field theory using discrete exterior calculus, high-order Whitney interpolation forms, and non-canonical Hamiltonian splitting method. The discretization preserves the non-canonical symplectic structure of the particle-field system, as well as the electromagnetic gauge symmetry. As a result, the algorithm is charge-conserving and possesses long-term conservation properties. Because drift wave turbulence and anomalous transport intrinsically involve multi time-scales, simulation studies using fully kinetic particle demand algorithms with long-term accuracy and fidelity. The structure-preserving geometric PIC algorithm developed adequately servers this purpose. The algorithm has been implemented in the \textsl{SymPIC} code, tested and benchmarked using the examples of ion Bernstein waves and drift waves. We apply the algorithm to study the Ion Temperature Gradient (ITG) instability and turbulence in a 2D slab geometry. Simulation results show that at the early stage of the turbulence, the energy diffusion is between the Bohm scaling and gyro-Bohm scaling. At later time, the observed diffusion is closer to the gyro-Bohm scaling, and density blobs generated by the rupture of unstable modes are the prominent structures of the fully developed ITG turbulence.