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We present a scheme that spatially couples two gyrokinetic codes using first-principles. Coupled equations are presented and a necessary and sufficient condition for ensuring accuracy is derived. This new scheme couples both the field and the particle distribution function. The coupling of the distribution function is only performed once every few time-steps, using a five-dimensional (5D) grid to communicate the distribution function between the two codes. This 5D grid interface enables the coupling of different types of codes and models, such as particle and continuum codes, or delta-f and total-f models. Transferring information from the 5D grid to the marker particle weights is achieved using a new resampling technique. Demonstration of the coupling scheme is shown using two XGC gyrokinetic simulations for both the core and edge. We also apply the coupling scheme to two continuum simulations for a one-dimensional advection–diffusion problem.

The vast separation dividing the characteristic times of energy confinement and turbulence in the core of toroidal plasmas makes first-principles prediction on long timescales extremely challenging. Here we report the demonstration of a multiple-timescale method that enables coupling global gyrokinetic simulations with a transport solver to calculate the evolution of the self-consistent temperature profile. This method, which exhibits resiliency to the intrinsic fluctuations arising in turbulence simulations, holds potential for integrating nonlocal gyrokinetic turbulence simulations into predictive, whole-device models., 7 pages, 3 figures

Here, COGENT is a continuum gyrokinetic code for edge plasma simulations being developed by the Edge Simulation Laboratory collaboration. The code is distinguished by application of a fourth-order finite-volume (conservative) discretization, and mapped multiblock grid technology to handle the geometric complexity of the tokamak edge. The distribution function F is discretized in v∥ – μ (parallel velocity – magnetic moment) velocity coordinates, and the code presently solves an axisymmetric full-f gyro-kinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. COGENT capabilities are extended by implementing the fully nonlinear Fokker-Plank operator to model Coulomb collisions in magnetized edge plasmas. The corresponding Rosenbluth potentials are computed by making use of a finite-difference scheme and multipole-expansion boundary conditions. Details of the numerical algorithms and results of the initial verification studies are discussed. (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

With the next generation of high-power laser facilities for inertial fusion coming online1,2, ensuring laser beam propagation through centimetre-scale plasmas is a key physics issue for reaching ignition. Existing experimental results3,4,5 including the most recent one6 are limited to small laser spots, low-interaction laser beam energies and small plasma volumes of 1–2 mm. Here, we demonstrate the propagation of an intense, high-energy, ignition-size laser beam through fusion-size plasmas on the National Ignition Facility (NIF) and find the experimental measurements to agree with full-scale modelling. Previous attempts to apply computer modelling as a predictive capability have been limited by the inherently multiscale description of the full laser–plasma interaction processes7,8,9,10,11. The findings of this study validate supercomputer modelling as an essential tool for the design of future ignition experiments.

The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five-dimensional (3d2v) gyrokinetic continuum edge-plasma code. (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential and mirror ratio, and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth–Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the plateau regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL.

A coupled mode system is derived to investigate a three-wave parametric instability leading to energy transfer between co-propagating laser beams crossing in a plasma flow. The model includes beams of finite width refracting in a prescribed transverse plasma flow with spatial and temporal gradients in velocity and density. The resulting paraxial light equations are discretized spatially with a Crank-Nicholson-type scheme, and these algebraic constraints are nonlinearly coupled with ordinary differential equations in time that describe the ion acoustic response. The entire nonlinear differential-algebraic system is solved using an adaptive, backward-differencing method coupled with Newton's method. A numerical study is conducted in two dimensions that compares the intensity gain of the fully time-dependent coupled mode system with the gain computed under the further assumption of a strongly damped ion acoustic response. The results demonstrate a time-dependent gain suppression when the beam diameter is commensurate with the velocity gradient scale length. The gain suppression is shown to depend on time-dependent beam refraction and is interpreted as a time-dependent frequency shift.

An analysis of the effects of ion trapping on ion acoustic waves excited by the stimulated Brillouin scattering of crossing intense laser beams is presented. Ion trapping alters the dispersion of ion acoustic waves by nonlinearly shifting the normal mode frequency and by reducing the ion Landau damping. This in turn can influence the energy transfer between two crossing laser beams in the presence of plasma flows such that stimulated Brillouin scattering (SBS) occurs. The same ion trapping physics can influence the saturation of SBS in other circumstances. A one-dimensional analytical model is presented along with reasonably successful comparisons of the theory to results from particle simulations and laboratory experiments. An analysis of the vulnerability of the National Ignition Facility Inertial Confinement Fusion point design [S. W. Haan et al., Fusion Sci. Technol. 41, 164 (2002)] is also presented.

We investigate the use of adaptive mesh refinement in the simulation of laser plasma filamentation. A numerical algorithm is constructed to solve model equations consisting of a fluid approximation of a quasineutral plasma combined with a paraxial light propagation model. The algorithm involves high-resolution plasma and light model discretizations on a block-structured, locally refined grid hierarchy, which is dynamically modified during the time integration to follow evolving fine-scale solution features. Comparisons of the efficiency of this approach to that of uniform grid calculations are presented.

The coupling of radiation transport and hydrodynamics is discussed for the Eulerian frame. The discussion is aimed at developing a suitable set of equations for nonrelativistic radiation hydrodynamics (RHD) that can be numerically integrated using high-resolution methods for conservation laws. We outline how numerical methods based on a wave decomposition may be developed, along with the importance of conservation, particularly in the equilibrium regime. The properties of the RHD equations are examined through asymptotic and dispersion analyses. The conditions required to obtain the classical equilibrium limit are rigorously studied. The results show that a simple coupling term developed recently by Morel, which retains a minimum of relativistic corrections, may be sufficient for nonrelativistic flows. We also give two constraints on the relativistic corrections that result in retaining terms on the order of the truncation. In addition, the dispersion results for the P1 approximation are studied in detail and are compared with both the exact-transport results and a full relativistic treatment. We also examine some nonintuitive behavior in the dispersion results.

The first 4D (axisymmetric) high-order continuum gyrokinetic transport simulations that span the magnetic separatrix of a tokamak are presented. The modeling is performed with the COGENT code, which is distinguished by fourth-order finite-volume discretization combined with mapped multiblock grid technology to handle the strong anisotropy of plasma transport and the complex X-point divertor geometry with high accuracy. The calculations take into account the effects of fully nonlinear Fokker-Plank collisions, electrostatic potential variations, and anomalous radial transport. Topics discussed include: (a) ion orbit loss and the associated toroidal rotation and (b) edge plasma relaxation in the presence of anomalous radial transport.

Geodesic acoustic modes (GAMs) are an important phenomenon in a tokamak edge plasma. They regulate turbulence in a low confinement (L-mode) regime and can play an important role in the low to high (L–H) mode transition. It is therefore of considerable importance to develop a detailed theoretical understanding of their dynamics and relaxation processes. The present work reports on the numerical modelling of collisionless GAM relaxation, including the effects of a strong radial electric field characteristic of a tokamak pedestal in a high confinement (H-mode) regime. The simulations demonstrate that the presence of a strong radial electric field enhances the GAM decay rate, and heuristic arguments elucidating this finding are provided. The numerical modelling is performed by making use of the continuum gyrokinetic code COGENT.

The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy. Topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.

Two-dimensional Vlasov simulations of nonlinear electron plasma waves are presented, in which the interplay of linear and nonlinear kinetic effects is evident. The plasma wave is created with an external traveling wave potential with a transverse envelope of width Δy such that thermal electrons transit the wave in a “sideloss” time, tsl~Δy/ve. Here, ve is the electron thermal velocity. The quasisteady distribution of trapped electrons and its self-consistent plasma wave are studied after the external field is turned off. In cases of particular interest, the bounce frequency, ωbe=keϕ/me, satisfies the trapping condition ωbetsl>2π such that the wave frequency is nonlinearly downshifted by an amount proportional to the number of trapped electrons. Here, k is the wavenumber of the plasma wave and ϕ is its electric potential. For sufficiently short times, the magnitude of the negative frequency shift is a local function of ϕ. Because the trapping frequency shift is negative, the phase of the wave on axis lags ...

We present edge gyrokinetic simulations of tokamak plasmas using the fully non-linear (full-f) continuum code TEMPEST. A non-linear Boltzmann model is used for the electrons. The electric field is obtained by solving the 2D gyrokinetic Poisson equation. We demonstrate the following. (1) High harmonic resonances (n > 2) significantly enhance geodesic-acoustic mode (GAM) damping at high q (tokamak safety factor), and are necessary to explain the damping observed in our TEMPEST q-scans and consistent with the experimental measurements of the scaling of the GAM amplitude with edge q 95 in the absence of obvious evidence that there is a strong q-dependence of the turbulent drive and damping of the GAM. (2) The kinetic GAM exists in the edge for steep density and temperature gradients in the form of outgoing waves, its radial scale is set by the ion temperature profile, and ion temperature inhomogeneity is necessary for GAM radial propagation. (3) The development of the neoclassical electric field evolves through different phases of relaxation, including GAMs, their radial propagation and their long-time collisional decay. (4) Natural consequences of orbits in the pedestal and scrape-off layer region in divertor geometry are substantial non-Maxwellian ion distributions and parallel flow characteristics qualitatively like those observed in experiments.