Volume-preserving particle integrator based on exact flow of velocity for nonrelativistic particle-in-cell simulations

We construct a particle integrator for nonrelativistic particles by means of the splitting method based on the exact flow of the equation of motion of particles in the presence of constant electric and magnetic field. This integrator is volume-preserving similar to the standard Boris integrator and is suitable for long-term integrations in particle-in-cell simulations. Numerical tests reveal that it is significantly more accurate than previous volume-preserving integrators with second-order accuracy. For example, in the $E times B$ drift test, this integrator is more accurate than the Boris integrator and the integrator based on the exact solution of gyro motion by three and two orders of magnitude, respectively. In addition, we derive approximate integrators that incur low computational cost and high-precision integrators displaying fourth- to tenth-order accuracy with the aid of the composition method. These integrators are also volume-preserving. It is also demonstrated that the Boris integrator is equivalent to the simplest case of the approximate integrators derived in this study.

Electron Surfing and Drift Accelerations in a Weibel-dominated High-Mach-number Shock

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 the shock transition region. The ion-Weibel turbulence is the key to the efficient nonthermal electron acceleration.

Particle acceleration and wave excitation in quasi-parallel high-Mach-number collisionless shocks: Particle-in-cell simulation

We herein investigate shock formation and particle acceleration processes for both protons and electrons in a quasi-parallel high-Mach-number collisionless shock through a long-term, large-scale particle-in-cell simulation. We show that both protons and electrons are accelerated in the shock and that these accelerated particles generate large-amplitude Alfv{e}nic waves in the upstream region of the shock. After the upstream waves have grown sufficiently, the local structure of the collisionless shock becomes substantially similar to that of a quasi-perpendicular shock due to the large transverse magnetic field of the waves. A fraction of protons are accelerated in the shock with a power-law-like energy distribution. The rate of proton injection to the acceleration process is approximately constant, and in the injection process, the phase-trapping mechanism for the protons by the upstream waves can play an important role. The dominant acceleration process is a Fermi-like process through repeated shock crossings of the protons. This process is a `fast process in the sense that the time required for most of the accelerated protons to complete one cycle of the acceleration process is much shorter than the diffusion time. A fraction of the electrons is also accelerated by the same mechanism, and have a power-law-like energy distribution. However, the injection does not enter a steady state during the simulation, which may be related to the intermittent activity of the upstream waves. Upstream of the shock, a fraction of the electrons is pre-accelerated before reaching the shock, which may contribute to steady electron injection at a later time.

Energy Loss of High-Energy Particles in Particle-in-Cell Simulation

When a charged particle moves through a plasma at a speed much higher than the thermal velocity of the plasma, it is subjected to the force of the electrostatic field induced in the plasma by itself and loses its energy. This process is well-known as the stopping power of a plasma. In this paper we show that the same process works in particle-in-cell (PIC) simulations as well and the energy loss rate of fast particles due to this process is mainly determined by the number of plasma electrons contained in the electron skin depth volume. However, since there are generally very few particles in that volume in PIC simulations compared with real plasmas, the energy loss effect can be exaggerated significantly and can affect the results. Therefore, especially for the simulations that investigate the particle acceleration processes, the number of particles used in the simulations should be chosen large enough to avoid this artificial energy loss.

Nonrelativistic collisionless shocks in weakly magnetized electron--ion plasmas: two-dimensional particle-in-cell simulation of perpendicular shock

A two-dimensional particle-in-cell simulation is performed to investigate weakly magnetized perpendicular shocks with a magnetization parameter of 6 x 10^-5, which is equivalent to a high Alfven Mach number M_A of ~130. It is shown that current filaments form in the foot region of the shock due to the ion-beam--Weibel instability (or the ion filamentation instability) and that they generate a strong magnetic field there. In the downstream region, these current filaments also generate a tangled magnetic field that is typically 15 times stronger than the upstream magnetic field. The thermal energies of electrons and ions in the downstream region are not in equipartition and their temperature ratio is T_e / T_i ~ 0.3 - 0.4. Efficient electron acceleration was not observed in our simulation, although a fraction of the ions are accelerated slightly on reflection at the shock. The simulation results agree very well with the Rankine-Hugoniot relations. It is also shown that electrons and ions are heated in the foot region by the Buneman instability (for electrons) and the ion-acoustic instability (for both electrons and ions). However, the growth rate of the Buneman instability is significantly reduced due to the relatively high temperature of the reflected ions. For the same reason, ion-ion streaming instability does not grow in the foot region.

Electrostatic and electromagnetic instabilities associated with electrostatic shocks: two-dimensional particle-in-cell simulation

A two-dimensional electromagnetic particle-in-cell simulation with the realistic ion-to-electron mass ratio of 1836 is carried out to investigate the electrostatic collisionless shocks in relatively high-speed (~3000 km s^-1) plasma flows and also the influence of both electrostatic and electromagnetic instabilities, which can develop around the shocks, on the shock dynamics. It is shown that the electrostatic ion-ion instability can develop in front of the shocks, where the plasma is under counter-streaming condition, with highly oblique wave vectors as was shown previously. The electrostatic potential generated by the electrostatic ion-ion instability propagating obliquely to the shock surface becomes comparable with the shock potential and finally the shock structure is destroyed. It is also shown that in front of the shock the beam-Weibel instability gradually grows as well, consequently suggesting that the magnetic field generated by the beam-Weibel instability becomes important in long-term evolution of the shock and the Weibel-mediated shock forms long after the electrostatic shock vanished. It is also observed that the secondary electrostatic shock forms in the reflected ions in front of the primary electrostatic shock.

Non-relativistic Collisionless Shocks in Unmagnetized Electron-Ion Plasmas

We show that the Weibel-mediated collisionless shocks are driven at non-relativistic propagation speed (0.1c < V < 0.45c) in unmagnetized electron-ion plasmas by performing two-dimensional particle-in-cell simulations. It is shown that the profiles of the number density and the mean velocity in the vicinity of the shock transition region, which are normalized by the respective upstream values, are almost independent of the upstream bulk velocity, i.e., the shock velocity. In particular, the width of the shock transition region is ~100 ion inertial length independent of the shock velocity. For these shocks the energy density of the magnetic field generated by the Weibel-type instability within the shock transition region reaches typically 1-2% of the upstream bulk kinetic energy density. This mechanism probably explains the robust formation of collisionless shocks, for example, driven by young supernova remnants, with no assumption of external magnetic field in the universe.

Relativistic Collisionless Shocks in Unmagnetized Electron-Positron Plasmas

It is shown that collisionless shock waves can be driven in unmagnetized electron-positron plasmas by performing a two-dimensional particle-in-cell simulation. At the shock transition region, strong magnetic fields are generated by a Weibel-like instability. The generated magnetic fields are strong enough to deflect the incoming particles from upstream of the shock at a large angle and provide an effective dissipation mechanism for the shock. The structure of the collisionless shock propagates at an almost constant speed. There is no linear wave corresponding to the shock wave and therefore this can be regarded as a kind of ``instability-driven shock wave. The generated magnetic fields rapidly decay in the downstream region. It is also observed that a fraction of the thermalized particles in the downstream region return upstream through the shock transition region. These particles interact with the upstream incoming particles and cause the generation of charge-separated current filaments in the upstream of the shock as well as the electrostatic beam instability. As a result, electric and magnetic fields are generated even upstream of the shock transition region. No efficient acceleration processes of particles were observed in our simulation.

Saturation mechanism of the Weibel instability in weakly magnetized plasmas

The saturation mechanism of the Weibel instability is investigated theoretically by considering the evolution of currents in numerous cylindrical beams that are generated in the initial stage of the instability. Based on a physical model of the beams, it is shown that the magnetic field strength attains a maximum value when the currents in the beams evolve into the Alfven current and that there exist two saturation regimes. The theoretical prediction of the magnetic field strength at saturation is in good agreement with the results of two-dimensional particle-in-cell simulations for a wide range of initial anisotropy.

Probabilistic description of the first-order Fermi acceleration in shock waves: Time-dependent solution by single-particle approach

We give a new coherent description of the first-order Fermi acceleration of particles in shock waves from the point of view of stochastic process of the individual particles, under the test particle approximation. The time development of the particle distribution function can be dealt with by this description, especially for relativistic shocks. We formulate the acceleration process of a particle as a two-dimensional Markov process in a logarithmic momentum-time space, and relate the solution of the Markov process with the particle distribution function at the shock front, for both steady and time-dependent case. For the case where the probability density function of the energy gain and cycle-time at each shock crossing of the particles obeys a scaling law in momentum, which is usually assumed in the literature, it is confirmed in more general form that the energy distribution of particles has the power-law feature in steady state. The equation to determine the exact power-law index which is applicable for any shock speed is derived and it is shown that the power-law index, in general, depends on the shape of the probability density function of the energy gain at each shock crossing; in particular for relativistic shocks, the dispersion of the energy gain can influence the power-law index. It is also shown that the time-dependent solution has a self-similarity for the same case.