Conservation of total energy and toroidal angular momentum in gyrokinetic particle-in-cell simulations

Predictive transport simulations of tokamak discharges are continuously improving thanks to increasing computational resources available and to the efforts made in code development. As an example of the progress, a recent work [1] with the Elmfire code demonstrates the ability of global gyrokinetic full f electrostatic particle simulations, including both neoclassical and turbulence physics, to reproduce experimental FT-2 tokamak measurements quantitatively. The validity of such first-principles simulations in a toroidal axisymmetric tokamak configuration requires naturally the conservation of total energy and total toroidal angular momentum, in order to model long-term neoclassical and turbulence physics correctly. The conservation of energy and toroidal angular momentum in gyrokinetic simulations necessitates the existence of corresponding conservation laws in the underlying theory and the application of numerical methods that realize the conservation laws in practice. For Elmfire, the gyrokinetic formalism in Ref. [2] provides the expressions for total energy K and total toroidal angular momentum L (explicit form given in Ref. [3]) with exact conservation up to the second order in the gyrokinetic expansion for a quasineutral plasma. Therefore, the conservation of these variables is analyzed in Elmfire simulations despite the low-order gyrokinetics may neglect important terms affecting, e.g., the expression for the conserved toroidal angular momentum [4]. The accuracy of the conservation of K and L is determined by the numerical methods applied in Elmfire, as shown in the following sections. The challenging features of Elmfire with respect to the conservation of energy and toroidal angular momentum are both that it is a particle-in-cell code and that it resolves the full 5D distribution of drift-kinetic electrons, in addition to gyrokinetic ions, under 3D electrostatic potential [5]. Under the circumstances, first, the simulated plasma is non-neutral in the space between the grid points although the charge neutrality at the grid points is enforced. This inhomogene-ity results in asymmetric electric force interpolation between ions and electrons and therefore in toroidal angular momentum drive unless the so-called momentum conserving interpolation scheme is employed. In the standard version of the scheme, the electric force is interpolated from the grid to the particle location by the same interpolation function as the charge is sampled to the grid. Ideally, the quasi-neutrality at the grid points guarantees then zero toroidal