Maxwellian Distribution Function Research Articles

Analytical study of whistler mode waves in presence of parallel DC electric field for relativistic plasma in the magnetosphere of Uranus

In present paper, field aligned whistler mode waves are analyzed, in the presence of DC field in background plasma having relativistic distribution function in the magnetosphere of Uranus. The work has been examined for relativistic Maxwellian and loss-cone distribution function. In both the cases, we have studied the effect of various plasma parameters on the growth rate of waves by using the method of characteristics and discussed using data provided by Voyager 2. Growth rate has increased by increasing the magnitude of electric field, temperature anisotropy, energy density and number density of particles for Maxwellian and loss-cone background. However, when relativistic factor λ=1-v2/c2 increases, growth rate decreases. The significant increase in real frequency of whistler waves can be observed. The results can be used for comparative study of planetary magnetospheres. The derivation can also be adapted to study various other instabilities in magnetosphere of Uranus.

Wavenumber shift due to nonlinear plasma and wave interaction

Wavenumber shift of the ion Bernstein wave has been observed in the particle-in-cell simulations when the input power of the injected wave is sufficiently large. It is demonstrated that the increase of the total kinetic energy of ions, including both the thermal energy related to the random thermal motion and the oscillation energy due to the coherent motion with the wave, gives rise to such change of the wavenumber. However, the velocity distribution function of the ions can approximately be fitted as a Maxwellian distribution function, and thus, the linear dispersion relation still holds, provided that the initial ion temperature is replaced by the effective temperature measured in the simulation.

Collisional damping rates for plasma waves

The distinction between the plasma dynamics dominated by collisional transport versus collective processes has never been rigorously addressed until recently. A recent paper [P. H. Yoon et al., Phys. Rev. E 93, 033203 (2016)] formulates for the first time, a unified kinetic theory in which collective processes and collisional dynamics are systematically incorporated from first principles. One of the outcomes of such a formalism is the rigorous derivation of collisional damping rates for Langmuir and ion-acoustic waves, which can be contrasted to the heuristic customary approach. However, the results are given only in formal mathematical expressions. The present brief communication numerically evaluates the rigorous collisional damping rates by considering the case of plasma particles with Maxwellian velocity distribution function so as to assess the consequence of the rigorous formalism in a quantitative manner. Comparison with the heuristic (“Spitzer”) formula shows that the accurate damping rates are much lower in magnitude than the conventional expression, which implies that the traditional approach over-estimates the importance of attenuation of plasma waves by collisional relaxation process. Such a finding may have a wide applicability ranging from laboratory to space and astrophysical plasmas.

Exact relativistic expressions for polarization of incoherent Thomson scattering

We present a derivation of the degree of polarization for incoherent Thomson scattering (TS) using Mueller matrix formalism. An exact analytic solution is obtained for spectrum-integrated matrix elements. The solution is valid for the full range of incident polarizations, scattering angles, and electron thermal motion from non-relativistic to ultra-relativistic. It is based on a newly developed theoretical model, a finite transit time (FTT) correction to previous theoretical work on TS polarization. The Mueller matrix elements are substantially different from previous calculations without the FTT correction, even to the lowest linear order in Te/mec2≪1. Mathematically, the derivation is a unique example of fully analytical integration of the 3D scattering operator over a relativistic Maxwellian distribution function; experimentally, the results have application to the use of the polarization properties of Thomson scattered light as a method of electron temperature measurement. The results can also be used as a reliable tool for benchmarking and verification of numerical codes for frequency resolved properties of TS polarization.

Excitation of dust kinetic Alfven waves by semi-relativistic ion beams

The growth rates for dust kinetic Alfvén wave (DKAW) based on semi-relativistic Maxwellian distribution function are investigated in a hot and magnetized plasma. The dispersion relation of DKAW is obtained on a dust acoustic velocity branch, and the kinetic instability due to cross-field semi-relativistic ion flow is examined by the effect of dust parameters. Analytical expressions are derived for various modes as a natural consequence of the form of the solution, and is shown through graphical representation that the presence of dust particles and the cross-field semi-relativistic ions sensibly modify the dispersion characteristics of low-frequency DKAW. The results are valid for a frequency regime well below the dust cyclotron frequency. We suggest that semi-relativistic particles are an important factor in the growth/damping of DKAWs. It is also found that relativistic effects appear with the dust lower hybrid frequency are more effective for dust kinetic Alfvén waves in the perpendicular component as compared to the parallel one. In particular, the relativistic effects associated with electrons suppress the instability while ions enhance the growth rates. The growth rates are significantly modified with dust parameters and streaming velocity of cross-field ions.

Debye screening under non-equilibrium plasma conditions

As has been revealed in a number of more recent astrophysical papers, in most of the tenuous space plasmas Maxwellian distribution functions cannot be expected for ions or electrons because of the lack of efficient relaxation processes. Many of the classical characteristics of plasmas, such as plasma frequency or Debye length, are calculated on the basis of the assumption, however, that Maxwellians prevail, which under most of the relevant astrophysical plasma conditions is not the case. We here therefore consider this specific problem of Debye shieldings of single charges in a plasma for the case of prevailing non-equilibrium distribution functions for ions and electrons. As typical non-equilibrium functions, so-called Kappa functions were considered with clear preference, and we therefore study here the Debye shielding in a plasma with Kappa-distributed electrons and ions. We show that the so-called Debye shielding increases with increasing extent of the high-velocity tail of the electron distribution function, or in other words, with lower Kappa index of the underlying Kappa function. In our calculations we demonstrate that the Debye lengths become enlarged by about a factor of 10 with respect to its classically expexted value if highly suprathermal electron distributions prevail with Kappa indices close to 1.5.

A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma

In order to enable kinetic simulation of non-thermal edge plasmas at a reduced computational cost, a new hybrid-Lagrangian δf scheme has been developed that utilizes the phase space grid in addition to the usual marker particles, taking advantage of the computational strengths from both sides. The new scheme splits the particle distribution function of a kinetic equation into two parts. Marker particles contain the fast space-time varying, δf, part of the distribution function and the coarse-grained phase-space grid contains the slow space-time varying part. The coarse-grained phase-space grid reduces the memory-requirement and the computing cost, while the marker particles provide scalable computing ability for the fine-grained physics. Weights of the marker particles are determined by a direct weight evolution equation instead of the differential form weight evolution equations that the conventional delta-f schemes use. The particle weight can be slowly transferred to the phase space grid, thereby reducing the growth of the particle weights. The non-Lagrangian part of the kinetic equation – e.g., collision operation, ionization, charge exchange, heat-source, radiative cooling, and others – can be operated directly on the phase space grid. Deviation of the particle distribution function on the velocity grid from a Maxwellian distribution function – driven by ionization, charge exchange and wall loss – is allowed to be arbitrarily large. The numerical scheme is implemented in the gyrokinetic particle code XGC1, which specializes in simulating the tokamak edge plasma that crosses the magnetic separatrix and is in contact with the material wall.

Radiation influence on the temperature-dependent parameters of fluids.

Based on the fundamental Bogolyubov chain of equations, a model relating the structural and thermophysical properties of the nonequilibrium liquid systems under irradiation in stationary state is introduced. The obtained results suggest that the thermophysical properties of the liquid systems under irradiation are defined by the "effective temperature" that can be calculated from the perturbed momentum distribution functions of the systems. It is shown that the structural changes in the liquid systems under irradiation are caused by the changes in the coefficients of the Maxwell distribution function due to the momentum exchange between the active particles and the particles forming the liquid. To confirm the theoretical predictions, a qualitative comparison of the model with the existing experimental data on irradiation influence on the surface tension coefficients of liquids is performed.

Determination of the electron temperature in plane-to-plane He dielectric barrier discharges at atmospheric pressure

Optical emission spectroscopy (OES) measurements coupled with a collisional-radiative model were used to characterize a plane-to-plane dielectric barrier discharge at atmospheric pressure operated in nominally pure helium. The model predicts the population densities for the n = 3 levels of He excited by electron impact processes from either ground or metastable states and takes into account excitation transfer processes between He n = 3 levels as well as all relevant radiative decays and quenching reactions. Time-resolved OES measurements indicate that line ratios from He n = 3 triplet states (for example, 587.5 nm-to-706.5 nm) and singlet states (for example, 667.8 nm-to-728.1 nm) first sharply rise as the discharge ignites and then slowly decrease as it extinguishes. Assuming that n = 3 levels are first populated only by electron impact on ground state He atoms and then only by electron impact on metastable He atoms as the discharge current and thus the metastable number density rise, triplet and singlet line ratios predicted by the model become in each opposite case solely dependent on the electron temperature Te (assuming Maxwellian electron energy distribution function). The values of Te deduced from the analysis of both ratios were relatively high early in the discharge cycle (around 1.0–1.4 eV) and then much lower near discharge extinction (around 0.15 eV). For analysis of time-integrated (or cycle-averaged) OES measurements, the electron temperatures were closer to the 0.15 eV values near the end of the discharge cycle, in good agreement with the values expected from theoretical predictions in the positive columns of He glow discharges at atmospheric pressure.

DISSIPATION OF PARALLEL AND OBLIQUE ALFVÉN-CYCLOTRON WAVES—IMPLICATIONS FOR HEATING OF ALPHA PARTICLES IN THE SOLAR WIND

We perform 2.5D hybrid simulations with massless fluid electrons and kinetic particle-in-cell ions to study the temporal evolution of ion temperatures, temperature anisotropies, and velocity distribution functions in relation to the dissipation and turbulent evolution of a broadband spectrum of parallel and obliquely propagating Alfven-cyclotron waves. The purpose of this paper is to study the relative role of parallel versus oblique Alfven-cyclotron waves in the observed heating and acceleration of alpha particles in the fast solar wind. We consider collisionless homogeneous multi-species plasma, consisting of isothermal electrons, isotropic protons, and a minor component of drifting α particles in a finite-β fast stream near the Earth. The kinetic ions are modeled by initially isotropic Maxwellian velocity distribution functions, which develop nonthermal features and temperature anisotropies when a broadband spectrum of low-frequency nonresonant, ω ≤ 0.34 Ωp, Alfven-cyclotron waves is imposed at the beginning of the simulations. The initial plasma parameter values, such as ion density, temperatures, and relative drift speeds, are supplied by fast solar wind observations made by the Wind spacecraft at 1 AU. The imposed broadband wave spectra are left-hand polarized and resemble Wind measurements of Alfvenic turbulence in the solar wind. The imposed magnetic field fluctuations for all cases are within the inertial range of the solar wind turbulence and have a Kraichnan-type spectral slope α = −3/2. We vary the propagation angle from θ = 0° to θ = 30° and θ = 60°, and find that the heating of alpha particles is most efficient for the highly oblique waves propagating at 60°, whereas the protons exhibit perpendicular cooling at all propagation angles.

Rate coefficients for electron impact excitation of N2

Rate coefficients for electron impact excitation of the nitrogen molecule have been determined. Calculations have been performed for all relevant non-elastic processes: rotational excitation, vibrational excitation, electronic excitation into singlet and triplet states and particularly for dissociation and ionization. The rate coefficients have been determined for both equilibrium and non-equilibrium conditions in the presence of electric and magnetic fields. For the equilibrium conditions the Maxwellian electron energy distribution function has been used for determining rate coefficients in the energy region from 0 to 17eV. In the case of the non-equilibrium conditions, where the homogenous electric and magnetic fields are present, we have calculated electron energy distribution functions needed for determining rate coefficients by use of a Monte Carlo simulation developed in our laboratory. The results for both equilibrium and non-equilibrium rate coefficients are presented in the paper and similarities and differences between them have been discussed.

Time-resolved study of the extreme-ultraviolet emission and plasma dynamics of a sub-Joule, fast capillary discharge

In this work, we discuss experimental observations on the dynamics of a fast, low energy capillary discharge when operated in argon and its properties as an intense source of extreme-ultraviolet (EUV) radiation. The discharge pre-ionization and self-triggering were accomplished by the use of the hollow cathode effect. This allowed a compact size and low inductance discharge with multi-kA current level and a quarter-period of ∼10 ns at sub-Joule energy level. We used the novel moiré and schlieren diagnostics with a 12 ps laser to obtain the time evolution of the line electron density and to study the plasma dynamics. EUV spectroscopy and filtered diodes were also implemented to estimate the plasma temperature and density throughout the evolution of the discharge. EUV source size was measured by using a filtered slit-wire camera. We observed that EUV emission starts from a compressed plasma on axis during the second quarter-period of the current and continues until the fifth quarter-period. Ionization levels from Ar VII to X were observed. By comparing the EUV emission spectra with synthetic spectra, we found that at the onset of emission (∼7 ns), the plasma is well fitted by a single Maxwellian electron distribution function with Te ∼ 12 eV and ne ∼ 1017 cm−3. Close to peak emission (∼13 ns), plasma temperature and density increase to ∼20 eV and ne ∼ 1018 cm−3, respectively. However, in order to successfully match the experimental data, a two component electron distribution function was necessary. Later in time, a smaller fraction in the high energy component and higher temperature suggests homogenization of the plasma. The moiré and schlieren diagnostics showed multiple radial compression-waves merging on axis throughout the discharge; they are an important heating mechanism that leads to a period of severe turbulence at peak EUV emission. It was also observed that emission ceases when the axial maximum of the electron density collapses.

Vasyliunas–Cairns distribution function for space plasma species

A more generalized form of non-Maxwellian distribution function (that can be named as Vasyliunas–Cairns distribution function) is introduced. Its basic properties are numerically analyzed by the variation of two important parameters, namely, α (which shows the amount of energetic particles present in the plasma system) and κ (which shows the superthermality of the plasma species). It has been observed that (i) for α→0 (κ→∞), the Vasyliunas–Cairns distribution function reduces to the Vasyliunas or κ (Cairns or nonthermal) distribution function; (ii) for α→0 and κ→∞, it reduces to the Maxwellian distribution function; and (iii) the effect of the parameter α (κ) significantly modifies the basic properties of the Vasyliunas (Cairns) distribution function. The applications of this generalized non-Maxwellian distribution function (Vasyliunas–Cairns distribution function) in different space plasma situations are briefly discussed.

Linear dispersion properties of ring velocity distribution functions

Linear properties of ring velocity distribution functions are investigated. The dispersion tensor in a form similar to the case of a Maxwellian distribution function, but for a general distribution function separable in velocities, is presented. Analytical forms of the dispersion tensor are derived for two cases of ring velocity distribution functions: one obtained from physical arguments and one for the usual, ad hoc ring distribution. The analytical expressions involve generalized hypergeometric, Kampé de Fériet functions of two arguments. For a set of plasma parameters, the two ring distribution functions are compared. At the parallel propagation with respect to the ambient magnetic field, the two ring distributions give the same results identical to the corresponding bi-Maxwellian distribution. At oblique propagation, the two ring distributions give similar results only for strong instabilities, whereas for weak growth rates their predictions are significantly different; the two ring distributions have different marginal stability conditions.

A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows

In this work, a truly three-dimensional (3D) flux solver is presented for simulation of inviscid compressible flows. Like the conventional multi-dimensional gas-kinetic scheme, in the present work, the local solution of 3D Boltzmann equation at the cell interface is used to evaluate the flux. On the other hand, different from most of the existing gas-kinetic schemes, which are constructed from Maxwellian distribution function, the present flux solver is derived from a simple distribution function defined on the spherical surface in the phase velocity space. As a result, the explicit expression of flux vector at the cell interface can be simply given. Since the simple distribution function is defined on the spherical surface, for simplicity, it is termed as sphere function hereafter. In addition, to simulate fluid flow problems with strong shock waves, the non-equilibrium part of the distribution function is regarded as numerical dissipation and involved in evaluating the inviscid flux at the cell interface. The weight of the non-equilibrium part is controlled by introducing a switch function which ranges from 0 to 1. In the smooth region, the switch function takes a value close to zero, while around the strong shock wave, it tends to one. To validate the proposed flux solver, several transonic, supersonic and hypersonic inviscid flows are simulated. Numerical results showed that the present solver can provide accurate numerical results for three-dimensional inviscid flows with strong shock waves.

Plasma immersion ion implantation characteristics with q-nonextensive electron velocity distribution

The plasma immersion ion implantation process is investigated in the presence ofq–nonextensive electrons by using a one-dimensional fluid model. The effect of the nonextensivity parameter,q, on the plasma parameters and sheath dynamics during the implantation process is studied. The results show that the implantation dose can be enhanced in the presence of energetic electrons at the tail of the distribution function. Different parameters of plasma such as sheath thickness, ion velocity and ion density show more change at the larger values of theq–parameter. Furthermore, the results of simulation tend to what is predicted by the Maxwellian electron distribution function (q= 1).

Observations in the E region ionosphere of kappa distribution functions associated with precipitating auroral electrons and discrete aurorae

AbstractPrecipitating auroral electrons can produce discrete auroral arcs that contain signatures of the magnetospheric auroral source region. Differential number flux observations over two discrete aurorae were obtained by the Auroral Currents and Electrodynamics Structure sounding rocket mission, which successfully launched in 2009. These observations were made at E region altitudes of approximately 130 km. A model of precipitating auroral electrons as described by Evans (1974) was fit to the electron differential number flux obtained by the payloads, and parameters from the model were used to infer properties of the auroral source region. It is shown that the field‐aligned precipitating electrons were better fit by a kappa distribution function versus a Maxwellian distribution function for the equatorward side of the first, quasi‐stable, auroral arc crossing. The latter half of the first auroral arc crossing and second auroral crossing show that the precipitating electrons were better fit by a Maxwellian distribution function, which provides additional observational confirmation of previous studies. The low‐energy electron population determined by the Evans (1974) model was within a factor of 2 of the observed differential number flux. The source region parameters determined from fitting the model to the data were compared with relevant studies from sounding rockets and satellites. Our observations are consistent with the results of Kletzing et al. (2003) that the plasma sheet electrons mapping to auroral zone invariant latitudes are characterized by kappa distribution functions.

Anomalous skin effects in a weakly magnetized degenerate electron plasma

Fully relativistic analysis of anomalous skin effects for parallel propagating waves in a weakly magnetized degenerate electron plasma is presented and a graphical comparison is made with the results obtained using relativistic Maxwellian distribution function [G. Abbas, M. F. Bashir, and G. Murtaza, Phys. Plasmas 18, 102115 (2011)]. It is found that the penetration depth for R- and L-waves for degenerate case is qualitatively small in comparison with the Maxwellian plasma case. The quantitative reduction due to weak magnetic field in the skin depth in R-wave for degenerate plasma is large as compared to the non-degenerate one. By ignoring the ambient magnetic field, previous results for degenerate field free case are salvaged [A. F. Alexandrov, A. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrodynamics (Springer-Verlag, Berlin/Heidelberg, 1984), p. 90].

Effect of the q-nonextensive electron velocity distribution on a magnetized plasma sheath

In this work, a sheath model has been developed to investigate the effect of the q-nonextensive electron velocity distribution on the different characteristics of a magnetized plasma. By using Segdeev potential method, a modified Bohm criterion for a magnetized plasma with the nonextensive electron velocity distribution is derived. The sheath model is then used to analyze numerically the sheath structure under different q, the parameter quantifying the nonextensivity degree of the system. The results show that as the q-parameter decreases, the floating potential becomes more negative. The sheath length increases at the lower values of the q-parameter due to the increase in the electron population at the high-energy tail of the distribution function. As q-parameter decreases, the effective temperature of the electrons increases which results in a more extended plasma sheath. The ion velocity and density profiles for the different nonextensivity degrees of the system reflect the gyro-motion of the ions in the presence of the magnetic field. Furthermore, the results coincide with those given by the Maxwellian electron distribution function, when q tends to 1.

Vlasov-kinetic computer simulations of electrostatic waves in dusty plasmas: an overview of recent results

A series of numerical simulations based on a recurrence-free Vlasov kinetic model using kinetic phase point trajectories are presented. Electron-ion plasmas and three-component (electron-ion-dust) dusty or complex plasmas are considered, via independent simulations. Considering all plasma components modeled through a kinetic approach, the linear and nonlinear behavior of ion-acoustic excitations is investigated. Maxwellian and kappa-type (superthermal) distribution functions are assumed, as initial conditions, in separate simulations for the sake of comparison. The focus is on the parametric dependence of ion-acoustic waves on the electron-to-ion temperature ratio and on the dust concentration.