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This article reports on 1D+2V heat pulse propagation studies using the COGENT guiding center kinetic code. The model uses magnetized kinetic ions and a simple Boltzmann electron model. Results agree with previous kinetic and fluid modeling benchmark studies that correspond to the parameters of edge localized modes (ELMs) observed on the JET tokamak. The plasma parameters for the edge pedestal and ensuing ELM dynamics are in the low collisionality regime. Hence, the dominant balance between the assumed Maxwellian ELM source and collisionless parallel advection causes the ion PDF to develop a significantly anisotropic velocity distribution. Adding nonlinear Coulomb ion-ion collisions to the model acts to smooth the sharp features of the ion distribution function, but the anisotropy remains robust due to the low collisionality. 2010 MSC: 00-01, 99-00, Kinetic theory, Vlasov equation, Magnetic fusion energy, Tokamak, Edge localized mode

The first continuum gyrokinetic calculations of electrostatic ion scale turbulence are presented for the case of a diverted tokamak geometry. The simulation model solves the long-wavelength limit of the full-F gyrokinetic equation for ion species coupled to the quasi-neutrality equation for electrostatic potential variations, where a fluid model is used for an electron response. In order to facilitate simulations of highly-anisotropic microturbulence in the presence of strong magnetic shear and a magnetic X-point, a numerical algorithm utilizing a locally field-aligned multiblock coordinate system has been developed and implemented in the 5D finite-volume code COGENT. In this approach, the toroidal direction is divided into blocks, such that within each block, the cells are field-aligned and a non-matching grid interface is allowed at block boundaries. The toroidal angle corresponds to the “coarse” field-aligned coordinate, whereas the poloidal cross section, comprised of the radial and poloidal directions, is finely gridded to resolve short-scale perpendicular turbulence structures and to support accurate re-mapping (interpolation) at block boundaries. The 5D simulations explore cross-separatrix ion scale turbulence in the presence of a self-consistent radial electric field and address the effects of magnetic-shear stabilization in the X-point region.

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)

In this study, COGENT is a continuum gyrokinetic code for edge plasmas being developed by the Edge Simulation Laboratory collaboration. The code is distinguished by application of the fourth order conservative discretization, and mapped multiblock grid technology to handle the geometric complexity of the tokamak edge. It is written in v∥-μ (parallel velocity – magnetic moment) velocity coordinates, and making use of the gyrokinetic Poisson equation for the calculation of a self-consistent electric potential. In the present manuscript we report on the implementation and initial testing of a succession of increasingly detailed collision operator options, including a simple drag-diffusion operator in the parallel velocity space, Lorentz collisions, and a linearized model Fokker-Planck collision operator conserving momentum and energy (© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider the treatment of electrons in kinetic simulations of edge plasmas. A fully kinetic treatment of electrons is expensive because of the short timescales associated with rapid streaming along field lines. Hence we survey a number of reduced options, of varying complexity, with particular attention to the “adiabatic model” commonly used for core plasmas. We note that the adiabatic model is the linear limit of a Boltzmann model, whose nonlinear form is more appropriate for the large potential variations in edge plasmas, and then consider extensions to the Boltzmann model which render it applicable to a plasma which includes end-loss in the scrape-off layer (© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourth-order accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge-Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.

Li-insertion-induced phase transformation in nanoscale olivine particles is studied by phase-field simulations in this paper. We show that the anisotropic growth morphology observed in experiments is thermodynamically controlled by the elastic energy arising from the misfit strain between the Li-rich and Li-poor olivine phases and kinetically influenced by the Li surface-reaction kinetics. The one-dimensional Li diffusivity inherent to the olivine structure is found to kinetically stabilize the phase boundary morphology after Li insertion termintates and facilitate ex-situ observation. Our calculations suggest that examination of the phase boundary morphology provides an effective approach to determine the limiting process of the Li intercalation kinetics in olivine nanoparticles.

We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in polycrystalline materials. The PFM system of equations includes a local order parameter, a quaternion representation of local orientation and a species composition parameter. The algorithm is based on the implicit integration of a semidiscretization of the PFM system using a backward difference formula (BDF) temporal discretization combined with a Newton-Krylov algorithm to solve the nonlinear system at each time step. The BDF algorithm is combined with a coordinate-projection method to maintain quaternion unit length, which is related to an important solution invariant. A key element of the Newton-Krylov algorithm is the selection of a preconditioner to accelerate the convergence of the Generalized Minimum Residual algorithm used to solve the Jacobian linear system in each Newton step. Results are presented for the application of the algorithm to 2D and 3D examples.

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.

The numerical simulation of plasmas is a critical tool for inertial confinement fusion (ICF). We have been working to improve the predictive capability of a continuum laser plasma interaction code pF3d, which couples a continuum hydrodynamic model of an unmagnetized plasma to paraxial wave equations modeling the laser light. Advanced numerical techniques such as local mesh refinement, multigrid, and multifluid Godunov methods have been adapted and applied to nonlinear heat conduction and to multifluid plasma models. We describe these algorithms and briefly demonstrate their capabilities.

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.

This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasmas under consideration are typical of those used for the processing of semiconductors. The governing equations consist of a drift?diffusion model of the electrons, together with an energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electric field, collisional, and source/sink terms. A discretization previously developed for a uniform spatial grid is generalized to enable local grid refinement. This extension involves the time integration of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. This approach represents an advancement of methodologies developed for neutral flows using block-structured adaptive mesh refinement (AMR) to include the significant additional effect of the electrostatic forces that couple the ion and electron fluid components. Numerical results that assess the accuracy and efficiency of the method and illustrate the importance of using adequate resolution are also presented.

This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level.

A strategy for implementing source iteration on massively parallel computers for use in solving multigroup discrete ordinates neutron transport equations on three-dimensional Cartesian grids is proposed and analyzed. Based on an analysis of the memory requirement and floating-point complexity of the formal matrix-vector multiplication effected by a single source iteration, a data decomposition and communication strategy is presented that is designed to achieve good scalability with respect to all phase-space variables, i.e., neutron position, energy, and direction. A performance model is developed to analyze the scalability properties of the algorithm and to provide computational and heuristic strategies for determining a data decomposition that minimizes wall clock execution time. Numerical results are presented to demonstrate the performance of a specific implementation of this approach on a 1,024-node nCUBE/2.

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.

With supercomputers anticipated to expand from thousands to millions of cores, one of the challenges facing scientists is how to effectively utilize this ever-increasing number. We report here an approach that creates a heterogeneous decomposition by partitioning effort according to the scaling properties of the component algorithms. We demonstrate our strategy by developing a capability to model hot dense plasma. We have performed benchmark calculations ranging from millions to billions of charged particles, including a 2.8 billion particle simulation that achieved 259.9 TFlop/s (26% of peak performance) on the 294,912 cpu JUGENE computer at the Julich Supercomputing Centre in Germany. With this unprecedented simulation capability we have begun an investigation of plasma fusion physics under conditions where both theory and experiment are lacking--in the strongly-coupled regime as the plasma begins to burn.Our strategy is applicable to other problems involving long-range forces (i.e., biological or astrophysical simulations). We believe that the flexible heterogeneous decomposition approach demonstrated here will allow many problems to scale across current and next-generation machines.

A preconditioner for the conjugate gradient solution of the linear system arising from a standard finite element discretization of a self-adjoint, second-order elliptic problem is proposed. To achieve an efficient implementation on multiprocessors, the preconditioner employs a domain decomposition strategy designed to substantially reduce the amount of interdomain (and therefore interprocessor) communication required to invert the preconditioner against the residual vector in each conjugate gradient iteration. A spectral equivalence result and some numerical experiments are presented.

Summary form only given. Experiments on the National Ignition Facility (NIF) have employed the first four beams to measure propagation and laser backscattering losses in large ignition-size plasmas. Gas-filled targets between 2 mm and 7 mm length have been heated from one side by overlapping the focal spots of the four beams from one quad operated at 351 nm with a total intensity of 2times1015 W cm-2. The targets were filled with 1 atm of CO2 producing of up to 7 mm long homogeneously heated plasmas with densities of ne = 5times1020 cm-3 and temperatures of Te = 2 keV. The high energy in a NIF quad of beams of 16 kJ, illuminating the target from one direction, creates unique conditions for the study of laser plasma interactions at scale lengths not previously accessible. The propagation through the large-scale plasma was measured with a gated X-ray imager that was filtered for 3.5 keV X-rays. These data indicate that the beams interact with the full length of this ignition-scale plasma during the last ~1 ns of the experiment when applying full laser beam smoothing consisting of phase plates, smoothing by spectral dispersion and polarization smoothing. Measurements that only apply phase plates show, laser beam filamentation and reduced propagation speed. These results demonstrate the NIF experimental capabilities and further provide a benchmark for three-dimensional modeling of the laser-plasma interactions at ignition-size scale lengths

The goal of this project was to investigate the utility of parallel adaptive mesh refinement (AMR) in the simulation of laser plasma interaction (LPI). The scope of work included the development of new numerical methods and parallel implementation strategies. The primary deliverables were (1) parallel adaptive algorithms to solve a system of equations combining plasma fluid and light propagation models, (2) a research code implementing these algorithms, and (3) an analysis of the performance of parallel AMR on LPI problems. The project accomplished these objectives. New algorithms were developed for the solution of a system of equations describing LPI. These algorithms were implemented in a new research code named ALPS (Adaptive Laser Plasma Simulator) that was used to test the effectiveness of the AMR algorithms on the Laboratory's large-scale computer platforms. The details of the algorithm and the results of the numerical tests were documented in an article published in the Journal of Computational Physics [2]. A principal conclusion of this investigation is that AMR is most effective for LPI systems that are ''hydrodynamically large'', i.e., problems requiring the simulation of a large plasma volume relative to the volume occupied by the laser light. Since the plasma-only regions require less more » resolution than the laser light, AMR enables the use of efficient meshes for such problems. In contrast, AMR is less effective for, say, a single highly filamented beam propagating through a phase plate, since the resulting speckle pattern may be too dense to adequately separate scales with a locally refined mesh. Ultimately, the gain to be expected from the use of AMR is highly problem-dependent. One class of problems investigated in this project involved a pair of laser beams crossing in a plasma flow. Under certain conditions, energy can be transferred from one beam to the other via a resonant interaction with an ion acoustic wave in the crossing region. AMR provides an effective means of achieving adequate resolution in the crossing region while avoiding the expense of using the same fine grid everywhere, including the region between the beams where no LPI occurs. We applied ALPS to a suite of problems modeling crossed beam experiments performed on the Omega laser at the University of Rochester. Our simulations contributed to the theoretical interpretation of these experiments, which was recently published in Physical Review Letters [4]. This project has advanced the Laboratory's computational capabilities in the area of AMR algorithms and their application to LPI problems. The knowledge gained and software developed will contribute to the computational tools available for use in the design and interpretation of experiments to be performed at the National Ignition Facility (NIF) in support of Laboratory missions in stockpile stewardship, energy research and high energy density science. « less

Experiments demonstrate energy and power transfer between copropagating, same frequency, beams crossing at a small angle in a plasma with a Mach 1 flow. The process is interpreted as amplification of the low intensity probe beam by the stimulated scatter of the high intensity pump beam. The observed probe amplification increases slowly with pump intensity and decreases with probe intensity, indicative of saturation limiting the energy and power transfer due to ion-wave nonlinearities and localized pump depletion. The results are consistent with numerical modeling including ion-wave nonlinearities.

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.

Two methods are presented for simulating the development of photolithographic profiles during the resist dissolution phase. These algorithms are the volume-of-fluid algorithm, and the steady level-set algorithm. These methods are compared with the ray-trace, cell and level- set techniques employed in SAMPLE-3D. The volume-of-fluid algorithm employs an Euclidean Grid with volume fractions. At each time step, the surface is reconstructed by computing an approximation of the tangent plane of the surface in each cell that contains a value between 0 and 1. The geometry constructed in this manner is used to determine flow velocity vectors and the flux across each edge. The material is then advanced by a split advection scheme. The steady level set algorithm is an extension of the iterative level set algorithm. The steady level set algorithm combines fast level set concepts and a technique for finding zero residual solutions to the eikonal function. The etch time for each cell is calculated in a time ordered manner. Use of heap sorting data structures allows the algorithm to execute extremely quickly. A similar technique was submitted by J. Sethian. Comparisons of the methods have been performed and the results are shown.

We are developing a new class of finite-volume methods on locally-refined and mapped grids, which are at least fourth-order accurate in regions where the solution is smooth. This paper discusses the implementation of such methods for time-dependent problems on both Cartesian and mapped grids with adaptive mesh refinement. We show 2D results with the Berger-Colella shock-ramp problem in Cartesian coordinates, and fourth-order accuracy of the solution of a Gaussian pulse problem in a polytropic gas in mapped coordinates.

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.

Experiments show that power is transferred between two copropagating 351nm laser beams crossing in an Al plasma when the frequency of the driven ion wave is shifted by a Mach 1 flow. The resonant amplification of a low-intensity (⩽2.5×1014W∕cm2) beam intersected by a high-intensity (7.0×1014W∕cm2) pump beam is determined by comparing the transmitted beam power to that measured in experiments where the plasma flow direction is reversed and the ion wave is evidently detuned. The polarization of the amplified light is also observed to align to the pump polarization consistent with ion-wave scattering. The amplification is found to reduce with probe-beam intensity demonstrating a nonlinear saturation mechanism that is effective when the ion-wave damping is weak, which is modeled with a calculation including both the nonlinear ion-wave frequency shifts due to ion trapping and whole-beam pump depletion.

A new method is presented for calculating the scattering of an arbitrary electromagnetic wave by a bounded, perfectly conducting body of general shape. The strategy is to replace the corresponding exterior boundary-value problem by an approximate problem on the boundary of the scattering body. This involves the introduction of a certain bilinear form and non-local boundary operator, together with the use of a special class of known solutions of the reduced Maxwell's equations satisfying the Sommerfeld radiation boundary conditions at infinity. Two computer programs implementing this method are described and numerical results showing the successful application of this method to some model problems are presented.

A new algorithm for modeling charged-particle transport in a fully ionized plasma is presented. A standard multigroup discretization of the Fokker-Planck-Boltzmann equation is transport-corrected to implicitly include the anisotropic effects of both coulomb scattering and nuclear reactions. This allows the subsequent application of the Levermore flux-limited diffusion theory, which was originally developed for isotropic radiative transfer calculations. A finite differencing of the resulting spatial transport operator is constructed so as to yield centered and upwinded operators in the diffusion and free-streaming limits, respectively. The time integration is performed by the general purpose ordinary differential equation solver TORANAGA. This approach results in a highly vectorizable algorithm that has been implemented on the CRAY-1. Some numerical results are presented that compare this algorithm to the corresponding, but far more expensive, Monte Carlo calculations.

In this paper we consider the numerical solution of the Helmholtz equation in an exterior domain in $\mathbb{R}^3 $. We introduce a new variational formulation of the problem involving a nonlocal boundary condition. We show that the discrete bilinear form associated with our variational formulation satisfies the “inf-sup” condition, and we give a quasi-optimal error estimate for the approximate solution. Moreover, we show a certain density property of our subspaces which plays a crucial role in the justification of the method.