We present results of a 2D3V kinetic Vlasov simulation of the Weibel instability. The kinetic Vlasov simulation allows us to investigate the velocity distribution of dilute plasmas, in which the effect of collisions between particles is negligible, and has the advantage that the accuracy of the calculated velocity distribution does not depend on the density of plasmas at each point in the physical space. We succeed in reproducing some features of the Weibel instability shown by other simulations, for example, the exponentially growing phase, the saturation of the magnetic field strength, the formation of filamentary structure, and the coalescence of the filaments. Especially, we concentrate on the behavior of the filaments after the saturation of the magnetic field strength and find that there is a kind of quasi-equilibrium states before the coalescence occurs. Furthermore, it is found that an analytical solution for stationary states of the 2D3V Vlasov-Maxwell system can reproduce some dominant features of the quasi-equilibrium, e.g, the configuration of the magnetic field and the velocity distribution at each point. The analytical expression could give a plausible model for the transition layer of a collisionless shock where a strong magnetic field generated by the Weibel instability provides an effective dissipation process instead of collisions between particles.
The electron beam Weibel instability in the electrostatic collisionless shock is studied via particle in cell simulation. When the non-relativistic incoming plasmas collide with cold dilute plasmas, an electrostatic shock forms near the interface. Following that, a filamentary out-of-plane magnetic field is formed as a result of the Weibel instability. It is demonstrated that the anisotropy of incoming hot electrons is insufficient to trigger the Weibel instability. And the Weibel instability is excited by cold electrons in dilute plasmas. After being accelerated to relativistic velocities by …
We present a particle-in-cell simulation of the generation of a collisionless turbulent shock in a dense plasma driven by an ultra-high-intensity laser pulse. From the linear analysis, we highlight the crucial role of the laser-heated and return-current electrons in triggering a strong Weibel-like instability, giving rise to a magnetic turbulence able to isotropize the target ions.
The formation of non-relativistic collisionless shocks in laboratory with ultrahigh intensity lasers is studied via \emph{ab initio} multi-dimensional particle-in-cell simulations. The microphysics behind shock formation and dissipation, and the detailed shock structure are analyzed, illustrating that the Weibel instability plays a crucial role in the generation of strong subequipartition magnetic fields that isotropize the incoming flow and lead to the formation of a collisionless shock, similarly to what occurs in astrophysical scenarios. The possibility of generating such collisionless shoc…
Astrophysical and high-energy-density laboratory plasmas often have large-amplitude, sub-Larmor-scale electromagnetic fluctuations excited by various kinetic-streaming or anisotropy-driven instabilities. The Weibel (or the filamentation) instability is particularly important because it can rapidly generate strong magnetic fields, even in the absence of seed fields. Particles propagating in collisionless plasmas with such small-scale magnetic fields undergo stochastic deflections similar to Coulomb collisions, with the magnetic pitch-angle diffusion coefficient representing the effective "colli…
We present a predictive model of the nonlinear phase of the Weibel instability induced by two symmetric, counter-streaming ion beams in the non-relativistic regime. This self-consistent model combines the quasilinear kinetic theory of Davidson et al. [Phys. Fluids 15, 317 (1972)] with a simple description of current filament coalescence. It allows us to follow the evolution of the ion parameters up to a stage close to complete isotropization, and is thus of prime interest to understand the dynamics of collisionless shock formation. Its predictions are supported by 2-D and 3-D particle-in-cell …
In this paper, we report our recent findings on the relativistic Weibel instability and its nonlinear saturation by performing numerical simulations of collisionless plasmas. Analysis of the obtained numerical results revealed that the nonlinear phase of the Weibel instability can be described by characteristic phases based on the Weibel filaments' current density in terms of particle and Alfvén limit currents. We also analyzed the relativistic kink instability based on the energy principle in the magnetohydrodynamic (MHD) regime, and found that the Weibel filaments do not suffer from the kink…
Collisionless shocks can be produced as a result of strong magnetic fields in a plasma flow, and therefore are common in many astrophysical systems. The Weibel instability is one candidate mechanism for the generation of sufficiently strong fields to create a collisionless shock. Despite their crucial role in astrophysical systems, observation of the magnetic fields produced by Weibel instabilities in experiments has been challenging. Using a proton probe to directly image electromagnetic fields, we present evidence of Weibel-generated magnetic fields that grow in opposing, initially unmagneti…
Observations of afterglows of gamma-ray bursts suggest (GRBs) that post-shock magnetic fields are strongly amplified to about 100 times the shock-compressed value. The Weibel instability appears to play an important role in generating of the magnetic field. However, recent simulations of collisionless shocks in homogeneous plasmas show that the magnetic field generated by the Weibel instability rapidly decays. There must be some density fluctuations in interstellar and circumstellar media. The density fluctuations are anisotropically compressed in the downstream region of relativistic shocks. …
An instability driven by the thermal anisotropy of a single electron species is investigated in a 2D particle-in-cell (PIC) simulation. This instability is the one considered by Weibel and it differs from the beam driven filamentation instability. A comparison of the simulation results with analytic theory provides similar exponential growth rates of the magnetic field during the linear growth phase of the instability. We observe in accordance with previous works the growth of electric fields during the saturation phase of the instability. Some components of this electric field are not account…
The time evolution and saturation of the Weibel instability at the ion Alfvén current are presented by ab initio particle-in-cell simulations. We found that the ion Weibel current in 3D could evolve into the Alfvén current where the magnetic field energy is sustained at 1.5\% of the initial beam kinetic energy. The current filaments are no longer isolated at saturation, but rather connected to each other to form a network structure. Electrons are continuously heated during the coalescence of the filaments, which is crucial for obtaining sustained magnetic fields with much stronger levels than …
We develop a comprehensive theoretical model of relativistic collisionless pair shocks mediated by the current filamentation instability. We notably characterize the noninertial frame in which this instability is of a mostly magnetic nature, and describe at a microscopic level the deceleration and heating of the incoming background plasma through its collisionless interaction with the electromagnetic turbulence. Our model compares well to large-scale 2D3V PIC simulations, and provides an important touchstone for the phenomenology of such plasma systems.
For a Weibel shock to form, two plasma shells have to collide and trigger the Weibel instability. At saturation, this instability generates in the overlapping region magnetic filaments with peak field $B_f$. In the absence of an external guiding magnetic field, these filaments can block the incoming flow, initiating the shock formation, if their size is larger than the Larmor radius of the incoming particles in the peak field. Here we show that this results still holds in the presence of an external magnetic field $B_0$, provided it is not too high. Yet, for $B_0 \gtrsim B_f/2$, the filaments …
We examine with particle-in-cell (PIC) simulations how a parallel shock in pair plasma reacts to upstream waves, which are driven by escaping downstream particles. Initially, the shock is sustained in the two-dimensional simulation by a magnetic filamentation (beam-Weibel) instability. Escaping particles drive an electrostatic beam instability upstream. Modifications of the upstream plasma by these waves hardly affect the shock. In time, a decreasing density and increasing temperature of the escaping particles quench the beam instability. A larger thermal energy along than perpendicular to the…
We investigate ion-scale kinetic plasma instabilities at the collisionless shock using linear theory and nonlinear Particle-in-Cell (PIC) simulations. We focus on the Alfvén-ion-cyclotron (AIC), mirror, and Weibel instabilities, which are all driven unstable by the effective temperature anisotropy induced by the shock-reflected ions within the transition layer of a strictly perpendicular shock. We conduct linear dispersion analysis with a homogeneous plasma model to mimic the shock transition layer by adopting a ring distribution with finite thermal spread to represent the velocity distributio…
We investigated the precursor wave emission efficiency in magnetized purely perpendicular relativistic shocks in pair plasmas. We extended our previous study to include the dependence of upstream magnetic field orientations. We performed two-dimensional particle-in-cell simulations and focused on two magnetic field orientations: the magnetic field to be in the simulation plane (i.e., in-plane configuration) and perpendicular to the simulation plane (i.e., out-of-plane configuration). Our simulations in the in-plane configuration demonstrated that not only extraordinary but also ordinary mode w…
The Weibel/filamentation instability is known to play a key role in the physics of weakly magnetized collisionless shock waves. From the point of view of high energy astrophysics, this instability also plays a crucial role because its development in the shock precursor populates the downstream with a small-scale magneto-static turbulence which shapes the acceleration and radiative processes of suprathermal particles. The present work discusses the physics of the dissipation of this Weibel-generated turbulence downstream of relativistic collisionless shock waves. It calculates explicitly the fi…
Collisionless shocks are ubiquitous in the Universe and often associated with strong magnetic field. Here we use large-scale particle-in-cell simulations of non-relativistic perpendicular shocks in the high-Mach-number regime to study the amplification of magnetic field within shocks. The magnetic field is amplified at the shock transition due to the ion-ion two-stream Weibel instability. The normalized magnetic-field strength strongly correlates with the Alfvénic Mach number. Mock spacecraft measurements derived from PIC simulations are fully consistent with those taken in-situ at Saturn's bo…
We present the results of theoretical analysis and numerical simulations of the Weibel instability in two counter-streaming hot relativistic plasma flows, e.g. flows of electron-proton plasma having rest-mass density $ρ\sim 10^{-4}\; \text{g}\, \text{cm}^{-3}$, Lorentz factors $Γ\sim 10$ and proper temperature $T \sim 10^{13}\; \text{K}$. The instability growth rate and the filament size at the linear stage are found analytically, and are in qualitative agreement with results of three-dimensional particle-in-cell simulations. In the simulations, incoherent synchrotron emission and pair photopr…
In this first paper of a series dedicated to the microphysics of unmagnetized, relativistic collisionless pair shocks, we discuss the physics of the Weibel-type transverse current filamentation instability (CFI) that develops in the shock precursor, through the interaction of an ultrarelativistic suprathermal particle beam with the background plasma. We introduce in particular the notion of "Weibel frame", or scattering center frame, in which the microturbulence is of mostly magnetic nature. We calculate the properties of this frame, using first a kinetic formulation of the linear phase of the…
Plasma instabilities (e.g., Buneman, Weibel and other two-stream instabilities) excited in collisionless shocks are responsible for particle (electron, positron, and ion) acceleration. Using a new 3-D relativistic particle-in-cell code, we have investigated the particle acceleration and shock structure associated with an unmagnetized relativistic electron-positron jet propagating into an unmagnetized electron-positron plasma. The simulation has been performed using a long simulation system in order to study the nonlinear stages of the Weibel instability, the particle acceleration mechanism, an…
The expansion of an initially unmagnetized planar rarefaction wave has recently been shown to trigger a thermal anisotropy-driven Weibel instability (TAWI), which can generate magnetic fields from noise levels. It is examined here if the TAWI can also grow in a curved rarefaction wave. The expansion of an initially unmagnetized circular plasma cloud, which consists of protons and hot electrons, into a vacuum is modelled for this purpose with a two-dimensional particle-in-cell (PIC) simulation. It is shown that the momentum transfer from the electrons to the radially accelerating protons can in…
The maximum magnetic field strength generated by Weibel-type plasma instabilities is estimated for typical conditions in the interstellar medium. The relevant kinetic dispersion relations are evaluated by conducting a parameter study both for Maxwellian and for suprathermal particle distributions showing that micro Gauss magnetic fields can be generated. It is shown that, depending on the streaming velocity and the plasma temperatures, either the longitudinal or a transverse instability will be dominant. In the presence of an ambient magnetic field, the filamentation instability is typically s…
A two-dimensional electromagnetic particle-in-cell simulation with the realistic ion-to-electron mass ratio of 1836 is carried out to investigate the electrostatic collisionless shocks in relatively high-speed (~3000 km s^-1) plasma flows and also the influence of both electrostatic and electromagnetic instabilities, which can develop around the shocks, on the shock dynamics. It is shown that the electrostatic ion-ion instability can develop in front of the shocks, where the plasma is under counter-streaming condition, with highly oblique wave vectors as was shown previously. The electrostatic…
The dynamics of two initially unmagnetized relativistic counter-streaming homogeneous ion-electron plasma beams are simulated in two dimensions using the particle-in-cell (PIC) method. It is shown that current filaments, which form due to the Weibel instability, develop a large scale longitudinal electric field in the direction opposite to the current carried by the filaments as predicted by theory. Fast moving ions in the current filaments decelerate due to this longitudinal electric field. The same longitudinal electric field, which is partially inductive and partially electrostatic, is iden…
Weibel instability turns out to be the a ubiquitous phenomenon in High-Energy Density environments, ranging from astrophysical sources, e.g., gamma-ray bursts, to laboratory experiments involving laser-produced plasmas. Relativistic particles (electrons) radiate in the Weibel-produced magnetic fields in the Jitter regime. Conventionally, in this regime, the particle deflections are considered to be smaller than the relativistic beaming angle of 1/$γ$ ($γ$ being the Lorentz factor of an emitting particle) and the particle distribution is assumed to be isotropic. This is a relatively idealized s…
A study of collisionless external shocks in gamma-ray bursts is presented. The shock structure, electromagnetic fields, and process of electron acceleration are assessed by performing a self-consistent 3D particle-in-cell (PIC) simulation. In accordance with hydrodynamic shock systems, the shock consists of a reverse shock (RS) and forward shock (FS) separated by a contact discontinuity (CD). The development and structure are controlled by the ion Weibel instability. The ion filaments are sources of strong transverse electromagnetic fields at both sides of the double shock structure over a len…
When two collisionless plasma shells collide, they interpenetrate and the overlapping region may turn Weibel unstable for some values of the collision parameters. This instability grows magnetic filaments which, at saturation, have to block the incoming flow if a Weibel shock is to form. In a recent paper [J. Plasma Phys. (2016), vol. 82, 905820403], it was found implementing a toy model for the incoming particles trajectories in the filaments, that a strong enough external magnetic field $\mathbf{B}_0$ can prevent the filaments to block the flow if it is aligned with. Denoting $B_f$ the peak …
Relativistic shocks are usually thought to occur in violent astrophysical explosions. These collisionless shocks are mediated by a plasma kinetic streaming instability, often loosely referred to as the Weibel instability, which generates strong magnetic fields "from scratch" very efficiently. In this review paper we discuss the shock micro-physics and present a recent model of "pre-conditioning" of an initially unmagnetized upstream region via the cosmic-ray-driven Weibel-type instability.
One of the key open questions in the study of relativistic jets is their interaction with the environment. Here, we study the initial evolution of both electron-proton and electron-positron relativistic jets, focusing on their lateral interaction with the ambient plasma. We trace the generation and evolution of the toroidal magnetic field generated by both kinetic Kelvin-Helmholtz (kKH) and Mushroom instabilities (MI). This magnetic field collimates the jet. We show that in electron-proton jet, electrons are perpendicularly accelerated with jet collimation. The magnetic polarity switches from …
We study the onset of vector instabilities in the post-inflationary epoch of the Universe as a mechanism for primordial magnetic fields amplification. We assume the presence of a charged spectator scalar field arbitrarily coupled to gravity during Inflation in its vacuum de Sitter state. Gravitational particle creation takes place at the transition from Inflation to the subsequent Reheating stage and thus the vacuum field state becomes an excited many particles one. Consequently this state can be described as a real fluid, and we build out the hydrodynamic framework using second order theories…
We show that the Weibel or currente filamentation instability can lead to the emission of circularly polarized radiation. Using particle-in-cell (PIC) simulations and a radiation post-processing numerical algorithm, we demonstrate that the level of circular polarization increases with the initial plasma magnetization, saturating at ~13% when the magnetization, given by the ratio of magnetic energy density to the electron kinetic energy density, is larger than 0.05. Furthermore, we show that this effect requires an ion-electron mass ratio greater than unity. These findings, which could also be …
The hierarchy of unstable modes when two counter-streaming pair plasmas interact over a flow-aligned magnetic field has been recently investigated [PoP \textbf{23}, 062122 (2016)]. The analysis is here extended to the case of an arbitrarily tilted magnetic field. The two plasma shells are initially cold and identical. For any angle $θ\in [0,π/2]$ between the field and the initial flow, the hierarchy of unstable modes is numerically determined in terms of the initial Lorentz factor of the shells $γ_0$, and the field strength as measured by a parameter denoted $σ$. For $θ=0$, four different kind…
Fully kinetic two-dimensional particle-in-cell simulations are used to study electron acceleration at high-Mach-number nonrelativistic perpendicular shocks. SNR shocks are mediated by the Weibel instability which is excited because of an interaction between shock-reflected and upstream ions. Nonlinear evolution of the Weibel instability leads to the formation of current sheets. At the turbulent shock ramp the current sheets decay through magnetic reconnection. The number of reconnection sites strongly depends on the ion-to-electron mass ratio and the Alfvénic Mach number of the simulated shock…
The external forward shock emitting the gamma-ray burst (GRB) afterglow is collisionless, and it is mediated by the Weibel instability which generates the magnetic field. The GRB afterglow shows that the magnetic field in the large downstream region is much stronger than the shock-compressed pre-shock field. However, particle-in-cell (PIC) simulations of relativistic shocks propagating into homogeneous media show that the Weibel generated field decays near the shock front. Some GRB observations and theoretical studies suggest that the preshock medium is inhomogeneous. We perform the PIC simula…
The filamentation instability (sometimes also referred to as "Weibel") is a key process in many astrophysical scenario. In the Fireball model for Gamma Ray Bursts, this instability is believed to mediate collisionless shock formation from the collision of two plasma shells. It has been known for long that a flow aligned magnetic field can completely cancel this instability. We show here that in the general case where there is an angle between the field and the flow, the filamentation instability can never be stabilized, regardless of the field strength. The presented model analyzes the stabili…
Magnetic field amplification by relativistic streaming plasma instabilities is central to a wide variety of high-energy astrophysical environments as well as to laboratory scenarios associated with intense lasers and electron beams. We report on a new secondary nonlinear instability which arises for relativistic dilute electron beams after the saturation of the linear Weibel instability. This instability grows due to the transverse magnetic pressure associated with the beam current filaments, which cannot be quickly neutralized due to the inertia of background ions. We show that it can amplify…
The filamentation (Weibel) instability plays a key role in the formation of collisionless shocks which are thought to produce Gamma-Ray-Bursts and High-Energy-Cosmic-Rays in astrophysical environments. While it has been known for long that a flow-aligned magnetic field can completely quench the instability, it was recently proved in 2D that in the cold regime, such cancelation is possible if and only if the field is perfectly aligned. Here, this result is finally extended to a 3D geometry. Calculations are conducted for symmetric and asymmetric counter-streaming relativistic plasma shells. 2D …
We study in a fully kinetic framework the generation of seed magnetic fields through the Weibel instability driven in an initially unmagnetized plasma by a large-scale shear force. We develop an analytical model that describes the development of thermal pressure anisotropy via phase mixing, the ensuing exponential growth of magnetic fields in the linear Weibel stage, and its saturation when the seed magnetic fields become strong enough to instigate gyromotion of particles and thereby inhibit their free-streaming. The predicted scaling dependencies of the saturated seed fields on key parameters…
We have investigated magnetic field generation in velocity shears via the kinetic Kelvin-Helmholtz instability (kKHI) using a relativistic plasma jet core and stationary plasma sheath. Our three-dimensional particle-in-cell simulations consider plasma jet cores with Lorentz factors of 1.5, 5, and 15 for both electron-proton and electron-positron plasmas. For electron-proton plasmas we find generation of strong large-scale DC currents and magnetic fields which extend over the entire shear-surface and reach thicknesses of a few tens of electron skin depths. For electron-positron plasmas we find …
Energetic electromagnetic emissions by astrophysical jets like those that are launched during the collapse of a massive star and trigger gamma-ray bursts (GRBs) are partially attributed to relativistic internal shocks. The shocks are mediated in the collisionless plasma of such jets by the filamentation instability of counterstreaming particle beams. The filamentation instability grows fastest only if the beams move at a relativistic relative speed. We model here with a particle-in-cell (PIC) simulation the collision of two cold pair clouds at the speed c/2 (c: speed of light). We demonstrate …
Fast radio bursts (FRBs) are bright, short-duration radio transients with very high brightness temperatures implying highly coherent emission. We suggest that the FRBs are caused by the self-focusing of an electron beam interacting with an ambient plasma right beyond the light cylinder radius of a neutron star. The magnetic field at the light cylinder radius is relatively high which can accommodate both young Crab-like systems and old millisecond pulsars addressing the diverse environments of FRBs. At the first stage, the intense pulsed-beam passing through the background plasma causes instabi…
Plasma streaming instabilities play an important role in magnetic field amplification and particle acceleration in relativistic shocks and their environments. However, in the far shock precursor region where accelerated particles constitute a highly relativistic and dilute beam, streaming instabilities typically become inefficient and operate at very small scales when compared to the gyroradii of the beam particles. We report on a plasma cavitation instability that is driven by dilute relativistic beams and can increase both the magnetic field strength and coherence scale by orders of magnitud…
We review recent progress on collisionless relativistic shocks. Kinetic instability theory is briefed including its predictions and limitations. The main focus is on numerical experiments in (i) pair and (ii) electron-nucleon plasmas. The main results are: (i) confirmation of shock evolution in non-magnetised relativistic plasma in 3D due to either the lepton-Weibel instability or the ion-Weibel instability; (ii) sensitive dependence on upstream magnetisation ; (iii) the sensitive dependence of particle dynamics on the upstream magnetic inclination angle $\thetabn$, where particles of $\thetab…
Using our new 3-D relativistic electromagnetic particle (REMP) code parallelized with MPI, we investigated long-term particle acceleration associated with a relativistic electron-positron jet propagating in an unmagnetized ambient electron-positron plasma. We have also performed simulations with electron-ion jets. The simulations were performed using a much longer simulation system than our previous simulations in order to investigate the full nonlinear stage of the Weibel instability for electron-positron jets and its particle acceleration mechanism. Cold jet electrons are thermalized and amb…
We study the interaction of relativistic jets with their environment, using 3-dimensional relativistic particle-in-cell simulations for two cases of jet composition: (i) electron-proton ($e^{-}-p^{+}$) and (ii) electron-positron ($e^{\pm}$) plasmas containing helical magnetic fields. We have performed simulations of "global" jets containing helical magnetic fields in order to examine how helical magnetic fields affect kinetic instabilities such as the Weibel instability, the kinetic Kelvin-Helmholtz instability and the Mushroom instability. We have found that these kinetic instabilities are su…
We demonstrate that cosmic rays form filamentary structures in the precursors of supernova remnant shocks due to their self-generated magnetic fields. The cosmic-ray filamentation results in the growth of a long wavelength instability, and naturally couples the rapid non-linear amplification on small scales to larger length scales. Hybrid magnetohydrodynamics--particle simulations are performed to confirm the effect. The resulting large scale magnetic field may facilitate the scattering of high energy cosmic rays as required to accelerate protons beyond the knee in the cosmic-ray spectrum at s…
Using our new 3-D relativistic electromagnetic particle (REMP) code parallelized with MPI, we investigated long-term particle acceleration associated with a relativistic electron-positron jet propagating in an unmagnetized ambient electron-positron plasma. The simulations were performed using a much longer simulation system than our previous simulations in order to investigate the full nonlinear stage of the Weibel instability and its particle acceleration mechanism. Cold jet electrons are thermalized and ambient electrons are accelerated in the resulting shocks. Acceleration of ambient electr…
