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 value of the field in the magnetic filaments, all test particles stream through them if $α=B_0/B_f > 1/2$. Here, this result is extended to the case of an oblique external field $B_0$ making an angle $θ$ with the flow. The result, numerically found, is simply $α> κ(θ)/\cosθ$, where $κ(θ)$ is of order unity. Noteworthily, test particles exhibit chaotic trajectories.
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 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 f…
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 …
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…
Counterstreaming plasma structures are widely present in laboratory experiments and astrophysical systems, and they are investigated either to prevent unstable modes arising in beam-plasma experiments or to prove the existence of large scale magnetic fields in astrophysical objects. Filamentation instability arises in a counterstreaming plasma and is responsible for the magnetization of the plasma. Filamentationally unstable mode is described by assuming that each of the counterstreaming plasmas has an isotropic Lorentzian (kappa) distribution. In this case, the filamentation instability growt…
Density filamentation has been observed in many beam-plasma simulations and experiments. Because current filamentation is a pure transverse mode, charge density filamentation cannot be produced directly by the current filamentation process. To explain this phenomenon, several mechanisms are proposed such as the coupling of the Weibel instability to the two-stream instability, coupling to the Langmuir wave, differences in thermal velocities between the beam and return currents, the magnetic pressure gradient force, etc. In this paper, it is shown that the gradient of the Lorentz factor can, in …
The Weibel instability could be responsible for the generation of magnetic fields in various objects such as gamma-ray bursts, jets from active galactic nuclei, and clusters of galaxies. Using numerical simulations, the development of the Weibel instability at a temperature gradient is studied. It is found that current sheets are first generated at the gradient, and then they are rounded off and turn into current filaments. During this process, return currents are generated around the filaments and they prevent filaments from merger. The magnetic fields around the filaments persist at least un…
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…
We present the first 2X2V continuum Vlasov-Maxwell simulations of interpenetrating, unmagnetized plasmas to study the competition between two-stream, Oblique, and filamentation modes in the weakly relativistic regime. We find that after nonlinear saturation of the fastest-growing two-stream and Oblique modes, the effective temperature anisotropy, which drives current filament formation via the secular Weibel instability, has a strong dependence on the internal temperature of the counter-streaming plasmas. The effective temperature anisotropy is significantly more reduced in colder than in hott…
Weibel-mediated filamentary structures observed in the ICF context
CEA, DAM, DIF（法国原子能和替代能源委员会，核能部，聚变研究部门）；Université Paris-Saclay, CEA, Laboratoire Matière en Conditions Extrêmes（巴黎萨克雷大学，法国原子能和替代能源委员会，极端条件物质实验室）；Center for Energy Research, University of California - San Diego（加州大学圣地亚哥分校能源研究中心）；Massachusetts Institute of Technology（麻省理工学院）
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 present experimental observations of electron-scale structures in an expanding high-energy-density (HED) plasma generated with a modest intensity $\sim 2 \times 10^{14}$ W/cm$^2$, $\sim1$ ns laser. The observed structures have wavelengths ($\sim 150-220 μm$) and growth rates ($\sim 0.4-1.0$ ns$^{-1}$) consistent with an electron-driven Weibel instability where the anisotropy in the electron distribution is small, $A\sim 0.002$. This instability is found to be a better match to the observed phenomena than other typical field-generation mechanisms found in HED plasmas, including counter-strea…
Many competing linear instabilities are likely to occur in astrophysical settings, and it is important to assess which one grows faster for a given situation. An analytical model including the main beam plasma instabilities is developed. The full 3D dielectric tensor is thus explained for a cold relativistic electron beam passing through a cold plasma, accounting for a guiding magnetic field, a return electronic current and moving protons. Considering any orientations of the wave vector allows to retrieve the most unstable mode for any parameters set. An unified description of the Filamentatio…
Filamentation due to the growth of a Weibel-type instability was observed in the interaction of a pair of counter-streaming, ablatively-driven plasma flows, in a supersonic, collisionless regime relevant to astrophysical collisionless shocks. The flows were created by irradiating a pair of opposing plastic (CH) foils with 1.8 kJ, 2-ns laser pulses on the OMEGA EP laser system. Ultrafast laser-driven proton radiography was used to image the Weibel-generated electromagnetic fields. The experimental observations are in good agreement with the analytical theory of the Weibel instability and with p…
When a relativistic intensity laser interacts with the surface of a solid density target, suprathermal electron currents are subject to Weibel instability filamentation when propagating through the thermal population of the bulk target. We present time resolved shadowgraphy of radial ionization front expansion and Weibel instability filamentation within a thin, sub-micron, sheet initiated by irradiation with a short pulse, high intensity laser. High temporal (100 fs) and spatial (1 $μ$m) resolution shadowgraphy of the interaction reveals a relativistic expansion of the ionization front within …
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 …
First-principles kinetic simulations are used to investigate magnetic field generation processes in expanding ablated plasmas relevant to laser-driven foils and hohlraums. In addition to Biermann-battery-generated magnetic fields, strong filamentary magnetic filaments are found to grow in the corona of single expanding plasma plumes; such filaments are observed to dominate Biermann fields at sufficiently large focal radius, reaching saturation values of $\sim$ 100 T at National Ignition Facility-like drive conditions. The filamentary fields result from the ion Weibel instability driven by rela…
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.
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 relativistic electron beam (REB) propagation in a plasma is fraught with beam plasma instabilities. The prominent amongst them being the collisionless Weibel destabilization which spatially separates the forward propagating REB and the return shielding currents. This results in the formation of REB current filaments which are typically of the size of electron skin depth during the linear stage of the instability. It has been observed that in the nonlinear stage the filaments size increases as they merge with each other. With the help of 2D PIC simulations in the plane perpendicular to the …
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…
We study the stability of current filaments produced by the Weibel, or current filamentation, instability in weakly magnetized counter-streaming plasmas. It is shown that a resonance exists between the current-carrying ions and a longitudinal drift-kink mode that strongly deforms and eventually breaks the current filaments. Analytical estimates of the wavelength, growth rate and saturation level of the resonant mode are derived and validated by three-dimensional particle-in-cell simulations. Furthermore, self-consistent simulations of counter-streaming plasmas indicate that this drift-kink mod…
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 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…
A Particle-in-Cell (PIC) numerical simulation of the electron Weibel instability is applied in a frame of Darwin (radiationless) approximation of the self-consistent fields of sparse plasma. As a result, we were able to supplement the classical picture of the instability and, in particular, to obtain the dependency of the basic characteristics (the time of development and the maximum field energy) of the thermal anisotropy parameter, to trace the dynamic restructuring of current filaments accompanying the nonlinear stage of the instability and to trace in detail the evolution of the initial an…
The motion of a particle in a spatially harmonic magnetic field is a basic problem involved, for example, in the mechanism of formation of a collisionless shock. In such settings, it is generally reasoned that particles entering a Weibel generated turbulence are trapped inside it, provided their Larmor radius in the peak field is smaller than the field coherence length. The goal of this work is to put this heuristic conclusion on firm ground by studying, both analytically and numerically, such motion. A toy model is analyzed, consisting of a relativistic particle entering a region of space occ…
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…
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…
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…
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 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…
The instabilities of relativistic ion beams in a relativistically hot electron background are derived for general propagation angles. It is shown that the Weibel instability in the direction perpendicular to the streaming direction is the fastest growing mode, and probably the first to appear, consistent with the aligned filaments that are seen in PIC simulations. Oblique, quasiperpendicular modes grow almost as fast, as the growth rate varies only moderately with angle, and they may distort or corrugate the filaments after the perpendicular mode saturates.
The nonlinear evolution of current filaments generated by the Weibel-type filamentation instability is a topic of prime interest in space and laboratory plasma physics. In this paper, we investigate the stability of a stationary periodic chain of nonlinear current filaments in counterstreaming pair plasmas. We make use of a relativistic four-fluid model and apply the Floquet theory to compute the two-dimensional unstable eigenmodes of the spatially periodic system. We examine three different cases, characterized by various levels of nonlinearity and asymmetry between the plasma streams: a weak…
The Weibel instability driven by two symmetric counter-streaming relativistic electron plasmas, also referred to as current-filamentation instability, is studied in a constant and uniform external magnetic field aligned with the plasma flows. Both the linear and non linear stages of the instability are investigated using analytical modeling and Particle-In-Cell (PIC) simulations. While previous studies have already described the stabilizing effect of the magnetic field, we show here that the saturation stage is only weakly affected. The different mechanisms responsible for the saturation are d…
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 …
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 …
Proton imaging is a powerful tool for probing electromagnetic fields in a plasma, providing a path-integrated map of the field topology. However, in cases where the field structure is highly inhomogeneous, inferring spatial properties of the underlying field from proton images can be difficult. This problem is exemplified by recent experiments which used proton imaging to probe the filamentary magnetic field structures produced by the Weibel instability in collisionless counter-streaming plasmas. In this paper, we perform analytical and numerical analysis of proton images of systems containing…
(Abridged) This paper addresses the issue of magnetic field generation in a relativistic shock precursor through micro-instabilities. The level of magnetization of the upstream plasma turns out to be a crucial parameter, notably because the length scale of the shock precursor is limited by the Larmor rotation of the accelerated particles in the background magnetic field and by the speed of the shock wave. We discuss in detail and calculate the growth rates of the following beam plasma instabilities seeded by the accelerated and reflected particle populations: for an unmagnetized shock, the Wei…
A two-dimensional particle-in-cell simulation is performed to investigate weakly magnetized perpendicular shocks with a magnetization parameter of 6 x 10^-5, which is equivalent to a high Alfvén Mach number M_A of ~130. It is shown that current filaments form in the foot region of the shock due to the ion-beam--Weibel instability (or the ion filamentation instability) and that they generate a strong magnetic field there. In the downstream region, these current filaments also generate a tangled magnetic field that is typically 15 times stronger than the upstream magnetic field. The thermal ener…
Collisionless shocks occur in various fields of physics. In the context of space and astrophysics they have been investigated for many decades. However, a thorough understanding of shock formation and particle acceleration is still missing. Collisionless shocks can be distinguished into electromagnetic and electrostatic shocks. Electromagnetic shocks are of importance mainly in astrophysical environments and they are mediated by the Weibel or filamentation instability. In such shocks, charged particles gain energy by diffusive shock acceleration. Electrostatic shocks are characterized by a str…
We present experimental measurements of the femtosecond time-scale generation of strong magnetic-field fluctuations during the interaction of ultrashort, moderately relativistic laser pulses with solid targets. These fields were probed using low-emittance, highly relativistic electron bunches from a laser wakefield accelerator, and a line-integrated $B$-field of $2.70 \pm 0.39\,\rm kT\,μm$ was measured. Three-dimensional, fully relativistic particle-in-cell simulations indicate that such fluctuations originate from a Weibel-type current filamentation instability developing at submicron scales …
Radiative diagnostics of high-energy density plasmas is addressed in this paper. We propose that the radiation produced by energetic particles in small-scale magnetic field turbulence, which can occur in laser-plasma experiments, collisionless shocks, and during magnetic reconnection, can be used to deduce some properties of the turbulent magnetic field. Particles propagating through such turbulence encounter locally strong magnetic fields, but over lengths much shorter than a particle gyroradius. Consequently, the particle is accelerated but not deviated substantially from a straight line pat…
The jets of compact accreting objects are composed of electrons and a mixture of positrons and ions. These outflows impinge on the interstellar or intergalactic medium and both plasmas interact via collisionless processes. Filamentation (beam-Weibel) instabilities give rise to the growth of strong electromagnetic fields. These fields thermalize the interpenetrating plasmas. Hitherto, the effects imposed by a spatial non-uniformity on filamentation instabilities have remained unexplored. We examine the interaction between spatially uniform background electrons and a minuscule cloud of electrons…
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…
Sustained interpenetrating plasma flows for the investigation of late time kinetic instability evolution
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Massachusetts Institute of Technology（麻省理工学院）；Dartmouth College（达特茅斯学院）；Instituto de Investigaciones Energéticas y Aplicaciones Industriales（能源研究与工业应用研究所）
Many problems in astrophysics involve relativistic outflows. The plasma dynamics in these scenarios is critical to determine the conditions for the self-consistent evolution of the fields and particle acceleration. Advances in computer power now allow for kinetic plasma simulations, based on the particle-in-cell (PIC) paradigm, capable of providing information about the role of plasma instabilities in relativistic outflows. A discussion of the key issues associated with PIC simulations is presented, along with some the most important results and open questions, with a particular emphasis on th…
The plasma dynamics resulting from the simultaneous impact, of two equal, ultra-intense laser pulses, in two spatially separated spots, onto a dense target is studied via particle-in-cell (PIC) simulations. The simulations show that electrons accelerated to relativistic speeds, cross the target and exit at its rear surface. Most energetic electrons are bound to the rear surface by the ambipolar electric field and expand along it. Their current is closed by a return current in the target, and this current configuration generates strong surface magnetic fields. The two electron sheaths collide a…
For parameters that are applicable to the conditions at young supernova remnants, we present results of 2D3V particle-in-cell simulations of a non-relativistic plasma shock with a large-scale perpendicular magnetic field inclined at 45-deg angle to the simulation plane to approximate 3D physics. We developed an improved clean setup that uses the collision of two plasma slabs with different density and velocity, leading to the development of two distinctive shocks and a contact discontinuity. The shock formation is mediated by Weibel-type filamentation instabilities that generate magnetic turbu…
