Velocity Distribution Function Research Articles

The excitation of whistler waves and wave-particle interaction in magnetic reconnection

A 2.5D PIC kinetic code is executed to simulate the excitation of electron whistlers and ion cyclotron whistler (L wave) during magnetic reconnection. Our simulation results suggest that quasi-parallel and anti-parallel whistler waves can be excited in the separatrix region near X-Line and in the outflow region, respectively. Electron whistlers play a crucial role in energy release and transfer, as the small mass of electrons allows them to move rapidly under the perturbation of a magnetic field during magnetic reconnection. The L wave excited near the separatrix region far from X-Line, it propagate obliquely to the local magnetic field. Velocity distribution functions (VDFs) indicate that these waves are associated with temperature anistropy instabilities of electron-beams (ion-beam). The electron whistler in the separatrix region is excited by cyclotron resonance due to the electron beam. The electron whistler observed in the outflow region is generated by Landau resonance owing to beam-mode, where the v∥ is close to the phase speed vph of the wave. Regarding L wave, their excitation mechanism is attributed to significant perpendicular temperature anisotropy instability (Ti⊥>Ti∥) of ion-beam, and satisfies cyclotron resonance condition. Meanwhile, our simulation results illustrate that the existence of energy conversion between particles and waves during the wave excitation by using the Wave-Particle Interaction Analyzer (WPIA).

Presheath-like structures and effusive particle losses for biased probes at and near I–V electron saturation

A theory for presheath-like structures near probes biased at and above the plasma potential is developed for collisionless plasmas with an electron-neutral mean free path on the order of the chamber scale. The theory predicts presheath-like perturbations to the plasma that result from the free streaming of electrons and an effusion loss process from the chamber at the electrode. For these situations, a loss-cone-like velocity distribution function for electrons is predicted where the loss angle of the depletion region corresponds to the angular size of the electrode at a specified distance. The angle of the loss cone becomes 180° at the sheath edge. In comparison with a previous collisional electron presheath model that required electrons satisfy a Bohm criterion at the sheath edge [Scheiner et al., Phys. Plasmas 22, 123520 (2015)], the present work suggests that no such condition is needed for collisionless low pressure plasmas in the ≲10 mTorr range. The theory predicts the generation of a density depletion of roughly 0.5ne and an electron velocity moment of tens of percent of the electron thermal speed by the sheath edge in a presheath with a potential drop of less than Ti/e. The range of this presheath perturbation is determined by the electrode geometry instead of the collisional mean free path. These predictions are tested against previously published particle in cell simulations and are found to be in good agreement.

Role of spontaneous thermal emissions in inflationary laser Raman instability

The role of thermal fluctuations on the stimulated Raman backscattering instability is investigated by means of Vlasov and particle-in-cell (PIC) simulations in a regime of strong linear Landau damping of the Langmuir wave. The instability is initially convective and amplifies thermal noise, leading to a low-amplitude back-scattered laser sideband. Linear Landau damping of the Langmuir sideband modifies and flattens the electron velocity distribution function at the resonant velocity, leading to a gradual decrease in the Landau damping rate and an increase in the convective amplification. The Langmuir wave traps electrons resulting in a rapid nonlinear absolute instability and large amplitude flashes of backscattered light off large amplitude Langmuir waves with trapped electrons, leading to the production of hot electrons. Conditions for simulating realistic thermal noise with Vlasov and PIC simulations are discussed and defined.

Effects of dust size distribution and non-Maxwellian electrons on shock waves in a dusty plasma

We present a study of dust acoustic shock waves in a non-Maxwellian plasma with dust charge fluctuations, which are seen to cause a dissipation term in fluid model, and consequently shocks are generated. In particular, we focus on dust acoustic waves as affected by various dust size distributions. Two distinct dust size distributions—the polynomial and the power law distributions—have been used. For analytical investigation of nonlinear wave propagation in complex plasmas, a reductive perturbation approach is used to obtain the Burgers equation. A dusty plasma system with non-Maxwellian Kappa distribution is considered and it is shown that the amplitude of a shock wave, for the dust size distribution is larger than that for the mono-sized counterpart, while the shock width manifests an opposite trend. Furthermore, the shock wave speed is also affected by the dust size distributions as well as by the nature of velocity distribution function. To benchmark our findings, we apply the proper limit on the spectral index, i.e., κ→∞, and retrieve the Maxwellian results. The current findings are crucial for comprehending respective shock distributions for a plasma system exhibiting non-thermal characteristics and having dust size distributions.

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.

Kinetic glass transition in granular gases and nonlinear molecular fluids.

In this paper, we investigate, both analytically and numerically, the emergence of a kinetic glass transition in two different model systems: a uniformly heated granular gas and a molecular fluid with nonlinear drag. Despite the profound differences between these two physical systems, their behavior in thermal cycles share strong similarities, which stem from the relaxation time diverging algebraically at low temperatures for both systems. When the driving intensity--for the granular gas-or the bath temperature-for the molecular fluid-is decreased to sufficiently low values, the kinetic temperature of both systems becomes "frozen" at a value that depends on the cooling rate through a power law with the same exponent. Interestingly, this frozen glassy state is universal in the following sense: for a suitable rescaling of the relevant variables, its velocity distribution function becomes independent of the cooling rate. Upon reheating, i.e., when either the driving intensity or the bath temperature is increased from this frozen state, hysteresis cycles arise and the apparent heat capacity displays a maximum. The numerical results obtained from the simulations are well described by a perturbative approach.

Dynamic acceleration of energetic protons by an interplanetary collisionless shock

Interplanetary collisionless shocks are known to be capable of accelerating charged particles up to hundreds of MeV. However, the underlying acceleration mechanisms are still under debate. We present the dynamic behaviors of energetic protons that are accelerated by an interplanetary shock that was observed with unprecedented high-resolution measurements by the Electron-Proton Telescope sensor of the Energetic Particle Detector suite on board the Solar Orbiter spacecraft on 2021 November 3. We constrain the potential acceleration mechanisms and processes. We first reconstructed the proton pitch-angle distributions (PADs) in the solar wind frame. Then, we examined the temporal flux profile, PAD, and the velocity distribution function of energetic protons close to the shock, and we qualitatively compared the observations with theoretical predictions. Moreover, we applied a velocity dispersion analysis (VDA) to an observed velocity dispersion event and derived the proton path length and release time at the shock. Then, we tested this derivation by comparing it with the shock motion and the magnetic field configuration. We find that sim 1000-4000 keV protons exhibit a rapid-rise, rapid-decay temporal flux profile with a clear velocity dispersion sim 2 min before the shock, similar to impulsive solar energetic particle events. The proton path length based on the VDA of this event is consistent with the length derived from the shock motion and magnetic field configuration. The peak spectrum in this event appears to be steeper than the spectrum at the shock. Furthermore, we find that sim 50-200 keV proton fluxes peak between sim 10 and sim 20 s before the shock, with an inverse velocity dispersion. The velocity dispersion event and the inverse velocity dispersion event are both accompanied by magnetic kinks or switchbacks. In addition, two distinct proton populations appear near the shock. The first population at energies below sim 300 keV is characterized by a power-law spectrum with an index of sim 6-7 and a flux profile that increases before and decreases after the shock. The other population at energies above sim 300 keV shows a long-lasting, anti-sunward-beamed PAD across the shock and a flux profile that remains relatively constant before and increases slightly after the shock. These results suggest that the shock acceleration of energetic protons is highly dynamic due to temporal and/or spatial variations at the shock front. The observation of the velocity dispersion event further suggests that shock acceleration can be impulsive and efficient, which may be due to the interaction between the shock and magnetic kinks or switchbacks. Moreover, these results may support shock-drift acceleration and diffusive shock acceleration as candidate acceleration mechanisms at interplanetary shocks.

Development of Nusselt number and friction factor models for rough rod bundles based on wall functions

The fuel assembly in the reactor undergo surface roughening due to fouling deposits and material corrosion, which significantly impact the pressure drop and convective heat transfer performance of the fuel assembly. In this study, new models were developed based on the wall velocity and temperature distribution functions. The Nusselt number and friction factor are obtained by integrating these functions over the flow cross-section. The findings demonstrate the accuracy of the method in predicting the friction factor and Nusselt number for both rough tube and smooth rod bundles. The models developed in this study can be utilized to predict the friction factor and Nusselt number in rod bundles with different relative roughness under turbulent flow states, considering square and hexagonal arrangements with P/D<1.5. The present study reveals that the roughness promotes convective heat transfer to a slightly less extent as the pressure drop.

Extended relativistic kinetic model composed of the scalar and two vector distribution functions: Application to the spin-electron-acoustic waves

Detailed deterministic derivation of kinetic equations for relativistic plasmas is given. Focus is made on the dynamic of one-coordinate distribution functions of various tensor dimensions, but the closed set of kinetic equations is constructed of three functions: the scalar distribution function, the vector distribution function of dipole moment, and the vector distribution function of velocity (or the dipole moment in the momentum space). All two-coordinate distribution functions are discussed as well. They are presented together with their limits existing in the self-consistent field approximation. The dynamics of the small amplitude spin-electron-acoustic waves in the dense degenerate plasmas is studied within the kinetic model. This work presents the deterministic approach to the derivation and interpretation of the kinetic equations. So, no probability is introduced during the transition from the level of individual particles to the collective functions. The problem of thermalization is not considered, but we can see that the structure of kinetic equations is kept for the system before and after thermalization. Hence, the kinetic equations can be used to approach this item.

Analytical irreversible thermodynamics in conduction-radiative heat transfer of neutral gases through a lens of thermal radiation fields

An analytical investigation has been conducted on the steady-state conduction and a diluted gas exposed to a thermal radiation field experiences radiative heat transfer. Analytical solutions have been derived for the transport Boltzmann BGK partial differential equations system. The investigation delves into the irreversible thermodynamic behavior of the system through the utilization of the Liu-Lees model, which employs two-stream Maxwellian distribution functions. The moment approach is utilized to predict the behavior of macroscopic gas parameters like temperature, concentration, and Fourier heat flux. This process entails substituting these parameters into the two-stream Maxwellian distribution functions to analyze the system's non-equilibrium thermodynamic characteristics, which encompass gas particles and the heated plate. Through this analysis, kinetic coefficients, entropy, entropy flux, entropy production, and thermodynamic forces are determined. The investigation assesses the validation of Onsager's reciprocity relation, the Boltzmann H-theorem, the Le Chatelier principle, and the second law of thermodynamics within the system. Furthermore, the Gibbs formula is employed to estimate the ratios between various contributions to internal energy changes, which are determined by the total derivatives of extensive parameters. The non-equilibrium velocity distribution function was calculated for the first time, and its behavior was compared with that of the equilibrium one using 3D graphics. The behavior of the computed variables is predicted through graphical analysis and discussion of the results.

RF antenna helicity dependent particle heating in a helicon source

Experiments have demonstrated that ion phenomena, such as the lower hybrid resonance, play an important role in helicon source operation. Damping of the slow branch of the bounded whistler wave at the edge of a helicon source (i.e. the Trivelpiece-Gould mode) has been correlated with the creation of energetic electrons, heating of ions at the plasma edge, and anisotropic ion heating. Here we present ion velocity distribution function measurements, electron density and temperature measurements, and magnetic fluctuation measurements on both sides of an m=|1| helical antenna in a helicon source as a function of the driving frequency, magnetic field strength, and magnetic field orientation relative to the antenna helicity. Significant electron and ion heating (up to two times larger) occurs on the side of the antenna consistent with the launch of the m=+1 mode. The electron and ion heating occurs within one electron skin depth of the plasma edge, where slow wave damping is expected. The source parameters for enhanced particle heating are also consistent with lower hybrid resonance effects, which can only occur for Trivelpiece-Gould wave excitation.

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.

Suprathermal Electron Transport in the Solar Wind: Effects of Coulomb Collisions and Whistler Turbulence

The nature and radial evolution of solar wind electrons in the suprathermal energy range are studied. A wave–particle interaction tensor and a Fokker–Planck Coulomb collision operator are introduced into the kinetic transport equation describing electron collisions and resonant interactions with whistler waves. The diffusion tensor includes diagonal and off-diagonal terms, and the Coulomb collision operator applies to arbitrary electron velocities describing collisions with both background protons and electrons. The background proton and electron densities and temperatures are based on previous turbulence models that mediate the supersonic solar wind. The electron velocity distribution functions and electron heat flux are calculated. Comparison and analysis of the numerical results with analytical solutions and observations in the near-Sun region are made. The numerical results reproduce well the creation of the sunward electron deficit observed in the near-Sun region. The deficit of the electron velocity distribution function below the core Maxwellian fit at low velocities results from Coulomb collisions, and the excess part above the core Maxwellian fit at high velocities is determined by strong wave–particle interactions.

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model.

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad's moment method and with computer simulations performed in the Brownian limiting case (m/mg→∞, where mg and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio m/mg increases.

The Regulation of the Solar Wind Electron Heat Flux by Wave–Particle Interactions

The solar wind electrons carry a significant heat flux into the heliosphere. The weakly collisional state of the solar wind implicates collisionless processes as the primary factor that constrains nonthermal features of the velocity distribution function (VDF), including the heat flux. Previous observational work suggests that the electron VDF sometimes becomes unstable to the whistler wave, but reliance on model VDFs (e.g., drifting bi-Maxwellians) has proven insufficient for an exact description of the behavior of the solar wind electrons—in particular, the regulation of the heat flux. The characterization of these processes requires methods to obtain fine details of the VDF and quantification of the impact of kinetic processes on the VDF. We employ measurements of the electron VDF by Solar Orbiter’s Solar Wind Analyser and of the magnetic field by the Radio and Plasma Waves instrument to study an unstable solar wind electron configuration. Through a Hermite–Laguerre expansion of the VDF, we implement a low-pass filter in velocity space to remove velocity space noise and obtain a VDF suitable for analysis. With our method, we directly measure the instability growth rate and the rate of change of the electron heat flux through wave–particle interactions.

Loss cone effects and monotonic sheath conditions of a partially magnetized plasma sheath

In this Letter, we propose the conditions for monotonic plasma sheaths adjacent to a floating wall in the presence of an applied, oblique magnetic field. The electron velocity distribution function (VDF) at the sheath edge obtained from a kinetic model exhibits a loss cone shaped truncation. Using an approximation of the truncated VDF, we derive an analytical framework of the sheath edge condition (i.e., Bohm condition), namely, the relation of ion injection velocity and sheath potential drop as a function of the magnetic field angle. The results show that the sheath edge velocity and total potential drop decrease for a steady-state sheath, which eventually collapses when the magnetic field lines become parallel to the wall.

Slow Electron Holes in the Earth's Magnetosheath

AbstractWe present a statistical analysis of electrostatic solitary waves observed aboard Magnetospheric Multiscale spacecraft in the Earth's magnetosheath. Applying single‐spacecraft interferometry to several hundred solitary waves collected in about 2‐minute interval, we show that almost all of them have the electrostatic potential of positive polarity and propagate quasi‐parallel to the local magnetic field with plasma frame velocities of the order of 100 km/s. The solitary waves have typical parallel half‐widths from 10 to 100 m that is between 1 and 10 Debye lengths and typical amplitudes of the electrostatic potential from 10 to 200 mV that is between 0.01% and 1% of local electron temperature. The solitary waves are associated with quasi‐Maxwellian ion velocity distribution functions, and their plasma frame velocities are comparable with ion thermal speed and well below electron thermal speed. We argue that the solitary waves of positive polarity are slow electron holes and estimate the time scale of their acceleration, which occurs due to interaction with ions, to be of the order of one second. The observation of slow electron holes indicates that their lifetime was shorter than the acceleration time scale. We argue that multi‐spacecraft interferometry applied previously to these solitary waves is not applicable because of their too‐short spatial scales. The source of the slow electron holes and the role in electron‐ion energy exchange remain to be established.

Local Proton Heating at Magnetic Discontinuities in Alfvénic and Non-Alfvénic Solar Wind

We investigate the local proton energization at magnetic discontinuities/intermittent structures and the corresponding kinetic signatures in velocity phase space in Alfvénic (high cross helicity) and non-Alfvénic (low cross helicity) wind streams observed by Parker Solar Probe. By means of the partial variance of increments method, we find that the hottest proton populations are localized around compressible, coherent magnetic structures in both types of wind. Analysis of parallel and perpendicular temperature distributions suggest that the Alfvénic wind undergoes preferential enhancements of T ∥ at such structures, whereas the non-Alfvénic wind experiences preferential T ⊥ enhancements. Although proton beams are present in both types of wind, the proton velocity distribution function displays distinct features. Hot beams, i.e., beams with beam-to-core perpendicular temperature T ⊥,b /T ⊥,c up to three times larger than the total distribution anisotropy, are found in the non-Alfvénic wind, whereas colder beams are in the Alfvénic wind. Our data analysis is complemented by 2.5D hybrid simulations in different geometrical setups, which support the idea that proton beams in Alfvénic and non-Alfvénic wind have different kinetic properties and different origins. The development of a perpendicular nonlinear cascade, favored in balanced turbulence, allows a preferential relative enhancement of the perpendicular plasma temperature and the formation of hot beams. Cold field-aligned beams are instead favored by Alfvén wave steepening. Non-Maxwellian distribution functions are found near discontinuities and intermittent structures, pointing to the fact that the nonlinear formation of small-scale structures is intrinsically related to the development of highly nonthermal features in collisionless plasmas. Our results contribute to understanding the role of different coherent structures in proton energization and their implication in collisionless energy dissipation processes in space plasmas.

Density and Magnetic Field Asymmetric Kelvin‐Helmholtz Instability

AbstractThe Kelvin‐Helmholtz (KH) instability can transport mass, momentum, magnetic flux, and energy between the magnetosheath and magnetosphere, which plays an important role in the solar‐wind‐magnetosphere coupling process for different planets. Meanwhile, strong density and magnetic field asymmetry are often present between the magnetosheath (MSH) and magnetosphere (MSP), which could affect the transport processes driven by the KH instability. Our magnetohydrodynamics simulation shows that the KH growth rate is insensitive to the density ratio between the MSP and the MSH in the compressible regime, which is different than the prediction from linear incompressible theory. When the interplanetary magnetic field (IMF) is parallel to the planet's magnetic field, the nonlinear KH instability can drive a double mid‐latitude reconnection (DMLR) process. The total double reconnected flux depends on the KH wavelength and the strength of the lower magnetic field. When the IMF is anti‐parallel to the planet's magnetic field, the nonlinear interaction between magnetic reconnection and the KH instability leads to fast reconnection (i.e., close to Petschek reconnection even without including kinetic physics). However, the peak value of the reconnection rate still follows the asymmetric reconnection scaling laws. We also demonstrate that the DMLR process driven by the KH instability mixes the plasma from different regions and consequently generates different types of velocity distribution functions. We show that the counter‐streaming beams can be simply generated via the change of the flux tube connection and do not require parallel electric fields.

Mesoscopic Kinetic Approach of Nonequilibrium Effects for Shock Waves.

A shock wave is a flow phenomenon that needs to be considered in the development of high-speed aircraft and engines. The traditional computational fluid dynamics (CFD) method describes it from the perspective of macroscopic variables, such as the Mach number, pressure, density, and temperature. The thickness of the shock wave is close to the level of the molecular free path, and molecular motion has a strong influence on the shock wave. According to the analysis of the Chapman-Enskog approach, the nonequilibrium effect is the source term that causes the fluid system to deviate from the equilibrium state. The nonequilibrium effect can be used to obtain a description of the physical characteristics of shock waves that are different from the macroscopic variables. The basic idea of the nonequilibrium effect approach is to obtain the nonequilibrium moment of the molecular velocity distribution function by solving the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann BGK) equations or multiple relaxation times Boltzmann (MRT-Boltzmann) equations and to explore the nonequilibrium effect near the shock wave from the molecular motion level. This article introduces the theory and understanding of the nonequilibrium effect approach and reviews the research progress of nonequilibrium behavior in shock-related flow phenomena. The role of nonequilibrium moments played on the macroscopic governing equations of fluids is discussed, the physical meaning of nonequilibrium moments is given from the perspective of molecular motion, and the relationship between nonequilibrium moments and equilibrium moments is analyzed. Studies on the nonequilibrium effects of shock problems, such as the Riemann problem, shock reflection, shock wave/boundary layer interaction, and detonation wave, are introduced. It reveals the nonequilibrium behavior of the shock wave from the mesoscopic level, which is different from the traditional macro perspective and shows the application potential of the mesoscopic kinetic approach of the nonequilibrium effect in the shock problem.