No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Energy and momentum preserving Coulomb collision model for kinetic Monte Carlo simulations of plasma steady states in toroidal fusion devices
Abstract
A kinetic Monte Carlo model suited for self-consistent transport studies is proposed and tested. The Monte Carlo collision operator is based on a widely used model of Coulomb scattering by a drifting Maxwellian and a new algorithm enforcing the momentum and energy conservation laws. The difference to other approaches consists in a specific procedure of calculating the background Maxwellian parameters, which does not require ensemble averaging and, therefore, allows for the use of single-particle algorithms. This possibility is useful in transport balance (steady state) problems with a phenomenological diffusive ansatz for the turbulent transport, because it allows a direct use of variance reduction methods well suited for single particle algorithms. In addition, a method for the self-consistent calculation of the electric field is discussed. Results of testing of the new collision operator using a set of 1D examples, and preliminary results of 2D modelling in realistic tokamak geometry, are presented.
... An initial prototype of this approach [15] relied on a formulation with orbits in axisymmetric geometry within the code K2D [16]. A single iteration step has been compared to an linear ideal MHD model for the pressure response in [9]. ...
... Using Nédélec elements of the lowest order for the vector potential and the test function , the weak formulation for Galerkin's method then reads For illustration purposes, we set a large maximum distance between contours, and used refinement only around three resonant surfaces, resulting in a coarser grid (left) than used for actual computations (right). MDEs (16) and (19) are solved on such contours of constant . Note that regions with a large magnitude of the perturbation field, i.e., in proximity to the RMP coils, are cut out in the right plot in order to keep the patterns in the inboard region discernible. ...
The class of plasma instabilities known as edge-localized modes (ELMs) is of special concern in tokamaks operating in high-confinement mode, such as ASDEX Upgrade and ITER. One strategy for ELM mitigation is the application of resonant magnetic perturbations (RMPs) via external coils. Kinetic modeling accurately describes the plasma response to these RMPs ab initio , particularly the parallel shielding currents at resonant surfaces. Away from resonant surfaces, ideal magnetohydrodynamics (iMHD) is expected to yield sufficiently accurate results, providing a computationally less expensive option that could complement kinetic modeling.
The code MEPHIT has been developed to solve the linearized iMHD equations in a way that is compatible with iterative kinetic modeling approaches. We consider an axisymmetric iMHD equilibrium in realistic tokamak geometry under the influence of a quasi-static non-axisymmetric external perturbation from ELM mitigation coils. The plasma responds to this external magnetic perturbation with a current perturbation, which in turn produces a magnetic field perturbation. The resulting fixed-point equation can be solved in a self-consistent manner by preconditioned iterations in which Ampère’s equation and the magnetic differential equations for pressure and current are solved in alternation until convergence is reached. After expansion in toroidal Fourier harmonics, these equations are solved on a triangular mesh in the poloidal plane using finite elements. These results are then benchmarked against established codes.
... where δf = f − f 0 is the perturbation of axisymmetric steady state distribution function f 0 , δV k = V k − V k 0 is the non-axisymmetric perturbation of the phase space velocity andL C is the linearized collision operator (in this case a Fokker-Planck operator described in Ref. [16]). Using for f 0 a local Maxwellian, f 0 = f M (ψ, W ) where ψ is the unperturbed poloidal magnetic flux and W is the total particle energy (such an f 0 model approximates the core plasma but is not suited for the scrape-off layer and private flux region where, in turn, plasma response currents are relatively small), and taking δV k in leading order over the Larmor radius (see, e.g., Ref. [8]), the r.h.s. of (3) is approximated as ...
... The Monte Carlo procedure for evaluation of the perturbed distribution (6) and corresponding plasma response currents has been realized on the basis of the K2D code [16] which employs a geometrical integrator [17] for advancing the guiding center orbits with a significantly lower CPU cost as compared to usual integration of guiding center equations. Various quantities are discretized in this code on the triangular mesh (left Fig. 1) covering the whole vacuum vessel volume including the divertor region and the scrape-off layer. ...
External resonant magnetic perturbations (RMPs) can modify the magnetic topology in a tokamak. In this case the magnetic field cannot generally be described by ideal MHD equilibrium equations in the vicinity of resonant magnetic surfaces where parallel and perpendicular relaxation timescales are comparable. Usually, resistive MHD models are used to describe these regions. In the present work, a kinetic model is used for this purpose. Within this model, plasma response, current and charge density are computed with help of a Monte Carlo method, where guiding center orbit equations are solved using a semianalytical geometrical integrator. Besides its higher efficiency in comparison to usual integrators this method is not sensitive to noise in field quantities. The computed charges and currents are used to calculate the electromagnetic field with help of a finite element solver. A preconditioned iterative scheme is applied to search for a self-consistent solution. The discussed method is aimed at the nonlinear kinetic description of RMPs in experiments on Edge Localized Mode (ELM) mitigation by external perturbation coil systems without simplification of the device geometry.
The fluid edge plasma Monte-Carlo code in three dimensions (EMC3) coupled to the kinetic (neutral particle) transport code EIRENE has demonstrated good performance in describing and even predicting the experimental trends of a wide range of stellarator and tokamak edge plasma configurations, under a certain range of relevant limiter and divertor scenarios. One major limitation so far, however, has been the restriction of EMC3 to hydrogen isotopes, although in the initial operation phase of the newly built, optimised stellarator Wendelstein 7-X, (and probably also in ITER during its initial low activation phase) helium plasmas are used. An approach is presented on how to extend EMC3 and expand the use of EIRENE features in plasma edge simulations for helium edge plasmas. The approach is based on modelling He⁺⁺ as a fluid, calculated by the plasma fluid code EMC3, and treating helium atoms and He⁺ ions as particles, calculated by the kinetic transport code EIRENE. The applicability, current limitations and future directions of this hybrid approach will be discussed. The first simulation results for Wendelstein 7-X helium edge plasma conditions demonstrate the feasibility of the present computational model.
A semi-analytical geometric integrator of guiding centre orbits in an axisymmetric tokamak is described. The integrator preserves all three invariants of motion up to computer accuracy at the expense of reduced orbit accuracy and it is roughly an order of magnitude more efficient than a direct solution of the equations of guiding centre motion with a standard high order adaptive ODE integrator.
The problem of describing charged particle transport in hot plasma under the conditions in which the ratio of the electron mean free path to the gradient length is not too small is one of the key problems of plasma physics. However, up to now, there was a deficit of the systematic interpretation of the current state of this problem, which, in most studies, is formulated as the problem of nonlocal transport. In this review, we fill this gap by presenting a self-consistent linear theory of nonlocal transport for small plasma perturbations and an arbitrary collisionality from the classical highly collisional hydrodynamic regime to the collisionless regime. We describe a number of nonlinear transport models and demonstrate the application of the nonclassical transport theory to the solution of some problems of plasma physics, first of all for plasmas produced by nanosecond laser pulses with intensities of 1013–1016 W/cm2.
In this work, we test the validity of several common simplifying assumptions
used in numerical neoclassical calculations for nonaxisymmetric plasmas. First,
neoclassical phenomena are computed for the LHD and W7-X stellarators using
several versions of the drift-kinetic equation, including the commonly used
incompressible-ExB-drift approximation and two other variants, corresponding to
different effective particle trajectories. It is found that for electric fields
below roughly one third of the resonant value, the different formulations give
nearly identical results, demonstrating the incompressible ExB-drift
approximation is quite accurate in this regime. However, near the electric
field resonance, the models yield substantially different results. We also
compare results for various collision operators, including the full linearized
Fokker-Planck operator. At low collisionality, the radial transport driven by
radial gradients is nearly identical for the different operators, while in
other cases it is found to be important that collisions conserve momentum.
Heat shock proteins (HSPs) are important molecules in the stress response of organisms from prokaryotes to mammals, and thus may be useful biomarkers for environmental stress. Here we characterize the functional roles of genes belonging to four distinct families of HSPs (hsp40, hsp60, hsp70, and hsp90) in the monogonont rotifer Brachionus manjavacas. Because B. manjavacas inhabits ponds of varying thermal regimes, including ephemeral ponds that may experience temperature fluctuations, HSP-mediated thermotolerance likely is important to its survival and adaptation. Using interference RNA (RNAi), we provide the first conclusive evidence that HSPs are required for rotifer survival following heat stress. Effective RNAi-mediated suppression of all hsp genes except hsp90 was verified via quantitative PCR. Hsp40, hsp60, and hsp70 are required for rotifer thermotolerance (P < 0.05); however, our data do not indicate hsp90 is essential. Quantitative PCR further revealed immediate up-regulation of hsp40 mRNA following heat stress. Additionally, we demonstrated expression of hsp40 mRNA in multiple tissues using fluorescent in situ hybridization. Our characterization of mRNA expression and functional roles for four distinct hsp genes provides a baseline for molecular-level comparisons of the stress response of rotifers with other taxonomic groups, and the technique for in-depth studies of the role of specific genes in rotifer stress responses. Considering the potential for ambient temperatures to impact species survival, competitive interactions, and body size of individuals, thermotolerance may be an important influence on zooplankton community structure.
NEO-2 is a code for the computation of neoclassical transport coefficients and current drive efficiency in toroidal devices which is based on the field line integration technique. The possibility to use the complete lin- earized collision integral is realized in this code. In this report the results of comparison of the full matrix of transport coefficients in a tokamak with analytical models are presented. Effects of simplifications of the lin- earized collision model (e.g., reduction to a Lorentz model) are studied in order to provide a comparison with various momentum correction techniques used for the computation of transport coefficients in stellarators.
Using an analytic solution of the kinetic equation in the 1/ν regime, a new formula for the neoclassical transport coefficients is obtained which takes into account all classes of trapped particles. This formula holds in any coordinate system and no simplifying assumptions about the magnetic field are needed. Therefore it is also applicable to complex magnetic fields given in real space coordinates. The method and the results can be used to optimize magnetic field configurations with respect to the 1/ν regime. The method is bench-marked against Monte Carlo calculation both for the l = 3 classical stellarator model and also for the original Helias configuration [J. Nührenberg and R. Zille, Phys. Lett. A 114, 129 (1986)] with a more complex magnetic field structure. Some features of transport for Helias are clarified by analyzing the bounce-averaged drift velocity. © 1999 American Institute of Physics.
The {delta}f approach is extended for simulating the transport time-scale evolution of near-Maxwellian distributions in collisional plasmas. This involves simultaneously advancing weighted marker particles for representing the intrinsically kinetic component {delta}f, and fluid equations for the parameters of the shifted Maxwellian background f{sub SM}. The issue of increasing numerical noise in a collisional {delta}f algorithm, due to marker particle weight spreading, is addressed in detail, and a solution to this problem is proposed. To obtain higher resolution in critical regions of phase space, a practical procedure for implementing sources and sinks of marker particles is developed. As a proof of principal, this set of methods is applied for computing electrical Spitzer conductivity as well as collisional absorption in a homogeneous plasma. {copyright} {ital 1999 American Institute of Physics.}
Effects of linear plasma response currents on non-axisymmetric magnetic field perturbations from the I-coil used for edge localized mode mitigation in DIII-D tokamak are analysed with the help of a kinetic plasma response model developed for cylindrical geometry. It is shown that these currents eliminate the ergodization of the magnetic field in the core plasma and reduce the size of the ergodic layer at the edge. A simple balance model is proposed which qualitatively reproduces the evolution of the plasma parameters in the pedestal region with the onset of the perturbation. It is suggested that the experimentally observed density pump-out effect in the long mean free path regime is the result of a combined action of ion orbit losses and magnetic field ergodization at the edge.
The Stochastic Mapping Technique SMT, a highly efficient method to solve the five-dimensional drift kinetic equation in the long-mean-free-path regime, is presented in an application to stellarators. Within this method, the dimensionality of the problem is reduced to four dimensions through a discretization in one dimension. Instead of tracing test particles in the whole phase space, test particles are followed on particular Poincaré cuts. With this approach, the computation time is reduced by a large factor compared to direct Monte Carlo methods. The SMT is applicable to stellarators with arbitrary magnetic field geometries and topologies. It can be used for any problem where currently conventional Monte Carlo methods are applied. In particular, it is well suited for modeling the distribution function of supra-thermal particles generated by auxiliary heating methods, for studies of stellarator transport properties and for a fast survey of a specific configuration in the whole phase space necessary for an estimation of -particle confinement. © 2002 American Institute of Physics.
We construct and investigate a set of stochastic differential equations that incorporate the physics of velocity-dependent small-angle Coulomb collisions among the plasma particles in a particle-in-cell simulation. Each particle is scattered stochastically from all the other particles in a simulation cell modeled as one or more Maxwellians. Total energy and momentum are conserved by linear transformation of the velocity increments. In two test simulations the proposed “particle-moment” collision algorithm performs well with time steps as large as 10% of the relaxation time – far larger than a particle-pairing collision algorithm, in which pairs of particles are scattered from one another, requires to achieve the same accuracy.
ICRF heating in the Large Helical Device is studied applying two global simulation codes; a drift kinetic equation solver, GNET, and a wave field solver, TASK/WM. Characteristics of energetic ion distributions in the phase space are investigated changing the resonance heating position; i.e. the on-axis and off-axis heating cases. A clear peak of the energetic ion distribution can be seen in the off-axis heating case because of the stable orbit of resonant energetic ions. The simulation results are also compared with experimental results evaluating the count number of the neutral particle analyzer and a relatively good agreement is obtained.
The radial electric field Er at the tokamak plasma edge is simulated both with a two-dimensional (2D) fluid code solving the most complete system of transport equations and with five-dimensional (three-dimensional in configuration space and 2D in velocity space) Monte Carlo particle following code. At low to high confinement transition conditions, the Er×B shearing rate is found to be high enough for turbulence suppression even though the field is essentially neoclassical. Here, B is the magnetic field. No bifurcation of Er is found.
The fluid simulation of a divertor tokamak edge plasma by the B2-SOLPS5.0 transport code gives the dependence of the radial electric field on the local and global plasma parameters. The shear of the radial electric field, which is responsible for the transition to an improved confinement regime, is a linear function of a local ion temperature and the local average toroidal velocity and is inversely proportional to the toroidal magnetic field. The scaling for the L-H transition threshold agrees with the experimental H-mode scaling of ASDEX Upgrade. The radial electric field shows no bifurcation and is close to the neoclassical electric field with the toroidal rotation contribution determined by the radial anomalous transport of the toroidal momentum. The fine structure of the electric field at the separatrix and its dependence on the toroidal magnetic field inversion is analysed.
The toroidal torque driven by external non-resonant magnetic perturbations (neoclassical toroidal viscosity) is an important momentum source affecting the toroidal plasma rotation in tokamaks. The well-known force-flux relation directly links this torque to the non-ambipolar neoclassical particle fluxes arising due to the violation of the toroidal symmetry of the magnetic field. Here, a quasilinear approach for the numerical computation of these fluxes is described, which reduces the dimension of a standard neoclassical transport problem by one without model simplifications of the linearized drift kinetic equation. The only limiting condition is that the non-axisymmetric perturbation field is small enough such that the effect of the perturbation field on particle motion within the flux surface is negligible. Therefore, in addition to most of the transport regimes described by the banana (bounce averaged) kinetic equation also such regimes as, e. g., ripple-plateau and resonant diffusion regimes are naturally included in this approach. Based on this approach, a quasilinear version of the code NEO-2 [W. Kernbichler et al., Plasma Fusion Res. 3, S1061 (2008).] has been developed and benchmarked against a few analytical and numerical models. Results from NEO-2 stay in good agreement with results from these models in their pertinent range of validity.
A method is developed for evaluating transport coefficients in asymmetric geometries using the Monte Carlo method. The method is applied to the stellarator.
This paper seeks to critique the theoretical components of everyday life information seeking through an exogenous incorporation of the sociology of everyday life. In particular, it focuses on the way in which everyday life spaces are increasingly being produced and sustained by practices of surveillance in order to structure information users into geo-demographic typologies of information consumers. The argument advanced is that an analysis of the production of space can enable a more theoretically solid groundwork for understanding the relationship between information users and the social contexts of everyday information seeking.
In the l = 3/m = 9 Uragan-3M (U-3M) torsatron (R0 = 1 m, ā≈0.12 m, B = 0.72 T, ι(ā)/2π≈0.4), an open helical divertor is realized. A hydrogen plasma with e≈2×1018 m-3, Te≈0.3 keV, Ti≈0.1 keV is produced and heated by RF fields (ω≈ωci). The flows of diverted plasma are detected by 78 plane Langmuir probes aligned poloidally in the spacings between the helical coils in two geometrically symmetric poloidal cross-sections of the torus. In measurements of the distributions of ambipolar (e.g. the ion saturation current Is) and non-ambipolar (e.g. the current to a grounded probe Ip) plasma flows, a strong vertical asymmetry of these distributions is observed, its main characteristics being a many-fold difference in the values of Is in the outgoing flows in the upper and lower parts of the torus and the opposite signs of Ip in these flows, with the positive current corresponding to the larger ambipolar flow of the diverted plasma. Reversal of the direction of the toroidal magnetic field results in the reversal of the asymmetry, with the larger flux (and Ip>0) always flowing in the ion B×∇B drift direction. On this basis, it is concluded that the asymmetry is related to direct (non-diffusive) losses of charged particles from the confinement volume. This conclusion is validated by numerical modelling of thermal and fast particle orbits in U-3M, where qualitative agreement has been revealed between the calculated distribution of the angular co-ordinates of lost particles and the measured poloidal distributions of the flows of diverted plasma.
In this paper we present a general description of the ISDEP code (Integrator of Stochastic Differential Equations for Plasmas) and a brief overview of its physical results and applications so far. ISDEP is a Monte Carlo code that calculates the distribution function of a minority population of ions in a magnetized plasma. It solves the ion equations of motion taking into account the complex 3D structure of fusion devices, the confining electromagnetic field and collisions with other plasma species. The Monte Carlo method used is based on the equivalence between the Fokker–Planck and Langevin equations. This allows ISDEP to run in distributed computing platforms without communication between nodes with almost linear scaling. This paper intends to be a general description and a reference paper in ISDEP.
A {delta}f Monte Carlo code has been developed which efficiently calculates plasma currents by finding the deviation, {cflx f}, of the exact distribution function, f, from the Maxwellian, {bold F}, with {psi} the toroidal magnetic flux enclosed by a pressure surface and H the Hamiltonian. The particles in the simulation are followed with a traditional Monte Carlo scheme consisting of an orbit step in which new values for the positions and momenta are obtained and a collision step in which a Monte Carlo equivalent of the Lorentz operator is applied to change the pitch of each particle. The dual-species code includes momentum conservation and was used to study the bootstrap and two Fourier components of the Pfirsch{endash}Schl{umlt u}ter current in nonaxisymmetric magnetic field configuration. The effects of the toroidal ripple on the orbits of the particles as well as on the bootstrap current were examined. {copyright} {ital 1997 American Institute of Physics.}
A brief review of nonlocal heat-flow models is presented. Numerical difficulties associated with their implementation, as recently demonstrated by Prasad and Kershaw [Phys. Fluids B 1, 2430 (1989)], are discussed and a simple solution is proposed. A new nonlocal heat-flow formula is developed, based on numerical simulations of the decay of linearized thermal waves, using the electron Fokker–Planck code spark. The formula is tested by modeling the full implosion of a CH shell driven by 351 nm laser irradiation. Results are shown to be in good agreement with spark simulations.
Characteristics of divertor plasmas are studied by using a one-dimensional electrostatic particle simulation code with a binary collision model. Dependence of plasma parameters, such as temperature, flow speed, pre-sheath potential, etc., on the collisionality is investigated in detail. Collisions play an important role in supplying electrons with large velocity parallel to the magnetic field, which can pass through the sheath potential barrier near the divertor plate. The pre-sheath with the scale length of the system size is formed by the collisional relaxation of the velocity distribution as well as by the particle source. The flow speed is increased by this potential and exceeds the sound speed.
Effects of radial electric field on scrape-off layer (SOL) plasma and sheath formation are studied by the use of a particle simulation code PARASOL. Previous theory for the magnetic presheath is confirmed by the simulation. The flow pattern and the density profile in the SOL plasma become asymmetric due to the Er × B drift. The condition of the sheath formation is clarified; at the sheath entrance, the flow velocity normal to the diverto plate is larger than a specific sound speed projected in the normal direction. The asymmetry of the SOL plasma is principally brought by this boundary condition. The asymmetry becomes large with the increase of Er × B drift velocity. When the velocity exceeds a critical value, the detached divertor with very low density is formed in a divertor region where Er × B drift flows out from the plate.
The plasma rotation is theoretically investigated in toroidal devices. The dependence of magnetic axis curvature and torsion on the longitudinal coordinate and magnetic field ripples are taken into account. The calculations are carried out within the large aspect ratio and circular magnetic surfaces approximation. General equations for the relaxation of poloidal and toroidal velocities are obtained. The analysis of these equations is completed for the strongly collisional regime (the Pfirsch–Schlu¨ter regime). It is shown that, as a result of the relaxation due to the ion parallel viscosity, there is an equilibrium with the ion toroidal velocity equal to zero. The general expression for the ion poloidal velocity in the Pfirsch–Schlu¨ter regime is obtained. As in the tokamak case, this velocity is proportional to the ion temperature gradient. It does not depend on the plasma density gradient and on the radial electric field. The problems considered in the paper are of interest, specifically for toroidal devices of the “Drakon” (“Dragon”) kind [V. M. Glagolev &etal;, in Proceedings of the X European Conference on Controlled Fusion and Plasma Physics, USSR, Moscow (European Physical Society, Petit-Lancy, 1981), Vol. 1, p. E8].
A 5D Monte Carlo particle simulation method for advancing rotating plasmas in tori is presented. The method exploits the neoclassical radial current balance (quasineutrality condition). Including the ion polarization current gives the time rate of change of the radial electric field and related evolution of the rotation velocity components. A special orbit initialization for a quiescent start and an efficient radial flux solving algorithm with reduced numerical noise are developed. Numerical stability of the method with respect to the strength of the perpendicular viscosity and Mach number of the poloidal rotation is investigated. This new approach enables one to separate the nonambipolar transport characteristics from the ambipolar ones. Because nonambipolar transport can support sheared flows, this model can provide a very efficient tool for studying transport barriers and related neoclassical mechanisms in toroidal plasmas.
In the l = 3/m = 9 Uragan-3M (U-3M) torsatron (R0 = 1 m, abar; ≈ 12 m, B = 0.7 T, ι(abar)/2π ≈ 0.4), an open helical divertor has been realized. Recently, under RF plasma production and heating conditions, a strong up–down asymmetry of diverted plasma flow has been observed as a result of measurements of distributions of this flow in two symmetric poloidal cross-sections of the U-3M torus. In many aspects, this asymmetry is similar to that observed in the l = 2 Heliotron E (H-E) heliotron/torsatron under neutral beam injection and electron cyclotron heating conditions. The main feature of the asymmetry is a predominant outflow of the diverted plasma in the ion toroidal drift direction. On this basis, the asymmetry can be related to non-uniformity of the distribution of direct charged particle losses in the minor azimuth. In the work reported, the magnitude of diverted plasma flow in U-3M and the degree of its vertical asymmetry are studied as functions of the heating parameter , P being the RF power absorbed in the plasma, and are juxtaposed with corresponding P-related changes in the density, , and suprathermal ion content in the plasma. As the heating power increases, both the temperature of the main ion group and the relative content of suprathermal ions increase. At the same time, a decrease in plasma density is observed, evidencing a rise of particle loss. The rise of particle loss with heating could result from both a shift of diffusion regime towards a lower collisionality and a rise of direct (non-diffusive) loss of high-energy particles. Outside the confinement volume, the total flow of diverted plasma increases together with an increase of vertical flow asymmetry towards the ion toroidal drift side. Such a mutual accordance between the processes in the confinement volume and in the divertor region validates the hypothesis on a dominating role of fast particle loss in the formation of vertical asymmetry of divertor flow in U-3M. In conclusion, the results obtained on U-3M are compared with those from similar research on H-E.
The B2-Eirene code package was developed to give better insight into the physics in the scrape-off layer (SOL), which is defined as the region of open field-lines intersecting walls. The SOL is characterised by the competition of parallel and perpendicular transport defining by this a 2D system. The description of the plasma-wall interaction due to the existence of walls and atomic processes are necessary ingredients for an understanding of the scrape-off layer. This paper concentrates on understanding the basic physics by combining the results of the code with experiments and analytical models or estimates. This work will mainly focus on divertor tokamaks, but most of the arguments and principles can be easily adapted also to other concepts like island divertors in stellarators or limiter devices.
A complete system of transport equations with all the important perpendicular currents
is derived for the simulation of tokamak edge plasma. These transport equations are implemented in
the B2.5 code and solved for the parameters of the ASDEX Upgrade tokamak. The relative roles of
different mechanisms of transverse conductivity in the formation of the potential profile are
studied. It is demonstrated that a reasonable potential distribution in the tokamak edge plasma can
be obtained without an ad hoc assumption of the existence of the anomalous perpendicular
conductivity. The role of E × B drifts in the redistribution of edge plasma and closing of the
currents in the plasma is analysed.
The algorithms, implementation details, and applications of VPIC, a state-of-the-art first principles 3D electromagnetic relativistic kinetic particle-in-cell code, are discussed. Unlike most codes, VPIC is designed to minimize data motion, as, due to physical limitations (including the speed of light!), moving data between and even within modern microprocessors is more time consuming than performing computations. As a result, VPIC has achieved unprecedented levels of performance. For example, VPIC can perform similar to 0.17 billion cold particles pushed and charge conserving accumulated per second per processor on IBM's Cell microprocessor-equivalent to sustaining Los Alamos's planned Roadrunner supercomputer at similar to 0.56 petaflop (quadrillion floating point operations per second). VPIC has enabled previously intractable simulations in numerous areas of plasma physics, including magnetic reconnection and laser plasma interactions; next generation supercomputers like Roadrunner will enable further advances. (C) 2008 American Institute of Physics.
A binary collision model for plasma particle simulation is re-examined with emphasis on the application to gyrokinetic plasma description and efficient implementation on presentday vector computers. The model conserves the number of particles, the total momentum and the total energy quasi-locally. Two efficient implementation schemes are presented. The results of measurement on various relaxation rates in velocity space are shown to agree well with test particle theory. Classical cross-field transport of particles and heat in an inhomogeneous plasma is examined using this model, and the results indicate that good agreement between continuous solution of particle and heat transport and particle simulation can be achieved.
Classical drifts offer potential explanations for a variety of physical phenomena such as poloidal asymmetries, possible extra pinch or outward flow of plasma depending on the Bt direction, non-ambipolarity of radial plasma flow and current flow towards the target. Their incorporation into 2D numerical codes is promising to greatly improve their predictive capacity. The paper contains basic analysis of drift and fluid flows, modified boundary conditions, ad-hoc models and summary of experimental results which are widely regarded as having their origin in drift motions. Unresolved issues of plasma transport are highlighted.
A binary collision model by the Monte Carlo method is proposed for plasma simulations with particle codes. The model describes a collision integral of the Landau form. Collisional effects in spatially uniform plasmas are simulated, and the results are in good agreement with theoretical ones.
In order to treat electron-electron and electron-ion collisions, an efficient mesh generated Langevin formulation is developed for a particle-in-cell plasma simulation code. The Langevin equation is used to formulate these collisions in a probabilistic way analogous to the Monte Carlo method that is used for non-coulomb scattering. The only difference is that the coulomb collisions are very frequent and weak so that the Langevin equation represents the net probabilistic effect of many small angle scatterings.
A new method is presented to model the intermediate regime between collisionless and Coulomb collision dominated plasmas in particle-in-cell codes. Collisional processes between particles of different species are treated through the concept of a grid-based “collision field,” which can be particularly efficient for multi-dimensional applications. In this method, particles are scattered using a force which is determined from the moments of the distribution functions accumulated on the grid. The form of the force is such to reproduce the multi-fluid transport equations through the second (energy) moment. Collisions between particles of the same species require a separate treatment. For this, a Monte Carlo-like scattering method based on the Langevin equation is used. The details of both methods are presented, and their implementation in a new hybrid (particle ion, massless fluid electron) algorithm is described. Aspects of the collision model are illustrated through several one- and two-dimensional test problems as well as examples involving laser produced colliding plasmas.
A binary Coulomb collision algorithm is developed for weighted particle simulations employing Monte Carlo techniques. Charged particles within a given spatial grid cell are pair-wise scattered, explicitly conserving momentum and implicitly conserving energy. A similar algorithm developed by Takizuka and Abe (1977) conserves momentum and energy provided the particles are unweighted (each particle representing equal fractions of the total particle density). If applied as is to simulations incorporating weighted particles, the plasma temperatures equilibrate to an incorrect temperature, as compared to theory. Using the appropriate pairing statistics, a Coulomb collision algorithm is developed for weighted particles. The algorithm conserves energy and momentum and produces the appropriate relaxation time scales as compared to theoretical predictions. Such an algorithm is necessary for future work studying self-consistent multi-species kinetic transport.
We identify, analyze, and propose remedies for a numerical instability responsible for the growth or decay of sums that should be conserved in Monte Carlo simulations of stochastically interacting particles. ``Noisy`` sums with fluctuations proportional to 1/ â{ital n} , where {ital n} is the number of particles in the simulation, provide feedback that drives the instability. Numerical illustrations of an energy loss or ``cooling`` instability in an Ornstein-Uhlenbeck process support our analysis. (c) 1995 The American Physical Society