# Electromagnetic full-$f$ gyrokinetics in the tokamak edge with discontinuous Galerkin methods

## TL;DR

This paper introduces an energy-conserving discontinuous Galerkin scheme for full-$f$ electromagnetic gyrokinetics in tokamak edge plasmas, successfully performing nonlinear simulations on open field lines without Ampère cancellation issues.

## Contribution

It develops the first nonlinear electromagnetic gyrokinetic simulation on open field lines using a novel discontinuous Galerkin method that conserves energy and avoids Ampère cancellation.

## Key findings

- Scheme successfully avoids Ampère cancellation problem.
- First nonlinear electromagnetic gyrokinetic simulation on open field lines.
- Comparison shows differences between electromagnetic and electrostatic results.

## Abstract

We present an energy-conserving discontinuous Galerkin scheme for the full-$f$ electromagnetic gyrokinetic system in the long-wavelength limit. We use the symplectic formulation and solve directly for $\partial A_\parallel/\partial t$, the inductive component of the parallel electric field, using a generalized Ohm's law derived directly from the gyrokinetic equation. Linear benchmarks are performed to verify the implementation and show that the scheme avoids the Amp\`ere cancellation problem. We perform a nonlinear electromagnetic simulation in a helical open-field-line system as a rough model of the tokamak scrape-off layer using parameters from the National Spherical Torus Experiment (NSTX). This is the first published nonlinear electromagnetic gyrokinetic simulation on open field lines. Comparisons are made to a corresponding electrostatic simulation.