A High-order piecewise field-aligned triangular finite element method for electromagnetic gyrokinetic particle simulations of tokamak plasmas with open field lines

TL;DR

The paper tackles the challenge of global electromagnetic gyrokinetic simulations in tokamaks with open field lines by developing a high-order piecewise field-aligned triangular finite element method implemented in cylindrical coordinates within the TRIMEG-C1 framework. This approach preserves field-line continuity while mitigating grid distortion, enabling whole-volume simulations that handle core, separatrix, and scrape-off-layer regions with either δf or full-f models. The authors introduce generalized mixed-variable pullback schemes and demonstrate the method on the TCV-X21 case, capturing ITG and KBM physics and showing ITG–KBM transitions as β is varied, as well as nonlinear multi-n mode turbulence without toroidal filtering. The work provides a robust, flexible tool for predictive, realistic tokamak simulations and sets the stage for including finite Larmor radius effects and detailed edge physics for validation against experiments.

Abstract

A high-order piecewise field-aligned triangular finite element method is developed and implemented for global electromagnetic gyrokinetic particle-in-cell simulations of tokamak plasmas with open field lines. The approach combines locally field-aligned finite element basis functions with unstructured $C^{1}$ triangular meshes in cylindrical coordinates, enabling whole-volume simulations with substantially reduced computational effort, while avoiding the grid distortion associated with globally field-aligned coordinates and the associated singularity at the separatrix of diverted plasmas. The formulation is compatible with both $δf$ and full-$f$ models and employs mixed-variable representations, along with a generalized pullback scheme, to control numerical cancellation in electromagnetic simulations. The method is implemented in the TRIMEG-C1 code and demonstrated using linear and nonlinear electromagnetic simulations of the TCV-X21 configuration. The results indicate that the approach accurately captures the key features of electromagnetic ion-temperature-gradient and kinetic ballooning mode physics, including the separatrix regions in the simulation, thereby providing a robust framework for whole-volume electromagnetic gyrokinetic simulations in realistic tokamak geometries.

Figures (8)

• Figure 1: The grid of the piecewise field-aligned coordinates $(R_{\rm FA},Z_{\rm FA})$ (Section \ref{['subsec:PFAFEM']}).
• Figure 2: The poloidal flux function $2\pi\psi_{\mathrm p}$ (upper left); the simulation domain with triangular meshes (upper right); the safety factor profile (lower left); the density and temperature ($n, T$) profiles along $R$ at $Z=Z_{\rm axis}$ for simulations of the core instability (lower middle) and the edge instability (lower right). The density and temperature are normalized to $n_{\rm ref}$ and $T_{\rm ref}$, respectively (Section \ref{['subsec:parameters']}).
• Figure 3: The 2D structures of $\delta\phi$ and $\delta A_\|$ for the $n=10$ ITG mode (Section \ref{['subsec:tcv_em_edge']}). Top row: case 1a; bottom row: case 1b.
• Figure 4: The growth rate (left) and the ratio of the field energy (right) across the spectrum of toroidal mode number $n$ for instabilities in the core (case 1a) and near the separatrix (case 1b) (Section \ref{['subsec:tcv_em_edge']}).
• Figure 5: The growth rate (left) and the ratio of the field energy (right) for different values of $\beta_{\rm ref}$ ($\beta$ scan based on case 1a) for the $n=10$ harmonic. (Section \ref{['subsec:tcv_em_beta']}).