Fokker-Planck Collision Operator Research Articles

Energetic particles transport in constants of motion space due to collisions in tokamak plasmas

Abstract The spatio-temporal evolution of the energetic particles in the transport time scale in tokamak plasmas is a key issue of the plasmas confinement, especially in burning plasmas. In order to include sources and sinks and collisional slowing down processes, a new solver, ATEP-3D was implemented to simulate the evolution of the EP distribution in the three-dimensional constants of motion (CoM) space. The Fokker-Planck collision operator represented in the CoM space is derived and numerically calculated. The collision coefficients are averaged over the unperturbed orbits to capture the fundamental properties of EPs. ATEP-3D is fully embedded in ITER IMAS framework and combined with the LIGKA/HAGIS codes. The finite volume method and the implicit Crank-Nicholson scheme are adopted due to their optimal numerical properties for transport time scale studies. ATEP-3D allows the analysis of the particle and power balance with the source and sink during the transport process to evaluate the EP confinement properties.

A unified description of atomic physics for electron Fokker–Planck calculations

Most realistic kinetic calculations for tokamak plasmas are now required to incorporate the effect of partially ionized high-Z elements arising either from uncontrolled influxes of metallic impurities, such as tungsten in high input power regimes or from mitigation of runaway electrons generated after possible major disruptions by massive gas injection. The usual electron–ion Fokker–Planck collision operator must therefore be modified, because all plasma atoms are not entirely ionized, as is the case for light elements. This represents a challenge, in order to perform fast but also accurate calculations, regardless of the type of element present in the plasma, but also their local levels of ionization while covering a wide range of electron energies in a consistent way, from a few keV to tens of MeV in plasmas whose electron temperature may itself vary from 10 eV to several keV. In this context, a unified description of the atomic models is proposed, based on a multi-Yukawa representation of the electrostatic potential calibrated against results obtained by advanced quantum calculations. Besides the possibility to improve the description of inner and outer atomic shells in the determination of the atomic form factor, this model allows one to derive analytical formulations for both elastic and inelastic scattering, which can then be easily incorporated in kinetic calculations. The impact of the number of exponentials in the description of the atomic potential is discussed, and a comparison with simple and advanced atomic models is also performed.

Fokker-Planck multi-species equations in the adiabatic asymptotics

The main concern of the present paper is the study of the multi-scale dynamics of thermonuclear fusion plasmas via a multi-species Fokker-Planck kinetic model. One of the goals is the generalization of the standard Fokker-Planck collision operator to a multi-species one, conserving mass, total momentum and energy, as well as satisfying Boltzmann's H-theorem. Secondly, we perform on one hand a mathematical asymptotic limit, letting the electron/ion mass ratio converging towards zero, to obtain a thermodynamic equilibrium state for the electrons (adiabatic regime), whereas the ions are kept kinetic. On the other hand, we develop a first numerical scheme, based on a Hermite spectral method, and perform numerical simulations to investigate in more details this asymptotic limit.

Nonlinear Fokker-Planck collision operator in Rosenbluth form for gyrokinetic simulations using discontinuous Galerkin method

A gyroaveraged nonlinear collision operator is formulated based on the Fokker-Planck operator in the Rosenbluth-MacDonald-Judd (RMJ) potential form and implemented for the gyrokinetic simulations with the discontinuous Galerkin scheme. The divergence structure of the original RMJ form is carefully preserved throughout the formulation to guarantee the density conservation while neglecting the finite Larmor radius effect. The B-spline finite element method is used to calculate the Rosenbluth potentials for the nonlinear collision operator. In addition to the nonlinear collision operator, linear and Dougherty collision models are also implemented to assess the benefits and drawbacks of each model. For the conservation of the parallel momentum and energy, we adopt a simple advection-diffusion model which numerically enforces the conservation of physical quantities. From bump-on-tail relaxation tests, the monotonically increasing entropy in time and conservation properties are demonstrated for the developed collision operator. Also, a few theoretical predictions for the neoclassical physics such as the neoclassical heat flux, poloidal flow and collisional damping of zonal flow are successfully reproduced by numerical simulations.

Kinetic simulations of collision-less plasmas in open magnetic geometries *

Laboratory plasmas in open magnetic geometries can be found in many different applications such as (a) scrape-of-layer (SOL) and divertor regions in toroidal confinement fusion devices, (b) linear divertor simulators, (c) plasma-based thrusters and (d) magnetic mirrors etc. A common feature of these plasma systems is the need to resolve, in addition to velocity space, at least one physical dimension (e.g. along flux lines) to capture the relevant physics. In general, this requires a kinetic treatment. Fully kinetic particle-in-cell (PIC) simulations can be applied but at the expense of large computational effort. A common way to resolve this is to use a hybrid approach: kinetic ions and fluid electrons. In the present work, the development of a hybrid PIC computational tool suitable for open magnetic geometries is described which includes (a) the effect of non-uniform magnetic fields, (b) finite fully-absorbing boundaries for the particles and (c) volumetric particle sources. Analytical expressions for the momentum transport in the paraxial limit are presented with their underlying assumptions and are used to validate the results from the PIC simulations. A general method is described to construct discrete particle distribution functions in a state of mirror-equilibrium. This method is used to obtain the initial state for the PIC simulation. Collisionless simulations in a mirror geometry are performed. The results show that the effect of magnetic compression is correctly described and momentum is conserved. The self-consistent electric field is calculated and is shown to modify the ion velocity distribution function in a manner consistent with analytic theory. Based on this analysis, the ion distribution function is understood in terms of a loss-cone distribution and an isotropic Maxwell-Boltzmann distribution driven by a volumetric plasma source. Finally, the inclusion of a Monte Carlo based Fokker-Planck collision operator is discussed in the context of future work.

Importance of gyrokinetic exact Fokker-Planck collisions in fusion plasma turbulence.

Gyrokinetic simulations of turbulence are fundamental to understanding and predicting particle and energy loss in magnetic fusion devices. Previous works have used model collision operators with approximate field-particle terms of unknown accuracy and/or have neglected collisional finite Larmor radius effects. This work moves beyond models to demonstrate important corrections using a gyrokinetic Fokker-Planck collision operator with the exact field-particle terms, in realistic simulations of turbulence in magnetically confined fusion plasmas. The exact operator shows significant corrections for temperature-gradient-driven trapped electron mode turbulence and zonal flow damping, and for microtearing modes in a Joint European Torus pedestal under ITER-like wall conditions. Analysis of the corrections using parameter scans motivates an accurate model which closely reproduces the exact results while reducing computational demands.

An analytic model for the collisional transport and poloidal asymmetry distribution of impurities in tokamak plasmas

The coupling between the poloidal distribution and the collisional flux of impurities can be exploited to derive a simplified analytical model covering toroidal rotation and electrostatic potential asymmetry effects, valid in the Pfirsh-Schlüter regime. This model is compared with earlier works and with the drift-kinetic code NEO, which includes the full Fokker-Planck collision operator. The low computational cost of the model, compared to NEO, is particularly adapted for fast integrated simulation purposes.

Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators

We present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $v^{2}$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.

First implementation of gyrokinetic exact linearized Landau collision operator and comparison with models

Gyrokinetic simulations are fundamental to understanding and predicting turbulent transport in magnetically confined fusion plasmas. Previous simulations have used model collision operators with approximate field-particle terms of unknown accuracy and/or have neglected collisional finite Larmor radius (FLR) effects. We have implemented the linearized Fokker–Planck collision operator with exact field-particle terms and full FLR effects in a gyrokinetic code (GENE). The new operator, referred to as “exact” in this paper, allows the accuracy of model collision operators to be assessed. The conservative Landau form is implemented because its symmetry underlies the conservation laws and the H-theorem, and enables numerical methods to preserve this conservation, independent of resolution. The implementation utilizes the finite-volume method recently employed to discretize the Sugama collision model in GENE, allowing direct comparison between the two operators. Results show that the Sugama model appears accurate for the growth rates of trapped electron modes (TEMs) driven only by density gradients, but appreciably underestimates the growth rates as the collisionality and electron temperature gradient increase. The TEM turbulent fluxes near the nonlinear threshold using the exact operator are similar to the Sugama model for the ηe=d ln Te/d ln ne=0 case, but substantially larger than the Sugama model for the ηe=1 case. The FLR effects reduce the growth rates increasingly with wavenumber, deepening a “valley” at the intermediate binormal wavenumber as the unstable mode extends from the TEM regime to the electron temperature gradient instability regime. Application to the Hinton–Rosenbluth problem shows that zonal flows decay faster as the radial wavenumber increases and the exact operator yields weaker decay rates.

Gyrokinetic study of collisional resonant magnetic perturbation (RMP)-driven plasma density and heat transport in tokamak edge plasma using a magnetohydrodynamic screened RMP field

The total-f gyrokinetic particle-in-cell code XGC is applied to study various aspects of collisional transport in the tokamak edge pedestal in the presence of resonant magnetic perturbations (RMPs) calculated with M3D-C1. Simulations including the separatrix and scrape-off layer are exercised for a model DIII-D H-mode plasma. Neutral particle recycling is modeled by means of ionization and charge exchange with the plasma. A fully nonlinear Fokker–Planck collision operator is utilized to account for non-Maxwellian edge plasma. The study yields kinetic evidence that (i) neglecting the non-axisymmetric component of the RMP-generated electrostatic potential perturbation could yield a fictitiously high particle pump-out rate in the edge pedestal; that (ii) the experimental level of particle pump-out from the pedestal cannot be provided by collisional transport across Kolmogorov–Arnold–Moser surfaces except across the magnetic separatrix where stochastic magnetic field lines exist; that (iii) the H-mode type electron heat barrier is retained in the steep slope area under the M3D-C1 calculated RMPs; and that (iv) the stochastic electron heat-transport rate around the magnetic separatrix surface is much lower than the prediction by Rechester–Rosenbluth (1978 Phys. Rev. Lett. 40 38–41) and is qualitatively consistent with the experimentally observed rate. Detailed kinetic physics analysis on how the transport fluxes, electrostatic potential perturbation, and -well structure respond to onset of RMPs is also reported.

Influence of 3D plasmoid dynamics on the transition from collisional to kinetic reconnection

Within the resistive magnetohydrodynamic model, high-Lundquist number reconnection layers are unstable to the plasmoid instability, leading to a turbulent evolution where the reconnection rate can be independent of the underlying resistivity. However, the physical relevance of these results remains questionable for many applications. First, the reconnection electric field is often well above the runaway limit, implying that collisional resistivity is invalid. Furthermore, both theory and simulations suggest that plasmoid formation may rapidly induce a transition to kinetic scales, due to the formation of thin current sheets. Here, this problem is studied for the first time using a first-principles kinetic simulation with a Fokker-Planck collision operator in 3D. The low-β reconnecting current layer thins rapidly due to Joule heating before the onset of the oblique plasmoid instability. Linear growth rates for standard (ky = 0) tearing modes agree with semicollisional boundary layer theory, but the angular spectrum of oblique (|ky|>0) modes is significantly narrower than predicted. In the nonlinear regime, flux-ropes formed by the instability undergo complex interactions as they are advected and rotated by the reconnection outflow jets, leading to a turbulent state with stochastic magnetic field. In a manner similar to previous 2D results, super-Dreicer fields induce a transition to kinetic reconnection in thin current layers that form between flux-ropes. These results may be testable within new laboratory experiments.

An integral transform technique for kinetic systems with collisions

The linearized Vlasov-Poisson system can be exactly solved using the G-transform, an integral transform introduced in Morrison and Pfirsch [Phys. Fluids B 4, 3038–3057 (1992)] and Morrison [Phys. Plasmas 1, 1447 (1994); Transp. Theory Stat. Phys. 29, 397 (2000)] that removes the electric field term, leaving a simple advection equation. We investigate how this integral transform interacts with the Fokker-Planck collision operator. The commutator of this collision operator with the G-transform (the “shielding term”) is shown to be negligible. We exactly solve the advection-diffusion equation without the shielding term. This solution determines when collisions dominate and when advection (i.e., Landau damping) dominates. This integral transform can also be used to simplify gyro-/drift-kinetic equations. We present new gyrofluid equations formed by taking moments of the G-transformed equation. Since many gyro-/drift-kinetic codes use Hermite polynomials as base elements, we include an explicit calculation of their G-transform.

Anomalous Diffusion Limit of Kinetic Equations in Spatially Bounded Domains

This paper is devoted to the anomalous diffusion limit of kinetic equations with a fractional Fokker–Planck collision operator in a spatially bounded domain. We consider two boundary conditions at the kinetic scale: absorption and specular reflection. In the absorption case, we show that the long time/small mean free path asymptotic dynamics are described by a fractional diffusion equation with homogeneous Dirichlet-type boundary conditions set on the whole complement of the spatial domain. On the other hand, specular reflections will give rise to a new operator which we call specular diffusion operator and write $${(-\Delta)_{{\rm SR}}^{s}}$$ . This non-local diffusion operator strongly depends on the geometry of the domain and includes in its definition the interaction between the diffusion and the boundary. We consider two types of domains: half-spaces and balls in $${\mathbb{R}^d}$$ . In these domains, we prove properties of the specular diffusion operator and establish existence and uniqueness of weak solutions to the associated heat-type equation.

An adaptive, implicit, conservative, 1D-2V multi-species Vlasov–Fokker–Planck multi-scale solver in planar geometry

We consider a 1D-2V Vlasov–Fokker–Planck multi-species ionic description coupled to fluid electrons. We address temporal stiffness with implicit time stepping, suitably preconditioned. To address temperature disparity in time and space, we extend the conservative adaptive velocity-space discretization scheme proposed in [Taitano et al., J. Comput. Phys., 318, 391–420, (2016)] to a spatially inhomogeneous system. In this approach, we normalize the velocity-space coordinate to a temporally and spatially varying local characteristic speed per species. We explicitly consider the resulting inertial terms in the Vlasov equation, and derive a discrete formulation that conserves mass, momentum, and energy up to a prescribed nonlinear tolerance upon convergence. Our conservation strategy employs nonlinear constraints to enforce these properties discretely for both the Vlasov operator and the Fokker–Planck collision operator. Numerical examples of varying degrees of complexity, including shock-wave propagation, demonstrate the favorable efficiency and accuracy properties of the scheme.

Positivity-preserving scheme for two-dimensional advection–diffusion equations including mixed derivatives

In this work, we propose a positivity-preserving scheme for solving two-dimensional advection–diffusion equations including mixed derivative terms, in order to improve the accuracy of lower-order methods. The solution of these equations, in the absence of mixed derivatives, has been studied in detail, while positivity-preserving schemes for mixed derivative terms have received much less attention. A two-dimensional diffusion equation, for which the analytical solution is known, is solved numerically to show the applicability of the scheme. It is further applied to the Fokker–Planck collision operator in two-dimensional cylindrical coordinates under the assumption of local thermal equilibrium. For a thermal equilibration problem, it is shown that the scheme conserves particle number and energy, while the preservation of positivity is ensured and the steady-state solution is the Maxwellian distribution.

Generalized fluid theory including non-Maxwellian kinetic effects

The results obtained by the plasma physics community for the validation and the prediction of turbulence and transport in magnetized plasmas come mainly from the use of very central processing unit (CPU)-consuming particle-in-cell or (gyro)kinetic codes which naturally include non-Maxwellian kinetic effects. To date, fluid codes are not considered to be relevant for the description of these kinetic effects. Here, after revisiting the limitations of the current fluid theory developed in the 19th century, we generalize the fluid theory including kinetic effects such as non-Maxwellian super-thermal tails with as few fluid equations as possible. The collisionless and collisional fluid closures from the nonlinear Landau Fokker–Planck collision operator are shown for an arbitrary collisionality. Indeed, the first fluid models associated with two examples of collisionless fluid closures are obtained by assuming an analytic non-Maxwellian distribution function (e.g. the INMDF (Izacard, O. 2016b Kinetic corrections from analytic non-Maxwellian distribution functions in magnetized plasmas. Phys. Plasmas 23, 082504) that stands for interpreted non-Maxwellian distribution function). One of the main differences with the literature is our analytic representation of the distribution function in the velocity phase space with as few hidden variables as possible thanks to the use of non-orthogonal basis sets. These new non-Maxwellian fluid equations could initiate the next generation of fluid codes including kinetic effects and can be expanded to other scientific disciplines such as astrophysics, condensed matter or hydrodynamics. As a validation test, we perform a numerical simulation based on a minimal reduced INMDF fluid model. The result of this test is the discovery of the origin of particle and heat diffusion. The diffusion is due to the competition between a growing INMDF on short time scales due to spatial gradients and the thermalization on longer time scales. The results shown here could provide the insights to break some of the unsolved puzzles of turbulence.

NORSE: A solver for the relativistic non-linear Fokker–Planck equation for electrons in a homogeneous plasma

Energetic electrons are of interest in many types of plasmas, however previous modeling of their properties has been restricted to the use of linear Fokker–Planck collision operators or non-relativistic formulations. Here, we describe a fully non-linear kinetic-equation solver, capable of handling large electric-field strengths (compared to the Dreicer field) and relativistic temperatures. This tool allows modeling of the momentum-space dynamics of the electrons in cases where strong departures from Maxwellian distributions may arise. As an example, we consider electron runaway in magnetic-confinement fusion plasmas and describe a transition to electron slide-away at field strengths significantly lower than previously predicted. Program summaryProgram title: NORSEProgram Files doi:http://dx.doi.org/10.17632/86wmgj758w.1Licensing provisions: GPLv3Programming language: MatlabNature of problem: Solves the Fokker–Planck equation for electrons in 2D momentum space in a homogeneous plasma (allowing for magnetization), using a relativistic non-linear electron–electron collision operator. Electric-field acceleration, synchrotron-radiation-reaction losses, as well as heat and particle sources are included. Scenarios with time-dependent plasma parameters can be studied.Solution method: The kinetic equation is represented on a non-uniform 2D finite-difference grid and is evolved using a linearly implicit time-advancement scheme. A mixed finite-difference-Legendre-mode representation is used to obtain the relativistic potentials (analogous to the non-relativistic Rosenbluth potentials) from the distribution.

Parallel Fokker–Planck-DSMC algorithm for rarefied gas flow simulation in complex domains at all Knudsen numbers

A major challenge for the conventional Direct Simulation Monte Carlo (DSMC) technique lies in the fact that its computational cost becomes prohibitive in the near continuum regime, where the Knudsen number (Kn)—characterizing the degree of rarefaction—becomes small. In contrast, the Fokker–Planck (FP) based particle Monte Carlo scheme allows for computationally efficient simulations of rarefied gas flows in the low and intermediate Kn regime. The Fokker–Planck collision operator—instead of performing binary collisions employed by the DSMC method—integrates continuous stochastic processes for the phase space evolution in time. This allows for time step and grid cell sizes larger than the respective collisional scales required by DSMC. Dynamically switching between the FP and the DSMC collision operators in each computational cell is the basis of the combined FP-DSMC method, which has been proven successful in simulating flows covering the whole Kn range. Until recently, this algorithm had only been applied to two-dimensional test cases. In this contribution, we present the first general purpose implementation of the combined FP-DSMC method. Utilizing both shared- and distributed-memory parallelization, this implementation provides the capability for simulations involving many particles and complex geometries by exploiting state of the art computer cluster technologies.

An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes

In this study, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth–Fokker–Planck (RFP) equation, which is exactly mass-, momentum-, and energy-conserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local thermal velocity, v t h ( t ) , and explicitly considers the resulting inertial terms in the Fokker–Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker–Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic v t h -ratio-based expansion of the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. In particular, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.

Gyrokinetic neoclassical study of the bootstrap current in the tokamak edge pedestal with fully non-linear Coulomb collisions

As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.