A conservative scheme for electromagnetic simulation of magnetized plasmas with kinetic electrons

Abstract

A conservative scheme has been formulated and verified for gyrokinetic particle simulations of electromagnetic waves and instabilities in magnetized plasmas. An electron continuity equation derived from drift kinetic equation is used to time advance electron density perturbation by using the perturbed mechanical flow calculated from the parallel vector potential, and the parallel vector potential is solved by using the perturbed canonical flow from the perturbed distribution function. In gyrokinetic particle simulations using this new scheme, shear Alfvé n wave dispersion relation in shearless slab and continuum damping in sheared cylinder have been recovered. The new scheme overcomes the stringent requirement in conventional perturbative simulation method that perpendicular grid size needs to be as small as electron collisionless skin depth even for the long wavelength Alfvé n waves. The new scheme also avoids the problem in conventional method that an unphysically large parallel electric field arises due to the inconsistency between electrostatic potential calculated from the perturbed density and vector potential calculated from the perturbed canonical flow. Finally, the gyrokinetic particle simulations of the Alfvé n waves in sheared cylinder have superior numerical properties compared with the fluid simulations, which suffer from numerical difficulties associated with singular mode structures.

Gyrokinetic simulations of nonlinear interactions between magnetic islands and microturbulence

“…to the above equations, we can get the linear dispersion relation of KAW as: The advantage of the conservative scheme [11,12] is that it can avoid the requirement of grid size being larger than electron skin depth by using the continuity equation to guarantee the consistency between the electron density and flow. To illustrate this, we compare the original method using kinetic markers to calculate electron density and the conservative scheme in this paper.…”

“…. To solve this, we use the conservative scheme [11,12], where the particle gathering process is only taken for the electrons pressure terms  dP e and dP e . dn e and  du e are solved from electron continuity equation and the Ampere's law for consistency.…”

“…Firstly, electron density and flow calculated from particles cannot accurately satisfy electron continuity equation due to particle noises. This inconsistency leads to an incomplete cancellation of electrostatic and inductive electric fields, and gives rise to an unphysical component in parallel electric field [11,12]. Secondly, it is difficult to analytically calculate the adiabatic response of electron in the presence of magnetic islands, where the zonal and non-zonal components of perturbed electromagnetic fields need to be defined with respect to the helical magnetic flux surface of the islands.…”

“…In this paper, we extend the conservative scheme of kinetic electrons in gyrokinetic simulation [11,12] to incorporate magnetic islands and verify this new method in the gyrokinetic toroidal code (GTC) [13][14][15]. Our conservative scheme uses electron continuity equation and the Ampere's law to evolve electron density and parallel flow, thus enforcing the consistency in the calculation of parallel electric field.…”

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

“…The so-called Ampère cancellation problem is one of the main numerical issues that has troubled primarily PIC codes (Reynders 1993;Cummings 1994). Various δf PIC schemes to address the cancellation problem have been developed and there are interesting recent advances in this area (Chen & Parker 2003;Mishchenko, Hatzky & Könies 2004;Hatzky, Könies & Mishchenko 2007;Mishchenko et al 2014;Startsev & Lee 2014;Bao, Lin & Lu 2018). Meanwhile, some continuum δf core codes avoided the cancellation problem completely (Rewoldt, Tang & Hastie 1987;Kotschenreuther et al 1995), while others had to address somewhat minor issues resulting from it (Jenko 2000;Candy & Waltz 2003).…”