Roman Hatzky

We introduce a smooth B-spline discretization in polar coordinates on the unit disc that corrects the loss of regularity present at the origin caused by the coordinate singularity in standard tensor-product B-spline formulations. The method constructs “smooth polar splines” via a Galerkin projection of harmonic polar functions S-m (r, θ) ∶= rlsin(mθ) and Sm (r, θ) ∶= rl cos(mθ), derived from the polar representation of Cartesian monomials, onto the central tensor-product B-spline basis in the innermost radial region. The radial component reproduces rl exactly for 0 ≤ l ≤ p, where p is the B-spline degree, satisfying the near-origin regularity condition. However, exact compatibility with C∞ -regularity at the origin is recovered only in the limit Δθ → 0, when the angular component resolves all angular harmonics accurately. The smooth polar splines are linear combinations of standard tensor-product B-splines and lie in the same function space, enabling mapping between the C∞-regular subspace and the original discretization space via an exact prolongation operator and a corresponding restriction operator acting on the discrete variables. They match standard tensor-product B-splines away from the origin, preserve orthogonality among the newly constructed origin-centered basis functions, and maintain local support and sparse matrices. This smoothness and locality improve the conditioning of mass and stiffness matrices, conserve charge, and reduce statistical errors in particle-in-cell simulations near the origin, while eliminating spurious eigenvalues in eigenvalue problems. The approach provides a robust, high-order, and efficient adaptation of tensor-product B-splines for polar coordinates in physics simulations.

In this work, we present a systematic analysis of the mixed variable-pullback scheme and the pure symplectic v∥ scheme used in electromagnetic gyrokinetic particle simulations. The non-ideal Ohm's law is integrated in the mixed variable-pullback scheme for the gyrokinetic particle simulations. This scheme captures the evolution of the symplectic solution of the gyrokinetic model accurately not only in the MHD limit but also in the electrostatic limit. This scheme also provides a pure symplectic (v∥) scheme for electromagnetic gyrokinetic particle simulations without causing the traditional cancelation problem in the pure Hamiltonian scheme. Various mixed variable schemes have been comprehensively analyzed for the 1D shear Alfvèn wave problem with kinetic electrons, with the connection to the traditional pure Hamiltonian scheme and the symplectic scheme. It is demonstrated that the pure v∥ form with the non-ideal Ohm's law has comparable performance to the widely used mixed variable-pullback scheme with the ideal Ohm's law. The mixed variable-pullback scheme without Ohm's law is also proposed as a feasible improvement of the traditional pure Hamiltonian scheme with minimum modification and considerable performance improvement in terms of a marker number reduction.

The use of gyrokinetics, wherein phase-space coordinate transformations result in a phase-space dimensionality reduction as well as the removal of fast time scales, has enabled the simulation of microturbulence in fusion devices. The state-of-the-art gyrokinetic models used in practice are parallel-only models wherein the perpendicular part of the vector potential is neglected. Such models are inherently not gauge-invariant. We generalise the work of Burby & Brizard (2019 Phys. Lett. A vol. 383, no. 18, pp. 2172–2175) by deriving a sufficient condition on the gyrocentre coordinate transformation that ensures gauge invariance. This leads to a parametrised family of gyrokinetic models for which we motivate a specific choice of parameters that results in the smallest gyrocentre coordinate transformation for which the resulting gyrokinetic model is consistent, gyro-phase independent, gauge-invariant and has an invariant magnetic moment. Due to gauge invariance, this model can be expressed directly in terms of the electromagnetic fields rather than the potentials, and the gyrokinetic model thereby results in the macroscopic Maxwell’s equations. For the linearised model, it is demonstrated that the shear and compressional Alfvén waves are present with the correct frequencies. The fast compressional Alfvén wave can be removed by making use of a Darwin approximation. This approximation retains the gauge invariance of the proposed model.

This paper is a written summary of an overview oral presentation given at the 1st Spanish Fusion High Performance Computer (HPC) Workshop that took place on the 27 November 2020 as an online event. Given that over the next few years ITER ²⁴ 24 ITER (‘The Way’ in Latin) is the world’s largest tokamak under construction in the south of France: a magnetic fusion device that has been designed to prove the feasibility of fusion as a large-scale and carbon-free source of energy (https://www.iter.org/). will move to its operation phase and the European-DEMO design will be significantly advanced, the EUROfusion consortium has initiated a coordination effort in theory and advanced simulation to address some of the challenges of the fusion research in Horizon EUROPE (2021–2027), i.e. the next EU Framework Programme for Research and Technological Development. This initiative has been called E-TASC, which stands for EUROfusion-Theory and Advanced Simulation Coordination. The general and guiding principles of E-TASC are summarized in this paper. In addition, an overview of the scientific results obtained in the pilot phase (2019–2020) of E-TASC are provided while highlighting the importance of the required progress in computational methods and HPC techniques. In the initial phase, five pilot theory and simulation tasks were initiated: towards a validated predictive capability of the low to high transition and pedestal physics; runaway electrons in tokamak disruptions in the presence of massive material injection; fast code for the calculation of neoclassical toroidal viscosity in stellarators and tokamaks; development of a neutral gas kinetics modular code; European edge and boundary code for reactor-relevant devices. In this paper, we report on recent progress made by each of these projects.

Magnetic reconnection in W7-X is studied by means of global numerical simulations in a series of models of increasing complexity. The magnetic geometry ranges from that of a cylinder to the full three-dimensional field of W7-X, and the equations solved range from ideal magnetohydrodynamics (MHD) to gyrokinetics. We simulate plasmas from the first operation phase with electron cyclotron current drive (ECCD). These are characterized by an equilibrium magnetic field featuring an ECCD-distorted ‘humped’ profile of the rotational transformι, withι= 1 in two radial locations. Such plasmas generally show sawtooth activity, hence motivating the present study. We pay particular attention to the role of equilibrium current density gradients in the destabilization of reconnecting modes. When the equilibrium temperature and density gradients are artificially suppressed (to eliminate the pressure gradient drive), the perturbed electrostatic potential is radially localized between the locations at whichι= 1. This is shown with a purely collisionless gyrokinetic model, in cylindrical geometry. In the real toroidal geometry of W7-X, for a non-ideal MHD model including a uniform resistivity, electron inertia and (numerical) viscosity, the same qualitative behaviour is observed. In particular, even if a resonant (m,n) = (1, −1) perturbation is initialized, the most unstable mode is the (m,n) = (−4, 4), wheremandnare the poloidal and toroidal mode numbers, respectively. Other modes are destabilized due to geometric coupling. The growth rate of this instability scales asη^1/3, whereηis the plasma resistivity, thus suggesting that ECCD drives ideal MHD stable W7-X plasmas towards non-ideal marginality. An ideal magnetohydrodynamic analysis confirms the result. A fluid-kinetic hybrid version of theEUTERPEcode shows that gyrokinetic ions have a stabilizing effect on these modes. For W7-X relevant collisionalities, the growth rate scales linearly with the electron skin depth,d_e. Implications of our results for sawtoothing W7-X operation are discussed.