Distribution Function In Velocity Space Research Articles

Predicting ion cyclotron emission from neutral beam heated plasmas in Wendelstein7-X stellarator

Measurements of ion cyclotron emission (ICE) are planned for magnetically confined fusion plasmas heated by neutral beam injection (NBI) in the Wendelstein 7-X stellarator (W7-X). Freshly injected NBI ions in the edge region, whose velocity-space distribution function approximates a delta-function, are potentially unstable against the magnetoacoustic cyclotron instability (MCI), which could drive a detectable ICE signal. Prediction of ICE from NBI protons in W7-X hydrogen plasmas is challenging, owing to the low ratio of the ions’ perpendicular velocity to the local Alfvén speed, v⊥(NBI)/VA≃0.14 . We address this from first principles, using the particle-in-cell kinetic code EPOCH. This self-consistently solves the Lorentz force equation and Maxwell’s equations for tens of millions of computational ions (both thermal majority and energetic NBI minority) and electrons, fully resolving gyromotion and hence capturing the cyclotron resonant phenomenology which gives rise to ICE. Our simulations predict an ICE signal which is predominantly electrostatic while incorporating a significant electromagnetic component. Its frequency power spectrum reflects novel MCI physics, reported here for the first time. The NBI ions relaxing under the MCI first drive broadband field energy at frequencies a little below the lower hybrid frequency ωLH, across the wavenumber range kωc/VA=40 –60, where ω c and VA denote ion cyclotron frequency and Alfvén velocity. Nonlinear coupling between these waves then excites spectrally structured ICE with narrow peaks, at much lower frequencies, typically the proton cyclotron frequency and its lower harmonics. The relative strength of these peaks depends on the specifics of the NBI ion velocity-space distribution and of the local plasma conditions, implying diagnostic potential for the predicted ICE signal from W7-X.

Non-Thermal Solar Wind Electron Velocity Distribution Function.

The quiet-time solar wind electrons feature non-thermal characteristics when viewed from the perspective of their velocity distribution functions. They typically have an appearance of being composed of a denser thermal "core" population plus a tenuous energetic "halo" population. At first, such a feature was empirically fitted with the kappa velocity space distribution function, but ever since the ground-breaking work by Tsallis, the space physics community has embraced the potential implication of the kappa distribution as reflecting the non-extensive nature of the space plasma. From the viewpoint of microscopic plasma theory, the formation of the non-thermal electron velocity distribution function can be interpreted in terms of the plasma being in a state of turbulent quasi-equilibrium. Such a finding brings forth the possible existence of a profound inter-relationship between the non-extensive statistical state and the turbulent quasi-equilibrium state. The present paper further develops the idea of solar wind electrons being in the turbulent equilibrium, but, unlike the previous model, which involves the electrostatic turbulence near the plasma oscillation frequency (i.e., Langmuir turbulence), the present paper considers the impact of transverse electromagnetic turbulence, particularly, the turbulence in the whistler-mode frequency range. It is found that the coupling of spontaneously emitted thermal fluctuations and the background turbulence leads to the formation of a non-thermal electron velocity distribution function of the type observed in the solar wind during quiet times. This demonstrates that the whistler-range turbulence represents an alternative mechanism for producing the kappa-like non-thermal distribution, especially close to the Sun and in the near-Earth space environment.

Design and Optimization of a High-Time-Resolution Magnetic Plasma Analyzer (MPA)

In-situ measurements of space plasma throughout the solar system require high time resolution to understand the plasma’s kinetic fine structure and evolution. In this context, research is conducted to design instruments with the capability to acquire the plasma velocity distribution and its moments with high cadence. We study a new instrument design, using a constant magnetic field generated by two permanent magnets, to analyze solar wind protons and α-particles with high time resolution. We determine the optimal configuration of the instrument in terms of aperture size, sensor position, pixel size and magnetic field strength. We conduct this analysis based on analytical calculations and SIMION simulations of the particle trajectories in our instrument. We evaluate the velocity resolution of the instrument as well as Poisson errors associated with finite counting statistics. Our instrument is able to resolve Maxwellian and κ-distributions for both protons and α-particles. This method retrieves measurements of the moments (density, bulk speed and temperature) with a relative error below 1%. Our instrument design achieves these results with an acquisition time of only 5 ms, significantly faster than state-of-the-art electrostatic analyzers. Although the instrument only acquires one-dimensional cuts of the distribution function in velocity space, the simplicity and reliability of the presented instrument concept are two key advantages of our new design.

Ionospheric Oxygen ions in the dayside magnetosphere

Prior to its discovery, O+ from the Earth's high-latitude ionosphere was not expected to be present in any significant amount in the magnetosphere. It is now known that O+ can dominate ionospheric outflow and dominate magnetotail composition during geomagnetically active times. Observations from the Magnetospheric Multiscale (MMS) mission are used here to illustrate properties of O+ in the dayside outer magnetosphere and to identify gaps in the knowledge of this ionospheric ion. When the interplanetary magnetic field (IMF) is southward, O+ from the high-latitude ionosphere convects to the magnetotail then from the magnetotail to the dayside. The two primary O+ populations observed in the dayside magnetosphere are the ring current and warm plasma cloak. They are sometimes distinguishable by their energy, although velocity-space distribution functions are often needed to make a true distinction. There are several sinks for these populations in the dayside magnetosphere. One important sink is the dayside magnetopause. While much is understood about O+ in the dayside magnetosphere, there are several open questions detailed in the final section of the paper.

Low- and high-frequency nature of oblique filamentation modes. I. Linear theory

The solution of the linear dispersion relation of electromagnetic oblique instabilities, for two counterstreaming electron beams, is investigated by using an extended fluid approach that includes the full dynamics of the pressure tensor. Numerical solutions of the simplified polynomial formulation so obtained are analyzed and compared to full kinetic solutions. They correspond to two classes of eigenmodes: low- and high-frequency oblique modes of resonant character. Coexistence of several oblique modes in neighboring regions of the wave vector plane, having close growth-rates, leads to the possibility of a transition starting from a low wave number mode to an oblique mode of high values in wave numbers. For such counterstreaming plasmas, the oblique instability may strengthen and amplify the filamentation process of the distribution function in velocity space, a property of the Vlasov equation. In addition to its simplicity, useful for solving the dispersion relation in the linear regime and for identifying kinetic solutions difficult to calculate otherwise, this extended fluid model is helpful in gaining insight into the fundamental properties of Vlasov theory, which are possibly relevant to kinetic heating processes.

Evidence of a new class of cnoidal electron holes exhibiting intrinsic substructures, its impact on linear (and nonlinear) Vlasov theories and role in anomalous transport

Novel types of solitary electron holes or in more general terms a novel class of cnoidal electron holes propagating in collisionless, driven plasmas is found numerically and described analytically. These structural modes are distinguished by substructures both in the electron density in real space and in the trapped electron distribution function in velocity space and are due to a two-fold trapping scenario; a regular one, called β-trapping, and a weakly singular one, called γ-trapping. Although the latter already appears early in O(ϕ) in the -expansion of the density, where ϕ is the electrostatic potential, it is the regular β-trapping effect, appearing in O(ϕ3/2), that remains the main cause of these new class of self-consistent structures. Through a proper participation of both trapping mechanisms the shortcomings (inconsistencies) of undamped linear wave theories (Landau, van Kampen) are nonlinearly rectified becoming true solutions of the Vlasov-Poisson system at this kinetic level of plasma description only. They hence provide the right key to the strictly nonlinear, kinetic world of undamped, coherent, small amplitude structures in driven, collisionless plasmas. A cusp-like slope singularity of the otherwise continuous electron distribution fe along the contour of a separatrix, on the other hand, exposes thereby a limit for the reliability of a kinetic nonlinear Vlasov-Poisson description and a simulation of structure formation in intermittent turbulent plasmas. It points to a new, deeper settled physics caused by a local enhancement of collisionality near separatrices through the generation of correlations. It is suggested that multiple trapping has the same origin than chaos in the particle trajectories near resonance, namely the non-integrability of discrete particle dynamics.

A multi-term spherical harmonic expansion of the Boltzmann equation for application to low-temperature collisional plasmas

The Boltzmann equation describes the evolution of the electron and ion distributions in a plasma over time through a six-dimensional phase space. For highly collisional plasmas, scattering collisions keep the distribution function nearly isotropic in velocity space with small perturbations created by the hydrodynamic and electromagnetic forces. For these plasmas, a spherical-harmonic expansion of the velocity-space distribution function is an effective technique for solving the Boltzmann equation. This paper examines each of the terms in the Boltzmann equation in detail to derive conditions where a spherical harmonic expansion is useful. Expressions for the matrix elements are presented which represent the projection of the various operators in the Boltzmann equation onto the spherical harmonics basis set. The resulting multiple-term spherical-harmonic expansion makes no assumptions about either the direction of the electric and magnetic fields or the magnitude of the spatial gradients and is appropriate for coupling with a Maxwell equation solver for the time- and spatially-dependent electromagnetic fields. When only the first two lowest-order terms are kept, it is shown that the resulting equations are very similar in form to the continuity and force-balance fluid equations. Additional kinetic terms appear in the continuity-like equation which are related to the changes in the energy distribution due to the electric field and collisions, including Ohmic heating. Two additional kinetic terms also appear in the force-balance-like equation. The collision term accounts for momentum-transfer during scattering collisions and the other accounts for the flow of energy in velocity space and is proportional to the derivative with respect to energy of the energy density.

Thermodynamic Nonequilibrium Features in Binary Diffusion**Supported by the MOST National Key Research and Development Programme under Grant No. 2016YFB0600805, the China Post-doctoral Science Foundation under Grant No. 2017M620757, the Center for Combustion Energy at Tsinghua University, Natural Science Foundation of Hebei Province under Grant Nos. A2017409014, ZD2017001 and A201500111, FJKLMAA, Fujian Normal University, and the UK Engineering and Physical Sciences

Diffusion is a ubiquitous physical phenomenon where thermodynamic nonequilibrium effects (TNEs) are outstanding issues. In this work, we employ the discrete Boltzmann method to investigate the TNEs in the dynamic process of binary diffusion. The main features of the distribution function in velocity space are recovered and discussed. It is found that, with the decreasing gradients of macroscopic quantities (such as density, concentration, velocity, etc.), both the local and global TNEs decrease with the time but increase with the relaxation time in a power law, respectively.

Force-free collisionless current sheet models with non-uniform temperature and density profiles

We present a class of one-dimensional, strictly neutral, Vlasov-Maxwell equilibrium distribution functions for force-free current sheets, with magnetic fields defined in terms of Jacobian elliptic functions, extending the results of Abraham-Shrauner [Phys. Plasmas 20, 102117 (2013)] to allow for non-uniform density and temperature profiles. To achieve this, we use an approach previously applied to the force-free Harris sheet by Kolotkov et al. [Phys. Plasmas 22, 112902 (2015)]. In one limit of the parameters, we recover the model of Kolotkov et al. [Phys. Plasmas 22, 112902 (2015)], while another limit gives a linear force-free field. We discuss conditions on the parameters such that the distribution functions are always positive and give expressions for the pressure, density, temperature, and bulk-flow velocities of the equilibrium, discussing the differences from previous models. We also present some illustrative plots of the distribution function in velocity space.

Nonlinear interaction between energetic particles and turbulence in gyro-kinetic simulations and impact on turbulence properties

The modification of radial structure, frequency and intensity of turbulent transport in the presence of energetic-particle-driven geodesic acoustic modes (EGAMs) is analysed by means of full-f global gyro-kinetic simulations using Gysela code. It is observed that turbulence leads to a smoother evolution of the distribution function, less pronounced flattening of the distribution function in velocity space during the nonlinear saturation of EGAMs and reduced saturation level of electrostatic potential with respect to the case where turbulence is artificially suppressed. It is shown that EGAMs are excited and impact turbulent transport in the region where the EP is localised, fading away the staircase structure observed in the absence of energetic particles. For the first time, evidences of a three-wave coupling between turbulent modes and EGAMs in gyro-kinetic simulations are provided by means of bispectral analysis using wavelet transform in time. The coupling evolves from the standard self-regulation of turbulence by the zero-frequency zonal component to a steady-state regime where turbulence dynamics is dominated by the zonal component oscillating at the EGAM frequency.

Reflection symmetries of Isolated Self-consistent Stellar Systems

Isolated, steady-state galaxies correspond to equilibrium solutions of the Poisson--Vlasov system. We show that (i) all galaxies with a distribution function depending on energy alone must be spherically symmetric and (ii) all axisymmetric galaxies with a distribution function depending on energy and the angular momentum component parallel to the symmetry axis must also be reflection-symmetric about the plane $z=0$. The former result is Lichtenstein's Theorem, derived here by a method exploiting symmetries of solutions of elliptic partial differential equations, while the latter result is new. These results are subsumed into the Symmetry Theorem, which specifies how the symmetries of the distribution function in configuration or velocity space can control the planes of reflection symmetries of the ensuing stellar system.

Non-Maxwellian electron distribution functions due to self-generated turbulence in collisionless guide-field reconnection

Non-Maxwellian electron velocity space distribution functions (EVDFs) are useful signatures of plasma conditions and non-local consequences of collisionless magnetic reconnection. In the past, EVDFs were obtained mainly for antiparallel reconnection and under the influence of weak guide-fields in the direction perpendicular to the reconnection plane. EVDFs are, however, not well known, yet, for oblique (or component-) reconnection in case and in dependence on stronger guide-magnetic fields and for the exhaust (outflow) region of reconnection away from the diffusion region. In view of the multi-spacecraft Magnetospheric Multiscale Mission (MMS), we derived the non-Maxwellian EVDFs of collisionless magnetic reconnection in dependence on the guide-field strength bg from small (bg≈0) to very strong (bg = 8) guide-fields, taking into account the feedback of the self-generated turbulence. For this sake, we carried out 2.5D fully kinetic Particle-in-Cell simulations using the ACRONYM code. We obtained anisotropic EVDFs and electron beams propagating along the separatrices as well as in the exhaust region of reconnection. The beams are anisotropic with a higher temperature in the direction perpendicular rather than parallel to the local magnetic field. The beams propagate in the direction opposite to the background electrons and cause instabilities. We also obtained the guide-field dependence of the relative electron-beam drift speed, threshold, and properties of the resulting streaming instabilities including the strongly non-linear saturation of the self-generated plasma turbulence. This turbulence and its non-linear feedback cause non-adiabatic parallel electron acceleration. We further obtained the resulting EVDFs due to the non-linear feedback of the saturated self-generated turbulence near the separatrices and in the exhaust region of reconnection in dependence on the guide field strength. We found that the influence of the self-generated plasma turbulence leads well beyond the limits of the quasi-linear approximation to the creation of phase space holes and an isotropizing pitch-angle scattering. EVDFs obtained by this way can be used for diagnosing collisionless reconnection by using the multi-spacecraft observations carried out by the MMS mission.

Kinetic transport simulation of energetic particles

A kinetic transport code (EPtran) is developed for the transport of the energetic particles (EPs). The EPtran code evolves the EP distribution function in radius, energy, and pitch angle phase space (r, E, λ) to steady state with classical slowing down, pitch angle scattering, as well as radial and energy transport of the injected EPs (neutral beam injection (NBI) or fusion alpha). The EPtran code is illustrated by treating the transport of NBI fast ions from high-n ITG/TEM micro-turbulence and EP driven unstable low-n Alfvén eigenmodes (AEs) in a well-studied DIII-D NBI heated discharge with significant AE central core loss. The kinetic transport code results for this discharge are compared with previous study using a simple EP density moment transport code ALPHA (R.E. Waltz and E.M. Bass 2014 Nucl. Fusion 54 104006). The dominant EP-AE transport is treated with a local stiff critical EP density (or equivalent pressure) gradient radial transport model modified to include energy-dependence and the nonlocal effects EP drift orbits. All previous EP transport models assume that the EP velocity space distribution function is not significantly distorted from the classical ‘no transport’ slowing down distribution. Important transport distortions away from the slowing down EP spectrum are illustrated by a focus on the coefficient of convection: EP energy flux divided by the product of EP average energy and EP particle flux.

Coupling of charged particles via Coulombic interactions: Numerical simulations and resultant kappa‐like velocity space distribution functions

AbstractA parametric study is performed using the electrostatic simulations of Randol and Christian () in which the number density, n, and initial thermal speed, θ, are varied. The range of parameters covers an extremely broad plasma regime, all the way from the very weak coupling of space plasmas to the very strong coupling of solid plasmas. The first result is that simulations at the same ΓD, where ΓD ( n1/3θ−2) is the plasma coupling parameter, but at different combinations of n and θ, behave exactly the same. As a function of ΓD, the form of p(v), the velocity distribution function of v, the magnitude of v, the velocity vector, is studied. For intermediate to high ΓD, heating is observed in p(v) that obeys conservation of energy, and a suprathermal tail is formed, with a spectral index that depends on ΓD. For strong coupling (ΓD≫1), the form of the tail is v−5, consistent with the findings of Randol and Christian (). For weak coupling (ΓD≪1), no acceleration or heating occurs, as there is no free energy. The dependence on N, the number of particles in the simulation, is also explored. There is a subtle dependence in the index of the tail, such that v−5 appears to be the limit.

Consistency between real and synthetic fast-ion measurements at ASDEX Upgrade

Internally consistent characterization of the properties of the fast-ion distribution from multiple diagnostics is a prerequisite for obtaining a full understanding of fast-ion behavior in tokamak plasmas. Here we benchmark several absolutely-calibrated core fast-ion diagnostics at ASDEX Upgrade by comparing fast-ion measurements from collective Thomson scattering, fast-ion spectroscopy, and neutron rate detectors with numerical predictions from the TRANSP/NUBEAM transport code. We also study the sensitivity of the theoretical predictions to uncertainties in the plasma kinetic profiles. We find that theory and measurements generally agree within these uncertainties for all three diagnostics during heating phases with either one or two neutral beam injection sources. This suggests that the measurements can be described by the same model assuming classical slowing down of fast ions. Since the three diagnostics in the adopted configurations probe partially overlapping regions in fast-ion velocity space, this is also consistent with good internal agreement among the measurements themselves. Hence, our results support the feasibility of combining multiple diagnostics at ASDEX Upgrade to reconstruct the fast-ion distribution function in 2D velocity space.

Quasilinear relaxation of a beam with power law injected electrons propagating through solar corona

It is assumed that the electron beam propagating thorough the Maxwellian solar corona plasma has a power law spectra. Using numerical simulations of the quasilinear equations, the effects of power law injected electrons on the generation of Langmuir waves are compared with a Maxwellian beam. It is found that the level of Langmuir waves increases in the presence of power law injected electrons. The average velocity of the beam propagation is constant for both Maxwellian and power law injected electrons but its value increases in the second case. The influence of the power law injected electrons on the evolution of gas-dynamical parameters such as the height of the plateau in the beam distribution function in velocity space, its upper and lower velocities boundary, and the local velocity of the beam and its spread is investigated. It is shown that the these parameters are dependent on the characteristics of the power law injected electrons, p (spectral index), and v0 (the break speed). The upper boundary of plateau decreases (increases) with the p(v0) but the lower boundary has inverse behavior. The height of plateau p(x,t) is a decreasing function of p and for a fixed value of p it has a maximum in a certain value of v0 at a given time and position.

Verification of continuum drift kinetic equation solvers in NIMROD

Verification of continuum solutions to the electron and ion drift kinetic equations (DKEs) in NIMROD [C. R. Sovinec et al., J. Comp. Phys. 195, 355 (2004)] is demonstrated through comparison with several neoclassical transport codes, most notably NEO [E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 54, 015015 (2012)]. The DKE solutions use NIMROD's spatial representation, 2D finite-elements in the poloidal plane and a 1D Fourier expansion in toroidal angle. For 2D velocity space, a novel 1D expansion in finite elements is applied for the pitch angle dependence and a collocation grid is used for the normalized speed coordinate. The full, linearized Coulomb collision operator is kept and shown to be important for obtaining quantitative results. Bootstrap currents, parallel ion flows, and radial particle and heat fluxes show quantitative agreement between NIMROD and NEO for a variety of tokamak equilibria. In addition, velocity space distribution function contours for ions and electrons show nearly identical detailed structure and agree quantitatively. A Θ-centered, implicit time discretization and a block-preconditioned, iterative linear algebra solver provide efficient electron and ion DKE solutions that ultimately will be used to obtain closures for NIMROD's evolving fluid model.

The Weibel instability in a strongly coupled plasma

In this paper, the growth rate of the Weibel instability is calculated for an energetic relativistic electron beam penetrated into a strongly coupled plasma, where the collision effects of background electron-ion scattering play an important role in equations. In order to calculate the growth rate of the Weibel instability, two different models of anisotropic distribution function are used. First, the distribution of the plasma and beam electrons considered as similar forms of bi-Maxwellian distribution. Second, the distribution functions of the plasma electrons and the beam electrons follows bi-Maxwellian and delta-like distributions, respectively. The obtained results show that the collision effect decreases the growth rate in two models. When the distribution function of electrons beam is in bi-Maxwellian form, the instability growth rate is greater than where the distribution function of beam electrons is in delta-like form, because, the anisotropic temperature for bi-Maxwellian distribution function in velocity space is greater than the delta-like distribution function.

Scaled Experiment to Investigate Auroral Kilometric Radiation Mechanisms in the Presence of Background Electrons

Auroral Kilometric Radiation (AKR) emissions occur at frequencies ~300kHz polarised in the X-mode with efficiencies ~1-2% [1,2] in the auroral density cavity in the polar regions of the Earth's magnetosphere, a region of low density plasma ~3200km above the Earth's surface, where electrons are accelerated down towards the Earth whilst undergoing magnetic compression. As a result of this magnetic compression the electrons acquire a horseshoe distribution function in velocity space. Previous theoretical studies have predicted that this distribution is capable of driving the cyclotron maser instability. To test this theory a scaled laboratory experiment was constructed to replicate this phenomenon in a controlled environment, [3-5] whilst 2D and 3D simulations are also being conducted to predict the experimental radiation power and mode, [6-9]. The experiment operates in the microwave frequency regime and incorporates a region of increasing magnetic field as found at the Earth's pole using magnet solenoids to encase the cylindrical interaction waveguide through which an initially rectilinear electron beam (12A) was accelerated by a 75keV pulse. Experimental results showed evidence of the formation of the horseshoe distribution function. The radiation was produced in the near cut-off TE01 mode, comparable with X-mode characteristics, at 4.42GHz. Peak microwave output power was measured ~35kW and peak efficiency of emission ~2%, [3]. A Penning trap was constructed and inserted into the interaction waveguide to enable generation of a background plasma which would lead to closer comparisons with the magnetospheric conditions. Initial design and measurements are presented showing the principle features of the new geometry.

Hole-Clump Pair Creation in the Evolution of Energetic-Particle-Driven Geodesic Acoustic Modes

Nonlinear frequency chirping of the energetic-particle-driven geodesic acoustic mode (EGAM) is investigated using a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic fluid. It is demonstrated in the simulation result that both frequency chirping up and chirping down take place in the nonlinear evolution of the EGAM. It is found that two hole-clump pairs are formed in the energetic particle distribution function in two-dimensional velocity space of pitch angle variable and energy. One pair is formed in the phase space region that destabilizes the instability, while the other is formed in the stabilizing region. The transit frequency of the hole (clump) in the destabilizing region chirps up (down), while in the stabilizing region the hole (clump) chirps down (up). The transit frequencies of particles in the holes and clumps are in good agreement with the chirping EGAM frequency indicating that the particles are kept resonant with the EGAM during the nonlinear frequency chirping. Continuous energy transfer takes place from the destabilizing phase space region to the stabilizing region during the spontaneous frequency chirping of the wave.