The field line map approach for simulations of magnetically confined plasmas

TL;DR

This paper introduces a novel field line map simulation approach for tokamak plasmas that uses a cylindrical grid with finite difference methods, enabling accurate modeling of edge regions without singularities at the separatrix.

Contribution

The paper presents a new discretisation scheme for parallel diffusion operators based on the support operator method, implemented in the GRILLIX code, avoiding flux-aligned grids and handling the separatrix effectively.

Findings

The scheme exhibits very low numerical perpendicular diffusion.

Extensive benchmarks confirm the approach's validity.

GRILLIX successfully simulates edge and scrape-off layer regions.

Abstract

In the presented field line map approach the simulation domain of a tokamak is covered with a cylindrical grid, which is Cartesian within poloidal planes. Standard finite-difference methods can be used for the discretisation of perpendicular (w.r.t.~magnetic field lines) operators. The characteristic flute mode property $\left(k_{\parallel}\ll k_{\perp}\right)$ of structures is exploited computationally by a grid sparsification in the toroidal direction. A field line following discretisation of parallel operators is then required, which is achieved via a finite difference along magnetic field lines. This includes field line tracing and interpolation or integration. The main emphasis of this paper is on the discretisation of the parallel diffusion operator. Based on the support operator method a scheme is constructed which exhibits only very low numerical perpendicular diffusion. The schemes are implemented in the new code GRILLIX, and extensive benchmarks are presented which show the validity of the approach in general and GRILLIX in particular. The main advantage of the approach is that it does not rely on field/flux-aligned, which become singular on the separatrix/X-point. Most tokamaks are based on the divertor concept, and the numerical treatment of the separatrix is therefore of importance for simulations of the edge and scrape-off layer.