based on talks given at KIAA "Cosmic ray" workshop, at SF2A2011 and at WISAP2011; 16 pages, 1 figure. To be published in the proceedings of the WISAP 2011 Eilat meeting, eds M. Mond and P.-L. Sulem
Relativistic sources, e.g. gamma-ray bursts, pulsar wind nebulae and powerful active galactic nuclei produce relativistic outflows that lead to the formation of collisionless shock waves, where particle acceleration is thought to take place. Our understanding of relativistic shock acceleration has improved in the past decade, thanks to the combination of analytical studies and high level numerical simulations. In ultra-relativistic shocks, particle acceleration is made difficult by the generically transverse magnetic field and large advection speed of the shocked plasma. Fast growing microturbulence is thus needed to make the Fermi process operative. It is thought, and numerical simulations support that view, that the penetration of supra-thermal particles in the shock precursor generates a magnetic turbulence which in turn produces the scattering process needed for particle acceleration through the Fermi mechanism. Through the comparison of the growth timescale of the microinstabilities in the shock precursor and the precursor crossing timescale, it is possible to delimit in terms of magnetization and shock Lorentz factor the region in which micro-turbulence may be excited, hence whether and how Fermi acceleration is triggered. These findings are summarized here and astrophysical consequences are drawn.
This paper summarizes recent progresses in our theoretical understanding of particle acceleration at relativistic shock waves and it discusses two salient consequences: (1) the maximal energy of accelerated particles; (2) the impact of the shock-generated micro-turbulence on the multi-wavelength light curves of gamma-ray burst afterglows.
The non-thermal acceleration of electrons and ions at an oblique, non-relativistic shock is studied using large scale particle-in-cell (PIC) simulations in one spatial dimension. Physical parameters are selected to highlight the role of electron pre-heating in injection and subsequent acceleration of particles at high Mach number shocks. Simulation results show evidence for the early onset of the diffusive shock acceleration process for both electrons and ions at a highly oblique subluminal shock. Ion acceleration efficiencies of $\lesssim 5\%$ are measured at late times, though this is not th…
The non-thermal particle spectra responsible for the emission from many astrophysical systems are thought to originate from shocks via a first order Fermi process otherwise known as diffusive shock acceleration. The same mechanism is also widely believed to be responsible for the production of high energy cosmic rays. With the growing interest in collisionless shock physics in laser produced plasmas, the possibility of reproducing and detecting shock acceleration in controlled laboratory experiments should be considered. The various experimental constraints that must be satisfied are reviewed.…
We reexamine nonlinear diffusive shock acceleration (DSA) at cosmological shocks in the large scale structure of the Universe, incorporating wave-particle interactions that are expected to operate in collisionless shocks. Adopting simple phenomenological models for magnetic field amplification (MFA) by cosmic-ray (CR) streaming instabilities and Alfv'enic drift, we perform kinetic DSA simulations for a wide range of sonic and Alfv'enic Mach numbers and evaluate the CR injection fraction and acceleration efficiency. In our DSA model the CR acceleration efficiency is determined mainly by the son…
Observations of large-scale radio emissions prove the existence of shock accelerated cosmic-ray electrons in galaxy clusters, while the lack of detected $γ$-rays limits the acceleration of cosmic-ray protons in galaxy clusters. This challenges our understanding of how diffusive shock acceleration works. In this work, we couple the most updated recipes for shock acceleration in the intracluster medium to state-of-the-art magneto-hydrodynamical simulations of massive galaxy clusters. Furthermore, we use passive tracer particles to follow the evolution of accelerated cosmic-rays. We show that whe…
We investigate shock acceleration in a realistic astrophysical environment with density inhomogeneities. The turbulence induced by the interaction of the shock precursor with upstream density fluctuations amplifies both upstream and downstream magnetic fields via the turbulent dynamo. The dynamo-amplified turbulent magnetic fields (a) introduce variations of shock obliquities along the shock face, (b) enable energy gain through a combination of shock drift and diffusive processes, (c) give rise to various spectral indices of accelerated particles, (d) regulate the diffusion of particles both p…
Ab-initio numerical study of collisionless shocks in electron-ion unmagnetized plasmas is performed with fully relativistic particle in cell simulations. The main properties of the shock are shown, focusing on the implications for particle acceleration. Results from previous works with a distinct numerical framework are recovered, including the shock structure and the overall acceleration features. Particle tracking is then used to analyze in detail the particle dynamics and the acceleration process. We observe an energy growth in time that can be reproduced by a Fermi-like mechanism with a re…
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…
In this Letter, we present a new self-consistent theory for the production of the relativistic outflows observed from radio-loud black hole candidates and active galaxies as a result of particle acceleration in hot, viscous accretion disks containing standing, centrifugally-supported isothermal shocks. This is the first work to obtain the structure of such disks for a relatively large value of the Shakura-Sunyaev viscosity parameter ($α=0.1$), and to consider the implications of the shock for the acceleration of relativistic particles in viscous disks. In our approach, the hydrodynamics and th…
Shock waves propagating in collisionless heliospheric and astrophysical plasmas have been studied extensively over the decades. One prime motivation is to understand the nonthermal particle acceleration at shocks. Although the theory of diffusive shock acceleration (DSA) has long been the standard for cosmic-ray acceleration at shocks, plasma physical understanding of particle acceleration remains elusive. In this review, we discuss nonthermal electron acceleration mechanisms at quasi-perpendicular shocks, for which substantial progress has been made in recent years. The discussion presented i…
We perform particle-in-cell simulations of perpendicular nonrelativistic collisionless shocks to study electron heating and pre-acceleration for parameters that permit extrapolation to the conditions at young supernova remnants. Our high-resolution large-scale numerical experiments sample a representative portion of the shock surface and demonstrate that the efficiency of electron injection is strongly modulated with the phase of the shock reformation. For plasmas with low and moderate temperature (plasma beta $β_{\rm p}=5\cdot 10^{-4}$ and $β_{\rm p}=0.5$), we explore the nonlinear shock stru…
We explore electron pre-acceleration at high Mach-number nonrelativistic perpendicular shocks at, e.g., young supernova remnants, which are a prerequisite of further acceleration to very high energies via diffusive shock acceleration. Using fully kinetic particle-in-cell simulations of shocks and electron dynamics in them, we investigate the influence of shock-surfing acceleration at the shock foot on the nonthermal population of electrons downstream of the shock. The shock-surfing acceleration is followed by further energization at the shock ramp where the Weibel instability spawns a type of …
A shock wave propagating perpendicularly to an ambient magnetic field accelerates particles considerably faster than in the parallel propagation regime. However, the perpendicular acceleration stops after the shock overruns a circular particle orbit. At the same time, it may continue in flows resulting from supersonically colliding plasmas bound by a pair of perpendicular shocks. Although the double-shock acceleration mechanism, which we consider in detail, is not advantageous for thermal particles, pre-energized particles may avoid the premature end of acceleration. We argue that if their gyr…
Particle acceleration and heating at mildly relativistic magnetized shocks in electron-ion plasma are investigated with unprecedentedly high-resolution two-dimensional particle-in-cell simulations that include ion-scale shock rippling. Electrons are super-adiabatically heated at the shock, and most of the energy transfer from protons to electrons takes place at or downstream of the shock. We are the first to demonstrate that shock rippling is crucial for the energization of electrons at the shock. They remain well below equipartition with the protons. The downstream electron spectra are approx…
Delayed radio flares of optical tidal disruption events (TDEs) indicate the existence of non-relativistic outflows accompanying TDEs. The interaction of TDE outflows with the surrounding circumnuclear medium creates quasi-perpendicular shocks in the presence of toroidal magnetic fields. Because of the large shock obliquity and large outflow velocity, we find that the shock acceleration induced by TDE outflows generally leads to a steep particle energy spectrum, with the power-law index significantly larger than the "universal" index for a parallel shock. The measured synchrotron spectral indic…
Using test particle simulations we study electron acceleration at collisionless shocks with a two-component model turbulent magnetic field with slab component including dissipation range. We investigate the importance of shock normal angle $θ_{Bn}$, magnetic turbulence level $\left(b/B_0\right)^2$, and shock thickness on the acceleration efficiency of electrons. It is shown that at perpendicular shocks the electron acceleration efficiency is enhanced with the decreasing of $\left(b/B_0\right)^2$, and at $\left(b/B_0\right)^2=0.01$ the acceleration becomes significant due to strong drift electr…
Supernova remnants (SNRs) are believed to produce the majority of galactic cosmic rays (CRs). SNRs harbor non-relativistic collisionless shocks responsible for acceleration of CRs via diffusive shock acceleration (DSA), in which particles gain their energies via repeated interactions with the shock front. As the DSA theory involves pre-existing mildly energetic particles, a means of pre-acceleration is required, especially for electrons. Electron injection remains one of the most troublesome and still unresolved issues and our physical understanding of it is essential to fully comprehend the p…
Electrons can be accelerated to ultrarelativistic energies at strong (high-Mach number) collisionless shock waves that form when stellar debris rapidly expands after a supernova. Collisionless shock waves also form in the flow of particles from the Sun (the solar wind), and extensive spacecraft observations have established that electron acceleration at these shocks is effectively absent whenever the upstream magnetic field is roughly parallel to the shock surface normal (quasi-parallel conditions). However, it is unclear whether this magnetic dependence of electron acceleration also applies t…
We investigate shock structure and particle acceleration in relativistic magnetized collisionless pair shocks by means of 2.5D and 3D particle-in-cell simulations. We explore a range of inclination angles between the pre-shock magnetic field and the shock normal. We find that only magnetic inclinations corresponding to "subluminal" shocks, where relativistic particles following the magnetic field can escape ahead of the shock, lead to particle acceleration. The downstream spectrum in such shocks consists of a relativistic Maxwellian and a high-energy power-law tail with exponential cutoff. For…
How electrons get accelerated to relativistic energies in a high-Mach-number quasi-perpendicular shock is presented by means of ab initio particle-in-cell simulations in three dimensions. We found that coherent electrostatic Buneman waves and ion-Weibel magnetic turbulence coexist in a strong-shock structure whereby particles gain energy during shock-surfing and subsequent stochastic drift accelerations. Energetic electrons that initially experienced the surfing acceleration undergo pitch-angle diffusion by interacting with magnetic turbulence and continuous acceleration during confinement in …
Radio relics could be generated by multiple shocks induced in the turbulent intracluster medium during galaxy mergers. Kang (2021) demonstrated that the re-acceleration of cosmic ray (CR) protons via diffusive shock acceleration (DSA) by multiple shocks could enhance the acceleration efficiency and flatten the CR spectrum, compared to a single episode of DSA. Here we examine the CR electron acceleration through multiple re-acceleration by considering energy losses and decompression of the particle distribution and magnetic fields in the postshock region between consecutive shock passages. We f…
The recent discoveries in the theory of diffusive shock acceleration (DSA) that stem from first-principle kinetic plasma simulations are discussed. When ion acceleration is efficient, the back-reaction of non-thermal particles and self-generated magnetic fields becomes prominent and leads to both enhanced shock compression and particle spectra significantly softer than those predicted by the standard test-particle DSA theory. These results are discussed in the context of the non-thermal phenomenology of astrophysical shocks, with a special focus on the remnant of SN1006.
We propose the particle acceleration model coupled with multiple plasmoid ejections in a solar flare. Unsteady reconnection produces plasmoids in a current sheet and ejects them out to the fast shocks, where particles in a plasmoid are reflected upstream the shock front by magnetic mirror effect. As the plasmoid passes through the shock front, the reflection distance becomes shorter and shorter driving Fermi acceleration, until it becomes proton Larmor radius. The fractal distribution of plasmoids may also have a role in naturally explaining the power-law spectrum in nonthermal emissions.
Electron acceleration to non-thermal energies in low Mach number (M<5) shocks is revealed by radio and X-ray observations of galaxy clusters and solar flares, but the electron acceleration mechanism remains poorly understood. Diffusive shock acceleration, also known as first-order Fermi acceleration, cannot be directly invoked to explain the acceleration of electrons. Rather, an additional mechanism is required to pre-accelerate the electrons from thermal to supra-thermal energies, so they can then participate in the Fermi process. In this work, we use two- and three-dimensional particle-in-ce…
The first-order Fermi acceleration of electrons requires an injection of electrons into a mildly relativistic energy range. However, the mechanism of injection has remained a puzzle both in theory and observation. We present direct evidence for a novel stochastic shock drift acceleration theory for the injection obtained with Magnetospheric Multiscale (MMS) observations at Earth's bow shock. The theoretical model can explain electron acceleration to mildly relativistic energies at high-speed astrophysical shocks, which may provide a solution to the long-standing issue of electron injection.
We investigate effects of upstream density fluctuations on the diffusive shock acceleration by Monte Carlo simulations. The simulations show that particles are reaccelerated in the shock downstream region by a sound wave generated at the shock front. The time scale of turbulent acceleration by the sound wave is estimated. We propose a new back reaction of particles accelerated around the shock front. The accelerated particles generate the upstream density fluctuations by the Drury instability, which are converted to the downstream sound waves by the shock. The downstream sound waves modify the…
The intracluster medium (ICM) is expected to experience on average about three passages of weak shocks with low sonic Mach numbers, $M\lesssim 3$, during the formation of galaxy clusters. Both protons and electrons could be accelerated to become high energy cosmic rays (CRs) at such ICM shocks via diffusive shock acceleration (DSA). We examine the effects of DSA by multiple shocks on the spectrum of accelerated CRs by including {\it in situ} injection/acceleration at each shock, followed by repeated re-acceleration at successive shocks in the test-particle regime. For simplicity, the accelerat…
Using an eigenfunction expansion to solve the transport equation, complemented by Monte-Carlo simulations, we show that ultrarelativistic shocks can be effective particle accelerators even when they fail to produce large amplitude turbulence in the downstream plasma. This finding contradicts the widely held belief that a uniform downstream magnetic field perpendicular to the shock normal inhibits acceleration by the first order Fermi process. In the ultrarelativistic limit, we find a stationary power-law particle spectrum of index s=4.17 for these shocks, close to that predicted for a strictly…
Electron acceleration to non-thermal energies is known to occur in low Mach number (M<5) shocks in galaxy clusters and solar flares, but the electron acceleration mechanism remains poorly understood. Using two-dimensional (2D) particle-in-cell (PIC) plasma simulations, we showed in Paper I that electrons are efficiently accelerated in low Mach number (M=3) quasi-perpendicular shocks via a Fermi-like process. The electrons bounce between the upstream region and the shock front, with each reflection at the shock resulting in energy gain via shock drift acceleration. The upstream scattering is pr…
The topic of this review paper is on the influence of solar wind turbulence on shock propagation and its consequence on the acceleration and transport of energetic particles at shocks. As the interplanetary shocks sweep through the turbulent solar wind, the shock surfaces fluctuate and ripple in a range of different scales. We discuss particle acceleration at rippled shocks in the presence of ambient solar-wind turbulence. This strongly affects particle acceleration and transport of energetic particles (both ions and electrons) at shock fronts. In particular, we point out that the effects of u…
We present a three-dimensional hybrid simulation of a collisionless perpendicular shock in a partially ionized plasma for the first time. In this simulation, the shock velocity and the upstream ionization fraction are Vsh ~ 1333 km/s and fi ~ 0.5, that are typical values for isolated young supernova remnants in the interstellar medium. We confirm previous two-dimensional simulation results that downstream hydrogen atoms leak into the upstream region, they are accelerated by the pickup process in the upstream region, and large magnetic field fluctuations are generated both in the upstream and d…
Relativistic particle acceleration in collisionless shocks of supernova remnants is accompanied by magnetic field amplification from cosmic ray (CR) driven plasma instabilities. Bell's fast CR-current instability is predicted to produce turbulence with a non-zero mean electric field in the shock precursor. We present a Monte Carlo model of Fermi shock acceleration explicitly taking into account an effective mean upstream electric field. Our model is nonlinear and includes the backreaction effects of efficient Fermi acceleration on the shock structure.
We show that the diffusion approximation breaks down for particle acceleration at oblique shocks with velocities typical of young supernova remnants. Higher order anisotropies flatten the spectral index at quasi-parallel shocks and steepen the spectral index at quasi-perpendicular shocks. We compare the theory with observed spectral indices.
Using large-scale fully-kinetic two-dimensional particle-in-cell simulations, we investigate the effects of shock rippling on electron acceleration at low-Mach-number shocks propagating in high-$β$ plasmas, in application to merger shocks in galaxy clusters. We find that the electron acceleration rate increases considerably when the rippling modes appear. The main acceleration mechanism is stochastic shock-drift acceleration, in which electrons are confined at the shock by pitch-angle scattering off turbulence and gain energy from the motional electric field. The presence of multi-scale magnet…
An extreme case of electron shock drift acceleration in low Mach number collisionless shocks is investigated as a plausible mechanism of initial acceleration of relativistic electrons in large-scale shocks in galaxy clusters where upstream plasma temperature is of the order of 10 keV and a degree of magnetization is not too small. One-dimensional electromagnetic full particle simulations reveal that, even though a shock is rather moderate, a part of thermal incoming electrons are accelerated and reflected through relativistic shock drift acceleration and form a local nonthermal population just…
We examine limits to the energy to which cosmic rays can be accelerated by relativistic shocks, showing that acceleration of light ions as high as 100 EeV is unlikely. The implication of our estimates is that if ultra-high energy cosmic rays are accelerated by shocks, then those shocks are probably not relativistic.
We present the results of large hybrid (kinetic ions - fluid electrons) simulations of particle acceleration at non-relativistic collisionless shocks. Ion acceleration efficiency and magnetic field amplification are investigated in detail as a function of shock inclination and strength, and compared with predictions of diffusive shock acceleration theory, for shocks with Mach number up to 100. Moreover, we discuss the relative importance of resonant and Bell's instability in the shock precursor, and show that diffusion in the self-generated turbulence can be effectively parametrized as Bohm di…
We show that the energy required to turbulently amplify magnetic field during cosmic ray (CR) acceleration by shocks extracts energy from the CR and steepens the CR energy spectrum.
We have performed 2D hybrid simulations of non-relativistic collisionless shocks in the presence of pre-existing energetic particles ("seeds"); such a study applies, for instance, to the re-acceleration of Galactic cosmic rays (CRs) in supernova remnant (SNR) shocks and solar wind energetic particles in heliospheric shocks. Energetic particles can be effectively reflected and accelerated regardless of shock inclination via a process that we call \emph{diffusive shock re-acceleration}. We find that reaccelerated seeds can drive the streaming instability in the shock upstream and produce effecti…
Collisionless shocks with low sonic Mach numbers, $M_{\rm s} \lesssim 4$, are expected to accelerate cosmic ray (CR) protons via diffusive shock acceleration (DSA) in the intracluster medium (ICM). However, observational evidence for CR protons in the ICM has yet to be established. Performing particle-in-cell simulations, we study the injection of protons into DSA and the early development of a nonthermal particle population in weak shocks in high $β$ ($\approx 100$) plasmas. Reflection of incident protons, self-excitation of plasma waves via CR-driven instabilities, and multiple cycles of sho…
We propose a novel electron acceleration mechanism, which we call stochastic shock drift acceleration (SSDA), that extends the standard shock drift acceleration (SDA) for low-energy electrons at a quasi-perpendicular shock to include the effect of stochastic pitch-angle scattering. We demonstrate that the steady-state energy spectrum of electrons accelerated within the shock transition region becomes a power-law in the limit of strong scattering. The spectral index is independent of the pitch-angle scattering coefficient. On the other hand, the maximum energy attainable through the mechanism s…
We study diffusive shock acceleration (DSA) of electrons in non-relativistic quasi-perpendicular shocks using self-consistent one-dimensional particle-in-cell (PIC) simulations. By exploring the parameter space of sonic and Alfvénic Mach numbers we find that high Mach number quasi-perpendicular shocks can efficiently accelerate electrons to power-law downstream spectra with slopes consistent with DSA prediction. Electrons are reflected by magnetic mirroring at the shock and drive non-resonant waves in the upstream. Reflected electrons are trapped between the shock front and upstream waves and …
Perpendicular shocks are shown to be rapid particle accelerators that perform optimally when the ratio $u_{\rm s}$ of the shock speed to the particle speed roughly equals the ratio $1/η$ of the scattering rate to the gyro frequency. We use analytical methods and Monte-Carlo simulations to solve the kinetic equation that governs the anisotropy generated at these shocks, and find, for $ηu_{\rm s}\approx1$, that the spectral index softens by unity and the acceleration time increases by a factor of two compared to the standard result of diffusive shock acceleration theory. These results provide a …
We use 2D and 3D hybrid (kinetic ions - fluid electrons) simulations to investigate particle acceleration and magnetic field amplification at non-relativistic astrophysical shocks. We show that diffusive shock acceleration operates for quasi-parallel configurations (i.e., when the background magnetic field is almost aligned with the shock normal) and, for large sonic and Alfvénic Mach numbers, produces universal power-law spectra proportional to p^(-4), where p is the particle momentum. The maximum energy of accelerated ions increases with time, and it is only limited by finite box size and ru…
Thermal electrons cannot directly participate in the process of diffusive acceleration at electron-ion shocks because their Larmor radii are smaller than the shock transition width: this is the well-known electron injection problem of diffusive shock acceleration. Instead, an efficient pre-acceleration process must exist that scatters electrons off of electromagnetic fluctuations on scales much shorter than the ion gyro radius. The recently found intermediate-scale instability provides a natural way to produce such fluctuations in parallel shocks. The instability drives comoving (with the upst…
A new numerical model of the nonlinear diffusive shock acceleration is presented. It is used for modeling of particle acceleration in supernova remnants. The model contains coupled spherically symmetric hydrodynamic equations and the transport equations for energetic protons, ions and electrons. The forward and reverse shocks are included in the consideration. The spectra of cosmic rays released into interstellar medium from a supernova remnant are determined. The role of the reverse shock in the production of CR ions and positrons is discussed.
Supernova Remnants (SNRs) shocks are believed to accelerate charged particles and to generate strong turbulence in the post-shock flow. From high-energy observations in the past decade, a magnetic field at SNR shocks largely exceeding the shock-compressed interstellar field has been inferred. We outline how such a field amplification results from a small-scale dynamo process downstream of the shock, providing an explicit expression for the turbulence back-reaction to the fluid whirling. The spatial scale of the $X-$ray rims and the short time-variability can be obtained by using reasonable par…
When two galaxy clusters encounter each other, the interaction results in a collisionless shock that is characterized by a low (1-4) sonic Mach number, and a high Alfv{é}nic Mach number. Our goal is to determine if, and to what extent, such shocks can accelerate particles to sufficient velocities that they can contribute to the cosmic ray spectrum. We combine two different computational methods, magnetohydrodynamics (MHD) and particle-in-cell (PIC) into a single code that allows us to take advantage of the high computational efficiency of MHD while maintaining the ability to model the behaviou…
This review presents the fundamentals of the particle acceleration processes active in interstellar medium (ISM), which are essentially based on the so-called Fermi mechanism theory. More specifically, the review presents here in more details the first order Fermi acceleration process -- also known as diffusive shock acceleration (DSA) mechanism. In this case, acceleration is induced by the interstellar (IS) shock waves. These IS shocks are mainly associated with emission nebulae (HII regions, planetary nebulae and supernova remnants). Among all types of emission nebulae, the strongest shocks …
The early acceleration of protons and electrons in the nonrelativistic collisionless shocks with three obliquities are investigated through 1D particle-in-cell simulations. In the simulations, the charged particles possessing a velocity of $0.2\, c$ flow towards a reflecting boundary, and the shocks with a sonic Mach number of $13.4$ and a Alfven Mach number of $16.5$ in the downstream shock frame are generated. In these quasi-parallel shocks with the obliquity angles $θ= 15^\circ$, $30^\circ$, and $45^\circ$, some of the protons and the electrons can be injected into the acceleration processe…
Recent observations in the quasi-parallel bow shock by the MMS spacecraft show rapid heating and acceleration of ions up to an energy of about 100 keV. It is demonstrated that a prominent acceleration mechanism is the nonlinear interaction with a spectrum of waves produced by gradient driven instabilities, including the lower hybrid drift (LHD) instability, modified two-stream (MTS) instability and electron cyclotron drift (ECD) instability. Test-particle simulations show that the observed spectrum of waves can rapidly accelerate protons up to a few hundreds keV by the ExB mechanism. The ExB w…
We study the physics of electron acceleration at collisionless shocks that move through a plasma containing large-scale magnetic fluctuations. We numerically integrate the trajectories of a large number of electrons, which are treated as test particles moving in the time dependent electric and magnetic fields determined from 2-D hybrid simulations (kinetic ions, fluid electron). The large-scale magnetic fluctuations effect the electrons in a number of ways and lead to efficient and rapid energization at the shock front. Since the electrons mainly follow along magnetic lines of force, the large…
