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GX is a code for solving the nonlinear gyrokinetic system for low-frequency turbulence in magnetized plasmas, particularly tokamaks and stellarators. In GX, our primary motivation and target is a fast gyrokinetic solver that can be used for fusion reactor design and optimization along with wide-ranging physics exploration. This has led to several code and algorithm design decisions, specifically chosen to prioritize time to solution. First, we have used a discretization algorithm that is pseudo-spectral in the entire phase-space, including a Laguerre-Hermite pseudo-spectral formulation of velocity space, which allows for smooth interpolation between coarse gyrofluid-like resolutions and finer conventional gyrokinetic resolutions and efficient evaluation of a model collision operator. Additionally, we have built GX to natively target graphics processors (GPUs), which are among the fastest computational platforms available today. Finally, we have taken advantage of the reactor-relevant limit of small $\rho_*$ by using the radially-local flux-tube approach. In this paper we present details about the gyrokinetic system and the numerical algorithms used in GX to solve the system. We then present several numerical benchmarks against established gyrokinetic codes in both tokamak and stellarator magnetic geometries to verify that GX correctly simulates gyrokinetic turbulence in the small $\rho_*$ limit. Moreover, we show that the convergence properties of the Laguerre-Hermite spectral velocity formulation are quite favorable for nonlinear problems of interest. Coupled with GPU acceleration, which we also investigate with scaling studies, this enables GX to be able to produce useful turbulence simulations in minutes on one (or a few) GPUs. GX is open-source software that is ready for fusion reactor design studies., Comment: 39 pages, 12 figures

Nonlinear multiscale gyrokinetic simulations of a Joint European Torus edge pedestal are used to show that electron-temperature-gradient (ETG) turbulence has a rich three-dimensional structure, varying strongly according to the local magnetic-field configuration. In the plane normal to the magnetic field, the steep pedestal electron temperature gradient gives rise to anisotropic turbulence with a radial (normal) wavelength much shorter than in the binormal direction. In the parallel direction, the location and parallel extent of the turbulence are determined by the variation in the magnetic drifts and finite-Larmor-radius (FLR) effects. The magnetic drift and FLR topographies have a perpendicular-wavelength dependence, which permits turbulence intensity maxima near the flux-surface top and bottom at longer binormal scales, but constrains turbulence to the outboard midplane at shorter electron-gyroradius binormal scales. Our simulations show that long-wavelength ETG turbulence does not transport heat efficiently, and significantly decreases overall ETG transport -- in our case by $\sim$40 \% -- through multiscale interactions., Comment: 17 pages, 14 figures

We are developing a new continuum gyrokinetic code, Gkeyll, for use in edge plasma simulations, and here present initial simulations of turbulence on open field lines with model sheath boundary conditions. The code implements an energy conserving discontinuous Galerkin scheme, applicable to a general class of Hamiltonian equations. Several applications to test problems have been done, including a calculation of the parallel heat-flux on divertor plates resulting from an ELM crash in JET, for a 1x/1v SOL scenario explored previously, where the ELM is modeled as a time-dependent intense upstream source. Here we present initial simulations of turbulence on open field lines in the LAPD linear plasma device. We have also done simulations in a helical open-field-line geometry. While various simplifications have been made at present, this still includes some of the key physics of SOL turbulence, such as bad-curvature drive for instabilities and rapid parallel losses with sheath boundary conditions. This is useful for demonstrating the overall feasibility of this approach and for initial physics studies of SOL turbulence. We developed a novel version of DG that uses Maxwellian-weighted basis functions while still preserving exact particle and energy conservation. The Maxwellian-weighted DG method achieves the same error with 4 times less computational cost in 1v, or 16 times lower cost in the 2 velocity dimensions of gyrokinetics (assuming memory bandwidth is the limiting factor)., Comment: Paper accepted for 26th IAEA Fusion Energy Conference (2016, Kyoto, Japan) proceedings

A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihin et al. 2009) for pressure-anisotropic plasmas, allowing for species drifts---a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas (e.g. intracluster medium). Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g., the Alfven ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. The main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvenic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvenic cascade is fluid, satisfying RMHD equations (with the Alfven speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping. Secondly, the organising principle of this turbulence is elucidated in the form of a generalised kinetic free-energy invariant. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses; these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics., Comment: 61 pages, accepted to Journal of Plasma Physics; Abstract abridged

In this paper, we extend the multiscale approach developed in [Abel et. al., Rep. Prog. Phys., submitted] by exploiting the scale separation between ions and the electrons. The gyrokinetic equation is expanded in powers of the electron to ion mass ratio, which provides a rigorous method for deriving the reduced electron model. We prove that ion-scale electromagnetic turbulence cannot change the magnetic topology, and argue that to lowest order the magnetic field lies on fluctuating flux surfaces. These flux surfaces are used to construct magnetic coordinates, and in these coordinates a closed system of equations for the electron response to ion-scale turbulence is derived. All fast electron timescales have been eliminated from these equations. We also use these magnetic surfaces to construct transport equations for electrons and for electron heat in terms of the reduced electron model., Comment: 62 pages, 1 figure

This paper presents a complete theoretical framework for plasma turbulence and transport in tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio of the gyroradius to the equilibrium scale length. Proceeding order-by-order in this expansion, a framework for plasma turbulence is developed. It comprises an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equillibrium is obtained from the Grad-Shafranov equation for a rotating plasma and the slow (resistive) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the high-flow gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local cascade of free energy. Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical corrections and the fluctuations act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived. Total energy is conserved and there is no net turbulent heating. Entropy is produced by the action of fluxes flattening gradients, Ohmic heating, and the equilibration of mean temperatures. Finally, this framework is condensed, in the low-Mach-number limit, to a concise set of equations suitable for numerical implementation., Comment: 113 pages, 3 figures

Beam Emission Spectroscopy (BES) measurements of ion-scale density fluctuations in the MAST tokamak are used to show that the turbulence correlation time, the drift time associated with ion temperature or density gradients, the particle (ion) streaming time along the magnetic field and the magnetic drift time are consistently comparable, suggesting a "critically balanced" turbulence determined by the local equilibrium. The resulting scalings of the poloidal and radial correlation lengths are derived and tested. The nonlinear time inferred from the density fluctuations is longer than the other times; its ratio to the correlation time scales as $\nu_{*i}^{-0.8\pm0.1}$, where $\nu_{*i}=$ ion collision rate/streaming rate. This is consistent with turbulent decorrelation being controlled by a zonal component, invisible to the BES, with an amplitude exceeding the drift waves' by $\sim \nu_{*i}^{-0.8}$., Comment: 6 pages, 4 figures, submitted to PRL

We present the Young Exoplanet Transit Initiative (YETI), in which we use several 0.2 to 2.6m telescopes around the world to monitor continuously young (< 100 Myr), nearby (< 1 kpc) stellar clusters mainly to detect young transiting planets (and to study other variability phenomena on time-scales from minutes to years). The telescope network enables us to observe the targets continuously for several days in order not to miss any transit. The runs are typically one to two weeks long, about three runs per year per cluster in two or three subsequent years for about ten clusters. There are thousands of stars detectable in each field with several hundred known cluster members, e.g. in the first cluster observed, Tr-37, a typical cluster for the YETI survey, there are at least 469 known young stars detected in YETI data down to R=16.5 mag with sufficient precision of 50 milli-mag rms (5 mmag rms down to R=14.5 mag) to detect transits, so that we can expect at least about one young transiting object in this cluster. If we observe 10 similar clusters, we can expect to detect approximately 10 young transiting planets with radius determinations. The precision given above is for a typical telescope of the YETI network, namely the 60/90-cm Jena telescope (similar brightness limit, namely within +/-1 mag, for the others) so that planetary transits can be detected. For planets with mass and radius determinations, we can calculate the mean density and probe the internal structure. We aim to constrain planet formation models and their time-scales by discovering planets younger than 100 Myr and determining not only their orbital parameters, but also measuring their true masses and radii, which is possible so far only by the transit method. Here, we present an overview and first results. (Abstract shortened), Comment: 15 pages, 10 figures, AN accepted 2011 June 15

Direct coupling between a transport solver and local, nonlinear gyrokinetic calculations using the multiscale gyrokinetic code TRINITY [M. Barnes, Ph.D. thesis, arxiv:0901.2868] is described. The coupling of the microscopic and macroscopic physics is done within the framework of multiscale gyrokinetic theory, of which we present the assumptions and key results. An assumption of scale separation in space and time allows for the simulation of turbulence in small regions of the space-time grid, which are embedded in a coarse grid on which the transport equations are implicitly evolved. This leads to a reduction in computational expense of several orders of magnitude, making first-principles simulations of the full fusion device volume over the confinement time feasible on current computing resources. Numerical results from TRINITY simulations are presented and compared with experimental data from JET and ASDEX Upgrade plasmas., Comment: 12 pages, 13 figures, invited paper for 2009 APS-DPP meeting, submitted to Phys. Plasmas

A new analytic expression is derived for the resonant drive of high n Alfvenic modes by particles accelerated to high energy by Ion Cyclotron Resonance Heating. This derivation includes finite orbit effects, and the formalism is completely non-perturbative. The high-n limit is used to calculate the complex particle response integrals along the orbits explicitly. This new theory is applied to downward sweeping Alfven Cascade quasimodes completing the theory of these modes, and making testable predictions. These predictions are found to be consistent with experiments carried out on the Joint European Torus [P. H. Rebut and B. E. Keen, Fusion Technol. 11, 13 (1987)]., Comment: 31 pages, 6 figures

A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a model collision operator satisfying these properties is described. This operator is based on the exact linearized test-particle collision operator, with approximations to the field-particle terms that preserve conservation laws and an H-Theorem. It includes energy diffusion, pitch-angle scattering, and finite Larmor radius effects corresponding to classical (real-space) diffusion. The numerical implementation in the continuum gyrokinetic code GS2 is fully implicit and guarantees exact satisfaction of conservation properties. Numerical results are presented showing that the correct physics is captured over the entire range of collisionalities, from the collisionless to the strongly collisional regimes, without recourse to artificial dissipation., Comment: 13 pages, 8 figures, submitted to Physics of Plasmas; typos fixed

A new analytically and numerically manageable model collision operator is developed specifically for turbulence simulations. The like-particle collision operator includes both pitch-angle scattering and energy diffusion and satisfies the physical constraints required for collision operators: it conserves particles, momentum and energy, obeys Boltzmann's H-theorem (collisions cannot decrease entropy), vanishes on a Maxwellian, and efficiently dissipates small-scale structure in the velocity space. The process of transforming this collision operator into the gyroaveraged form for use in gyrokinetic simulations is detailed. The gyroaveraged model operator is shown to have more suitable behavior at small scales in phase space than previously suggested models. A model operator for electron-ion collisions is also presented., Comment: revtex, 12 pages