Kinetic Plasma Simulations Research Articles

Fully kinetic simulation of coupled plasma and neutral particles in scrape-off layer plasmas of fusion devices

Fluid descriptions of plasmas, which are usually applied to a collisional plasma, can only be justified for very small Coulomb Knudsen numbers. However, the scrape-off layer (SOL) plasmas of experimental magnetic confinement fusion devices tend to have operational regimes characterized by a Coulomb Knudsen number around 0.1. In interesting detached regimes of an SOL plasma in a tokamak, when the plasma detaches from the limiters or divertors, this number may increase along with the local plasma gradients. Plasma gradients are also known to increase (and thus drive non-local effects) in inertial confinement fusion. Neutrals, which are being produced owing to plasma recombination at the plasma–divertor interface, may be in a mixed collisional regime as well. Thus simultaneous kinetic treatments of plasma and neutral particles with self-consistent evaluation of boundary conditions at the material walls are required. We present a physical model and a numerical scheme, and discuss results of purely kinetic simulations of plasmas and neutrals for actual conditions in the Alcator C-Mod and Tokamak-de-Varennes experimental tokamaks. Results for both steady-state and transient regimes of SOL plasma flow are presented. Our approach, unlike particle-in-cell and Monte Carlo methods, is free from statistical noise.

Spices1 — A smart-particle code for kinetic plasma simulation

Spices1 (Smart Particle In Cell ElectroStatic 1D) simulates 1D plasmas governed by the Vlasov equation. A linear Fokker—Planck operator can be used to represent collisions between plasma species and a given background. The electrostatic approximation is used and no magnetic field is present. Spices1 uses computational particles with internal degrees of freedom (smart particles) to represent better the evolution of the many physical particles allocated to one single computational particle, employing a modified version of the Blob method. The Blob approach leads to inexpensive low noise calculations and has the additional advantage to allow an accurate and deterministic treatment of collision operators. The present work describes the steps taken to implement the Blob method to create Spices1. All the instructions required to use Spices1 are given in details. Several test cases and paradigmatic applications are presented to illustrate the capabilities of Spices1 and to provide benchmarks for testing purposes.

Vlasov Simulations Using Velocity-Scaled Hermite Representations

The efficiency, accuracy, and stability of two different pseudo-spectral methods using scaled Hermite basis and weight functions, applied to the nonlinear Vlasov–Poisson equations in one dimension (1d-1v), are explored and compared. A variable velocity scaleUis introduced into the Hermite basis and is shown to yield orders of magnitude reduction in errors, as compared to linear kinetic theory, with no increase in workload. A set of Fourier–Hermite coefficients, representing a periodic Gaussian distribution function, are advanced through time with anO(Δt2)-accurate splitting method. Within this splitting scheme, the advection and acceleration terms of the Vlasov equation are solved separately using anO(Δt4)-accurate Runge–Kutta method. The asymmetrically weighted (AW) Hermite basis, which has been used previously by many authors, conserves particles and momentum exactly and total energy toO(Δt3); however, the AW Hermite method doesnotconserve the square integral of the distribution and is, in fact, numerically unstable. The symmetrically weighted (SW) Hermite algorithm, applied here to the Vlasov system for the first time, can either conserve particles and energy (forNueven) or momentum (forNuodd) as Δ t → 0, whereNuis the largest Hermite mode number. The SW Hermite method conserves the square integral of the distribution and, therefore, remains numerically stable. In addition, careful velocity scaling improves the conservation properties of the SW Hermite method. Damping and growth rates, oscillation frequencies, E-field saturation levels, and phase-space evolution are seen to be qualitatively correct during simulations. Relative errors with respect to linear Landau damping and linear bump-on-tail instability are shown to be less than 1% using only 64 velocity-scaled Hermite functions. Comparisons to particle-in-cell (PIC) simulations show that as the number of particles increases to more than 106, the PIC solutions converge to scaled SW Hermite solutions that were found in only 1/10 of the run-time. The SW Hermite method with velocity scaling is well-suited to kinetic simulations of warm plasmas.

A nonspecialist's guide to kinetic simulations of space plasmas

The basic concepts involved in formulating and analyzing simulations of kinetic space plasmas are presented for readers with little background in simulation methods. Issues related to choosing the appropriate physics models and numerical algorithms, such as the correct representation for the plasma, proper description of the electromagnetic fields, geometric effects, initial and boundary conditions, and diagnostics, are discussed. Some of the strengths and limitations of the various approaches are also described. Explicit examples involving the simulation of collisionless shocks are presented to illustrate these points. Questions related to numerical accuracy and verification of the results are also examined and again illustrated through appropriate examples.

Kinetic theory of charged particles of variable shape

Recently a new kind of computational technique for kinetic plasma simulation has been developed, making use of particles of variable shape (blobs). The aim of the present work is to derive a kinetic equation governing the evolution of the blobs and solve it in the case of small perturbations. Results are compared with those obtained for the real plasma and for a plasma of particles of fixed shape (the model usually employed in popular particle-in-cell simulations). The comparison is useful to find out whether the additional cost of dealing with more degrees of freedom is balanced by the superior accuracy provided by the new technique in describing the phase-space distribution.

Implicit-moment, partially linearized particle simulation of kinetic plasma phenomena.

An algorithm for kinetic simulation of electron and ion plasmas is introduced that combines both implicit and perturbative methods to deal efficiently with disparate time scales and signal-to-noise issues associated with particle discreteness. The algorithm is introduced for an electrostatic model in a strong magnetic field with gyrokinetic ions and drift-kinetic electrons. Some of the numerical dispersion properties are analyzed and confirmed in test simulations. Two-dimensional applications of the algorithm to drift-wave instabilities excited either by an ion temperature gradient (ion-temperature-gradient instability) or by electron inverse Landau damping in the presence of a density gradient (collisionless drift instability) are presented that illustrate the usefulness of the method. Clear gains in resolution and computational efficiency over previous methods are achieved. {copyright} {ital 1996 The American Physical Society.}

Dynamic and Selective Control of the Number of Particles in Kinetic Plasma Simulations

An algorithm for the dynamic control of the number of particles in a particle-in-cell (PIC), plasma simulation is presented. The algorithm selectively splits and coalesces particles to control the number of particles in each grid cell. It is designed for multiple-length scale problems where an adaptive grid can be applied and for PIC simulations on parallel computers where a constant number of particles per cell is useful for computational efficiency. The algorithm preserves the charge assignment at grid points while splitting one particle into two, or while coalescing two particles into one. Errors in momentum or energy conservation are controlled by a simple, a priori test. The accuracy of the algorithm is demonstrated in several simulations in one dimension, including a collisionless slow-shock, where an adaptive grid calculation with dynamic number control gives comparable accuracy to a uniformly zoned calculation without dynamic control with just 50% of the effort.

Radiation trapping simulations using the propagator function method: complete and partial frequency redistribution

An integral method for solving the problem of imprisonment of resonance radiation based on propagator functions is further developed. Earlier work was restricted to plane parallel and spherical geometries, to a Lorentz lineshape, and to the approximation of complete frequency redistribution. This work extends the method to cylindrical geometry, to a Voigt lineshape, and to include the effects of partial frequency redistribution. The method is ideal for calculating both the time-dependent and the steady-state densities of resonance atoms which result from an arbitrary production rate per unit volume. An emission spectrum is also generated in calculations involving partial frequency redistribution. The propagator function method is at least 50 times faster than the Monte Carlo method. The greater speed of the propagator function method makes it well suited to fully self-consistent kinetic simulations of glow discharge plasmas.

Radiation trapping simulations using the propagator function method

An integral method of solving the Holstein-Biberman equation based on a propagator function is described. This method is used to solve the equation with a Lorentz lineshape in an infinite plane parallel geometry and a hollow spherical geometry. The method is ideal for solving for both the time dependent and the steady state density of resonance atoms which results from an arbitrary production rate per unit volume. The propagator function method is 100 times faster than the Monte Carlo method. The greater speed of the propagator function method makes it well suited to fully self-consistent kinetic simulations of glow discharge plasmas.

Macroscale implicit electromagnetic particle simulation of magnetized plasmas

An electromagnetic and multi-dimensional macroscale particle simulation code (MACROS) is presented which enables us a large time and spatial scale kinetic simulation of magnetized plasmas. Particle ions, finite mass electrons with the guiding-center approximation, and a complete set of Maxwell equations are employed. Implicit field particle coupled equations are derived in which a time-decentered (slightly backward) finite differential scheme is used to achieve stability for large time and spatial scales. It is shown analytically that the present simulation scheme suppresses high frequency electromagnetic waves and that it accurately reproduces low frequency waves in the plasma. These properties are verified by numerical examination of eigenmodes in a 2D thermal equilibrium plasma and by that of the kinetic Alfven wave.

Kinetic theory and simulation of multispecies plasmas in tokamaks excited with electromagnetic waves in the ion-cyclotron range of frequencies

A description of a bounce-averaged Fokker–Planck quasilinear model for the kinetic description of tokamak plasmas is presented. The nonlinear collision and quasilinear resonant diffusion operators are represented in a form conducive to numerical solution with specific attention to the treatment of the boundary layer separating trapped and passing orbit regions of velocity space. The numerical techniques employed are detailed insofar as they constitute significant departure from those used in the conventional uniform magnetic field case. Examples are given to illustrate the combined effects of collisional and resonant diffusion.

Implicit time integration for plasma simulation

To increase the size of the time step, and thereby extend the applicability of kinetic plasma simulation, analysis of the stability and accuracy of implicit time integration schemes for plasma particle-in-cell simulation and the synthesis of new algorithms have been undertaken. Three classes of implicit algorithms are considered in general form. Stability and accuracy calculations provide guidelines for the design and application of implicit simulation algorithms, as illustrated in the construction of new difference schemes with desirable properties. Of particular interest- are the relaxation of stability constraints on the time step, strong damping of unwanted high-frequency modes, accuracy of the simulation of low-frequency phenomena, numerical secular acceleration, and numerical stability of fast and slow space-charge waves. The potency of higher order accurate differencing schemes in reducing undesirable numerical effects is demonstrated.