Constructing Field Aligned Coordinate Systems for Gyrokinetic Simulations of Tokamaks in X-point Geometries

TL;DR

This work addresses the challenge of applying field-aligned coordinates to tokamak geometries with magnetic X-points, where the poloidal field vanishes and the coordinate Jacobian diverges. It introduces a multi-block field-aligned grid anchored in Clebsch coordinates $(\psi,\alpha,\theta)$ with a carefully chosen Jacobian $J_c$ that aligns the magnetic field with the $z^3$ direction and avoids computing geometric quantities at the X-point, enabling stable 2D axisymmetric GK simulations. A Discontinuous Galerkin discretization is used to advance the GK equation, with interior and surface quadrature points ensuring geometric consistency across block interfaces and excluding the X-point from flux evaluations. The grid is generated from G-EQDSK equilibrium data, mapping a rectangular computational grid to physical space, and the approach is demonstrated in a STEP geometry, showing well-behaved core-SOL dynamics near the X-point and setting the stage for future 3D extensions and integrating more physics such as turbulenc e and neutrals.

Abstract

Structures in tokamak plasmas are elongated along the direction of the magnetic field and short in the directions perpendicular to the magnetic field. Many tokamak simulation codes take advantage of this by using a field aligned coordinate system. However, field aligned coordinate systems have a coordinate singularity at magnetic X-points where the poloidal magnetic field vanishes, which makes it difficult to use field aligned coordinate systems when simulating the core and scrape-off layer (SOL) simultaneously. Here we present an algorithm for computing geometric quantities in a standard field aligned coordinate system that avoids the singularity and allows one to conduct 2D axisymmetric simulations in X-point geometries. We demonstrate the efficacy of this algorithm with an example simulation of the Spherical Tokamak for Energy Production (STEP).

Figures (9)

• Figure 1: Schematic for field line tracing in a double null (a) and single null (b) configuration.
• Figure 2: In (a) we show the interior, surface, and corner points on the unit cell. In (b) we show these points mapped to the physical domain for cells abutting the X-point. The cell in physical space is not rectuangluar, allowing for an accurate representation of the flux-surface geometry. The surface and interior nodes used for the evaluation of geometric quantities do not lie directly on the X-point and are thus well defined.
• Figure 3: Block layout and grid for the Spherical Tokamak for Energy Production in a double null configuration with different colors indicating different blocks and a number 1-12 labeling each block. The full grid is shown in (a), (b) shows a close-up of the grid near the upper X-point, and (c) shows a close-up of the grid near the upper outer divertor plate (red).
• Figure 4: Grid for ASDEX-Upgrade in a single null configuration with different colors indicating different blocks and a number 1-6 labeling each block. The full grid is shown in (a) and (b) shows a close-up of the grid near the X-point.
• Figure 5: Simulation results from a 2D, axisymmetric simulation of the Spherical Tokamak for Energy Production. The poloidal projection of the electron density and temperature are shown in (a) and (b) respectively. A close-up of the electron density is shown in (c) and a close-up of the electron temperature is shown in (d).