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The energy densities of the magnetic (left) and electric field (right) for the different initial field strengths B 0 .
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The impact of a flow-aligned and spatially homogeneous magnetic field on the filamentation instability (FI) is examined in a system of two equal counterstreaming non-relativistic cool electron beams. Particle-in-cell simulations that represent the plane perpendicular to the flow velocity vector confirm the reduction of the linear growth rate by the...
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... are solved with p cp m cp v cp , where v cp is the velocity of a computational particle. The code we employ fulfills ∇ · E = ρ / ε 0 and ∇ · B = 0 to round-off precision. The boundary conditions of the simulations are periodic in all directions. Fig. 2 shows the energy densities of the magnetic and electric field. The linear phase is well distinguishable from the non-linear phase and the same amplification level is reached for all B 0 . Due to the saturation mechanism, the amplification of ε E is delayed towards the amplification of ε B . ε B grows also for B 0 = B c and a detailed analysis has shown that the amplification behaves completely different compared with the cases B 0 < B c . This might be due to a temperature effect or a non-linear effect which allows particle diffusion across the magnetic field. A Fourier transform of the field data allows a determination of the maximum linear growth rate. Fig. 3 shows the power spectra of the perpendicular magnetic field component for a constant wave number k max , which shows the fastest growth of the instability during the linear phase, as B 2 ( k , t ) ∝ exp ( 2 σ t ) . Straight lines are fitted to the simulation data in order to demonstrate the exponential be- haviour. The growth rates of the simulations have been compared to the analytic estimated (Eq. (2)) and yield a deviance of 10% at most. The saturation mechanism is a result of the electron displacement by the magnetic pressure gradient, which has already been observed in a 1D simulation [7]. The amplification of the magnetic field is related to a magnetic pressure P b = B 2 ⊥ / 2 μ 0 in the perpendicular plane. The pressure gradient ∇ P b = B ⊥ ∇ B ⊥ / μ 0 pushes away the electrons, resulting in a space charge separation. This induces a restoring force which is proportional to the perpendicular electric field. For a constant wave number this results in ∇ P b ∝ E ⊥ ∝ exp ( 2 σ t ) . A cut through the perpendicular plane shows the minima of the gradient of | B | , and thus ...
Citations
... This was later able to be identified in laser-plasma experiments [8]. Different Weibel instability (WI) configurations have been investigated for unmagnetized plasma [9][10] and magnetized plasma [12][13][14]. Weibel instability (WI) is used to describe the existence of strong magnetic fields inside of plasma. An intense magnetic field (up to 10 8 G) is produced when highly intense ultra-short laser pulses are bombarded on the solid target [15]. ...
This paper investigates how an electrostatic Langmuir wave having large amplitude affects the Weibel instability (WI) in the existence of ions and an electron beam. Two Langmuir side band waves are produced by coupling of EM perturbation to the Langmuir wave. The Langmuir wave (LW) increases the growth rate beyond its linear value. Here, we noticed that the growth rate Γ(sec⁻¹) scales linearly with the electron beam velocity vbe and 1/2 power of the electron beam density nbe. As we increase the density of ions inside the plasma, the growth rate stabilizes. Additionally, we find that the growth rate is very sensitive to the plasma frequency of ions. Therefore, our work finds an application in space, galactic cosmic rays and supernovas. Also, our work covers a range of application from the development of fusion power to understand the various astrophysical phenomena.
... 27 For example, two-stream and Buneman instabilities have an electrostatic nature, while the filamentation and Weibel instabilities are types of electromagnetic instabilities. [27][28][29][30][31][32][33][34][35][36] It is worthy to note that the microwave generation process in HPM sources can be understood in terms of resonant interactions between the normal modes of a cavity or waveguide and the natural modes of oscillations in the REBs. 37 The interactions leading to microwave generation in HPM devices produce waves that grow up from small perturbations on an initial equilibrium state. ...
The charged particle beams, such as electrons, ions, and plasma compression flow, have received considerable attention due to their
applications in science and technology; therefore, studying the stability of these beams is of particular importance. Here, we examine
theoretically the stability properties of a cold relativistic electron beam with a transverse velocity shear and non-uniform density profile. We
consider a plane-parallel beam propagating along an external magnetic field and evaluate its macroscopic equilibrium state. We derive the
dispersion relation of the slipping instability based on the linear electrodynamics of an inhomogeneous plasma and kinetic theory. In this
model, the oscillation spectrum and the growth rate are derived by using the eikonal equation and the quasi-classical quantization rule. A linear
velocity shear and a non-linear density gradient are assumed. Furthermore, we analyze numerically the dispersion relation of the slipping
instability. The impacts of the inhomogeneity parameter and the relativistic factor on the properties of the slipping instability are discussed.
... Let us estimate the amplitude of the beam-aligned magnetic field B c that would suppress the filamentation instability of counterstreaming electron beams. [29][30][31][32] We obtain eB c =m e x pe ¼ v b =c in physical units for our nonrelativistic beam speed v b and equal densities of the counterstreaming electron beams. 30 Consider a plasma with the density 10 cm À1 and magnetic field strength 5 nT of the solar wind near the Earth's orbit. ...
... [29][30][31][32] We obtain eB c =m e x pe ¼ v b =c in physical units for our nonrelativistic beam speed v b and equal densities of the counterstreaming electron beams. 30 Consider a plasma with the density 10 cm À1 and magnetic field strength 5 nT of the solar wind near the Earth's orbit. The magnetic field of the interstellar medium has a similar amplitude, and we may expect that the magnetic fields of stellar winds of black hole companions are usually not much stronger. ...
... The values k % 3:3; v b ¼ 0:3 and B k % 0:1 [see Fig. 1(e)] give x B % 0:3, which matches the growth rate of the filamentation instability between unmagnetized cold electron beams of equal density for a beam speed v b . 30 The E y field slows down the beam particles and transfers some of their kinetic energy to the magnetic B z component, which lets its energy density / B 2 z grow. The magnetic field is not at rest in the simulation frame and its motion along y induces a convective electric field E x . ...
Particle-in-cell simulations of jets of electrons and positrons in an ambient electron–proton plasma have revealed an acceleration of positrons at the expense of electron kinetic energy. We show that a filamentation instability, between an unmagnetized ambient electron–proton plasma at rest and a beam of pair plasma that moves through it at a non-relativistic speed, indeed results in preferential positron acceleration. Filaments form that are filled predominantly with particles with the same direction of their electric current vector. Positron filaments are separated by electromagnetic fields from beam electron filaments. Some particles can cross the field boundary and enter the filament of the other species. Positron filaments can neutralize their net charge by collecting the electrons of the ambient plasma, while protons cannot easily follow the beam electron filaments. Positron filaments can thus be compressed to a higher density and temperature than the beam electron filaments. Filament mergers, which take place after the exponential growth phase of the instability has ended, lead to an expansion of the beam electron filaments, which amplifies the magnetic field they generate and induces an electric field in this filament. Beam electrons lose a substantial fraction of their kinetic energy to the electric field. Some positrons in the beam electron filament are accelerated by the induced electric field to almost twice their initial speed. The simulations show that a weaker electric field is induced in the positron filament and particles in this filament hardly change their speed.
... Let us estimate the amplitude of the beam-aligned magnetic field B c that would suppress the filamentation instability of counterstreaming electron beams. [30][31][32][33] We obtain eB c /m e ω pe = v b /c in physical units for our nonrelativistic beam speed v b and equal densities of the counterstreaming electron beams. 31 Consider a plasma with the density 10 cm −1 and magnetic field strength 5 nT of the solar wind near the Earth's orbit. ...
... [30][31][32][33] We obtain eB c /m e ω pe = v b /c in physical units for our nonrelativistic beam speed v b and equal densities of the counterstreaming electron beams. 31 Consider a plasma with the density 10 cm −1 and magnetic field strength 5 nT of the solar wind near the Earth's orbit. The magnetic field of the interstellar medium has a similar amplitude and we may expect that the magnetic fields of stellar winds of black hole companions are usually not much stronger. ...
... give ω B ≈ 0.3, which matches the growth rate of the filamentation instability between unmagnetized cold electron beams of equal density for a beam speed v b . 31 The E y field slows down the beam particles and transfers some of their kinetic energy to the magnetic B z component, which lets its energy density ∝ B 2 z grow. The magnetic field is not at rest in the simulation frame and its motion along y induces a convective electric field E x . ...
Particle-in-cell (PIC) simulations of collisionless jets of electrons and positrons in an ambient electron-proton plasma have revealed an acceleration of positrons at the expense of electron kinetic energy. The dominant instability within the jet was a filamentation instability between electrons, protons and positrons. In this work we show that a filamentation instability, between an initially unmagnetized ambient electron-proton plasma at rest and a beam of pair plasma that moves through it at a non-relativistic speed, indeed results in preferential positron acceleration. Filaments form that are filled predominantly with particles with the same direction of their electric current vector. Positron filaments are separated by electromagnetic fields from beam electron filaments. Some particles can cross the field boundary and enter the filament of the other species. Positron filaments can neutralize their net charge by collecting the electrons of the ambient plasma while protons cannot easily follow the beam electron filaments. Positron filaments can thus be compressed to a higher density and temperature than the beam electron filaments. Filament mergers, which take place after the exponential growth phase of the instability has ended, lead to an expansion of the beam electron filaments, which amplifies the magnetic field they generate and induces an electric field in this filament. Beam electrons lose a substantial fraction of their kinetic energy to the electric field. Some positrons in the beam electron filament are accelerated by the induced electric field to almost twice their initial speed. The simulations show that a weaker electric field is induced in the positron filament and particles in this filament hardly change their speed.
... Notorious in unmagnetized plasmas are the so-called Weibel or magnetic instabilities, which can be induced by the kinetic anisotropies of plasma populations, e.g. temperature anisotropy or counter-beaming populations [6,7], and are frequently invoked to explain the origin of cosmological magnetic field seeds, e.g. in the early Universe [8,9], and the filamentation of energetic plasma beams [5,10,11]. However, the influence of a guiding stationary magnetic field on filamentation instability is not clear yet, Vlasov and particle-incell (PIC) simulations showing contradictory results [5,10]. ...
... temperature anisotropy or counter-beaming populations [6,7], and are frequently invoked to explain the origin of cosmological magnetic field seeds, e.g. in the early Universe [8,9], and the filamentation of energetic plasma beams [5,10,11]. However, the influence of a guiding stationary magnetic field on filamentation instability is not clear yet, Vlasov and particle-incell (PIC) simulations showing contradictory results [5,10]. Instead, the aperiodic mirror [2,12,13] and firehose instabilities [1,14,15] may develop efficiently in finite beta plasmas, constraining any anisotropic temperature [13,16,17] induced by magnetic compression or adiabatic expansion along the magnetic field lines (e.g. ...
... These are zero-frequency (ω = 0) waves with spatial propagation, i.e., with finite wave-numbers (k = 0), but locally their amplitude is purely growing in time with a rate (γ > 0), usually much higher than that of the periodic modes. In unmagnetized plasmas notorious are the so-called Weibel or magnetic instabilities, which can be induced by the kinetic anisotropies of plasma populations, e.g., temperature anisotropy or counter-beaming populations [6,7], and are frequently invoked to explain the origin of cosmological magnetic field seeds, e.g., in the early Universe [8,9], and the filamentation of energetic plasma beams [5,10,11]. However, the influence of a guiding stationary magnetic field on filamentation instability is not clear yet, Vlasov and particle-incell (PIC) simulations showing contradictory results [5,10]. ...
... In unmagnetized plasmas notorious are the so-called Weibel or magnetic instabilities, which can be induced by the kinetic anisotropies of plasma populations, e.g., temperature anisotropy or counter-beaming populations [6,7], and are frequently invoked to explain the origin of cosmological magnetic field seeds, e.g., in the early Universe [8,9], and the filamentation of energetic plasma beams [5,10,11]. However, the influence of a guiding stationary magnetic field on filamentation instability is not clear yet, Vlasov and particle-incell (PIC) simulations showing contradictory results [5,10]. Instead, the aperiodic mirror [2,12,13] and firehose instabilities [1,14,15] may develop efficiently in finite beta plasmas, constraining any anisotropic temperature [13,16,17] induced by magnetic compression or adiabatic expansion along the magnetic field lines (e.g., solar outflows in the heliosphere). ...
Depending on the physical conditions involved the beam plasma systems may reveal new unstable regimes triggered by the wave instabilities of different nature. We show through linear theory and numerical simulations the existence of an aperiodic electromagnetic instability which solely develops and control the stability of two symmetric plasma populations counter-moving along the regular magnetic field with a relative drift, $v_d$, small enough to not exceed the particle thermal speed, $\alpha_e$. Emerging at highly oblique angles this mode resembles properties of the aperiodic firehose instability driven by temperature anisotropy. The high growth rates achieved with increasing the relative drift or/and decreasing the plasma beta parameter lead to significant saturation levels of the fluctuating magnetic field power, which explain the relative fast relaxation of electrons. For $v_d>\alpha_e$ this instability can coexist with the electrostatic two-stream instability, dominating the long-term dynamics of the plasma as soon as $v_d$ has relaxed to values smaller than the thermal speed.
... Observationally, the orientation of magnetic field is closely related to the jet sidedness [22]; while theoretically, it leads to mode coupling of different instabilities [23,24] and alters the particle acceleration [25][26][27][28]. For example, as for the parallel component, it suppresses the growth rate of Weibel instability (WBI) [29,30], reduces the saturation level [31], and prevents the collisionless shock formation [32,33]; while as for the perpendicular one, ultra-strong magnetic field supports the relativistic jet transport in a relay manner by quantum electrodynamics effects [34]. ...
We present results from fully kinetic particle-in-cell simulations of the transport of astrophysical relativistic jets in magnetized intergalactic medium. As opposed to magnetohydrodynamic simulations, the results show that a strong charge-separation electric field, induced by the different responses between jet electrons and ions to the magnetic fields, significantly enhances the energy exchange between different species of charged particles and electromagnetic fields, thus playing a key role in determining the collimation and shape of the jet spectral energy distribution (SED). It is found that when the magnetic field strength increases, the jet collimation also increases while the power-law slope of the jet SED decreases; this provides potential enlightenment on related astrophysical observations.
... The Weibel instability has been at the center of several works and different configurations have been investigated for unmagnetized plasmas, considering as initial unstable conditions both counter-streaming plasmas (Pegoraro et al., 1996;Califano et al., 1997Califano et al., , 1998Bret et al., 2010b,a) or a plasma distribution function with temperature anisotropy (Morse and Nielson, 1971;Davidson et al., 1972;Palodhi et al., 2009;Stockem et al., 2010). More recent studies focus on magnetized scenarios, considering counter-streaming flows in a macroscopic external magnetic fields, perpendicular or parallel to the plasma flows (Ji-Wei and Wen-Bing, 2005; Stockem et al., 2007Stockem et al., , 2008, and with a general orientation (Bret, 2014;Bret and Dieckmann, 2017). In magnetized plasmas, the linear phase of the instability in the cold beam approximation is well characterized. ...
... Indeed, Stockem et al. (2008) suggested that trapping can not be the saturation mechanism in the magnetized case and a detailed investigation of different processes that might be responsible for saturation has been carried out by Dieckmann (2009);Dieckmann et al. (2009);Dieckmann and Bret (2010). Moreover, the possibility to reach saturation via the Larmor/Alfvén mechanism in magnetized plasma has been questioned by Bret (2016). ...
... The bouncing frequency in this case becomes 2.19) with ω b defined in Eq. (3.2.18). Considering that saturation is reached when the bouncing frequency equates the growth rate of the instability (Γ approximately the asymptotic value reported in Eq. (3.2.3) for large k), the expected saturation level would depend on the strength of the external magnetic field (see Stockem et al., 2008). ...
The work presented in this thesis belongs to the general framework of Laboratory Astrophysics. We address various aspects of the physics of collisionless shocks developing in the presence of relativistic plasma flows, in configurations of interest for the astrophysical and the laser-plasma interaction (LPI) communities. The approach used throughout this thesis relied on both analytical modeling and high-performance kinetic simulations, a central tool to describe LPI processes as well as the non-linear physics behind shock formation. The PIC code SMILEI has been widely used and developed during this work. Three physical configurations are studied. First we consider the Weibel instability driven by two counter-streaming electron beams aligned with an external magnetic field. The linear and non-linear phases are explained using theoretical models confirmed by simulations.Then the generation of non-collisional shocks during the interaction of two relativistic plasma pairs is studied in the presence of a perpendicular magnetic field. We focus on the comparison of theoretical predictions for macroscopic variables with the simulation results, as well as on the definition and measurement of the shock formation time, all of which are of great importance for future experiments.Finally, we proposed a scheme to produce, in the laboratory, the ion-Weibel-instability with the use of an ultra-high-intensity laser. The produced flows are faster and denser than in current experiments, leading to a larger growth rate and stronger magnetic fields. These results are important for the LPI at very high intensity.
... The relativistic factors of the internal shocks in GRB jets are probably of the order of a few. A wide range of theoretical and numerical studies have addressed the collision of lepton clouds at relativistic speeds and the instabilities that sustain the shock and thermalize the plasma that crosses it (Kazimura et al. 1998;Medvedev & Loeb 1999;Brainerd 2000;Sakai et al. 2000;Silva et al. 2003;Haruki & Sakai 2003;Jaroschek et al. 2004;Medvedev et al. 2005;Milosavljevic & Nakar 2006;Chang et al. 2008;Stockem et al. 2008;Bret et al. 2008Bret et al. , 2013Sironi & Giannios 2014;Marcowith et al. 2016). Such shocks thermalize plasma via the magnetic fields that are driven by the filamentation instability of counter-streaming beams of charged particles, which is also known as the beam-Weibel instability. ...
We examine with a particle-in-cell (PIC) simulation the collision of two equally dense clouds of cold pair plasma. The clouds interpenetrate until instabilities set in, which heat up the plasma and trigger the formation of a pair of shocks. The fastest-growing waves at the collision speed c/5 and low temperature are the electrostatic two-stream mode and the quasi-electrostatic oblique mode. Both waves grow and saturate via the formation of phase space vortices. The strong electric fields of these nonlinear plasma structures provide an efficient means of heating up and compressing the inflowing upstream leptons. The interaction of the hot leptons, which leak back into the upstream region, with the inflowing cool upstream leptons continuously drives electrostatic waves that mediate the shock. These waves heat up the inflowing upstream leptons primarily along the shock normal, which results in an anisotropic velocity distribution in the post-shock region. This distribution gives rise to the Weibel instability. Our simulation shows that even if the shock is mediated by quasi-electrostatic waves, strong magnetowaves will still develop in its downstream region.
... Present studies Refs. [18][19][20] related to the magne-tized scenarios, leave many open questions, in particular regarding the saturation mechanisms at play. ...
... Using both analytic modeling and Particle-In-Cell (PIC) simulations, we highlight the effect of the external magnetic field on the linear and nonlinear phases of the instability. In particular we show that, the well-known result [18][19][20] that the growth rate of the instability is reduced in the presence of a flow-aligned external magnetic field does not imply that the latter has a stabilizing effect in the nonlinear phase. Furthermore, we generalize previously proposed saturation mechanisms to account for the presence of an external magnetic field. ...
... Considering that saturation is reached when the bouncing frequency equates the growth rate of the instability Γ, the expected saturation level would depend on the strength of the external magnetic field (see e.g. Ref. [20]). ...
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 discussed in detail in the relativistic cold fluid framework considering a single unstable mode. The application of an external field leads to a slighlt increase of the saturation level for large wavelengths, while it does not affect the small wavelengths. Multi-mode and temperature effects are then investigated. While at large temperature the saturation level is independent of the external magnetic field, at small but finite temperature the competition between different modes in the presence of an external magnetic field leads to a saturation level lower with respect to the unmagnetized case.
