Particle Velocity Distribution Function Research Articles

On the analytical description of the nonlinear stage of the Weibel instability in collisionless anisotropic plasma

On the basis of the energy invariants for the Weibel instability in a collisionless non-relativistic plasma, an analytical relation is obtained between the instantaneous values of the space-averaged energy density of the magnetic field, its dominant wavenumber and the average plasma anisotropy parameter. The relation is valid for arbitrary particle velocity distribution functions, including ones that vary with time at the nonlinear stage of instability. It is obtained under the assumption that the plasma is homogeneous along a certain axis, determined, for example, by an external magnetic field, and taking into account only the modes with wavevectors orthogonal to this axis. We give an estimate of the maximum magnetic field energy achievable during the Weibel instability and show that its ratio to the initial longitudinal energy of particles, even for a large initial plasma anisotropy parameter, cannot exceed the value approximately equal to 0.2. The obtained analytical relation is verified using two-dimensional particle-in-cell simulations.

Ion-driven Instabilities in the Inner Heliosphere. II. Classification and Multidimensional Mapping

Linear theory is a well-developed framework for characterizing instabilities in weakly collisional plasmas, such as the solar wind. In the previous installment of this series, we analyzed ∼1.5M proton and α particle velocity distribution functions (VDFs) observed by Helios I and II to determine the statistical properties of the standard instability parameters such as the growth rate, frequency, the direction of wave propagation, and the power emitted or absorbed by each component, as well as to characterize their behavior with respect to the distance from the Sun and collisional processing. In this work, we use this comprehensive set of instability calculations to train a machine-learning algorithm consisting of three interlaced components that: (1) predict if an interval is unstable from observed VDF parameters; (2) predict the instability properties for a given unstable VDF; and (3) classify the type of the unstable mode. We use these methods to map the properties in multidimensional phase space to find that the parallel-propagating, proton-core-induced ion cyclotron mode dominates the young solar wind, while the oblique fast magnetosonic mode regulates the proton beam drift in the collisionally old plasma.

14-moment maximum-entropy modeling of collisionless ions for Hall thruster discharges

Ions in Hall effect thrusters are often characterized by a low collisionality. In the presence of acceleration fields and azimuthal electric field waves, this results in strong deviations from thermodynamic equilibrium, introducing kinetic effects. This work investigates the application of the 14-moment maximum-entropy model to this problem. This method consists in a set of 14 partial differential equations (PDEs) for the density, momentum, pressure tensor components, heat flux vector, and fourth-order moment associated with the particle velocity distribution function. The model is applied to the study of collisionless ion dynamics in a Hall thruster-like configuration, and its accuracy is assessed against different models, including the Vlasov kinetic equation. Three test cases are considered: a purely axial acceleration problem, the problem of ion-wave trapping, and finally the evolution of ions in the axial-azimuthal plane. Most of this work considers ions only, and the coupling with electrons is removed by prescribing reasonable values of the electric field. This allows us to obtain a direct comparison among different ion models. However, the possibility to run self-consistent plasma simulations is also briefly discussed, considering quasi-neutral or multi-fluid models. The maximum-entropy system appears to be a robust and accurate option for the considered test cases. The accuracy is improved over the simpler pressureless gas model (cold ions) and the Euler equations for gas dynamics, while the computational cost shows to remain much lower than direct kinetic simulations.

Gas kinetic principles in Navier–Stokes finite-volume solvers

We present two simplified gas kinetic models for the computation of the Navier–Stokes equations for compressible flows. Upon starting from the non-linear BGK kinetic model, we derive a simplified procedure for computing the normalized particle velocity distribution function, ψ.From a first-order approximation in time and a directional splitting of the BGK equation, the function ψ is computed over each time-step and at each cell interface of a finite-volume algorithm, as a convex combination between its value at the beginning of each time-step and an equilibrium state of the gas.Hence, depending on the definition of this equilibrium state, we can generate two kinetic theory inspired schemes.The first scheme (SCG scheme), is designed by computing the initial equilibrium state from two Maxwellian distributions of the gas, at each cell interface. The second scheme (ASCGE scheme) is designed by introducing acoustic principles into the definition of this initial equilibrium state. Then, the resulting flux function for convective terms is computed from the gas distribution function, ψ, at each time-step.Viscous fluxes are weakly coupled with convective fluxes by using kinematic features coming from both sides of the generic cell face.Finally, these convective and diffusive fluxes are introduced into a fourth-order MUSCL-like finite-volume procedure for modeling compressible flows.These two resulting simplified gas kinetic models are tested and compared with an HLLC Riemann solver scheme and recent simplified gas kinetic schemes over a variety of physical problems including inviscid/viscous and steady/unsteady problems for a compressible flow.

Effective interaction potentials and the scattering cross sections of the Lorentzian dense semiclassical plasma particles

Modified versions of the dynamic effective interaction potentials for the charged particles in dense semiclassical plasma are presented. These models consider dynamic screening, quantum mechanical effect of diffraction and particle velocity distribution function with non-Maxwellian suprathermal tail. Such distributions are well described by the so-called kappa-distribution and can be used in different space plasma and in some plasma installations where the suprathermal particles take place. Based on this modified dynamic interaction potentials the differential scattering cross sections of the electron-electron and electron-ion collisions were investigated. The obtained results for different values of the density parameter, coupling parameter and spectral index κ are presented.

Resolving Space Plasma Species With Electrostatic Analyzers

Electrostatic analyzers resolve the energy-per-charge distributions of charged plasma particles. Some space plasma instruments use electrostatic analyzers among other units, such as aperture deflectors and position sensitive detectors, in order to resolve the three-dimensional energy (velocity) distribution functions of plasma particles. When these instruments do not comprise a mass analyzer unit, different species can be resolved only if there are measurable differences in their energy-per-charge distributions. This study examines the ability of single electrostatic analyzer systems in resolving co-moving plasma species with different mass-per-charge ratios. We consider examples of static plasma consisting of two species of heavy negative ions measured by a typical electrostatic analyzer design, similar to the electron spectrometer on board Cassini spacecraft. We demonstrate an appropriate modeling technique to simulate the basic features of the instrument response in the specific plasma conditions and we quantify its ability to resolve the key species as a function of the spacecraft speed and the plasma temperature. We show that for the parameter range we examine, the mass resolution increases with increasing spacecraft speed and decreasing plasma temperature. We also demonstrate how our model can analyze real measurements and drive future instrument designs.

Quantifying Wave–Particle Interactions in Collisionless Plasmas: Theory and Its Application to the Alfvén-mode Wave

Wave–particle interactions can induce energy transfer at different timescales in collisionless plasmas, which leads to the reshaping of the particle velocity distribution function. Therefore, how to quantify wave–particle interactions is one of the fundamental problems in the heliosphere and in astrophysical plasmas. This study proposes a systematic method to quantify linear wave–particle interactions based on the Vlasov–Maxwellian model. We introduce energy transfer rates with various expressions by using perturbed electric fields and perturbed particle velocity distribution functions. Then, we use different expressions of the energy transfer rate to perform a comprehensive investigation of wave–particle interactions of the Alfvén-mode wave. We clarify the physical mechanisms responsible for the damping of the Alfvén-mode wave in wavevector space. Moreover, this study exhibits for the first time evident signatures of wave–particle interactions between Alfvén-mode waves and resonant/nonresonant particles in the velocity space. These resonant and nonresonant particles can induce energy transfer in opposite directions, which leads to self-regulation of the particle velocity distribution function. Furthermore, this study exhibits a comprehensive dependence of wave–particle interactions of the Alfvén-mode wave on the wavenumber and plasma beta (the ratio between the plasma thermal pressure and the magnetic pressure). These results illustrate that the proposed method would be very useful for quantifying different types of linear wave–particle interactions of an arbitrary wave mode.

An atomistic approach to study the dynamic and structural response in 2D nanofiller reinforced polyethylene nanocomposites under ultra-short shock pulse loading

An atomistic model was developed to systematically investigate the shock propagation and its attenuation behavior in two-dimensional (2D) nanofiller reinforced polyethylene (PE) nanocomposites. Under the framework of non-equilibrium molecular dynamics, an explicit ultra-short shock pulse with an impact velocity of 1.0 km/s was applied on the mono- and bi-crystalline 2D nanofiller reinforced PE nanocomposites. Their dynamic and structural responses were captured in terms of particle velocity, mean pressure, kinetic energy, and radial distribution function (RDF). The trends depicted that the energy dissipated from graphene (GNS) and boron nitride nanosheet (BNNS) reinforced PE nanocomposites were 30.69% and 24.92%, respectively, whereas the energy dissipation improves up to 45.33% and 41.77% for bi-crystalline GNS and bi-crystalline BNNS, respectively. Higher energy dissipation in GNS reinforced PE nanocomposites can be attributed to extreme impedance mismatch and superior interfacial interaction energy between GNS and PE chains.

Nonequilibrium effects of reactive flow based on gas kinetic theory* *This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51806116 and 11875001.

How to accurately probe chemically reactive flows with essential thermodynamic nonequilibrium effects is an open issue. Via the Chapman–Enskog analysis, the local nonequilibrium particle velocity distribution function is derived from the gas kinetic theory. It is demonstrated theoretically and numerically that the distribution function depends on the physical quantities and derivatives, and is independent of the chemical reactions directly as the chemical time scale is longer than the molecular relaxation time. Based on the simulation results of the discrete Boltzmann model, the departure between equilibrium and nonequilibrium distribution functions is obtained and analyzed around the detonation wave. In addition, it has been verified for the first time that the kinetic moments calculated by summations of the discrete distribution functions are close to those calculated by integrals of their original forms.

Kinetic Heating by Alfvén Waves in Magnetic Shears.

With first-principles kinetic simulations, we show that a large-scale Alfvén wave (AW) propagating in an inhomogeneous background decays into kinetic Alfvén waves (KAWs), triggering ion and electron energization. We demonstrate that the two species can access unequal amounts of the initial AW energy, experiencing differential heating. During the decay process, the electric field carried by KAWs produces non-Maxwellian features in the particle velocity distribution functions, in accordance with space observations. The process we present solely requires the interaction of a large-scale AW with a magnetic shear and may be relevant for several astrophysical and laboratory plasmas.

Ion-driven Instabilities in the Inner Heliosphere. I. Statistical Trends

Abstract Instabilities described by linear theory characterize an important form of wave–particle interaction in the solar wind. We diagnose unstable behavior of solar wind plasma between 0.3 and 1 au via the Nyquist criterion, applying it to fits of ∼1.5M proton and α particle Velocity Distribution Functions (VDFs) observed by Helios I and II. The variation of the fraction of unstable intervals with radial distance from the Sun is linear, signaling a gradual decline in the activity of unstable modes. When calculated as functions of the solar wind velocity and Coulomb number, we obtain more extreme, exponential trends in the regions where collisions appear to have a notable influence on the VDF. Instability growth rates demonstrate similar behavior, and significantly decrease with Coulomb number. We find that for a nonnegligible fraction of observations, the proton beam or secondary component might not be detected, due to instrument resolution limitations, and demonstrate that the impact of this issue does not affect the main conclusions of this work.

Assessment of shutdown dose rates in the ITER Collective Thomson Scattering system and in equatorial port plug 12

The ITER Collective Thomson Scattering (CTS) system will be the main diagnostic responsible for measuring the velocity distribution function of fusion-born alpha particles in the plasma. As the CTS diagnostic is integrated in the equatorial port plug 12 (drawer 3), with direct apertures to the port interspace where maintenance hands-on operation will be carried out, it is essential to assess the shutdown dose rates (SDDR) in these maintenance areas. In this work, the D1S-UNED3.1.4 Monte-Carlo transport code, based on the implementation of the direct-one-step methodology in MCNP5 v1.60, was used to estimate the dose rate level 12 days (106 s) after shutdown in the port interspace. The results show that the CTS system does not contribute significantly to the SDDR in the area where hands-on maintenance is foreseen with contribution to dose rates less than 1 µSv/h. This is consistent with previous estimates, although with the most recent model of the CTS design there is a slight increase of the SDDR values. This deviation can be attributed to design changes and improved shielding modelling and/or most importantly, to statistical fluctuations of the D1S simulations. From a neutronics point of view, the increase in the SDDR falls within the range of the statistical fluctuations, and the design is still compliant with the radiation safety ALARA principle aiming at minimizing radiation doses, and there is no requirement for further design optimizations.

High-cadence measurements of electron pitch-angle distributions from Solar Orbiter SWA-EAS burst mode operations

Small-scale kinetic processes associated with turbulence, plasma instabilities, magnetic reconnection, etc., play a major role in dissipating energy and governing the large-scale evolution of the solar wind. However, a general impediment to improving the understanding of the kinetic physics of the solar wind is in the limitations on measurement cadences of particle instruments, which are usually several orders of magnitude below the equivalent cadences of field instruments. Nevertheless, knowledge of the details of the particle velocity distribution functions (VDFs) at sub-second cadence is required to make progress in this area. This is particularly true for the electron VDFs, which play a significant role in the overall energetics of the solar wind through their transmission of heat flux from the Sun. In this paper, we detail and illustrate a novel measurement scheme deployed on Solar Orbiter’s Solar Wind Analyser Electron Analyser System (SWA-EAS), which allows for 2D pitch angle distributions (PAD) to be returned over short periods (5–10 min) at a cadence of 0.125 s. This is achieved through the use of a B-field vector shared by the magnetometer (MAG) instrument to steer the SWA-EAS system to record only that part of the full SWA-EAS field-of-view needed to construct the PAD. We provide an example of early observations using this scheme to illustrate that it is working well. Given that the electrons are usually gyrotropic, these measurements provide a new tool with which to derive details of the electron VDFs at high cadence for the study of the solar wind’s kinetic processes.

An extended hydrodynamics model for inertial confinement fusion hohlraums

In some inertial confinement fusion hohlraum designs, the inside plasma is not sufficiently collisional to be satisfactorily described by the Euler equations implemented in hydrodynamic simulation codes, particularly in converging regions of the expanding plasma flow. To better treat that situation, this paper presents an extended hydrodynamics model including higher moments of the particle velocity distribution function, together with physically justified closure assumptions and relaxation terms. A preliminary one-dimensional numerical implementation of the model is shown to give satisfactory results in a test case involving a high-velocity collision of two plasma flows. Paths to extend that model to three dimensions as needed for an actual hohlraum geometry are briefly discussed.

Nonequilibrium characteristics and spatiotemporal long-range correlations in dense gas-solid suspensions

The nonequilibrium properties and strong spatiotemporal long-range correlations in gas-solid suspensions were analyzed systematically by discrete particle simulation in periodic domains. The effective interphase slip velocity is significantly larger than the particle terminal velocity, the particle velocity distribution function is non-Maxwellian with a high-energy tail and the solid concentration distribution function is far from the discrete Poisson distribution, all of which manifest strong deviation from the equilibrium or homogeneous state in gas-solid flow. The solid concentration, gas and solid velocities are spatiotemporally correlated and show a clear scale-dependence. More importantly, all characteristic correlation lengths scale linearly with the size of simulation domain, although the correlation length of solid concentration is much smaller than those of gas and solid velocities. On the other hand, the characteristic correlation times at first increase with increasing domain size and then seem to approach to asymptotic values. Those results indicate that the heterogeneity and spatiotemporal long-range correlations should be considered properly in continuum theory of heterogeneous gas-solid flow.

Eulerian formulation for the triboelectric charging of polydisperse powder flows

We propose an Eulerian formulation for the triboelectric charging of powder flows. Recently, continuum descriptions of the charging of monodisperse and bidisperse powder appeared. Now, we propose the first Eulerian formulation for this type of flows that can fully account for polydisperse particle size distributions. To this end, the joint particle size, velocity, and charge distribution function are solved through the direct quadrature method of moments. The newly established approach includes the uptake of electrostatic charge by the particles when contacting a solid wall and the exchange of charge in-between particles during collisions. The electrostatic field induced by the charge of the particles and the drag forces of the surrounding gas affects the particle dynamics. Test cases of two-dimensional steady channel flows demonstrate the charging of the flow from the walls toward the center and in the downstream direction. Further, the method predicts the effect of variations of the particle size and initial velocity distribution and the charge diffusion coefficient. The ability to handle polydispersity is a step toward the simulation of the electric charge buildup of real powder flows in full-scale technical applications.

Dissipation measures in weakly collisional plasmas

ABSTRACT The physical foundations of the dissipation of energy and the associated heating in weakly collisional plasmas are poorly understood. Here, we compare and contrast several measures that have been used to characterize energy dissipation and kinetic-scale conversion in plasmas by means of a suite of kinetic numerical simulations describing both magnetic reconnection and decaying plasma turbulence. We adopt three different numerical codes that can also include interparticle collisions: the fully kinetic particle-in-cell vpic, the fully kinetic continuum Gkeyll, and the Eulerian Hybrid Vlasov–Maxwell (HVM) code. We differentiate between (i) four energy-based parameters, whose definition is related to energy transfer in a fluid description of a plasma, and (ii) four distribution function-based parameters, requiring knowledge of the particle velocity distribution function. There is an overall agreement between the dissipation measures obtained in the PIC and continuum reconnection simulations, with slight differences due to the presence/absence of secondary islands in the two simulations. There are also many qualitative similarities between the signatures in the reconnection simulations and the self-consistent current sheets that form in turbulence, although the latter exhibits significant variations compared to the reconnection results. All the parameters confirm that dissipation occurs close to regions of intense magnetic stresses, thus exhibiting local correlation. The distribution function-based measures show a broader width compared to energy-based proxies, suggesting that energy transfer is co-localized at coherent structures, but can affect the particle distribution function in wider regions. The effect of interparticle collisions on these parameters is finally discussed.

Performance and simulated moment uncertainties of an ion spectrometer with asymmetric 2π field of view for ion measurements in space.

Space plasma instruments provide 3D particle velocity distribution functions. Because of telemetry limitations, these cannot be transmitted in high time resolution and the plasma needs to be characterized by moments of the velocity distribution function. These moment uncertainties have vital effects on the reliability and accuracy of onboard plasma moments. We assess the measurement accuracy for magnetosheath and solar wind ions using an ion spectrometer with an asymmetric field of view designed for the all-sky measurement of low-energy ions in the magnetosheath and solar wind. We focus on moment uncertainties for the ideal spectrometer, not considering the background counts, which may have considerable effects on the uncertainties in real life. To obtain number density, bulk velocity, and temperature, different orders of moments are integrated assuming a Maxwellian velocity distribution. Based on the design specifications, we use simulations to estimate systematic and random errors for typical plasma conditions. We find that the spectrometer resolution is adequate for determining the density of solar wind (∼7% error) and magnetosheath ions (∼4% error). The resolution is also adequate for determining the temperature of solar wind (∼10% error) and magnetosheath ions (∼2% error). For high speed flows with a bulk velocity of 750 km/s and a temperature of 20 eV, the maximum density and temperature errors become 9% and 7%, respectively. The bulk velocity errors are less than 2% for all cases. The contributions of heavy ions to the systematic errors are less than 5% for magnetosheath ions and less than 8% for solar wind ions.

Meshfree method based on discrete gas-kinetic scheme to simulate incompressible/compressible flows

A meshfree method based on the discrete gas-kinetic scheme (DGKS) (called the meshfree-DGKS) for simulation of incompressible/compressible flows is proposed in this work. In this approach, the governing equations are discretized using the meshfree method based on the least squares-based finite difference approach. To simulate compressible problems with discontinuities, the virtual mid-points between adjacent nodes, which are regarded as Riemann discontinuities, are established. Then, the concept of numerical flux is introduced, which enables computing both compressible and incompressible problems. The fluxes at the mid-points are calculated using the DGKS based on the discrete particle velocity model. The corresponding particle velocity components and distribution functions are integrated based on moment relations to obtain the flux. The meshfree-DGKS maintains the advantages of the meshless method as it is implemented at arbitrarily distributed nodes. This breaks through the limitations of the grid topology and is suitable to handle complex geometries. More importantly, the fluxes at the mid-point are reconstructed with the DGKS using the local solution of the Boltzmann equation, which can describe its physical properties well, thus easily and stably capturing the shock wave. In addition, the DGKS can simultaneously calculate inviscid and viscous fluxes when simulating viscous flow problems, which gives an improved algorithm consistency. Several representative examples, such as shock tube problems, implosion problem, couette flow, lid-driven cavity flow, flow in a channel with a backward-facing step, supersonic flow around a ramp segment, and flow around staggered NACA0012 biplane configuration, are simulated to validate the proposed meshfree-DGKS.

Multiscale kinetic theory for heterogeneous granular and gas-solid flows

It has been recognized that the particle phase stress model derived from classical kinetic theory is valid only when sufficient scale resolution is offered to explicitly resolve the heterogeneous structures during numerical simulations, however, industrial applications prefer to use coarse computational grids where the heterogeneous structures are not explicitly resolved but implicitly modeled. Unfortunately, in this case a kinetic theory for heterogeneous granular and gas–solid flows is not available yet. To this end, an attempt was made to tracking this challenge: The single particle velocity distribution function at the nonequilibrium stationary state with heterogeneous structures was firstly derived by combining the idea of doubly stochastic Poisson processes or superstatistics with the concept of compromise in competition in the EMMS (Energy Minimization Multi-Scale) theory, the standard Chapman-Enskog method was then used to develop the constitutive relations of heterogeneous continuum theory. It was found that (i) seven state variables are needed to quantify the heterogeneous structures as compared to three for homogeneous systems; (ii) the constitutive relations not only include the contribution from microscale particle–particle interactions but also those due to the interactions between mesoscale structures; and (iii) the resultant constitutive relations are much more complex than those of homogeneous systems due to the simultaneous consideration of microscale and mesoscale contributions and the appearance of cross-coupling effects, but they correctly contain the constitutive relations of homogeneous systems as a limiting case. Finally, the theory was coupled with an EMMS drag model to offer a preliminary validation and to provide a unified EMMS-based constitutive relations for heterogeneous gas–solid flows.