For small-shear helical-axis stellarators, linear ideal-magnetohydrodynamic (MHD) stability calculations and full-torus, nonlinear, electromagnetic gyrokinetic (GK) simulations (the latter with this unprecedented combination of objectives in stellarator GKs) in their linear phase are shown to yield well agreeing spatio-temporal structures of unstable, globally extended perturbations. Likewise, good agreement is found for their dependence on the plasma pressure and the vacuum-field magnetic well in plasma equilibria with identical gradient lengths of the temperature and density profiles. In the nonlinear phase, these perturbations with MHD signatures entail deformations of the magnetic surfaces, growing magnetic islands which rotate in the electron diamagnetic direction and, eventually, lead to ergodisation of a larger part of the magnetic surfaces.

The $\unicode[STIX]{x1D6FF}f$-PIC method is widely used for electrostatic particle-in-cell (PIC) simulations. Its basic idea is the ansatz $f=f_{0}+\unicode[STIX]{x1D6FF}f$ ($\unicode[STIX]{x1D6FF}f$-ansatz) where the particle distribution function $f$ is split into a usually time-independent background $f_{0}$ and a time-dependent perturbation $\unicode[STIX]{x1D6FF}f$. 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 $\unicode[STIX]{x1D6FF}f$-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.