Flux-coordinate Independent Approach Research Articles

Landau-fluid simulations of edge-SOL turbulence with GRILLIX

The Landau-fluid closure for parallel heat fluxes is implemented in the edge turbulence fluid code GRILLIX, replacing the previously used collisional Braginskii closure (with limiters). This extends the validity of the model toward lower collisionality, introduces non-local effects, and leads to a more realistic and self-consistent limiting of heat fluxes. Turbulence simulations comparing the Landau-fluid with the Braginskii closure in realistic divertor geometry are carried out. Clear differences between the simulations are observed, most pronounced a spurious up-down ion temperature asymmetry emerges for a strongly limited Braginskii case. For the Landau-fluid case, we demonstrate the presence and relevance of non-local heat fluxes in full-scale turbulence simulations and show that this behavior could only hardly be reproduced with simple flux-limited models. The implementation of the Landau-fluid closure within the flux-coordinate independent approach employed by GRILLIX results in a set of 3D elliptic problems, where magnetic flutter can be incorporated naturally. On reusing the existing solver in GRILLIX, only a moderate additional computational effort is necessary for the higher fidelity Landau-fluid closure.

A finite volume flux coordinate independent approach

We present a novel family of schemes as the merging between a one-dimensional advection scheme with the flux coordinate independent approach. The scheme can be used to discretize the field-aligned Navier-Stokes equations in three dimensions. Our approach consists of three major steps: (i) the formulation of the one-dimensional scheme in a locally field-aligned coordinate system, (ii) a numerical evaluation of the surface integrals over the field-aligned finite volumes and (iii) the introduction of smoothing into the numerical transformation operators to ensure stability of the resulting scheme.We study this approach at the example of a staggered finite volume scheme with a field-aligned cylinder as initial condition. We show superior stability and conservative properties over previous direct discretizations. In particular, the relative mass conservation is improved by several orders of magnitude. Without smoothing in the transformation operators the scheme is prone to oscillations in both parallel and perpendicular directions. In the presence of strong perpendicular gradients, additional parallel diffusion is needed to control spurious oscillations in the perpendicular planes.We provide parallel implementations for various platforms including GPUs freely in the C++ library Feltor

Analysis of locally-aligned and non-aligned discretisation schemes for reactor-scale tokamak edge turbulence simulations

Edge turbulence codes will play a key role in the interpretation of ITER data and for reliable predictions of EU-DEMO. At present, such codes are not yet capable of routine simulations at reactor scale, and instead focus on smaller experiments like TCV or ASDEX Upgrade. Numerical methods have to be identified that scale to reactor size, and are able to cope with the highly anisotropic turbulent structures and simultaneously with the complex magnetic geometry in the edge with X-point(s), the vessel wall and divertor targets present. Particularly for such conditions two main approaches, non-aligned discretisation schemes and a locally-aligned discretisation scheme (commonly referred to as Flux-Coordinate Independent approach (FCI)), have emerged. We analyse both schemes concerning their applicability and scalability to next generation fusion reactors. We find that the ratio of the computational cost of a non-aligned scheme compared to aligned scheme scales as ∝(R0qρs)2 for shear Alfvén dynamics, and as (R0qρs)3 for electron heat conduction, where R0 is the major radius of a tokamak, q an estimate for the safety factor at the edge and ρs is the drift scale representing the typical size of turbulent structures to be resolved. Locally-aligned schemes can therefore be considered strongly favourable for reactor scale tokamaks concerning computational performance. On the other hand, locally-aligned schemes suffer from a more complex treatment of boundary conditions, for which an immersed boundary approach (IBA) was recently proposed. We demonstrate numerically the validity of this method in combination with the FCI approach. Finally, we present a first attempt of an ITER edge turbulence simulation with the FCI code GRILLIX at realistic parameters and in realistic geometry.

The role of neutral gas in validated global edge turbulence simulations

To make predictions for and design fusion reactors, a multitude of physical processes must be considered. In the edge and scrape-off layer (SOL), turbulent fluctuations intertwine with the plasma background, which is largely determined by neutral gas, and magnetic geometry plays an important role. A diffusive neutrals model has now been implemented in the global Braginskii edge turbulence code GRILLIX. The code is based on the flux-coordinate independent (FCI) approach, which allows efficient turbulence simulations in diverted equilibria. We validate simulations across the ASDEX Upgrade edge and SOL against measurements in discharge #36190, and find much better agreement thanks to the neutrals. Disentangling the effects of the neutral gas, we find that it affects the plasma in several ways. Firstly, the ionization of neutrals modifies the radial profiles of plasma density and temperature, leading to a transition of the turbulence drive from the general ballooning type to the ion temperature gradient type. Secondly, strong poloidal asymmetries can be induced due to divertor recycling, depending on the ionization pattern. As ballooned perpendicular plasma transport is stronger at the low-field side, neutrals penetrate deeper into the plasma at the high-field side, leading to significant ionization and radiation there. With increasing divertor neutrals density, the targets cool down while plasma density increases, more strongly at the high-field side. Much of the dynamics take place directly around the X-point and along the separatrix, which can be resolved by the FCI approach. Potential remains in extending the model and the code, but our results build confidence that predictive capability is within reach for major design questions for fusion reactors, such as the near SOL fall-off length.

GENE-X: A full-f gyrokinetic turbulence code based on the flux-coordinate independent approach

Understanding and predicting plasma turbulence in the scrape-off layer of a magnetic confinement fusion device is a key open problem in modern plasma physics. The transitional region between the core and scrape-off layer poses a difficult problem for turbulence simulations. The poloidal magnetic field vanishes at the X-point of a fusion device, which introduces a coordinate singularity in the commonly used field-aligned coordinates. In the present work, we present a full-f gyrokinetic code based on a locally field-aligned coordinate system that is flux-coordinate independent and free of singularities. The coordinate system, as well as the equations and numerical methods are described. In addition, careful numerical and physical verifications in closed magnetic flux surfaces are included.

On the nature of blob propagation and generation in the large plasma device: Global GRILLIX studies

The appearance of blobs, filamentary structures featuring large intermittent perturbations, is characteristic for the scrape-off layer of magnetic fusion devices. Therefore, a global model, which does not rely on assumptions about the fluctuation level, is necessary to model blobs accurately. Whereas GRILLIX, a global 3D fluid turbulence code, is originally designed to handle complex geometries via the flux-coordinate independent approach, the analysis here focuses on a thorough verification, validation, and identification of basic phenomena in simplified slab geometry. As such the impact of the routinely employed Boussinesq approximation is studied systematically, revealing that not only the density amplitude itself matters, but also the blob regime that is also influenced by temperature. This points out that the validity of the Boussinesq approximation cannot generally be taken for granted. Furthermore, GRILLIX is validated against the large plasma device experiment and the formation of blobs is studied. A cross-phase evaluation suggests as candidates for the blob drive mechanism the rotational interchange instability.

Global turbulence simulations of the tokamak edge region with GRILLIX

Turbulent dynamics in the scrape-off layer of magnetic fusion devices is intermittent with large fluctuations in density and pressure. Therefore, a model is required that allows perturbations of similar or even larger magnitude to the time-averaged background value. The fluid-turbulence code GRILLIX is extended to such a global model, which consistently accounts for large variation in plasma parameters. Derived from the drift reduced Braginskii equations, the new GRILLIX model includes electromagnetic and electron-thermal dynamics, retains global parametric dependencies and the Boussinesq approximation is not applied. The penalization technique is combined with the flux-coordinate independent approach [F. Hariri and M. Ottaviani, Comput. Phys. Commun. 184, 2419 (2013) and A. Stegmeir et al., Comput. Phys. Commun. 198, 139 (2016)], which allows to study realistic diverted geometries with X-point(s) and general boundary contours. We characterize results from turbulence simulations and investigate the effect of geometry by comparing simulations in circular geometry with toroidal limiter against realistic diverted geometry at otherwise comparable parameters. Turbulence is found to be intermittent with relative fluctuation levels of up to 40% showing that a global description is indeed important. At the same time via direct comparison, we find that the Boussinesq approximation has only a small quantitative impact in a turbulent environment. In comparison to circular geometry, the fluctuations are reduced in diverted geometry, which is related to a different zonal flow structure. Moreover, the fluctuation level has a more complex spatial distribution in diverted geometry. Due to local magnetic shear, which differs fundamentally in circular and diverted geometries, turbulent structures become strongly distorted in the perpendicular direction and are eventually damped away toward the X-point.

GRILLIX: a 3D turbulence code based on the flux-coordinate independent approach

The GRILLIX code is presented with which plasma turbulence/transport in various geometries can be simulated in 3D. The distinguishing feature of the code is that it is based on the flux-coordinate independent approach (FCI) (Hariri and Ottaviani 2013 Comput. Phys. Commun. 184 2419; Stegmeir et al 2016 Comput. Phys. Commun. 198 139). Cylindrical or Cartesian grids are used on which perpendicular operators are discretised via standard finite difference methods and parallel operators via a field line tracing and interpolation procedure (field line map). This offers a very high flexibility with respect to geometry, especially a separatrix with X-point(s) or a magnetic axis can be treated easily in contrast to approaches which are based on field aligned coordinates and suffer from coordinate singularities. Aiming finally for simulation of edge and scrape-off layer (SOL) turbulence, an isothermal electrostatic drift-reduced Braginskii model (Zeiler et al 1997 Phys. Plasmas 4 2134) has been implemented in GRILLIX. We present the numerical approach, which is based on a toroidally staggered formulation of the FCI, we show verification of the code with the method of manufactured solutions and show a benchmark based on a TORPEX blob experiment, previously performed by several edge/SOL codes (Riva et al 2016 Plasma Phys. Control. Fusion 58 044005). Examples for slab, circular, limiter and diverted geometry are presented. Finally, the results show that the FCI approach in general and GRILLIX in particular are viable approaches in order to tackle simulation of edge/SOL turbulence in diverted geometry.

Advances in the flux-coordinate independent approach

The flux-coordinate independent approach (FCI) offers a promising solution to deal with a separatrix and X-point(s) in diverted tokamaks. Whereas the discretisation of perpendicular operators (with respect to magnetic field) is straight forward, the major complexity lies in the discretisation of parallel operators, for which field line tracing and interpolation is employed. A discrete version for the parallel diffusion operator was proposed in Stegmeir et al. (2016), which maintains the self-adjointness property on the discrete level and exhibits only very low numerical perpendicular diffusion/pollution. However, in situations where the field line map is strongly distorted this scheme revealed its limits. Moreover, the appearance of small scale corrugations deteriorated the convergence order with respect to spatial resolution (Held et al., 2016). In this paper we present an extension to the scheme where the parallel gradient is reformulated via a combination of integration and interpolation. It is shown that the resultant scheme finally combines many good numerical properties, i.e. it is self-adjoint on the discrete level, it has very low numerical perpendicular diffusion, it can cope with strongly distorted maps and exhibits optimal convergence. Another subtle issue in the FCI approach is the treatment of boundary conditions, especially where magnetic field lines intersect with material plates. We present a solution based on ghost points, whose value can be set in a flexible way according to Taylor expansion around the boundary.

Dirichlet boundary conditions for arbitrary-shaped boundaries in stellarator-like magnetic fields for the Flux-Coordinate Independent method

We present a technique for handling Dirichlet boundary conditions with the Flux Coordinate Independent (FCI) parallel derivative operator with arbitrary-shaped material geometry in general 3D magnetic fields. The FCI method constructs a finite difference scheme for ∇∥ by following field lines between poloidal planes and interpolating within planes. Doing so removes the need for field-aligned coordinate systems that suffer from singularities in the metric tensor at null points in the magnetic field (or equivalently, when q→∞). One cost of this method is that as the field lines are not on the mesh, they may leave the domain at any point between neighbouring planes, complicating the application of boundary conditions.The Leg Value Fill (LVF) boundary condition scheme presented here involves an extrapolation/interpolation of the boundary value onto the field line end point. The usual finite difference scheme can then be used unmodified. We implement the LVF scheme in BOUT++ and use the Method of Manufactured Solutions to verify the implementation in a rectangular domain, and show that it does not modify the error scaling of the finite difference scheme. The use of LVF for arbitrary wall geometry is outlined.We also demonstrate the feasibility of using the FCI approach in non-axisymmetric configurations for a simple diffusion model in a “straight stellarator” magnetic field. A Gaussian blob diffuses along the field lines, tracing out flux surfaces. Dirichlet boundary conditions impose a last closed flux surface (LCFS) that confines the density. Including a poloidal limiter moves the LCFS to a smaller radius.The expected scaling of the numerical perpendicular diffusion, which is a consequence of the FCI method, in stellarator-like geometry is recovered. A novel technique for increasing the parallel resolution during post-processing, in order to reduce artefacts in visualisations, is described.

Verification of BOUT++ by the method of manufactured solutions

BOUT++ is a software package designed for solving plasma fluid models. It has been used to simulate a wide range of plasma phenomena ranging from linear stability analysis to 3D plasma turbulence and is capable of simulating a wide range of drift-reduced plasma fluid and gyro-fluid models. A verification exercise has been performed as part of a EUROfusion Enabling Research project, to rigorously test the correctness of the algorithms implemented in BOUT++, by testing order-of-accuracy convergence rates using the Method of Manufactured Solutions (MMS). We present tests of individual components including time-integration and advection schemes, non-orthogonal toroidal field-aligned coordinate systems and the shifted metric procedure which is used to handle highly sheared grids. The flux coordinate independent approach to differencing along magnetic field-lines has been implemented in BOUT++ and is here verified using the MMS in a sheared slab configuration. Finally, we show tests of three complete models: 2-field Hasegawa-Wakatani in 2D slab, 3-field reduced magnetohydrodynamics (MHD) in 3D field-aligned toroidal coordinates, and 5-field reduced MHD in slab geometry.

Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids

We present and discuss three discontinuous Galerkin (dG) discretizations for the anisotropic heat conduction equation on non-aligned cylindrical grids. Our non-aligned scheme relies on a self-adjoint local dG (LDG) discretization of the elliptic operator. It conserves the energy exactly and converges with arbitrary order. The pollution by numerical perpendicular heat fluxes decreases with superconvergence rates. We compare this scheme with aligned schemes that are based on the flux-coordinate independent approach for the discretization of parallel derivatives. Here, the dG method provides the necessary interpolation. The first aligned discretization can be used in an explicit time-integrator. However, the scheme violates conservation of energy and shows up stagnating convergence rates for very high resolutions. We overcome this partly by using the adjoint of the parallel derivative operator to construct a second self-adjoint aligned scheme. This scheme preserves energy, but reveals unphysical oscillations in the numerical tests, which result in a decreased order of convergence. Both aligned schemes exhibit low numerical heat fluxes into the perpendicular direction and are superior for flute-modes with finite parallel gradients. We build our argumentation on various numerical experiments on all three schemes for a general axisymmetric magnetic field, which is closed by a comparison to the aligned finite difference (FD) schemes of Stegmeir et al. (2014) and Stegmeir et al. (submitted for publication).

Plasma turbulence simulations with X-points using the flux-coordinate independent approach

In this work, the flux-coordinate independent (FCI) approach to plasma turbulence simulations is formulated for the case of generic, static magnetic fields, including those possessing stochastic field lines. It is then demonstrated that FCI is applicable to nonlinear turbulent problems with and without X-point geometry. In particular, by means of simulations with the FENICIA code, it is shown that the standard features of ion temperature gradient (ITG) modes are recovered with reduced toroidal resolution. Finally, the impact of ITG turbulent transport on the temperature profile is studied under the influence of a static island, with ITER-like parameters. Results show the wide range of applicability of the method.

The flux-coordinate independent approach applied to X-point geometries

A Flux-Coordinate Independent (FCI) approach for anisotropic systems, not based on magnetic flux coordinates, has been introduced in Hariri and Ottaviani [Comput. Phys. Commun. 184, 2419 (2013)]. In this paper, we show that the approach can tackle magnetic configurations including X-points. Using the code FENICIA, an equilibrium with a magnetic island has been used to show the robustness of the FCI approach to cases in which a magnetic separatrix is present in the system, either by design or as a consequence of instabilities. Numerical results are in good agreement with the analytic solutions of the sound-wave propagation problem. Conservation properties are verified. Finally, the critical gain of the FCI approach in situations including the magnetic separatrix with an X-point is demonstrated by a fast convergence of the code with the numerical resolution in the direction of symmetry. The results highlighted in this paper show that the FCI approach can efficiently deal with X-point geometries.