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We propose a dynamical low-rank algorithm for a gyrokinetic model that is used to describe strongly magnetized plasmas. The low-rank approximation is based on a decomposition into variables parallel and perpendicular to the magnetic field, as suggested by the physics of the underlying problem. We show that the resulting scheme exactly recovers the dispersion relation even with rank 1. We then perform a simulation of kinetic shear Alfvén waves and show that using the proposed dynamical low-rank algorithm a drastic reduction (multiple orders of magnitude) in both computational time and memory consumption can be achieved. We also compare the performance of robust first and second-order projector splitting, BUG (also called unconventional), and augmented BUG integrators as well as a FFT-based spectral and Lax–Wendroff discretization. • First dynamical low-rank framework for strongly magnetized plasmas. • Drastic reduction in the memory and computational cost due to a decomposition motivated by the underlying physical problem. • We show that exact dispersion relation is recovered with rank 1. • We provide a comparison of novel robust integrators and two space/time discretization strategies. [ABSTRACT FROM AUTHOR]

A novel approach to global gyrokinetic simulation is implemented in the flux-tube code stella. This is done by using a subsidiary expansion of the gyrokinetic equation in the perpendicular scale length of the turbulence, originally derived by Parra and Barnes (2015) [23] , which allows the use of Fourier basis functions while enabling the effect of radial profile variation to be included in a perturbative way. Radial variation of the magnetic geometry is included by utilizing a global extension of the Grad-Shafranov equation and the Miller equilibrium equations which is obtained through Taylor expansion. Radial boundary conditions that employ multiple flux-tube simulations are also developed, serving as a more physically motivated replacement to the conventional Dirichlet radial boundary conditions that are used in global simulation. It is shown that these new boundary conditions eliminate much of the numerical artefacts generated near the radial boundary when expressing a non-periodic function using a spectral basis. We then benchmark the new approach both linearly and non-linearly using a number of standard test cases. • New radial boundary conditions coupling multiple flux-tube simulations are developed. • A subsidiary expansion of the gyrokinetic and quasineutrality equations is performed. • Global physics are included as a next-order effect in the gyrokinetic expansion. • Profile variation of the gyroaveraging operator is included, a novel feature for our approach. • Magnetic profile variation is given by an expansion of the Grad-Shafranov equation. [ABSTRACT FROM AUTHOR]

A new hybrid kinetic electron model is developed for electrostatic full- f gyrokinetic simulations of the ion temperature gradient driven trapped electron mode (ITG-TEM) turbulence at the ion scale. In the model, a full kinetic electron model is applied to the full- f gyrokinetic equation, the multi-species linear Fokker–Planck collision operator, and an axisymmetric part of the gyrokinetic Poisson equation, while in a non-axisymmetric part of the gyrokinetic Poisson equation, turbulent fluctuations are determined only by kinetic trapped electrons responses. By using this approach, the so-called ω H mode is avoided with keeping important physics such as the ITG-TEM, the neoclassical transport, the ambipolar condition, and particle trapping and detrapping processes. The model enables full- f gyrokinetic simulations of ITG-TEM turbulence with a reasonable computational cost. Comparisons between flux driven ITG turbulence simulations with kinetic and adiabatic electrons are presented. Although the similar ion temperature gradients with nonlinear upshift from linear critical gradients are sustained in quasi-steady states, parallel flows and radial electric fields are qualitatively different with kinetic electrons. [ABSTRACT FROM AUTHOR]

Finite-grid (aliasing) instabilities are known to place severe limitations on momentum-conserving particle-in-cell (PIC) methods applied to models including charge separation effects by requiring the resolution of the Debye length. Gyrokinetic models, on the other hand, generally enforce quasi-neutrality thereby removing the Debye length analytically. Recent studies with momentum-conserving PIC applied to gyrokinetic models, however, show that a manifestation of this instability exists in certain physical parameter regimes for arbitrary spatial resolution. In this paper, we show that a simple reformulation of the discrete equations, making use of a co-located discretization of the continuity equation, eliminates this instability for all practical purposes. We perform numerical dispersion analyses for both the original and reformulated schemes, including the effects of finite mean parallel velocity and finite β. We demonstrate that the reformulated scheme is numerically stable for stationary plasmas at any spatial resolution and for plasmas in which the electron mean parallel velocity is smaller than the electron thermal velocity (generally ensured by ambipolarity in plasmas of interest). This reformulation may be particularly useful for codes with complicated meshes, where high-order shape functions or energy-conserving schemes are difficult to implement. The analysis presented here may also be useful for explaining the success of previously considered methods to remove particle instabilities, such as the split-weight scheme. • A detailed numerical analysis for finite-grid instabilities in gyrokinetic particle-in-cell simulations is performed. • A reformulation of the discrete equations is proposed, which eliminates finite-grid instabilities for all practical purposes. • The simplicity of the proposed reformulation may be particularly useful when complicated meshes are involved. • The analysis presented may be useful for understanding the success of certain previously proposed schemes. [ABSTRACT FROM AUTHOR]

In this work, we report the development of a new gyrokinetic code for full-f simulation of electrostatic turbulence in general tokamak geometry. Backward semi-Lagrangian scheme is employed for noise-free simulation of the gyrokinetic Vlasov equation with finite Larmor radius effects. Grid systems and numerical interpolations are implemented to deal with arbitrary equilibrium information given in a GEQDSK format file. In particular, we introduce an adaptive interpolation technique for fluctuating quantities, which have elongated structures along equilibrium magnetic fields. This field-aligned interpolation can reduce the required number of grid points to represent the fluctuating quantities. Also, it is shown that the new interpolation allows us to choose bigger time sizes with better simulation accuracy. Several benchmark simulation results are presented for comparison with previously known cases. It is demonstrated that the new code can reproduce the well known results of zonal flow and linear ITG instabilities in concentric circular equilibrium. It is also shown that the code can capture the effects of plasma shaping on the zonal flow and ITG instabilities, and the stabilization effects of the shaping result in significant up-shifts of the threshold of ion temperature gradient for ITG in nonlinear simulation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present two new codes devoted to the study of ion temperature gradient (ITG) driven plasma turbulence in cylindrical geometry using a drift-kinetic multi-water-bag model for ion dynamics. Both codes were developed to complement the Runge–Kutta semi-lagrangian multi-water-bag code GMWB3D-SLC described in [1]. The CYLGYR code is an eigenvalue solver performing linear stability analysis from given mean radial profiles. It features three resolution schemes and three parallel velocity response models (fluid, multi-water-bag, continuous Maxwellian). The QUALIMUWABA quasilinear code is an initial value code allowing the study of zonal flow influence on drift-waves dynamics. Cross-validation test performed between the three codes show good agreement on both temporal and spatial characteristics of unstable modes in the linear growth phase. [Copyright &y& Elsevier]

Abstract: In this work, a Fourier solver [B.F. McMillan, S. Jolliet, A. Bottino, P. Angelino, T.M. Tran, L. Villard, Comp. Phys. Commun. 181 (2010) 715] is implemented in the global Eulerian gyrokinetic code GT5D [Y. Idomura, H. Urano, N. Aiba, S. Tokuda, Nucl. Fusion 49 (2009) 065029] and in the global Particle-In-Cell code ORB5 [S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T.M. Tran, B.F. McMillan, O. Sauter, K. Appert, Y. Idomura, L. Villard, Comp. Phys. Commun. 177 (2007) 409] in order to reduce the memory of the matrix associated with the field equation. This scheme is verified with linear and nonlinear simulations of turbulence. It is demonstrated that the straight-field-line angle is the coordinate that optimizes the Fourier solver, that both linear and nonlinear turbulent states are unaffected by the parallel filtering, and that the k ∥ spectrum is independent of plasma size at fixed normalized poloidal wave number. [Copyright &y& Elsevier]

Abstract: The understanding and prediction of transport due to plasma microturbulence is a key open problem in modern plasma physics, and a grand challenge for fusion energy research. Ab initio simulations of such small-scale, low-frequency turbulence are to be based on the gyrokinetic equations, a set of nonlinear integro-differential equations in reduced (five-dimensional) phase space. In the present paper, the extension of the well-established and widely used gyrokinetic code GENE [F. Jenko, W. Dorland, M. Kotschenreuther, B.N. Rogers, Electron temperature gradient driven turbulence, Phys. Plasmas 7 (2000) 1904–1910] from a radially local to a radially global (nonlocal) version is described. The necessary modifications of both the basic equations and the employed numerical methods are detailed, including, e.g., the change from spectral methods to finite difference and interpolation techniques in the radial direction and the implementation of sources and sinks. In addition, code verification studies and benchmarks are presented. [Copyright &y& Elsevier]

Abstract: Plasmas in modern tokamak experiments contain a significant fraction of impurity ion species in addition to main deuterium background. A new unlike-particle collision operator for δf particle simulation has been developed to self-consistently study the non-local effects of impurities on neoclassical transport in toroidal plasmas. A new algorithm for simulation of cross-collisions between different ion species includes test-particle and conserving field-particle operators. The field-particle operator is designed to enforce conservation of number, momentum and energy. It was shown that the new operator correctly simulates the thermal equilibration of different plasma components. It was verified that the ambipolar radial electric field reaches steady state when the total radial guiding center particle current vanishes. [Copyright &y& Elsevier]

Abstract: A linear kinetic stability code for tokamak plasmas: AEGIS-K (Adaptive EiGenfunction Independent Solutions-Kinetic), is described. The AEGIS-K code is based on the newly developed gyrokinetic theory [L.J. Zheng, M.T. Kotschenreuther, J.W. Van Dam, Phys. Plasmas 14 (2007) 072505]. The success in recovering the ideal magnetohydrodynamics (MHD) from this newly developed gyrokinetic theory in the proper limit leads the AEGIS-K code to be featured by being fully kinetic in essence but hybrid in appearance. The radial adaptive shooting scheme based on the method of the independent solution decomposition in the MHD AEGIS code [L.J. Zheng, M.T. Kotschenreuther, J. Comp. Phys. 211 (2006) 748] is extended to the kinetic calculation. A numerical method is developed to solve the gyrokinetic equation of lowest order for the response to the independent solutions of the electromagnetic perturbations, with the quasineutrality condition taken into account. A transform method is implemented to allow the pre-computed Z-function (i.e., the plasma dispersion function) to be used to reduce the integration dimension in the moment calculation and to assure the numerical accuracy in determining the wave–particle resonance effects. Periodic boundary condition along the whole banana orbit is introduced to treat the trapped particles, in contrast to the usual reflection symmetry conditions at the banana tips. Due to the adaptive feature, the AEGIS-K code is able to resolve the coupling between the kinetic resonances and the shear Alfvén continuum damping. Application of the AEGIS-K code to compute the resistive wall modes in ITER is discussed. [Copyright &y& Elsevier]

Abstract: In a plasma with a population of super-thermal particles generated by heating or fusion processes, kinetic effects can lead to the additional destabilisation of MHD modes or even to additional energetic particle modes. In order to describe these modes, a new linear gyrokinetic MHD code has been developed and tested, LIGKA (linear gyrokinetic shear Alfvén physics) [Ph. Lauber, Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks, Ph.D. Thesis, TU München, 2003; Ph. Lauber, S. Günter, S.D. Pinches, Phys. Plasmas 12 (2005) 122501], based on a gyrokinetic model [H. Qin, Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks, Ph.D. Thesis, Princeton University, 1998]. A finite Larmor radius expansion together with the construction of some fluid moments and specification to the shear Alfvén regime results in a self-consistent, electromagnetic, non-perturbative model, that allows not only for growing or damped eigenvalues but also for a change in mode-structure of the magnetic perturbation due to the energetic particles and background kinetic effects. Compared to previous implementations [H. Qin, mentioned above], this model is coded in a more general and comprehensive way. LIGKA uses a Fourier decomposition in the poloidal coordinate and a finite element discretisation in the radial direction. Both analytical and numerical equilibria can be treated. Integration over the unperturbed particle orbits is performed with the drift-kinetic HAGIS code [S.D. Pinches, Ph.D. Thesis, The University of Nottingham, 1996; S.D. Pinches et al., CPC 111 (1998) 131] which accurately describes the particles’ trajectories. This allows finite-banana-width effects to be implemented in a rigorous way since the linear formulation of the model allows the exchange of the unperturbed orbit integration and the discretisation of the perturbed potentials in the radial direction. Successful benchmarks for toroidal Alfvén eigenmodes (TAEs) and kinetic Alfvén waves (KAWs) with analytical results, ideal MHD codes, drift-kinetic codes and other codes based on kinetic models are reported. [Copyright &y& Elsevier]

• Development and verification of gyrokinetic plasma turbulence code GENE-3D. • GENE-3D is able to consider plasma turbulence in three-dimensional magnetic equilibria. • First code to consider multiple gyrokinetic species (incl. electrons) in a 3D geometry. • GENE-3D is massively parallelized suitable for present day supercomputers. • First turbulence simulation results of the Wendelstein 7-X stellarator are presented. A detailed description of GENE-3D, a newly developed global stellarator version of the well established gyrokinetic turbulence code GENE, is provided – along with some initial simulation results. First, the underlying gyrokinetic equations and the use of field-aligned coordinates in non-axisymmetric magnetohydrodynamic equilibria are discussed. Then, various aspects regarding the numerical implementation in GENE-3D are described. Finally, the code is validated and its parallel performance is assessed, along with the influence of numerical precision. [ABSTRACT FROM AUTHOR]