non-Maxwellian Velocity Distribution Function Research Articles

Ten-moment fluid model with heat flux closure for gasdynamic flows

The gasdynamic 10-moment equations are revisited, applying a Chapman-Enskog type expansion to achieve the closure for heat flux. The resulting gradient-based closure performs favorably against previously employed closures in regions of shocks when compared to the kinetic results obtained from Direct Simulation Monte Carlo simulations. The 10-moment model can capture finite kinetic effects due to non-Maxwellian velocity distribution functions, as compared to the 5-moment Navier-Stokes equations, and remains hyperbolic-parabolic with real wave speeds under all initial conditions. Self-consistent kinetic boundary conditions are derived without consideration of a distinct Knudsen layer. The model is applied to a number of canonical gasdynamics problems in one and two dimensions such as steady normal and oblique shocks, as well as the Sod shock tube problem.

Non-Maxwellian electron effects on the macroscopic response of a Hall thruster discharge from an axial–radial kinetic model

A 2D axial–radial particle-in-cell (PIC) model of a Hall thruster discharge has been developed to analyze (mainly) the fluid equations satisfied by the azimuthally-averaged slow dynamics of electrons. Their weak collisionality together with a strong interaction with the thruster walls lead to a non-Maxwellian velocity distribution function (VDF). Consequently, the resulting macroscopic response differs from a conventional collisional fluid. First, the gyrotropic (diagonal) part of the pressure tensor is anisotropic. Second, its gyroviscous part, although small, is relevant in the azimuthal momentum balance, where the dominant contributions are orders of magnitude lower than in the axial momentum balance. Third, the heat flux vector does not satisfy simple laws, although convective and conductive behaviors can be identified for the parallel and perpendicular components, respectively. And fourth, the electron wall interaction parameters can differ largely from the classical sheath theory, based on near Maxwellian VDF. Furthermore, these effects behave differently in the near-anode and near-exit regions of the channel. Still, the profiles of basic plasma magnitudes agree well with those of 1D axial fluid models. To facilitate the interpretation of the plasma response, a quasiplanar geometry, a purely-radial magnetic field, and a simple empirical model of cross-field transport were used; but realistic configurations and a more elaborate anomalous diffusion formulation can be incorporated. Computational time was controlled by using an augmented vacuum permittivity and a stationary depletion law for neutrals.

Recovering non-Maxwellian particle velocity distribution functions from collective Thomson-scattered spectra

Collective optical Thomson scattering (TS) is a diagnostic commonly used to characterize plasma parameters. These parameters are typically extracted by a fitting algorithm that minimizes the difference between a measured scattered spectrum and an analytic spectrum calculated from the velocity distribution function (VDF) of the plasma. However, most existing TS analysis algorithms assume that the VDFs are Maxwellian, and applying an algorithm that makes this assumption does not accurately extract the plasma parameters of a non-Maxwellian plasma due to the effect of non-Maxwellian deviations on the TS spectra. We present new open-source numerical tools for forward modeling analytic spectra from arbitrary VDFs and show that these tools are able to more accurately extract plasma parameters from synthetic TS spectra generated by non-Maxwellian VDFs compared to standard TS algorithms. Estimated posterior probability distributions of fits to synthetic spectra for a variety of example non-Maxwellian VDFs are used to determine uncertainties in the extracted plasma parameters and show that correlations between parameters can significantly affect the accuracy of fits in plasmas with non-Maxwellian VDFs.

Inertial and anisotropic pressure effects on cross-field electron transport in low-temperature magnetized plasmas

In this paper, a one-dimensional (1D) particle-in-cell Monte Carlo collision (PIC-MCC) model is developed to investigate the effects of anisotropic pressure and inertial terms due to non-Maxwellian velocity distribution functions on cross-field electron transport. The conservation of momentum is evaluated by taking the moments of the first-principles gas-kinetic equation. A steady-state discharge is obtained without any low-frequency ionization oscillations by considering an anomalous electron scattering profile. The results obtained from the 1D PIC-MCC model are compared with fluid models, including the quasi-neutral drift-diffusion (DD), non-neutral DD, and full fluid moment models. The discharge current obtained from the PIC-MCC model is in good agreement with the fluid models. The cross-field electron transport due to the inertial terms, i.e. the gradient of axial and azimuthal drift, is evaluated. Moreover, PIC-MCC simulation results show non-zero, anisotropic, off-diagonal pressure tensor terms due to asymmetric non-Maxwellian electron velocity distribution function, potentially contributing to cross-field electron transport.

Macroscopic plasma analysis from 1D-radial kinetic results of a Hall thruster discharge

A radial particle-in-cell model of the weakly-collisional plasma discharge in a Hall thruster, provides the non-Maxwellian velocity distribution functions (VDF) of ions and electrons. The model considers a radial magnetic field, secondary electron emission from the two walls, and phenomenological models of anomalous electron scattering. The electron VDF is used to assess the different terms in the macroscopic momentum and energy equations, identifying those differing from the standard fluid model for a near-Maxwellian VDF. The pressure tensor consists of an anisotropic gyrotropic part and a small gyroviscous part. Nonetheless, the gradient of this last one affects the cross-field electron current density, generating radial undulations that resemble those reported for near-wall conductivity. A gyroviscous energy flux is identified too. The heat flux parallel to the magnetic lines does not follow a conductive-type law but a convective-type one, already found in other weakly-collisional plasmas. The tails of the electron velocity distribution function are partially depleted due to wall collection, leading to reduced electron fluxes of particles and energy, which are characterized with parameters useful for fluid models. Differences in the plasma response for annular and planar channel geometries are highlighted. The levels of replenishment of the electron VDF and of the asymmetries in radial profiles differ for isotropic and anisotropic anomalous scattering models.

Free Energy Sources in Current Sheets Formed in Collisionless Plasma Turbulence

Abstract Collisionless dissipation of macroscopic energy into heat is an unsolved problem of space and astrophysical plasmas, e.g., solar wind and Earth’s magnetosheath. The most viable process under consideration is the turbulent cascade of macroscopic energy to kinetic scales where collisionless plasma processes dissipate the energy. Space observations and numerical simulations show the formation of kinetic scale current sheets in turbulent plasmas. Instabilities in these current sheets (CS) can provide collisionless dissipation and influence the turbulence. Spatial gradients of physical quantities and non-Maxwellian velocity distribution functions provide the free energy sources for CS plasma instabilities. To determine the free energy sources provided by the spatial gradients of plasma density and electron/ion bulk velocities in CS formed in collisionless turbulent plasmas with an external magnetic field B 0, we carried out two-dimensional particle-in-cell-hybrid simulations and interpret the results within the limitations of the simulation model. We found that ion-scale CS in a collisionless turbulent plasma are formed primarily by electron shear flows, i.e., electron bulk velocity inside CS is much larger than ion bulk velocity while the density variations through the CS are relatively small (<10%). The electron bulk velocity and, thus, the current density inside the sheets are directed mainly parallel to B 0. The shear in the perpendicular electron and ion bulk velocities generates parallel electron and ion flow vorticities. Inside CS, parallel electron flow vorticity exceeds the parallel ion flow vorticity, changes sign around the CS centers, and peaks near the CS edges. An ion temperature anisotropy develops near CS during the CS formation. It has a positive correlation with the parallel ion and electron flow vorticities. Theoretical estimates support the simulation results.

Nonlinear Evolution of Ion Kinetic Instabilities in the Solar Wind

In-situ observations of the solar wind (SW) plasma from 0.29 to 1AU show that the protons and alpha particles are often non-Maxwellian, with evidence of kinetic instabilities, temperature anisotropies, differential ion streaming, and associated magnetic fluctuations spectra. The kinetic instabilities in the SW multi-ion plasma can lead to preferential heating of alpha particles and the dissipation of magnetic fluctuation energy, affecting the kinetic and global properties of the SW. Using for the first time a three-dimensional hybrid model, where ions are modeled as particle using the Particle-In-Cell (PIC) method and electrons are treated as fluid, we study the onset, nonlinear evolution and dissipation of ion kinetic instabilities. The Alfven/ion-cyclotron, and the ion drift instabilities are modeled in the region close to the Sun (~10R_s). Solar wind expansion is incorporated in the model. The model produces self-consistent non-Maxwellian velocity distribution functions (VDFs) of unstable ion populations, the associated temperature anisotropies, and wave spectra for several typical SW instability cases in the nonlinear growth and saturation stage of the instabilities. The 3D hybrid modeling of the multi-ion SW plasma could be used to study the SW acceleration region close to the Sun that will be explored by the Parker Solar Probe mission.

On distribution function of free and bound electrons in equilibrium Coulomb system

In classical thermodynamics, the velocity distribution function of particles is always Maxwell distributionfor any density. This is due to the fact that the dependences on the pulses and coordinates in theexpression for the total energy are separated. Integration over coordinates leads to the appearance of aconfiguration integral, and the remaining part is divided into the product of Maxwell distributionfunctions. In the case of formation of bound states (molecules) in an atomic gas, the full phase space ofthe relative motion of two particles is divided into two parts. The first corresponds to negative energies ofrelative motion (molecular component), and the second to positive (free atoms). The velocity distributionfunction remains Maxwellian, if we ignore the fact of separation of the phase space. It can be assumedthat for free atoms the velocity (kinetic energies) distribution may be different from Maxwell. Forplasmas, the assumption of the non-Maxwellian velocity distribution function of free electrons was made.The influence of the non-Maxwell electron distribution function on the recombination coefficient isestimated.

Suprathermal electron distributions in the solar transition region

Suprathermal tails are a common feature of solar wind electron velocity distributions, and are expected in the solar corona. From the corona, suprathermal electrons can propagate through the steep temperature gradient of the transition region towards the chromosphere, and lead to non-Maxwellian electron velocity distribution functions (VDFs) with pronounced suprathermal tails. We calculate the evolution of a coronal electron distribution through the transition region in order to quantify the suprathermal electron population there. A kinetic model for electrons is used which is based on solving the Boltzmann-Vlasov equation for electrons including Coulomb collisions with both ions and electrons. Initial and chromospheric boundary conditions are Maxwellian VDFs with densities and temperatures based on a background fluid model. The coronal boundary condition has been adopted from earlier studies of suprathermal electron formation in coronal loops. The model results show the presence of strong suprathermal tails in transition region electron VDFs, starting at energies of a few 10 eV. Above electron energies of 600 eV, electrons can traverse the transition region essentially collision-free. The presence of strong suprathermal tails in transition region electron VDFs shows that the assumption of local thermodynamic equilibrium is not justified there. This has a significant impact on ionization dynamics, as is shown in a companion paper.

Simulated ion velocity distribution and its incoherent scattering spectrum in the high-latitude ionosphere

If the relatively larger convection electric field and small the collision frequency between ions and neutrals in the high-latitude ionosphere are considered, the integral solution of the non-Maxwellian ion velocity distribution function for the relaxation collision model is simplified, and a torus distribution function of the ion velocity is obtained. Also, the feature of the torus velocity distribution function is analyzed and simulated with different parameters in high-latitude ionosphere. Furthermore, by using the electromagnetic radiation theory of Sheffield and taking into account the ion temperature anisotropy and its variation with the electric field, we simulate the incoherent scattering spectrum of ions in different directions. The research shows that the ionospheric convection electric field, the ion-neutral collision frequency and the ion anisotropic temperature distribution all have an effect on the incoherent scattering spectrum.

Radiative transfer of emission lines with non-Maxwellian velocity distribution function: Application to Mercury D2 sodium lines

We describe the theory and the numerical model used to simulate Doppler-broadened resonant emission lines for any type of velocity function distribution. The field of application of this theoretical model of radiative transfer is particularly well suited for the study of weakly dense atmospheres which are far from local thermodynamic equilibrium (as is the case for most planetary upper atmospheres/exospheres). This model is applied to study the potential effects of radiative transfer and non-Maxwellian distributions on the spectral shape of the D2 sodium emission line in Mercury’s exosphere. The small (but not negligible) optical thickness of the D2 sodium emission of an exosphere like Mercury’s (with a peak optical thickness of ∼2) can result in an increase of the observed spectral width by up to a few tens of percent. Combined with the non-Maxwellian nature of the exospheric velocity distribution, it may lead to an increase in the spectral width by a factor of up to 2 with respect to the width of an optically thin emission and a Maxwellian distribution. This model has been used to analyze new THEMIS observations of Mercury’s exosphere obtained at very high spectral resolution in a companion paper (Leblanc, F., Chaufray, J.-Y., Doressoundiram, A., Berthelier, J.-J., Mangano, V., Lopez-Ariste, A., Borin, P. [2013]).

Landau damping of Langmuir waves in non-Maxwellian plasmas

As free electrons move in the nearest neighbour ion’s potential well, the equilibrium velocity departs from Maxwell distribution. The effect of the non-Maxwellian velocity distribution function (NMVDF) on many properties of the plasma such as the transport coefficients, the kinetic energy, and the degree of ionization is found to be noticeable. A correction to the Langmuir wave dispersion relation is proved to arise due to the NMVDF as well [Phys. Plasmas 17, 052105 (2010)]. The study is extended hereafter to include the effect of NMVDF on the Landau damping of Langmuir wave.

Kinetic Models for Whistler Wave Scattering of Electrons in the Solar Corona and Wind

Kinetic models are necessary to describe the physical processes associated with non-Maxwellian velocity distribution functions (VDFs) of electrons or ions in the solar corona and wind. It is shown that pitch-angle scattering of electrons in the solar wind needs to be considered in kinetic solar wind models. Coulomb collisions are not efficient enough to provide this scattering, but resonant interaction with whistler waves is. A solar wind model for undisturbed fast wind is presented, and the influence of scattering on flare electron propagation is investigated. Furthermore, it is found that resonant interaction of electrons with whistler waves is capable of producing suprathermal tails of electron distributions even under quiet conditions without flare activity.

Gas kinetics and dust dynamics in low-density comet comae

Extensive regions of low-density cometary comae are characterized by important deviations from the Maxwell–Boltzmann velocity distribution, i.e. breakdown of thermodynamic equilibrium. The consequences of this on the shapes of emission and absorption lines, and for the acceleration of solid bodies due to gas drag, have rarely been investigated. These problems are studied here to aid in the development of future coma models, and in preparation for observations of Comet 67P/Churyumov–Gerasimenko from the ESA Rosetta spacecraft. Two topics in particular, related to Rosetta, are preparation for in situ observations of water, carbon monoxide, ammonia, and methanol emission lines by the mm/sub-mm spectrometer MIRO, as well as gas drag forces on dust grains and on the Rosetta spacecraft itself. Direct Simulation Monte Carlo (DSMC) modeling of H 2O/CO mixtures in spherically symmetric geometries at various heliocentric distances are used to study the evolution of the (generally non-Maxwellian) velocity distribution function throughout the coma. Such distribution functions are then used to calculate Doppler broadening profiles and drag forces. It is found that deviation from thermodynamic equilibrium indeed is commonplace, and already at 2.5 AU from the Sun the entire comet coma displays manifestations of such breakdown, e.g., non-equal partitioning of energy between kinetic and rotational modes, causing substantial differences between translational and rotational temperatures. We exemplify how deviations from thermodynamic equilibrium affect the properties of Doppler broadened line profiles. Upper limits on the size of liftable dust grains as well as terminal grain velocities are presented. Furthermore, it is demonstrated that the drag-to-gravity force ratio is likely to decrease with decreasing cometocentric distance, which may be of relevance both for Rosetta and for the lander probe Philae.

Electrical Conductivity in a Non-Maxwellian Plasma with Adjustable High-Energy-Tail Distribution

In this paper, a solution to the Fokker–Planck equation is presented, which is extended to the field particles' high-energy-tail non-Maxwellian velocity distribution function in transport theory. Based on the correct physical concept of collision intensity, introduced by CHANG and LI, the electrical conductivities for like-particles collisions are obtained in different conditions. The modified Fokker–Planck coefficients for non-Maxwellian scattering are applied in the study. It is found that the parallel part of the collision operator plays an important role. The non-Maxwellian scattering will stimulate the transport processes in various degrees with mutative deviation parameters.

The Modified Friction and Diffusion Coefficients of Fokker–Planck Equation and Relaxation Rates for Non-Maxwellian Scattering

In this paper, a new method to derive the Fokker–Planck coefficients defined by a non-Maxwellian velocity distribution function for the field particles is presented. The threefold integral and the new Debye cutoff parameter, which were introduced by CHANG and LI, are applied. Therefore, divergence difficulties and the customary replacement of relative velocity g by thermal velocity vth are naturally avoided. The probability function P(v,Δv) for non-Maxwellian scattering is derived by the method of choosing velocity transfer Δv, which is a true measure of collision intensity, as an independent variable. The method enables the difference between small-angle scattering and small-momentum-transfer collisions of the inverse-square force to be well clarified. With the help of the probability function, the Fokker–Planck coefficients are obtained by a normal original Fokker–Planck approach. The friction and diffusion coefficients of the Fokker–Planck equation are modified for non-Maxwellian scattering and are used to investigate the relaxation processes for the weakly coupled plasma. The profiles of the relaxation rates show that the slowing down and deflection processes are weakened in the conditions of non-Maxwellian scattering.

Perpendicularly propagating electromagnetic modes in a strongly magnetized hot plasma with non-Maxwellian distribution function

Electromagnetic modes (ordinary and extraordinary) for strongly magnetized plasma are studied and their damping factors γor and γex are calculated using non-Maxwellian velocity distribution function. It is observed that for moderate values of the spectral indices r and q [used in (r,q) distribution functions], both the damping decrements show substantial change. As the value of the spectral index r increases for a fixed value of q, the damping increases for the O mode but decreases for the X mode. In the limiting case of r=0, q→∞, the damping factors reduce to the standard Maxwellian values.

Laser-Induced Fluorescence Measurement of the Ion-Energy-Distribution Function in a Collisionless Reconnection Experiment

Observations in space and laboratory plasmas suggest magnetic reconnection as a mechanism for ion heating and formation of non-Maxwellian ion velocity distribution functions (IVDF). Laser-induced fluorescence measurements of the IVDF parallel to the X line of a periodically driven reconnection experiment are presented. A time-resolved analysis yields the evolution of the IVDF within a reconnection cycle. It is shown that reconnection causes a strong increase of the ion temperature, where the strongest increase is found at the maximum reconnection rate. Monte Carlo simulations demonstrate that ion heating is a consequence of the in-plane electric field that forms around the X line in response to reconnection.

Modelling the solar wind interaction with Mercury by a quasi-neutral hybrid model

Abstract. Quasi-neutral hybrid model is a self-consistent modelling approach that includes positively charged particles and an electron fluid. The approach has received an increasing interest in space plasma physics research because it makes it possible to study several plasma physical processes that are difficult or impossible to model by self-consistent fluid models, such as the effects associated with the ions’ finite gyroradius, the velocity difference between different ion species, or the non-Maxwellian velocity distribution function. By now quasi-neutral hybrid models have been used to study the solar wind interaction with the non-magnetised Solar System bodies of Mars, Venus, Titan and comets. Localized, two-dimensional hybrid model runs have also been made to study terrestrial dayside magnetosheath. However, the Hermean plasma environment has not yet been analysed by a global quasi-neutral hybrid model. In this paper we present a new quasi-neutral hybrid model developed to study various processes associated with the Mercury-solar wind interaction. Emphasis is placed on addressing advantages and disadvantages of the approach to study different plasma physical processes near the planet. The basic assumptions of the approach and the algorithms used in the new model are thoroughly presented. Finally, some of the first three-dimensional hybrid model runs made for Mercury are presented. The resulting macroscopic plasma parameters and the morphology of the magnetic field demonstrate the applicability of the new approach to study the Mercury-solar wind interaction globally. In addition, the real advantage of the kinetic hybrid model approach is to study the property of individual ions, and the study clearly demonstrates the large potential of the approach to address these more detailed issues by a quasi-neutral hybrid model in the future.Key words. Magnetospheric physics (planetary magnetospheres; solar wind-magnetosphere interactions) – Space plasma physics (numerical simulation studies)

Kinetic Theory of Electron-Plasma and Ion-Acoustic Waves in Nonuniformly Heated Laser Plasmas

Using Fokker-Planck simulations, we have studied the evolution of different charge state plasmas heated by lasers having transverse spatial modulations (speckle patterns) and hot spots. The kinetic response of the plasma is computed using the resulting strongly non-Maxwellian electron velocity distribution functions. Significant reductions in the electron plasma wave Landau damping rates and large modulations in the local ion acoustic frequencies are found which impact the behavior of all parametric instabilities in such plasmas. Various experimental consequences are given.