A novel radially local approximation of the drift kinetic equation is presented. The new drift kinetic equation that includes both ${\bf E} \times {\bf B}$ and tangential magnetic drift terms is written in the conservative form and it has favorable properties for numerical simulation that any additional terms for particle and energy sources are unnecessary for obtaining stationary solutions under the radially local approximation. These solutions satisfy the intrinsic ambipolarity condition for neoclassical particle fluxes in the presence of quasisymmetry of the magnetic field strength. Also, another radially local drift kinetic equation is presented, from which the positive definiteness of entropy production due to neoclassical transport and Onsager symmetry of neoclassical transport coefficients are derived while it sacrifices the ambipolarity condition for neoclassical particle fluxes in axisymmetric and quasi-symmetric systems.
The collisional drift wave instability in a straight magnetic field configuration is studied within a full-F gyro-fluid model, which relaxes the Oberbeck-Boussinesq (OB) approximation. Accordingly, we focus our study on steep background density gradients. In this regime we report on corrections by factors of order one to the eigenvalue analysis of former OB approximated approaches as well as on spatially localised eigenfunctions, that contrast strongly with their OB approximated equivalent. Remarkably, non-modal phenomena arise for large density inhomogeneities and for all collisionalities. As…
Existing high-order variational drift kinetic theories contain unphysical rapidly varying modes that are not seen at low-orders. These unphysical modes, which may be rapidly oscillating, damped, or growing, are ushered in by a failure of conventional high-order drift kinetic theory to preserve the structure of its parent model's initial value problem (Vlasov-Poisson for electrostatics, Vlasov-Darwin or Vlasov-Maxwell for electromagnetics.) In short, the system phase space is unphysically enlarged in conventional high-order variational drift kinetic theory. I present an alternative, "renormaliz…
The linear non-modal kinetic theory of the kinetic drift-Alfven instability of plasma shear flows reveals the temporal non-modal growth with growing with time growth rate. The turbulent scattering of the sheared modes on ions accelerates this growth. The instability ceases its growth when the coupling of the drift and Alfven waves violates due to the changing with time frequencies of the drift and Alfven waves in shear flow.
Conventional radially-local neoclassical calculations become inadequate if the radial gradient scale lengths of the H-mode pedestal become as small as the poloidal ion gyroradius. Here, we describe a radially global $δf$ continuum code that generalizes neoclassical calculations to allow stronger gradients. As with conventional neoclassical calculations, the formulation is time-independent and requires only the solution of a single sparse linear system. We demonstrate precise agreement with an asymptotic analytic solution of the radially global kinetic equation in the appropriate limits of aspe…
The non-modal kinetic theory of the kinetic drift instability of plasma shear flows [Phys.Plasmas, 18, 062103 (2011)] is extended to the investigation of the long-time evolution of the hydrodynamic ion temperature gradient and resistive drift instabilities in plasma shear flow. We find, that these hydrodynamic instabilities passed in their temporal evolution in shear flow through the kinetic stage of the evolution. In linear theory, this evolution involves the time dependent, due to flow shear, effects of the finite Larmor radius, which resulted in the non-modal effect of the decrease with tim…
The linear and renormalized nonlinear kinetic theory of drift instability of plasma shear flow across the magnetic field, which has the Kelvin's method of shearing modes or so-called non-modal approach as its foundation, is developed. The developed theory proves that the time-dependent effect of the finite ion Larmor radius is the key effect, which is responsible for the suppression of drift turbulence in an inhomogeneous electric field. This effect leads to the non-modal decrease of the frequency and growth rate of the unstable drift perturbations with time. We find that turbulent scattering …
We formulate a finite-size particle numerical model of strongly magnetized plasmas in the drift-kinetic approximation. We use the phase space action as an alternative to previous variational formulations based on Low's Lagrangian or on a Hamiltonian with a non-canonical Poisson bracket. The useful property of this variational principle is that it allows independent transformations of particle coordinates and velocities, i.e., transformations in particle phase space. With such transformations, a finite degree-of-freedom drift-kinetic action is obtained through time-averaging of the finite degre…
We present the TRIDK code, which is written in Python, that solves Triad Reduced Interactions for a Drift Kinetic system of equations. The four-dimensional drift kinetic system of equations captures nonlinear electromagnetic (or electrostatic) interactions for a magnetized plasma at scales larger than the ion gyroradius. In the TRIDK code, the nonlinear interaction are modelled using a compound of nested icosahedra-dodecohedra shells in wave space, which allows to perform computationally efficient simulations over a large span of scales. This reduced approach captures essential nonlinear dynam…
The kinetic effects of electrons are important to long wavelength magnetohydrodynamic(MHD)instabilities and short wavelength drift-Alfvenic instabilities responsible for turbulence transport in magnetized plasmas, since the non-adiabatic electron can interact with, modify and drive the low frequency instabilities. A novel conservative split weight scheme is proposed for the electromagnetic simulation with drift kinetic electrons in tokamak plasmas, which shows great computational advantages that there is no numerical constrain of electron skin depth on the perpendicular grid size without sacri…
A drift-kinetic model to describe the plasma dynamics in the scrape-off layer region of tokamak devices at arbitrary collisionality is derived. Our formulation is based on a gyroaveraged Lagrangian description of the charged particle motion, and the corresponding drift-kinetic Boltzmann equation that includes a full Coulomb collision operator. Using a Hermite-Laguerre velocity space decomposition of the gyroaveraged distribution function, a set of equations to evolve the coefficients of the expansion is presented. By evaluating explicitly the moments of the Coulomb collision operator, distribu…
A kinetic consideration of an axisymmetric equilibrium with vanishing electric field near the magnetic axis shows that del f should not vanish on axis within the framework of Vlasov theory while it can either vanish or not in the framework of both a drift kinetic and a gyrokinetic theories (f is either the pertinent particle or the guiding center distribution function). This different behavior, relating to the reduction of phase space which leads to the loss of a Vlasov constant of motion, may result in the construction of different currents in the reduced phase space than the Vlasov ones. Thi…
A four-dimensional plasma model able to describe the scrape-off layer region of tokamak devices at arbitrary collisionality is derived in the drift-reduced limit. The basis of the model is provided by a drift-kinetic equation that retains the full non-linear Coulomb collision operator and describes arbitrarily far from equilibrium distribution functions. By expanding the dependence of distribution function over the perpendicular velocity in a Laguerre polynomial basis and integrating over the perpendicular velocity, a set of four-dimensional moment equations for the expansion coefficients of t…
Transfer of free energy from large to small velocity-space scales by phase mixing leads to Landau damping in a linear plasma. In a turbulent drift-kinetic plasma, this transfer is statistically nearly canceled by an inverse transfer from small to large velocity-space scales due to "anti-phase-mixing" modes excited by a stochastic form of plasma echo. Fluid moments (density, velocity, temperature) are thus approximately energetically isolated from the higher moments of the distribution function, so phase mixing is ineffective as a dissipation mechanism when the plasma collisionality is small.
The benchmarks of the neoclassical transport codes based on the several local drift-kinetic models are reported here. Here, the drift-kinetic models are ZOW, ZMD, DKES-like, and global, as classified in [Matsuoka et al., Physics of Plasmas 22, 072511 (2015)]. The magnetic geometries of HSX, LHD, and W7-X are employed in the benchmarks. It is found that the assumption of $\boldsymbol E \times \boldsymbol B$ incompressibility causes discrepancy of neoclassical radial flux and parallel flow among the models, when $\boldsymbol E \times \boldsymbol B$ is sufficiently large compared to the magnetic …
We study a quasi-two-dimensional electrostatic drift kinetic system as a model for near-marginal ion temperature gradient (ITG) driven turbulence. A proof is given of the nonlinear stability of this system under conditions of linear stability. This proof is achieved using a transformation that diagonalizes the linear dynamics and also commutes with nonlinear $E\times B$ advection. For the case when linear instability is present, a corollary is found that forbids nonlinear energy transfer between appropriately defined sets of stable and unstable modes. It is speculated that this may explain the…
After reviewing the variational approach to splitting mean flow and fluctuation kinetics in the standard Vlasov theory, the same method is applied to the drift-kinetic equation from Littlejohn's theory of guiding-center motion. This process sheds a new light on drift-ordered fluid (drift-fluid) models, whose anisotropic pressure tensor is then considered in detail. In addition, current drift-fluid models are completed by the insertion of magnetization terms ensuring momentum conservation. Magnetization currents are also shown to lead to challenging aspects when drift-fluid models are coupled t…
We present a drift kinetic model for the free expansion of a thermal plasma out of a magnetic nozzle. This problem relates to plasma space propulsion systems, natural environments such as the solar wind, and end losses from the expander region of mirror magnetically confined fusion concepts such as the Gas Dynamic Trap. The model incorporates trapped and passing orbit types encountered in the mirror expander geometry and maps to an upstream thermal distribution. This boundary condition and quasineutrality require the generation of an ambipolar potential drop of $\sim5 T_e/e$, forming a thermal…
For a magnetised plasma in a straight magnetic guide field, we derive a set of four-dimensional kinetic equations, which can capture electromagnetic turbulence in the drift kinetic limit. To do so, we start from the gyrokinetic equations, employ a Laguerre decomposition in the perpendicular velocity direction, retain only the dominant gyroaverage contributions and only the first two Laguerre moments that source the electromagnetic fluctuations. The model conserves free energy, and can describe electromagnetic turbulence for a plasma at the transition between fluid and gyrokinetic regimes ($k_\…
Implicit discretization schemes for full-kinetic ion and drift-kinetic electron simulations
Southwestern Institute of Physics（西南物理研究院）；Department of Physics, University of Colorado at Boulder（科罗拉多大学博尔德分校物理系）；Institute of Plasma Physics, Chinese Academy of Science（中国科学院等离子体物理研究所）
The goal of this study is to develop an approximate self-consistent description of particle motion in strongly magnetised solar corona. We derive a set of reduced drift-kinetic equations based on the assumption that the gyro-velocity distribution is Maxwellian. The equations are tested using simple 1D models.
Inhomogeneous drift-wave turbulence can be modeled as an effective plasma where drift waves act as quantumlike particles and the zonal-flow velocity serves as a collective field through which they interact. This effective plasma can be described by a Wigner-Moyal equation (WME), which generalizes the quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves the wavelength scale. Unlike waves governed by manifestly quantumlike equations, whose WMEs can be borrowed from quantum mechanics and are commonly known, drift waves have Hamiltonians very different from those of conv…
KNOSOS (KiNetic Orbit-averaging SOlver for Stellarators) is a freely available, open-source code (\href{https://github.com/joseluisvelasco/KNOSOS}{https://github.com/joseluisvelasco/KNOSOS}) that calculates neoclassical transport in low-collisionality plasmas of three-dimensional magnetic confinement devices by solving the radially local drift-kinetic and quasineutrality equations. The main feature of KNOSOS is that it relies on orbit-averaging to solve the drift-kinetic equation very fast. KNOSOS treats rigorously the effect of the component of the magnetic drift that is tangent to magnetic s…
This work presents transport coefficients of electrons (bulk drift velocity, longitudinal diffusion coefficient, and effective ionization frequency) in CO2 measured under time-of-flight conditions over a wide range of the reduced electric field, 15Td <= E/N <= 2660Td in a scanning drift tube apparatus. The data obtained in the experiments are also applied to determine the effective steady-state Townsend ionization coefficient. These parameters are compared to the results of previous experimental studies, as well as to results of various kinetic computations: solutions of the electron Boltzmann…
Hamiltonian models for the first three moments of the drift-kinetic distribution function, namely the density, the fluid velocity and the parallel pressure, are derived from the Hamiltonian structure of the drift-kinetic equations. The link with the water-bag closure is established, showing that, unlike the one-dimensional Vlasov equations, these solutions are the only Hamiltonian fluid reductions for the drift-kinetic equation. These models are discussed through their equations of motion and their Casimir invariants.
In evaluating neoclassical transport by radially-local simulations, the magnetic drift tangential to a flux surface is usually ignored in order to keep the phase-space volume conservation. In this paper, effect of the tangential magnetic drift on the local neoclassical transport are investigated. To retain the effect of the tangential magnetic drift in the local treatment of neoclassical transport, a new local formulation for the drift kinetic simulation is developed. The compressibility of the phase-space volume caused by the tangential magnetic drift is regarded as a source term for the drif…
Upcoming tokamak experiments fueled with deuterium and tritium are expected to have large alpha particle populations. Such experiments motivate new attention to the theory of alpha particle confinement and transport. A key topic is the interaction of alphas with perturbations to the tokamak fields, including those from ripple and magnetohydrodynamic modes like Alfvén eigenmodes. These perturbations can transport alphas, leading to changed localization of alpha heating, loss of alpha power, and damage to device walls. Alpha interaction with these perturbations is often studied with single parti…
In this paper a conservation equation is derived for the radially dependent entropy in toroidal geometry using the local approximation of the gyro-kinetic framework. This equation naturally leads to an operative definition for the turbulence intensity. It is shown that the conservation equation can be split in two separate conservation equations, one describing the dynamics of the zonal modes and one for the non-zonal modes. In essence the paper provides an operative tool for both analytic as well as numeric studies of the radial propagation of turbulence in tokamak plasmas.
We present the derivation of kinetic formulas modeling the microscopic interaction of a charged particle withing a molecular gas under effect of thermal motion. Both elastic and inelastic processes are taken in account. The results were verified to reproduce the non-thermal formulas when the target molecule velocity is set to zero. A set of simulation is provided to highlight the effects in Argon and Carbon tetrafluoride. Our results can applied in Monte-Carlo simulation of particle drift at energies of the same order of the thermal kinetic energy of the buffer gas.
The Eulerian variational principle for the Vlasov-Poisson-Ampère system of equations in a general coordinate system is presented. The invariance of the action integral under an arbitrary spatial coordinate transformation is used to obtain the momentum conservation law and the symmetric pressure in a more direct way than using the translational and rotational symmetries of the system. Next, the Eulerian variational principle is given for the collisionless drift kinetic equation, where particles' phase-space trajectories in given electromagnetic fields are described by Littlejohn's guiding cente…
In this paper, we derive a new drift-kinetic method for estimating gradients in the plasma properties through a velocity space distribution at a single point. The gradients are intrinsically related to agyrotropic features of the distribution function. This method predicts the gradients in the magnetized distribution function, and can predict gradients of arbitrary moments of the gyrotropic background distribution function. The method allows for estimates on density and pressure gradients on the scale of a Larmor radius, proving to resolve smaller scales than any method currently available to …
We derive the conditions under which the fluid models obtained from the first two moments of Hamiltonian drift-kinetic systems of interest to plasma physics, preserve a Hamiltonian structure. The adopted procedure consists of determining closure relations that allow to truncate the Poisson bracket of the drift-kinetic system, expressed in terms of the moments, in such a way that the resulting operation is a Poisson bracket for functionals of the first two fluid moments. The analysis is carried out for a class of full drift-kinetic equations and also for drift-kinetic systems in which a splitti…
A new constant of motion for helically symmetric equilibria in the vicinity of the magnetic axis is obtained in the framework of Vlasov theory. In view of this constant of motion the Vlasov theory is compared with drift kinetic and gyrokinetic theories near axis. It turns out that as in the case of axisymmetric equilibria [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas 18, 064507 (2011)] the Vlasov current density thereon can differ appreciably from the drift kinetic and gyrokinetic current densities. This indicates some limitation on the implications of reduced kinetic theories, in partic…
We propose an asymptotic-preserving (AP), uniformly convergent numerical scheme for the relativistic collisional Drift-Kinetic Equation (rDKE) to simulate runaway electrons in axisymmetric toroidal magnetic field geometries typical of tokamak devices. The approach is derived from an exact Green's function solution with numerical approximations of quantifiable impact, and results in a simple, two-step operator-split algorithm, consisting of a collisional Eulerian step, and a Lagrangian orbit-integration step with analytically prescribed kernels. The AP character of the approach is demonstrated …
The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for a Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle ($θ$) with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes ($θ\neq 0$). We find th…
We present particle-in-cell (PIC) simulations of minority energetic protons in deuterium plasmas, which demonstrate a collective instability responsible for emission near the lower hybrid frequency and its harmonics. The simulations capture the lower hybrid drift instability in a regime relevant to tokamak fusion plasmas, and show further that the excited electromagnetic fields collectively and collisionlessly couple free energy from the protons to directed electron motion. This results in an asymmetric tail antiparallel to the magnetic field. We focus on obliquely propagating modes under cond…
The formation of zonal flows from inhomogeneous drift-wave (DW) turbulence is often described using statistical theories derived within the quasilinear approximation. However, this approximation neglects wave--wave collisions. Hence, some important effects such as the Batchelor--Kraichnan inverse-energy cascade are not captured within this approach. Here we derive a wave kinetic equation that includes a DW collision operator in the presence of zonal flows. Our derivation makes use of the Weyl calculus, the quasinormal statistical closure, and the geometrical-optics approximation. The obtained …
The linearized Vlasov-Poisson system can be exactly solved using the $G$-transform, an integral transform introduced in Refs. 1-3 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…
Fluid models that approximate kinetic effects have received attention recently in the modelling of large scale plasmas such as planetary magnetospheres. In three-dimensional reconnection, both reconnection itself and current sheet instabilities need to be represented appropriately. We show that a heat flux closure based on pressure gradients enables a ten moment fluid model to capture key properties of the lower-hybrid drift instability (LHDI) within a reconnection simulation. Characteristics of the instability are examined with kinetic and fluid continuum models, and its role in the three-dim…
Based on the drift-kinetic theory, we develop a model for particle acceleration and transport in solar flares. The model describes the evolution of the particle distribution function by means of a numerical simulation of the drift-kinetic Vlasov equation, which allows us to directly compare simulation results with observations within an actual parameter range of the solar corona. Using this model, we investigate the time evolution of the electron distribution in a flaring region. The simulation identifies two dominant mechanisms of electron acceleration. One is the betatron acceleration at the…
Basic physics of drift-wave turbulence and zonal flows has long been studied within the framework of wave-kinetic theory. Recently, this framework has been re-examined from first principles, which has led to more accurate yet still tractable "improved" wave-kinetic equations. In particular, these equations reveal an important effect of the zonal-flow "curvature" (the second radial derivative of the flow velocity) on dynamics and stability of drift waves and zonal flows. We overview these recent findings and present a consolidated high-level picture of (mostly quasilinear) zonal-flow physics wi…
The integration of kinetic effects in fluid models is important for global simulations of the Earth's magnetosphere. We use a two-fluid ten moment model, which includes the pressure tensor and has been used to study reconnection, to study the drift kink and lower hybrid drift instabilities. Using a nonlocal linear eigenmode analysis, we find that for the kink mode, the ten moment model shows good agreement with kinetic calculations with the same closure model used in reconnection simulations, while the electromagnetic and electrostatic lower hybrid instabilities require modeling the effects of…
Two-dimensional and three-dimensional kinetic simulation results reveal the importance of the Lower-Hybrid Drift Instability LHDI to the onset of magnetic reconnection. Both explicit and implicit kinetic simulations show that the LHDI heats electrons anisotropically and increases the peak current density. Linear theory predicts these modifications can increase the growth rate of the tearing instability by almost two orders of magnitude and shift the fastest growing modes to significantly shorter wavelengths. These predictions are confirmed by nonlinear kinetic simulations in which the growth a…
This work presents swarm parameters of electrons (the bulk drift velocity, the bulk longitudinal component of the diffusion tensor, and the effective ionization frequency) in C$_2$H$_n$, with $n =$ 2, 4 and 6, measured in a scanning drift tube apparatus under time-of-flight conditions over a wide range of the reduced electric field, 1 Td $\leq\,E/N\,\leq$ 1790 Td (1 Td = $10^{-21}$ Vm$^2$). The effective steady-state Townsend ionization coefficient is also derived from the experimental data. A kinetic simulation of the experimental drift cell allows estimating the uncertainties introduced in t…
This paper proposes a charge-conserving, variational, spatio-temporal discretization for the drift-kinetic Vlasov-Maxwell system, utilizing finite-elements for the electromagnetic fields and the particle-in-cell approach for the Vlasov distribution. The proposed scheme is fully electromagnetic, dealing with fields instead of potentials, and includes the effects of polarization and magnetization in the Gauss and Ampère-Maxwell laws, a consequence of reducing the full particle dynamics to drift-center dynamics. There is, however, no need to invert the Gauss law: it is satisfied automatically at …
