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Shockwaves in plasma are usually dealt with using Magnetohydrodynamics (MHD). Yet, MHD entails the assumption of a short mean free path, which is not fulfilled in a collisionless plasma. Recently, for pair plasmas, we devised a model allowing to account for kinetic effects within an MHD-like formalism. Its relies on an estimate of the anisotropy generated when crossing the front, with a subsequent assessment of the stability of this anisotropy in the downstream. We solved our model for parallel, perpendicular and switch-on shocks. Here we bridge between all these cases by treating the problem of an arbitrarily, but coplanar, oriented magnetic field. Even though the formalism presented is valid for anisotropic upstream temperatures, only the case of a cold upstream is solved. We find extra solutions which are not part of the MHD catalog, and a density jump that is notably less in the quasi parallel, highly magnetized, regime. Given the complexity of the calculations, this work is mainly devoted to the presentation of the mathematical aspect of our model. A forthcoming article will be devoted to the physics of the shocks here defined., 16 pages, 8 figures, to appear in Journal of Plasma Physics

Shocks waves are a ubiquitous feature of many astrophysical plasma systems, and an important process for energy dissipation and transfer. The physics of these shock waves are frequently treated/modeled as a collisional, fluid MHD discontinuity, despite the fact that many shocks occur in the collisionless regime. In light of this, using fully kinetic, 3D simulations of non-relativistic, parallel propagating collisionless shocks comprised of electron-positron plasma, we detail the deviation of collisionless shocks form MHD predictions for varying magnetization/Alfv��nic Mach numbers, with particular focus on systems with Alf��nic Mach numbers much smaller than sonic Mach numbers. We show that the shock compression ratio decreases for sufficiently large upstream magnetic fields, in agreement with the predictions of Bret & Narayan (2018). Additionally, we examine the role of magnetic field strength on the shock front width. This work reinforces a growing body of work that suggest that modeling many astrophysical systems with only a fluid plasma description omits potentially important physics., 7 pages, 5 figures + a diagram, submitted to MNRAS

The properties of collisionless shocks, like the density jump, are usually derived from magnetohydrodynamics (MHD), where isotropic pressures are assumed. Yet, in a collisionless plasma, an external magnetic field can sustain a stable anisotropy. In \cite{BretJPP2018}, we devised a model for the kinetic history of the plasma through the shock front, allowing to self-consistently compute the downstream anisotropy, hence the density jump, in terms of the upstream parameters. This model dealt with the case of a parallel shock, where the magnetic field is normal to the front both in the upstream and the downstream. Yet, MHD also allows for shock solutions, the so-called switch-on solutions, where the field is normal to the front only in the upstream. This article consists in applying our model to these switch-on shocks. While MHD offers only 1 switch-on solution within a limited range of Alfv\'{e}n Mach numbers, our model offers 2 kinds of solutions within a slightly different range of Alfv\'{e}n Mach numbers. These 2 solutions are most likely the outcome of the intermediate and fast MHD shocks under our model. While the intermediate and fast shocks merge in MHD for the parallel case, they do not within our model. For simplicity, the formalism is restricted to non-relativistic shocks in pair plasmas where the upstream is cold., Comment: 18 pages. 6 figures. To appear in Journal of Plasma Physics

In a collisionless shock, there are no binary collisions to isotropize the flow. It is therefore reasonable to ask to which extent the magnetohydrodynamics (MHD) jump conditions apply. Following up on recent works which found a significant departure from MHD in the case of parallel collisionless shocks, we here present a model allowing to compute the density jump for collisionless shocks. Because the departure from MHD eventually stems from a sustained downstream anisotropy that the Vlasov equation alone cannot specify, we hypothesize a kinetic history for the plasma, as it crosses the shock front. For simplicity, we deal with non-relativistic pair plasmas. We treat the cases of parallel and perpendicular shocks. Non-MHD behavior is more pronounced for the parallel case where, according to MHD, the field should not affect the shock at all.

In a recent work [Bret, EPL \textbf{135} (2021) 35001], quantum electrodynamic (QED) effects were evaluated for the two-stream instability. It pertains to the growth of perturbations with a wave vector oriented along the flow in a collisionless counter-streaming system. Here, the analysis is extended to every possible orientation of the wave vector. The previous result for the two-stream instability is recovered, and it is proved that even within the framework of a 3D analysis, this instability remains fundamentally 1D even when accounting for QED effects. The filamentation instability, found for wave vectors normal to the flow, is weakly affected by QED corrections. As in the classical case, its growth rate saturates at large $k_\perp$. The saturation value is found independent of QED corrections. Also, the smallest unstable $k_\perp$ is independent of QED corrections. Surprisingly, unstable modes found for oblique wave vectors do \emph{not} follow the same pattern. For some, QED corrections do reduce the growth rate. But for others, the same corrections increase the growth rate instead. The possibility for QED effects to play a role in un-magnetized systems is evaluated. Pair production resulting from gamma emission by particles oscillating in the exponentially growing fields, is not accounting for., Comment: 19 pages, 7 figures, to appear in Physical Review E

Within the context of Magnetohydrodynamics (MHD), the properties of a parallel shock do not depend on the field strength, as the field and the fluid are disconnected for such a geometry. However, in the collisionless case, the field can sustain a stable anisotropy in the downstream, triggering a departure from the expected MHD behavior. In a recent work [A. Bret and R. Narayan, J. Plasma Phys. \textbf{84}, 905840604 (2018)], a theoretical model was presented allowing to derive the density ratio of a non-relativistic parallel collisionless shock in an electron/positron plasma, as a function of the field. Here we derive the entropy, pressure and temperature jumps stemming from this model. It is found to offer a transition between a 3D and a 1D downstream for the jumps in density, entropy, parallel temperature and parallel pressure., 6 pages, 4 figure. To appear in Physics of Plasmas

In shockwave theory, the density, velocity and pressure jumps are derived from the conservation equations. Here, we address the physics of a weak shock the other way around. We first show that the density profile of a weak shockwave in a fluid can be expressed as a sum of linear acoustic modes. The shock so built propagates at the speed of sound and matter is exactly conserved at the front crossing. Yet, momentum and energy are only conserved up to order 0 in powers of the shock amplitude. The density, velocity and pressure jumps are similar to those of a fluid shock, and an equivalent Mach number can be defined. A similar process is possible in magnetohydrodynamic. Yet, such a decomposition is found impossible for collisionless shocks due to the dispersive nature of ion acoustic waves. Weakly nonlinear corrections to their frequency do not solve the problem. Weak collisionless shocks could be inherently nonlinear, non-amenable to any linear superposition. Or they could be nonexistent, as hinted by recent works., Comment: 8 pages, 2 figures, to appear in Journal of Plasma Physics

Collisionless shocks have been the subject on many studies in recent years, due to their ability to accelerate particles. In order to do so, a shock must fulfill three criteria. First, it must be strong enough to accelerate particles efficiently. Second, both the upstream and the downstream must be collisionless. Third, the shock front must be surrounded by electromagnetic turbulence capable of scattering particles back and forth. We here consider the encounter of two identical plasma shells with initial density, temperature, and velocity n0, T0, v0, respectively. We translate the three criteria to the corresponding requirements on these parameters. A non-trivial map of the allowed region for particle acceleration emerges in the (n0, T0, v0) phase space, especially at low velocities or high densities. We first assess the case of pair plasma shells, before we turn to electrons/protons.

In plasmas where the mean-free-path is much larger than the size of the system, shock waves can arise with a front much shorter than the mean-free-path. These so-called “collisionless shocks” are mediated by collective plasma interactions. Studies conducted so far on these shocks found that although binary collisions are absent, the distribution functions are thermalized downstream by scattering on the fields, so that magnetohydrodynamics prescriptions may apply. Here we show a clear departure from this pattern in the case of Weibel shocks forming over a flow-aligned magnetic field. A micro-physical analysis of the particle motion in the Weibel filaments shows how they become unable to trap the flow in the presence of too strong a field, inhibiting the mechanism of shock formation. Particle-in-cell simulations confirm these results.

When applied to compute the density jump of a shock, the standard magnetohydrodynamic (MHD) formalism assumes, 1) that all the upstream material passes downstream, together with the momentum and energy it carries, and 2) that pressures are isotropic. In a collisionless shock, shock accelerated particles going back and forth around the front can invalid the first assumption. In addition, an external magnetic field can sustain stable pressure anisotropies, invaliding the second assumption. It is therefore unclear whether the density jump of a collisionless shock fulfils the MHD jump or not. Here we try to clarify this issue. A literature review is conducted on 68 articles dealing with Particle-In-Cell simulations of collisionless shocks. We analyze the factors triggering departure from the MHD density jump and quantify their influence on $\Delta_{RH}$, the relative departure from the Rankine-Hugoniot jump. For small departures we propose $\Delta_{RH} = + \mathcal{O}(10^{-1-3.7\kappa})t^\kappa - \sigma \mathcal{O}(1)$ where $t$ is the timescale of the simulation, $\sigma$ the magnetization parameter and $\kappa$ a constant of order unity. The first term stems from the energy leakage into accelerated particle. The second term stems from the downstream anisotropy triggered by the field (assuming an isotropic upstream). This relation allows to assess to which extent a collisionless shock fulfils the RH density jump. In the strong field limit and for parallel shocks, the departure caused by the field saturates at a finite, negative, value. For perpendicular shocks, the departure goes to zero at small and high $\sigma$'s so that we find here a departure window. The results obtained have to be checked against full 3D simulations., Comment: To appear to ApJ. 13 pages, 5 figures

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 $\alpha=B_0/B_f > 1/2$. Here, this result is extended to the case of an oblique external field $B_0$ making an angle $\theta$ with the flow. The result, numerically found, is simply $\alpha > \kappa(\theta)/\cos\theta$, where $\kappa(\theta)$ is of order unity. Noteworthily, test particles exhibit chaotic trajectories., Comment: 10 pages, 6 figures, to appear in Journal of Plasma Physics

In the absence of frequent binary collisions to isotropize the plasma, the fulfillment of the magnetohydrodynamic (MHD) Rankine-Hugoniot jump conditions by collisionless shocks is not trivial. In particular, the presence of an external magnetic field can allow for stable anisotropies, implying some departures from the isotropic MHD jumps. The functional dependence of such anisotropies in terms of the field is yet to be determined. By hypothesizing a kinetic history of the plasma through the shock front, we recently devised a theory of the downstream anisotropy, hence of the density jump, in terms of the field strength for a parallel shock [J. Plasma Phys. (2018), vol. 84, 905840604]. Here, we extend the analysis to the case of a perpendicular shock. We still find that the field reduces the density jump, but the effect is less pronounced than in the parallel case., Comment: 17 pages, 9 figures, to appear in Physics of Plasmas

Collisionless shocks follow the Rankine-Hugoniot jump conditions to a good approximation. However, for a shock propagating parallel to a magnetic field, magnetohydrodynamics states that the shock properties are independent of the field strength, whereas recent Particle-in-Cell simulations reveal a significant departure from magnetohydrodynamics behavior for such shocks in the collisionless regime. This departure is found to be caused by a field-driven anisotropy in the downstream pressure, but the functional dependence of this anisotropy on the field strength is yet to be determined. Here, we present a non-relativistic model of the plasma evolution through the shock front, allowing for a derivation of the downstream anisotropy in terms of the field strength. Our scenario assumes double adiabatic evolution of a pair plasma through the shock front. As a result, the perpendicular temperature is conserved. If the resulting downstream is firehose stable, then the plasma remains in this state. If unstable, it migrates towards the firehose stability threshold. In both cases, the conservation equations, together with the relevant hypothesis made on the temperature, allows a full determination of the downstream anisotropy in terms of the field strength., Comment: To appear in Journal of Plasma Physics, 16 pages, 7 figures

When two plasmas collide, their interaction can be mediated by collisionless plasma instabilities or binary collisions between particles of each shell. By comparing the maximum growth rate of the collisionless instabilities with the collision frequency between particles of the shells, we determine the critical density separating the collisionless formation from the collisional formation of the resulting shock waves. This critical density is also the density beyond which the shock downstream is field free, as plasma instabilities do not have time to develop electromagnetic patterns. We further determine the conditions on the shells initial density and velocity for the downstream to be collisional. If these quantities fulfill the determined conditions, the collisionality of the downstream also prevents the shock from accelerating particles or generating strong magnetic fields. We compare the speed of sound with the relative speed of collision between the two shells, thus determining the portion of the parameters space where strong shock formation is possible for both classical and degenerate plasmas. Finally, we discuss the observational consequences in several astrophysical settings., Comment: To appearin Journal of Plasma Physics - 35 pages, 11 figures

Energetic electromagnetic emissions by astrophysical jets like those that are launched during the collapse of a massive star and trigger gamma-ray bursts 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 simulation, the collision of two cold pair clouds at the speed c/2 (c: speed of light). We demonstrate that the two-stream instability outgrows the filamentation instability for this speed and is thus responsible for the shock formation. The incomplete thermalization of the upstream plasma by its quasi-electrostatic waves allows other instabilities to grow. A shock transition layer forms, in which a filamentation instability modulates the plasma far upstream of the shock. The inflowing upstream plasma is progressively heated by a two-stream instability closer to the shock and compressed to the expected downstream density by the Weibel instability. The strong magnetic field due to the latter is confined to a layer 10 electron skin depths wide. Funding agencies: Ministerio de Educacion y Ciencia, Spain [ENE2013-45661-C2-1-P, ENE2016-75703-R]; Junta de Comunidades de Castilla-La Mancha [PEII-2014-008-P]; Grand Equipement National de Calcul Intensif (GENCI) [x2016046960]

The expansion of a radial blast shell into an ambient plasma is modeled with a particle-in-cell (PIC) simulation. The unmagnetized plasma consists of electrons and protons. The formation and evolution of an electrostatic shock is observed, which is trailed by ion-acoustic solitary waves that grow on the beam of the blast shell ions in the post-shock plasma. In spite of the initially radially symmetric outflow, the solitary waves become twisted and entangled and, hence, they break the radial symmetry of the flow. The waves and their interaction with the shocked ambient ions slows down the blast shell protons and brings the post-shock plasma closer to an equilibrium., Comment: 5 pages, 4 figures, accepted by Physics of Plasmas

According to magnetohydrodynamics (MHD), the encounter of two collisional magnetized plasmas at high velocity gives rise to shock waves. Investigations conducted so far have found that the same conclusion still holds in the case of collisionless plasmas. For the case of a flow-aligned field, MHD stipulates that the field and the fluid are disconnected, so that the shock produced is independent of the field. We present a violation of this MHD prediction when considering the encounter of two cold pair plasmas along a flow-aligned magnetic field. As the guiding magnetic field grows, isotropization is progressively suppressed, resulting in a strong influence of the field on the resulting structure. A micro-physics analysis allows us to understand the mechanisms at work. Particle-in-cell simulations also support our conclusions and show that the results are not restricted to a strictly parallel field.

Collisionless shocks are key processes in astrophysics where the energy dissipation at the shock front is provided by collective plasma effects rather than particle collisions. While numerous simulations and laser-plasma experiments have shown they can result from the encounter of two plasma shells, a first principle theory of the shock formation is still lacking. In this respect, a series of 2D Particle-In-Cells simulations have been performed of two identical cold colliding pair plasmas. The simplicity of this system allows for an accurate analytical tracking of the physics. To start with, the Weibel-filamentation instability is triggered in the overlapping region, which generates a turbulent region after a saturation time τs. The incoming flow then piles-up in this region, building-up the shock density region according to some nonlinear processes, which will be the subject of future works. By evaluating the seed field giving rise to the instability, we derive an analytical expression for τs in good agreement with simulations. In view of the importance of the filamentation instability, we show a static magnetic field can cancel it if and only if it is perfectly aligned with the flow.

Collisionless shocks are ubiquitous in astrophysics and in the laboratory. Recent numerical simulations and experiments have shown how these can arise from the encounter of two collisionless plasma shells. When the shells interpenetrate, the overlapping region turns unstable, triggering the shock formation. As a first step toward a microscopic understanding of the process, we here analyze in detail the initial instability phase. On the one hand, 2D relativistic PIC simulations are performed where two unmagnetized, symmetric, and initially cold pair plasmas collide. On the other hand, the instabilities at work are analyzed, as well as the field at saturation and the seed field which gets amplified. For mildly relativistic motions and onward, Weibel modes with ω=0+iδ govern the linear phase. We derive an expression for the duration of the linear phase in reasonable agreement with the simulations.

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 $\theta \in [0,\pi/2]$ 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 $\gamma_0$, and the field strength as measured by a parameter denoted $\sigma$. For $\theta=0$, four different kinds of mode are likely to lead the linear phase. The hierarchy simplifies for larger $\theta$'s, partly because the Weibel instability can no longer be cancelled in this regime. For $\theta>0.78$ (44$^\circ$) and in the relativistic regime, the Weibel instability always govern the interaction. In the non-relativistic regime, the hierarchy becomes $\theta$-independent because the interaction turns to be field-independent. As a result, the two-stream instability becomes the dominant one, regardless of the field obliquity., Comment: To appear in Physics of Plasmas

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., Comment: 17 pages, 9 figures, accepted for publication by the Journal of Plasma Physics

We report on the experimental observation of the instability of a plasma shell, which formed during the expansion of a laser-ablated plasma into a rarefied ambient medium. By means of a proton radiography technique, the evolution of the instability is temporally and spatially resolved on a timescale much shorter than the hydrodynamic one. The density of the thin shell exceeds that of the surrounding plasma, which lets electrons diffuse outward. An ambipolar electric field grows on both sides of the thin shell that is antiparallel to the density gradient. Ripples in the thin shell result in a spatially varying balance between the thermal pressure force mediated by this field and the ram pressure force that is exerted on it by the inflowing plasma. This mismatch amplifies the ripples by the same mechanism that drives the hydrodynamic nonlinear thin-shell instability (NTSI). Our results thus constitute the first experimental verification that the NTSI can develop in colliding flows. Funding agencies: EPSRC [EP/I031766/1, EP/K022415/1, EP/I029206/1, SFB-TR18, GRK1203, ENE2013-45661-C2-1-P, PEII-2014-008-P]; Vetenskapsradet [Dnr 2010-4063]; Triangle de la Physique RTRA network (ULIMAC)

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 become unable to stop any particle, regardless of its initially velocity., Comment: 11 pages, 6 figures. To appear in Journal of Plasma Physics

Beam-plasma instabilities are a key physical process in many astrophysical phenomena. Within the fireball model of Gamma ray bursts, they first mediate a relativistic collisionless shock before they produce upstream the turbulence needed for the Fermi acceleration process. While non-relativistic systems are usually governed by flow-aligned unstable modes, relativistic ones are likely to be dominated by normally or even obliquely propagating waves. After reviewing the basis of the theory, results related to the relativistic kinetic regime of the poorly-known oblique unstable modes will be presented. Relevant systems besides the well-known electron beam-plasma interaction are presented, and it is shown how the concept of modes hierarchy yields a criterion to assess the proton to electron mass ratio in Particle in cell simulations.

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 Filamentation (Weibel), Two-Stream, Buneman, Bell instabilities (and more) is thus provided, allowing for the exact determination of their hierarchy in terms of the system parameters. For relevance to both real situations and PIC simulations, the electron-to-proton mass ratio is treated as a parameter, and numerical calculations are conducted with two different values, namely 1/1836 and 1/100. In the system parameters phase space, the shape of the domains governed by each kind of instability is far from being trivial. For low density beams, the ultra-magnetized regime tends to be governed by either the Two-Stream or the Buneman instabilities. For beam densities equalling the plasma one, up to four kinds of modes are likely to play a role, depending of the beam Lorentz factor. In some regions of the system parameters phase space, the dominant mode may vary with the electron-to-proton mass ratio. Application is made to Solar Flares, Intergalactic Streams and Relativistic shocks physics., To Appear in ApJ

Ultra-cold plasmas obtained by ionization of atomic Rydberg states are qualified as classical and strongly coupled electron fluids. They are shown to share several common trends with ultra-cold electron flows used for ion-beam cooling. They exhibit specific stopping behaviour to charged particle beams, which may be used for diagnostic purposes. Ultra-cold plasmas are easily strongly magnetized. Then, one expects a strongly anisotropic behaviour of low ion velocity slowing down when the target electron cyclotron radius becomes smaller than the corresponding Debye length.

We evaluate the correlated stopping power of a swift chain of N charges aligned with its (non-relativistic) velocity. We find that the distance required between two charges for the chain to act as a separated set of isolated charges is much larger than the dynamical screening length because the wake effect is very important when the chain elements are situated close to each other.

Besides the possibility of hydrogen-driven vehicles, purely electrical vehicles, where electricity is stored in batteries, can equally be contemplated.

The instabilities triggered when two counter-streaming pair beams collide are analyzed. A guiding magnetic field is accounting for, while both beams are considered identical and cold. The instability analysis is conducted over the full \textbf{k}-spectrum, allowing to derive the hierarchy map of the dominant unstable modes, in terms of the initial beams energy $\gamma_0$ and a magnetic field strength parameter $\Omega_B$. Four different regions of the $(\Omega_B,\gamma_0)$ phase space are identified, each one governed by a different kind of mode. The analysis also unravels the existence of a "triple point", where 3 different modes grow exactly the same rate. A number of analytical expressions can be derived, either for the modes growth-rates, or for the frontiers between the 4 regions., Comment: To appear in Physics of Plasmas

Collisionless shocks, that is shocks mediated by electromagnetic processes, are customary in space physics and in astrophysics. They are to be found in a great variety of objects and environments: magnetospheric and heliospheric shocks, supernova remnants, pulsar winds and their nebul\ae, active galactic nuclei, gamma-ray bursts and clusters of galaxies shock waves. Collisionless shock microphysics enters at different stages of shock formation, shock dynamics and particle energization and/or acceleration. It turns out that the shock phenomenon is a multi-scale non-linear problem in time and space. It is complexified by the impact due to high-energy cosmic rays in astrophysical environments. This review adresses the physics of shock formation, shock dynamics and particle acceleration based on a close examination of available multi-wavelength or in-situ observations, analytical and numerical developments. A particular emphasize is made on the different instabilities triggered during the shock formation and in association with particle acceleration processes with regards to the properties of the background upstream medium. It appears that among the most important parameters the background magnetic field through the magnetization and its obliquity is the dominant one. The shock velocity that can reach relativistic speeds has also a strong impact over the development of the micro-instabilities and the fate of particle acceleration. Recent developments of laboratory shock experiments has started to bring some new insights in the physics of space plasma and astrophysical shock waves. A special section is dedicated to new laser plasma experiments probing shock physics, Comment: This is an author-created, un-copyedited version of an article accepted for publication/published in Reports on progress in physics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/0034-4885/79/4/046901

Collisionless shocks are shocks in which the mean-free path is much larger than the shock front. They are ubiquitous in astrophysics and the object of much current attention as they are known to be excellent particle accelerators that could be the key to the cosmic rays enigma. While the scenario leading to the formation of a fluid shock is well known, less is known about the formation of a collisionless shock. We present theoretical and numerical results on the formation of such shocks when two relativistic and symmetric plasma shells (pair or electron/proton) collide. As the two shells start to interpenetrate, the overlapping region turns Weibel unstable. A key concept is the one of trapping time τp, which is the time when the turbulence in the central region has grown enough to trap the incoming flow. For the pair case, this time is simply the saturation time of the Weibel instability. For the electron/proton case, the filaments resulting from the growth of the electronic and protonic Weibel instabilities, need to grow further for the trapping time to be reached. In either case, the shock formation time is 2τp in two-dimensional (2D), and 3τp in 3D. Our results are successfully checked by particle-in-cell simulations and may help designing experiments aiming at producing such shocks in the laboratory.

About two-thirds of the primary energy today is used directly for transportation or as heating fuels J.M. Ogden, Prospects for building a hydrogen energy infrastructure. Ann. Rev. Energy Environ. 24(1), 227–279 (1999).

An electrostatic instability driven by counter-propagating tenuous proton beams that traverse a bulk plasma consisting of electrons and protons is considered. The system is spatially homogeneous and is evolved in time with a one-dimensional particle-in-cell simulation, which allows for a good statistical plasma representation. Mildly and highly relativistic beam speeds are modeled. The proton beams with a speed of 0.9c result in waves that saturate by the trapping of electrons. The collapse of the phase space holes in the electron distribution scatters these to a flat-top momentum distribution. The final electric fields are weak and the proton beams are weakly modulated. No secondary instabilities are likely to form that could thermalize the proton beams. The proton beams moving with 0.99c initially heat the bulk plasma through a three-wave interaction. Coalescing phase space holes in the bulk proton distribution arising from the saturation of ion acoustic waves transport wave energy to low wavenumbers. Highly relativistic phase space holes form in the electron distribution, which are not spatially homogeneous. The spatial envelope of these electron phase space holes interacts with the fluctuations driven by the phase space holes in the bulk protons, triggering a modulational instability. A Langmuir wave condensate forms that gives rise to strong and long electrostatic wave packets, as well as to a substantial modulation of the proton beams. The final state of the system with the highly relativistic proton beams is thus more unstable to further secondary instabilities that may transfer a larger beam energy fraction to the electrons and thermalize the proton beams more rapidly.

The Weibel instability is prompted by a temperature anisotropy within a plasma. We investigate its growth rate for wave vectors making an arbitrary angle with the high temperature axis. We use a two temperatures waterbags model and derive stability conditions depending on both temperatures and wave vector orientation. It is found that the growth rate is maximum for wave vectors normal to the high temperature axis. Also, a critical angle is evidenced in the k space in which direction modes are unstable at high k although the growth rate decreases quickly in this direction. Exact results are derived in most cases. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A thorough analysis of the electromagnetic instabilities encountered in the beam plasma interaction physics shows that the most unstable modes are not the ones usually studied. We characterize these most unstable modes and determine the patterns they create. A fluid model able to reproduce the kinetic results in the small temperature regime is presented, as well as exact kinetic calculations in the high temperature limit.

The Weibel instability is prompted by a temperature anisotropy within a plasma. We present an analytical model based on a non-relativistic two temperatures waterbag distribution function. The model is exactly solvable in terms of the stability condition, the growth rate for any wave vector and temperature parameters, the maximum growth rate and the wave number for which it is reached.

In the fast ignition scenario for inertial fusion, a relativistic electron beam is supposed to travel from the side of the fusion pellet to its core. One one hand, a relativistic electron beam passing through a plasma is a highly unstable system. On the other hand, the pellet core is denser than its side by four orders of magnitude so that the beam makes its way through a important density gradient. We here investigate the effect of this gradient on the instabilities. It is found that they should develop so early that gradient effects are negligible in the linear phase.