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Pair cascades in polar cap regions of neutron stars are considered to be an essential process in various models of coherent radio emissions of pulsars. The cascades produce pair plasma bunch discharges in quasi-periodic spark events. The cascade properties, and therefore also the coherent radiation, depend strongly on the magnetospheric plasma properties and vary significantly across and along the polar cap. Importantly, where the radio emission emanates from in the polar cap region is still uncertain. We investigate the generation of electromagnetic waves by pair cascades and their propagation in the polar cap for three representative inclination angles of a magnetic dipole, $0^\circ$, $45^\circ$, and $90^\circ$. We use two-dimensional particle-in-cell simulations that include quantum-electrodynamic pair cascades in a charge-limited flow from the star surface. We find that the discharge properties are strongly dependent on the magnetospheric current profile in the polar cap and that transport channels for high intensity Poynting flux are formed along magnetic field lines where the magnetospheric currents approach zero and where the plasma cannot carry the magnetospheric currents. There, the parallel Poynting flux component is efficiently transported away from the star and may eventually escape the magnetosphere as coherent radio waves. The Poynting flux decreases with increasing distance from the star in regions of high magnetospheric currents. Our model shows that no process of energy conversion from particles to waves is necessary for the coherent radio wave emission. Moreover, the pulsar radio beam does not have a cone structure; rather, the radiation generated by the oscillating electric gap fields directly escapes along open magnetic field lines in which no pair creation occurs., Comment: 18 pages, 11 figures, 2 tables

Coherent radiation of pulsars, magnetars, and fast radio bursts could, in theory, be interpreted as radiation from solitons and soliton-like waves. The solitons are meant to contain a large number of electric charges confined on long time-scales and may radiate strongly by coherent curvature emission. However, solitons are also known to undergo a wave collapse, which may cast doubts on the correctness of the soliton radio emission models of neutron stars. We investigate the evolution of the caviton type of solitons self-consistently formed by the relativistic streaming instability and compare their apparent stability in 1D calculations with more generic 2D cases, in which the solitons are seen to collapse. Three representative cases of beam Lorentz factors and plasma temperatures are studied to obtain soliton dispersion properties. We utilized 1D electrostatic and 2D electromagnetic relativistic particle-in-cell simulations at kinetic microscales. We found that no solitons are generated by the streaming instability in the 2D simulations. Only superluminal L-mode (relativistic Langmuir) waves are produced during the saturation of the instability, but these waves have smaller amplitudes than the waves in the 1D simulations. The amplitudes tend to decrease after the instability has saturated, and only waves close to the light line, $\omega = c k$, remain. Solitons in the 1D approach are stable for $\gamma_\mathrm{b} \gtrsim 60$, but they disappear for low beam Lorentz factor $\gamma_\mathrm{b} < 6$. Our examples show that the superluminal soliton branch that is formed in 1D simulations will not be generated by the relativistic streaming instability when more dimensional degrees of freedom are present - unless one can show that there are alternative plasma mechanisms for the soliton generation., Comment: Astronomy & Astrophysics, 12 pages, 6 figures

The effects of the electron inertia on the current sheets that are formed out of kinetic turbulence are relevant to understand the importance of coherent structures in turbulence and the nature of turbulence at the dissipation scales. We investigate this problem by carrying out 3D hybrid-kinetic Particle-in-Cell (PIC) simulations of decaying kinetic turbulence with our CHIEF code. The main distinguishing feature of this code is an implementation of the electron inertia without approximations. Our simulation results show that the electron inertia plays an important role in regulating and limiting the largest values of current density in both real and wavenumber Fourier space, in particular near and, unexpectedly, even above electron scales. In addition, the electric field associated to the electron inertia dominates most of the strongest current sheets. The electron inertia is thus important to accurately describe the properties of current sheets formed in turbulence at electron scales., Comment: 34 pages, 10 figures. Revised version. Published in Physics of Plasmas

Hybrid-kinetic simulations describe ion-scale kinetic phenomena in space plasmas by considering ions kinetically, i.e. as particles, while electrons are modelled as a fluid. Most of the existing hybrid-kinetic codes neglect the electron mass (see chapter 3) for a simplified calculation of the electromagnetic fields. There are, however, situations in which delay in the electrons response due to the electron inertia matters. This chapter concentrates on hybrid-kinetic simulation models which take the finite mass of the electron fluid into account. First a review is given of the history of including the finite electron mass in hybrid-kinetic models. Then the equations are discussed which additionally have to be solved compared to the mass-less hybrid-kinetic models. For definiteness their numerical implementation without additional approximations is illustrated by describing a hybrid-kinetic code, CHIEF. The importance of the consideration of the finite electron mass are discussed for typical applications (magnetic reconnection, plasma turbulence, collisionless shocks and global magnetospheric simulations). In particular the problem of guide field magnetic reconnection is addressed in some detail. Possible next steps towards further improvements of hybrid-kinetic simulations with finite electron mass are suggested., Comment: To appear as a chapter in the book titled "Space and Astrophysical Plasma Simulation - Methods, Algorithms, and Applications" edited by J. B\"uchner

The dissipation mechanism of the magnetic energy in turbulent collisionless space and astrophysical plasmas is still not well understood. Its investigation requires efficient kinetic simulations of the energy transfer in collisionless plasma turbulence. In this respect, hybrid-kinetic simulations, in which ions are treated as particles and electrons as an inertial fluid, have begun to attract a significant interest recently. Hybrid-kinetic models describe both ion- and electron scale processes by ignoring electron kinetic effects so that they are computationally much less demanding compared to fully kinetic plasma models. Hybrid-kinetic codes solve either the Vlasov equation for the ions (Eulerian Vlasov-hybrid codes) or the equations of motion of the ions as macro-particles (Lagrangian Particle-in-Cell (PIC)-hybrid codes). They consider the inertia of the electron fluid using different approximations. We check the validity of these approximations by employing our recently massively parallelized three-dimensional PIC-hybrid code CHIEF which considers the electron inertia without any of the common approximations. In particular we report the results of simulations of two-dimensional collisionless plasma turbulence. We conclude that the simulation results obtained using hybrid-kinetic codes which use approximations to describe the electron inertia need to be interpreted with caution. We also discuss the parallel scalability of CHIEF, to the best of our knowledge, the first PIC-hybrid code which without approximations describes the inertial electron fluid., Comment: 9 figures, submitted

Linear acceleration emission is one of the mechanisms that might explain intense coherent emissions of radio pulsars. This mechanism is not well understood, however, because the effects of collective plasma response and nonlinear plasma evolution on the resulting emission power must be taken into account. In addition, details of the radio emission properties of this mechanism are unknown, which limits the observational verification of the emission model. By including collective and nonlinear plasma effects, we calculate radio emission power properties by the linear acceleration emission mechanism that occurs via the antenna principle for two instabilities in neutron star magnetospheres: 1) the relativistic streaming instability, and 2) interactions of plasma bunches. We used 1D electrostatic relativistic particle-in-cell simulations to evolve the instabilities self-consistently. From the simulations, the power properties of coherent emission were obtained by novel postprocessing of electric currents. We found that the total radio power by plasma bunch interactions exceeds the power of the streaming instability by eight orders of magnitude. The wave power generated by a plasma bunch interaction can be as large as $2.6\times10^{16}$ W. The number of bunch interactions that are required to explain the typical pulsar power, $10^{18}$-$10^{22}$ W, depends on how the coherent emissions of bunches are added up together. Although $\sim$$4\times (10^1-10^5)$ simultaneously emitting bunches are necessary for an incoherent addition of their radiation power, $\gtrsim 6-600$ bunches can explain the total pulsar power if they add up coherently. The radio spectrum of the plasma bunch is characterized by a flatter profile for low frequencies and by a power-law index up to $\approx-1.6 \pm 0.2$ for high frequencies. The plasma bunches simultaneously radiate in a wide range of frequencies., Comment: 19 pages, 13 figures, 1 table

Magnetic reconnection in solar flares can efficiently generate non-thermal electron beams. The energetic electrons can, in turn, cause radio waves through microscopic plasma instabilities as they propagate through the ambient plasma along the magnetic field lines. We aim at investigating the wave emission caused by fast moving electron beams (FEBs) with characteristic non-thermal electron velocity distribution functions (EVDFs) generated by kinetic magnetic reconnection: two-streaming EVDFs along the separatrices and in the diffusion region, and perpendicular crescent-shaped EVDFs closer to the diffusion region. For this purpose, we utilized 2.5D fully kinetic Particle-In-Cell (PIC) code simulations in this study. We found that:(1) the two-streaming EVDFs plus the background ions are unstable to electron/ion (streaming) instabilities which cause ion acoustic waves and Langmuir waves due to the net current. This can lead to multiple harmonic plasma emission in the diffusion region and the separatrices of reconnection. (2) The perpendicular crescent-shaped EVDFs can cause multiple harmonic electromagnetic electron cyclotron waves through the electron cyclotron maser instabilities in the diffusion region of reconnection. Our results are applicable to diagnose the plasma parameters which control magnetic reconnection in solar flares by means of radio waves observations., Comment: 12 figures, 3 tables. This article has been accepted for publication in The Astrophysical Journal (ApJ)

Electromagnetic waves due to electron-positron clouds (bunches), created by cascading processes in pulsar magnetospheres, have been proposed to explain the pulsar radio emission. In order to verify this hypothesis, we utilized for the first time Particle-in-Cell (PIC-) code simulations to study the nonlinear evolution of electron-positron bunches in dependence on the relative drift speeds of electrons and positrons, on the initial plasma temperature, and on the initial distance between the bunches. For this sake, we utilized the PIC-code ACRONYM with a high-order field solver and particle weighting factor, appropriate to describe relativistic pair plasmas. We found that the bunch expansion is mainly determined by the relative electron-positron drift speed. Finite drift speeds were found to cause the generation of strong electric fields that reach up to $E \sim 7.5 \times 10^{5}$ V/cm ($E / (m_\mathrm{e} c \omega_\mathrm{p} e^{-1}) \sim 4.4$) and strong plasma heating. As a result, up to 15~\% of the initial kinetic energy is transformed into the electric field energy. Assuming the same electron- and positron-distributions we found that the fastest (in the bunch reference frame) particles of consecutively emitted bunches eventually overlap in the momentum (velocity) space. This overlap causes two-stream instabilities that generate (electrostatic) subluminal L-mode waves with electric field amplitudes reaching up to $E \sim 1.9\times 10^{4}$ V/cm ($E / (m_\mathrm{e} c \omega_\mathrm{p} e^{-1}) \sim 0.11$). We found that the interaction of electron-position bunches leads to plasma heating, to the generation of strong electric fields and of intense superluminal L-mode waves which, in principle, can be behind the observed electromagnetic emissions of pulsars in the radio wave range., Comment: 18 pages, 13 figures

Magnetic reconnection can convert magnetic energy into kinetic energy of non-thermal electron beams. Those accelerated electrons can, in turn, cause radio emission in astrophysical plasma environments such as solar flares via micro-instabilities. The properties of the electron velocity distribution functions (EVDFs) of those non-thermal beams generated by reconnection are, however, still not well understood. In particular properties that are necessary conditions for some relevant micro-instabilities. We aim at characterizing the EVDFs generated in 3D magnetic reconnection by means of fully kinetic particle-in-cell (PIC) code simulations. In particular, our goal is to identify the possible sources of free energy offered by the generated EVDFs and their dependence on the strength of the guide field. By applying a machine learning algorithm on the EVDFs, we find that: (1) electron beams with positive gradients in their 1D parallel (to the local magnetic field direction) velocity distribution functions are generated in both diffusion region and separatrices. (2) Electron beams with positive gradients in their perpendicular (to the local magnetic field direction) velocity distribution functions are observed in the diffusion region and outflow region near the reconnection midplane. In particular, perpendicular crescent-shaped EVDFs (in the perpendicular velocity space) are mainly observed in the diffusion region. (3) As the guide field strength increases, the number of locations with EVDFs featuring a perpendicular source of free energy significantly decreases. The formation of non-thermal electron beams in the field-aligned direction is mainly due to magnetized and adiabatic electrons, while in the direction perpendicular to the local magnetic field it is attributed to unmagnetized electrons., Comment: 11 figures, 1 table. This article has been published in the journal Physics of Plasmas (PoP)

A number of possible pulsar radio emission mechanisms are based on streaming instabilities in relativistically hot electron-positron pair plasmas. At saturation the unstable waves can form, in principle, stable solitary waves which could emit the observed intense radio signals. We searched for the proper plasma parameters which would lead to the formation of solitons, investigated their properties and dynamics as well as the resulting oscillations of electrons and positrons possibly leading to radio wave emission. We utilized a one-dimensional version of the relativistic Particle-in-Cell code ACRONYM initialized with an appropriately parameterized one-dimensional Maxwell-J\"uttner velocity space particle distribution to study the evolution of the resulting streaming instability in a pulsar pair plasma. We found that strong electrostatic superluminal L-mode solitons are formed for plasmas with normalized inverse temperatures $\rho \geq 1.66$ or relative beam drift speeds with Lorentz factors $\gamma > 40$. The parameters of the solitons fulfill the wave emission conditions. For appropriate pulsar parameters the resulting energy densities of superluminal solitons can reach up to $1.1 \times 10^5$ erg$\cdot$cm$^{-3}$, while those of subluminal solitons reach only up to $1.2 \times 10^4$ erg$\cdot$cm$^{-3}$. Estimated energy densities of up to $7 \times 10^{12}$ erg$\cdot$cm$^{-3}$ suffice to explain pulsar nanoshots., Comment: 20 pages, 15 figures, 1 table

Several important mechanisms that explain the coherent pulsar radio emission rely on streaming (or beam) instabilities of the relativistic pair plasma in a pulsar magnetosphere. However, it is still not clear whether a streaming instability by itself is sufficient to explain the observed coherent radio emission. Due to the relativistic conditions that are present in the pulsar magnetosphere, kinetic instabilities could be quenched. Moreover, uncertainties regarding specific model-dependent parameters impede conclusions concerning this question. We aim to constrain the possible parameter range for which a streaming instability could lead to pulsar radio emission, focusing on the transition between strong and weak beam models, beam drift speed, and temperature dependence of the beam and background plasma components. We solve a linear relativistic kinetic dispersion relation appropriate for pulsar conditions in a more general way than in previous studies, considering a wider parameter range. The analytical results are validated by comparison with relativistic kinetic particle-in-cell (PIC) numerical simulations. We obtain growth rates as a function of background and beam densities, temperatures, and streaming velocities while finding a remarkable agreement of the linear dispersion predictions and numerical simulation results in a wide parameter range. Monotonous growth is found when increasing the beam-to-background density ratio. With growing beam velocity, the growth rates firstly increase, reach a maximum and decrease again for higher beam velocities. A monotonous dependence on the plasma temperatures is found, manifesting in an asymptotic behaviour when reaching colder temperatures. We show that the generated waves are phase-coherent by calculating the fractional bandwidth. We provide an explicit parameter range of plasma conditions for efficient pulsar radio emission., Comment: 15 pages, 7 figures. An abridged version of the abstract is shown here. Accepted in A&A

Type III radio bursts are radio emissions associated with solar flares. They are considered to be caused by electron beams traveling from the solar corona to the solar wind. Magnetic reconnection is a possible accelerator of electron beams in the course of solar flares since it causes unstable distribution functions, and density inhomogeneities (cavities). The properties of radio emission by electron beams in an inhomogeneous environment are still poorly understood. We capture the non-linear kinetic plasma processes of generation of beam-related radio emissions in inhomogeneous plasmas by utilizing fully-kinetic Particle-In-Cell (PIC) code numerical simulations. Our model takes into account initial electron velocity distribution functions (EVDFs) as they are supposed to be created by magnetic reconnection. We focus our analysis on low-density regions with strong magnetic fields. The assumed EVDFs allow two distinct mechanisms of radio wave emissions: plasma emissions due to wave-wave interactions and so-called electron cyclotron maser emissions (ECME) due to direct wave-particle interactions. We investigate the effects of density inhomogeneities on the conversion of free energy from the electron beams into the energy of electrostatic and electromagnetic waves via plasma emission and ECME, as well as the frequency shift of electron resonances caused by perpendicular gradients in the beam EVDFs. Our most important finding is that the number of harmonics of Langmuir waves increases due to the presence of density inhomogeneities. The additional harmonics of Langmuir waves are generated by a coalescence of beam-generated Langmuir waves and their harmonics., Comment: Accepted by Journal of Plasma Physics

The formation of collisionless shock fronts is an ubiquitous phenomenon in space plasma environments. In the solar wind shocks might accompany coronal mass ejections, while even more violent events, such as supernovae, produce shock fronts traveling at relativistic speeds. While the basic concepts of shock formation and particle acceleration in their vicinity are known, many details on a micro-physical scope are still under discussion. In recent years the hybrid kinetic simulation approach has allowed to study the dynamics and acceleration of protons and heavier ions in great detail. However, Particle-in-Cell codes allow to study the process including also electron dynamics and the radiation pressure. Additionally a further numerical method allows for crosschecking results. We therefore investigate shock formation and particle acceleration with a fully kinetic particle-in-cell code. Besides electrons and protons we also include helium and carbon ions in our simulations of a quasi-parallel shock. We are able to reproduce characteristic features of the energy spectra of the particles, such as the temperature ratios of the different ion species in the downstream which scale with the ratio of particle mass to charge. We also find that approximately 12-15% of the energy of the unperturbed upstream is transferred to the accelerated particles escaping the shock., Comment: 17 pages, 9 figures

Three dimensional electron-magnetohydrodynamic (EMHD) simulations of electron current sheet dynamics in a background of stationary and unmagnetized ions and the subsequent generation of electromagnetic fluctuations are carried out. The physical parameters and initial magnetic configuration in the simulations are chosen to be similar to those in the \textsc{Vineta}.II magnetic reconnection experiment. Consistent with the experimental results, our 3D EMHD simulations show the formation of an elongated electron scale current sheet together with the excitation of electromagnetic fluctuations within this sheet. The fluctuations in the simulations are generated by an electron shear flow instability growing on the in-plane (perpendicular to the direction of the main current in the sheet) electron shear flow (or current) developed during the current sheet evolution. Similar to the experiments, the magnetic field fluctuations perpendicular to the guide magnetic field exhibit a broadband frequency spectrum following a power law and a positive correlation with the axial current density. Although the experimental results show that ions influence the spectral properties of the fluctuations, the simulations suggest that the electron dynamics, even in the absence of ion motion, primarily determines the formation of the current sheet and the generation of electromagnetic fluctuations observed in the experiments., Comment: 9 figures, submitted

A large number of wave modes exist in a magnetized plasma. Their properties are determined by the interaction of particles and waves. In a simulation code, the correct treatment of field quantities and particle behavior is essential to correctly reproduce the wave properties. Consequently, plasma waves provide test problems that cover a large fraction of the simulation code. The large number of possible wave modes and the freedom to choose parameters make the selection of test problems time consuming and comparison between different codes difficult. This paper therefore aims to provide a selection of test problems, based on different wave modes and with well defined parameter values, that is accessible to a large number of simulation codes to allow for easy benchmarking and cross validation. Example results are provided for a number of plasma models. For all plasma models and wave modes that are used in the test problems, a mathematical description is provided to clarify notation and avoid possible misunderstanding in naming.