Reduction of the statistical error in electromagnetic gyrokinetic particle-in-cell simulations

The δf-PIC method is widely used for electrostatic particle-in-cell (PIC) simulations. Its basic idea is the ansatz f=f₀+δf (δf-ansatz) where the particle distribution function f is split into a usually time-independent background f₀ and a time-dependent perturbation δf. In principle, it can also be used for electromagnetic gyrokinetic PIC simulations, but the required number of markers can be so large that PIC simulations become impractical. The reason is a decreasing efficiency of the δf-ansatz for the so-called `Hamiltonian formulation' using p_\Vert as a dynamic variable. As a result, the density and current moment of the distribution function develop large statistical errors. To overcome this obstacle we propose to solve the potential equations in the symplectic formulation using v_\Vert as a dynamic variable. The distribution function itself is still evolved in the Hamiltonian formulation which is better suited for the numerical integration of the parallel dynamics. The contributions from the full Jacobian of phase space, a finite velocity sphere of the simulation domain and a shifted Maxwellian as a background are considered. Special care has been taken at the discretisation level to make damped magnetohydrodynamics (MHD) mode simulations within a realistic gyrokinetic model feasible. This includes devices like e.g. large tokamaks with a small aspect ratio.