The field line map approach for simulations of magnetically confined plasmas
Introduction
The modelling of the edge and scrape-off layer of tokamaks is in many ways more difficult than the core [1]. However, this region is of high importance since it may have a significant influence also on the core region, e.g. plasma and impurity densities are often largely set by edge conditions and in important operating conditions the edge plays a key role in the improvement of confinement [2]. Moreover, a prediction of heat fluxes on the divertor plates for future tokamaks is of high importance from the engineering point of view [3], [4].
A major complexity at the modelling is introduced by the complex geometry of diverted machines. Field-aligned coordinates are often employed in simulations, since they allow for a convenient way to computationally exploit the characteristic flute mode property of the structures. However, field-aligned coordinates become singular on the separatrix and simulations cannot span a domain across it. Any set of poloidal and toroidal straight field line angles has to satisfy along magnetic field lines the condition [5]: At the X-point the poloidal magnetic field vanishes and therefore on the separatrix the safety factor diverges. The straight field line angles, which have to satisfy condition (1), cannot span the whole separatrix. As exemplified in Fig. 1(a) the contours of are sucked into the X-point (see also [6]).
Also often employed are coordinates, where the field-alignment property is given up, but which is still aligned with flux surfaces, i.e. is retained as a radial coordinate. However, flux-aligned coordinate systems are still singular on points, where , i.e. at O-points and X-points [7]. Although these singularities can be cured numerically, O- and X-points remain somewhat exceptional points of the numerical grid (see Fig. 1(b)). This could in the worst case even lead to numerical artefacts. Moreover, structured flux-aligned meshes have a huge resolution imbalance within the poloidal plane due to the flux expansion near the X-point [8], [9]. Simulations might suffer from this as perpendicular operators arising in practically any plasma model (e.g. ) act approximately isotropically within poloidal planes of tokamaks.
The field line map approach is presented in Section 2. Although field/flux-aligned coordinates may become singular, the operators appearing in plasma models are still well defined, of course. The idea behind the approach is that the flute mode property can also be exploited at the discretisation step without any need for construction of a field/flux-aligned coordinate system. The approach consists of a cylindrical or Cartesian grid with a field line following discretisation for parallel operators. A separatrix can be treated as well as a magnetic axis, where X/O-points are treated like any other grid point and no resolution imbalance arises. The result is very similar to the flux-coordinate independent (FCI) [10], [11], [12] approach, and the derivation is here sketched without any reference to field- or flux-aligned coordinates.
As the discretisation of perpendicular operators in the field line map approach is straight forward, the main emphasis in this paper is on the discretisation of parallel operators. A hyperbolic problem has already been considered in [10], [11], and this work is mainly devoted to the discretisation of the parallel diffusion operator in Section 2.4. Since an interpolation or integration is involved at the discretisation, parallel operators exhibit also numerical perpendicular ‘diffusion’. Motivated by previous work from [13], [14], a numerical scheme is developed which exhibits very low numerical diffusion. The discussion extends previous work from [15]. Several model problems are also discussed in the Appendix.
The developed numerical methods are implemented in the new code GRILLIX. In Section 3 extensive benchmarks performed with GRILLIX are presented, which show the validity of the field line map approach in general and GRILLIX in particular.
The paper is concluded with a summary and final remarks in Section 4.
Section snippets
Overview
The field line map approach is described in the following for the case of a toroidal configuration , but it can be applied also to axial periodic configurations , where is the axial coordinate. The transition should be trivial.
For a tokamak a cylindrical coordinate system is well defined everywhere, except for the toroidal symmetry axis which is outside the domain of interest. We span the simulation domain with a cylindrical grid . (For axial configurations a Cartesian
Benchmarks and examples
The numerical methods presented in Section 2 are implemented in the new code GRILLIX. In this section we present benchmarks, which shall show the validity of the field line map approach in general and GRILLIX in particular. As a model problem the parallel diffusion equation is considered. Space scales are normalised to in toroidal geometries respectively in axial geometries, where is the axial periodicity length. Time is measured in respectively .
Conclusion and final remarks
In the flux-coordinate independent or field line map approach field/flux-aligned coordinates, which become singular on the separatrix/X-point, are avoided. The concept is based on a cylindrical grid, which is sparse in the toroidal direction, and a field line following discretisation for parallel operators to exploit the characteristic flute mode property of the solutions. The discretisation of perpendicular operators is straight forward and simple, whereas in the discretisation of parallel
Acknowledgements
The authors want to thank Markus Held and Matthias Wiesenberger from University of Innsbruck for contributing to this work with fruitful comments.
The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 277870.
A part of this work was carried out using the HELIOS supercomputer system at Computational Simulation Centre of International Fusion Energy Research Centre
References (23)
- et al.
Fusion Eng. Des.
(1995) Phys. Lett. A
(2011)- et al.
Comput. Phys. Comm.
(2013) - et al.
J. Comput. Phys.
(2005) - et al.
J. Comput. Phys.
(2007) - et al.
J. Comput. Phys.
(1995) Comput. Geosci.
(1989)- et al.
Tokamak experiments
- et al.
Nucl. Fusion
(1990) - K. Dietz, P.-H. Rebut, The ITER divertor, in: 15th IEEE/NPSS Symposium on Fusion Engineering, vol. 2, 1993, p....
Cited by (36)
Global fluid simulations of edge plasma turbulence in tokamaks: a review
2024, Computers and FluidsAnalysis of locally-aligned and non-aligned discretisation schemes for reactor-scale tokamak edge turbulence simulations
2023, Computer Physics CommunicationsFilamentary transport in global edge-SOL simulations of ASDEX Upgrade
2023, Nuclear Materials and EnergyA partially mesh-free scheme for representing anisotropic spatial variations along field lines: Conservation, quadrature, and the delta property
2023, Computer Physics CommunicationsGENE-X: A full-f gyrokinetic turbulence code based on the flux-coordinate independent approach
2021, Computer Physics CommunicationsCitation Excerpt :GENE uses a combination of a shock-stable finite volume and a finite difference scheme, and GYSELA a semi-Lagrangian method. To allow for continuum turbulence simulations in X-point geometries, locally field-aligned methods based on the flux-coordinate independent approach (FCI) [26,15] have been developed and implemented in fluid codes [15,27,18,28]. Along with fully non-aligned methods [29], FCI methods have been established as an effective and efficient numerical method for studying edge and SOL turbulence in the fluid community.
A high-order non field-aligned approach for the discretization of strongly anisotropic diffusion operators in magnetic fusion
2020, Computer Physics CommunicationsCitation Excerpt :Schemes based on non-aligned discretizations [14,15] improve the computation of the fluxes by a proper choice of the stencil with respect to a naive discretization, or rely on high-order finite-difference schemes to reduce the numerical diffusion, such as in [16]. On the other hand, schemes based on aligned discretizations [17–20] employ interpolations on the magnetic field lines to perform finite-difference discretizations along the anisotropy directions. In the perpendicular plane, such aligned discretizations seem so far to be limited to structured meshes for simplicity and efficiency of implementation.