Gyrokinetic Fokker-Planck collision operator

The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects in both field-particle and test-particle terms are evaluated. When implemented in gyrokinetic simulations, these effects can be precomputed. The field-particle operator at each time step requires the evaluation of a single two-dimensional integral, and is not only more accurate, but appears to be less expensive to evaluate than conserving model operators.     

Nonlinear collisional monte carlo simulations for high-temperature SOL plasma

A nonlinear Monte Carlo collisional model is applied to to investigate scrape-off layer (SOL) plasmas with high temperatures. In the proposed SOL modeling, A steady state SOL plasma, which satisfies the particle and energy balances and neutrality constraint, is determined in terms of total particle and heat fluxes across the separatrix, the edge plasma temperature, the secondary electron emission coefficient, and the SOL size. A conductive heat flux into the SOL is effectively modeled via random exchange of source particles and the SOL plasma particles. It is found that the potential drop and the electron transmission factor in the collisional SOL plasma are in good agreement with the theoretical prediction. The cooling effect of secondary electrons in the high temperature divertor operation is investigated. In such a collisionless plasma, the present nonlinear collision model is useful because the electron distribution function deviates far from a Maxwellian distribution. In the presence of strong secondary electron emission, the electron sheath energy transmission factor in the collisionless regime is found to be significantly smaller than that in the collisional regime. This fact suggests that a high-temperature divertor operation can be possible.     

Core-edge coupling and the effect of the EDGE on overall plasma performance

Several attempts to model the entire plasma cross section have been reported in the last few years. Two possibilities are to either couple a core code to a scrape-off layer (SOL) code at a specified interface or to extend the computational region of an SOL-code all the way to the plasma centre. The most advanced global code is the code COCONUT which is based on the former principle and comprises the Monte-Carlo code NIMBUS, the 2D scrape-off layer code EDGE2D, the core transport code JETTO and the core impurity transport code SANCO. A main feature of COCONUT is its modular structure which ensures a high degree of flexibility and the capability to cover a large range of time-scales. The influence of the SOL on the core is lllustrated with a range of global simulations carried out with COCONUT. The simulations show that the primary effect of the SOL is the control of the particle sources and sinks with a secondary effect on plasma dilution, radiation and perhaps pedestal temperatures.     

Implementation of an inelastic collision operator into KIPP‐SOLPS coupling and its effects on electron parallel transport in the scrape‐off layer plasmas

This paper develops an inelastic collision operator for the Kinetic Code for Plasma Periphery (KIPP) code to investigate the kinetic effects of electron cooling due to inelastic collisions. It is fully tested based on the self‐consistent KIPP‐SOLPS coupling algorithm by being compared to the ADAS database. The collisional radiative rate coefficients from the ADAS database for deuterium atomic physics can be recovered using the inelastic collision operator with assuming Maxwellian electrons, which shows that the inelastic collision operator works well for various plasma conditions. Across a wide range of plasma conditions in the scrape‐off layer, KIPP‐SOLPS coupling simulation results with the implementation of an inelastic collision operator are not significantly different from results using a simpler uniform cooling scheme. The uniform scheme is thus recommended rather than including computationally intensive inelastic collision physics.     

Progress with the 5D full-F continuum gyrokinetic code COGENT

COGENT is an Eulerian gyrokinetic code being developed for edge plasma modelling. The code is distinguished by the use of a high-order finite-volume (conservative) discretization combined with mapped multi-block grid technology. Our recent work is focused on the development of a 5D full-F COGENT version. A numerical algorithm utilizing locally a field-aligned multi-block coordinate system is implemented to facilitate simulations of highly anisotropic microturbulence in the presence of a strong magnetic shear. In this approach, the toroidal direction is divided into blocks such that, within each block, the cells are field-aligned and a non-matching (non-conformal) grid interface is allowed at the block boundaries. Here we report on details of the numerical implementation and present preliminary results of verification studies performed for the case of the ion temperature gradient (ITG) instability in a sheared toroidal annulus geometry.     

Energy transfer and dual cascade in kinetic magnetized plasma turbulence

The question of how nonlinear interactions redistribute the energy of fluctuations across available degrees of freedom is of fundamental importance in the study of turbulence and transport in magnetized weakly collisional plasmas, ranging from space settings to fusion devices. In this Letter, we present a theory for the dual cascade found in such plasmas, which predicts a range of new behavior that distinguishes this cascade from that of neutral fluid turbulence. These phenomena are explained in terms of the constrained nature of spectral transfer in nonlinear gyrokinetics. Accompanying this theory are the first observations of these phenomena, obtained via direct numerical simulations using the gyrokinetic code AstroGK. The basic mechanisms that are found provide a framework for understanding the turbulent energy transfer that couples scales both locally and nonlocally.     

Accurate models of collisions in glow discharge simulations

Very detailed, self-consistent kinetic glow discharge simulations are used to examine the effect of various models of collisional processes. The effects of allowing anisotropy in elastic electron collisions with neutral atoms instead of using the momentum transfer cross-section, the effects of using an isotropic distribution in inelastic electron-atom collisions, and the effects of including a Coulomb electron-electron collision operator are all described. It is shown that changes in any of the collisional models, especially the second and third described above, can make a profound difference in the simulation results. This confirms that many discharge simulations have great sensitivity to the physical and numerical approximations used. Our results reinforce the importance of using a kinetic theory approach with highly realistic models of various collisional processes     

Dimits shift in realistic gyrokinetic plasma-turbulence simulations

In simulations of turbulent plasma transport due to long wavelength (k perpendicular rhoi < or = 1) electrostatic drift-type instabilities, we find a persistent nonlinear up-shift of the effective threshold. Next-generation tokamaks will likely benefit from the higher effective threshold for turbulent transport, and transport models should incorporate suitable corrections to linear thresholds. The gyrokinetic simulations reported here are more realistic than previous reports of a Dimits shift because they include nonadiabatic electron dynamics, strong collisional damping of zonal flows, and finite electron and ion collisionality together with realistic shaped magnetic geometry. Reversing previously reported results based on idealized adiabatic electrons, we find that increasing collisionality reduces the heat flux because collisionality reduces the nonadiabatic electron microinstability drive.     

Landau Damping for the Linearized Vlasov Poisson Equation in a Weakly Collisional Regime

In this paper, we consider the linearized Vlasov–Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter \({\varepsilon }\) in front of the collision operator which will tend to 0. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker–Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter \({\varepsilon }\) as it goes to 0.     

Cyclokinetic models and simulations for high-frequency turbulence in fusion plasmas

Gyrokinetics is widely applied in plasma physics. However, this framework is limited to weak turbulence levels and low drift-wave frequencies because high-frequency gyro-motion is reduced by the gyro-phase averaging. In order to test where gyrokinetics breaks down, Waltz and Zhao developed a new theory, called cyclokinetics [R. E. Waltz and Zhao Deng, Phys. Plasmas 20, 012507 (2013)]. Cyclokinetics dynamically follows the high-frequency ion gyro-motion which is nonlinearly coupled to the low-frequency drift-waves interrupting and suppressing gyro-averaging. Cyclokinetics is valid in the high-frequency (ion cyclotron frequency) regime or for high turbulence levels. The ratio of the cyclokinetic perturbed distribution function over equilibrium distribution function δf/F can approach 1.This work presents, for the first time, a numerical simulation of nonlinear cyclokinetic theory for ions, and describes the first attempt to completely solve the ion gyro-phase motion in a nonlinear turbulence system. Simulations are performed [Zhao Deng and R. E. Waltz, Phys. Plasmas 22(5), 056101 (2015)] in a local flux-tube geometry with the parallel motion and variation suppressed by using a newly developed code named rCYCLO, which is executed in parallel by using an implicit time-advanced Eulerian (or continuum) scheme [Zhao Deng and R. E. Waltz, Comp. Phys. Comm. 195, 23 (2015)]. A novel numerical treatment of the magnetic moment velocity space derivative operator guarantee saccurate conservation of incremental entropy.By comparing the more fundamental cyclokinetic simulations with the corresponding gyrokinetic simulations, the gyrokinetics breakdown condition is quantitatively tested. Gyrokinetic transport and turbulence level recover those of cyclokinetics at high relative ion cyclotron frequencies and low turbulence levels, as required. Cyclokinetic transport and turbulence level are found to be lower than those of gyrokinetics at high turbulence levels and low-Ω* values with stable ion cyclotron modes. The gyrokinetic approximation is found to break down when the density perturbation exceeds 20%, or when the ratio of nonlinear E×B frequency over ion cyclotron frequency exceeds 20%. This result indicates that the density perturbation of the Tokamak L-mode near-edge is not sufficiently large for breaking the gyro-phase averaging. For cyclokinetic simulations with sufficiently unstable ion cyclotron (IC) modes and sufficiently low Ω* ～10, the high-frequency component of the cyclokinetic transport can exceed that of the gyrokinetic transport. However, the low-frequency component of the cyclokinetic transport does not exceed that of the gyrokinetic transport. For higher and more physically relevant Ω* ?50 values and physically realistic IC driving rates, the low-frequency component of the cyclokinetic transport remains smaller than that of the gyrokinetic transport. In conclusion, the “L-mode near-edge short-fall” phenomenon, observed in some low-frequency gyrokinetic turbulence transport simulations, does not arise owing to the nonlinear coupling of high-frequency ion cyclotron motion to low-frequency drift motion.     

A Vlasov–Fokker–Planck code for high energy density physics

OSHUN is a parallel relativistic 2D3P Vlasov–Fokker–Planck code, developed primarily to study electron transport and instabilities pertaining to laser-produced—including laser-fusion—plasmas. It incorporates a spherical harmonic expansion of the electron distribution function, where the number of terms is an input parameter that determines the angular resolution in momentum-space. The algorithm employs the full 3D electromagnetic fields and a rigorous linearized Fokker–Planck collision operator. The numerical scheme conserves energy and number density. This enables simulations for plasmas with temperatures from MeV down to a few eV and densities from less than critical to more than solid. Kinetic phenomena as well as electron transport physics can be recovered accurately and efficiently.     

Electromagnetic transport from microtearing mode turbulence

This Letter presents nonlinear gyrokinetic simulations of microtearing mode turbulence. The simulations include collisional and electromagnetic effects and use experimental parameters from a high-β discharge in the National Spherical Torus Experiment. The predicted electron thermal transport is comparable to that given by experimental analysis, and it is dominated by the electromagnetic contribution of electrons free-streaming along the resulting stochastic magnetic field line trajectories. Experimental values of flow shear can significantly reduce the predicted transport.     

Quantum Energy-Transport and Drift-Diffusion Models

We show that Quantum Energy-Transport and Quantum Drift-Diffusion models can be derived through diffusion limits of a collisional Wigner equation. The collision operator relaxes to an equilibrium defined through the entropy minimization principle. Both models are shown to be entropic and exhibit fluxes which are related with the state variables through spatially non-local relations. Thanks to an h expansion of these models, h² perturbations of the Classical Energy-Transport and Drift-Diffusion models are found. In the Drift-Diffusion case, the quantum correction is the Bohm potential and the model is still entropic. In the Energy-Transport case however, the quantum correction is a rather complex expression and the model cannot be proven entropic.This revised version was published online in March 2005 with corrections to the title. In the previous version, the author names were missing.     

COLLISIONAL EFFECTS IN CLOUD-IN-CELL SCHEME

In this paper, a physical model and numerical method in cloud-in-cell(C1C) scheme for simulation of weak collisional effects are proposed.Collisions are introduced at reconstruction time in terms of a one-dimensional Fokker-Planck operator., The friction term of the collision,operator is equivalent to the introduction of a velocity-dependent acceleration a(u)=-uy(u) into the equation of motion of the particles. The diffusion term is represented by finite differences over the nonuniform velocity grid. Col lisional relaxation of two Maxwellian beams is studied.The numerical results are in good agreement. with the analgtical values.     

Entanglement dephasing of spatially separated multiparticle GHZ states coupled to a squeezed environment

In tokamak plasmas, it is recognized that ITG (ion temperature gradient instability) and trapped electron modes (TEM) are held responsible for turbulence giving rise to anomalous transport. The present work focuses on the building of a model including trapped kinetic ions and trapped kinetic electrons. For this purpose, the dimensionality is reduced by averaging the motion over the cyclotron motion and the “banana” orbits, according to the fact that the instabilities are characterized by frequencies of the order of the low trapped particle precession frequency. Moreover, a set of action-angle variables is used. The final model is 4D (two-dimensional phase space parametrized by the two first adiabatic invariants namely the particle energy and the trapping parameter). In this paper, the trapped ion and electron modes (TIM and TEM) are studied by using a linear analysis of the model. This work is currently performed in order to include trapped electrons in an existing semi lagrangian code for which TIM modes are already taken into account. This study can be considered as a first step in order to include kinetic trapped electrons in the 5D gyrokinetic code GYSELA [J. Abiteboul et al., ESAIM Proc. 32, 103 (2011)].     

The global version of the gyrokinetic turbulence code GENE

The understanding and prediction of transport due to plasma microturbulence is a key open problem in modern plasma physics, and a grand challenge for fusion energy research. Ab initio simulations of such small-scale, low-frequency turbulence are to be based on the gyrokinetic equations, a set of nonlinear integro-differential equations in reduced (five-dimensional) phase space. In the present paper, the extension of the well-established and widely used gyrokinetic code GENE [F. Jenko, W. Dorland, M. Kotschenreuther, B.N. Rogers, Electron temperature gradient driven turbulence, Phys. Plasmas 7 (2000) 1904–1910] from a radially local to a radially global (nonlocal) version is described. The necessary modifications of both the basic equations and the employed numerical methods are detailed, including, e.g., the change from spectral methods to finite difference and interpolation techniques in the radial direction and the implementation of sources and sinks. In addition, code verification studies and benchmarks are presented.     

Diffusion in a collisional standard map

Test particle evaluation of the diffusion coefficient in the presence of magnetic field fluctuations and binary collisions is presented. Chaotic magnetic field lines originate from resonant magnetic perturbations (RMPs). To lowest order, charged particles follow magnetic field lines. Drifts and interaction (collisions) with other particles decorrelate particles from the magnetic field lines. We model the binary collision process by a constant collision frequency. The magnetic field configuration including perturbations on the integrable Hamiltonian part is such that the single particle motion can be followed by a collisional version of a Chirikov-Taylor (standard) map. Frequent collisions are allowed for. Scaling of the diffusion beyond the quasilinear and subdiffusive behaviour is investigated in dependence on the strength of the magnetic perturbations and the collision frequency. The appearance of the so called Rechester-Rosenbluth regime is verified. It is further shown that the so called Kadomtsev-Pogutse diffusion coefficient is the strong collisional limit of the Rechester-Rosenbluth formula. The theoretical estimates are supplemented by numerical simulations.     

On the collision operator for quantum systems with discrete spectrum

The existence of the asymptotic collision operator ψ(0) is established for a class of observables. Sufficient conditions are given for the vanishing of ψ(0) and for the existence of collisional and non-collisional invariants.     

Lattice-Boltzmann Simulations of Particle-Fluid Suspensions

This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,^{(69, 70)} taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.     

Two-Dimensional MHD Simulations of Magnetic Field Penetration in the Plasma Opening Switch

Three mechanisms for anomalous magnetic field penetration have been investigated in computer simulations of Sandia's plasma opening switch (POS). The POS simulations have been performed using the two-dimensional (2-D) two-temperature single-fluid magnetohydrodynamic (MHD) code HAM [1], [2]. The three penetration mechanisms considered are 1) the Chodura model based on the ion-acoustic instability where the saturated value of the anomalous collision frequency is approximated by the ion plasma frequency; 2) a model based on the lower hybrid instability in which the anomalous collision frequency is proportional to both the ion plasma frequency and the electron drift velocity; and 3) a model that limits the ion drift velocity to the plasma influx velocity. Two-dimensional MHD calculations of the POS will be presented which show these models to be qualitatively similar for densities above a few 1013 cm-3, though at lower densities they can be quite different. The calculations are compared to experiments, and some agreement is seen with the lower hybrid model. The other models compare only marginally to experimental results.