Physical problems (microphysics) in relativistic plasma flows

Many problems in astrophysics involve relativistic outflows. The plasma dynamics in these scenarios is critical to determine the conditions for the self-consistent evolution of the fields and particle acceleration. Advances in computer power now allow for kinetic plasma simulations, based on the particle-in-cell (PIC) paradigm, capable of providing information about the role of plasma instabilities in relativistic outflows. A discussion of the key issues associated with PIC simulations is presented, along with some the most important results and open questions, with a particular emphasis on the long time evolution of the filamentation, or Weibel, instability, and on the possible collisionless mechanisms for particle acceleration arising in the collision of relativistic plasma shells. RELATIVISTIC OUTFLOWS AND GAMMA-RAY BURSTS Relativistic outflows are pervasive in astrophysics, from jets to gamma-ray bursts, from accretion disks to supernovae explosions. Some of the underlying physical mechanisms in these flows have strong connections with the dynamics and the physics of plasmas, and many outstanding problems in astrophysics are closely associated with scenarios where the collisionless dynamics of plasmas can play an important role. The obvious examples are the formation of relativistic shocks (and particle acceleration in these structures), magnetic field generation due to kinetic plasma instabilities, or non-thermal particle acceleration. The onset of these processes is associated with collisionless plasma instabilities. There is a wealth of theoretical work on these instabilities, going back to the early days of plasma physics, but only now, with the advent of massivelly parallel computing, it is possible to perform realistic detailed numerical simulations of these instabilities, in order to understand not only the linear, transient, stage of these instabilities but also the long time saturated behavior of the scenarios where such instabilities can occur. This is opening the way to establish connections between the plasma dynamics at the kinetic level and its consequences in different astrophysical phenomena. A clear example of the importance of plasma instabilities in relativistic ouflows is associated with the fireball model of gamma rays bursts (GRBs) (cf. R. Sari, this conference [1], and [2], and references therein). In the fireball model of GRBs [3], relativistic plasma shells collide/overtake each other, leading to the rapid variability of the observed radiation. The radiation is believed to be from synchrotron origin, which requires sub-equipartition magnetic fields to be generated in the collision of the plasma shells, and to survive for time scales much longer than the collisionless time scales. One possible mechanism that can explain the generation of magnetic fields at these levels in GRBs is the Weibel instability [4, 5]. Recently, numerical simulations have strengthened this conclusion [6, 7, 8, 9, 10]. In this particular problem, many questions remain to be fully addressed, whose answers are general enough to be of interest to other problems in astrophysics. What is the long time evolution of the magnetic fields generated via collisionless instabilities? What are the consequences of the field structure generated by collisionless instabilities to the radiation observed from these objects? How are particles accelerated in the fields resulting from the collision of relativistic plasma flows? Can the fields lead to the formation of relativistic shocks? How are particles accelerated in relativistic shocks? While some answers have already been proposed, kinetic simulations, regarded as a numerical laboratory for astrophysics, combined with relativistic kinetic theory, can lead to significant progress in solving some of these open questions. In this paper, I will review the basic concepts behind kinetic plasma simulations, and their most common paradigm, pointing out some of the advantages and some of the difficulties of this technique. Relevance will be given to the possibility to probe the microphysics on the time scale of the electron collective dynamics, and to obtatin detailed information about the structure of the fields and the distribution function of the particles. In Section III, the most recent developments on collisionless instabilities in unmagnetized plasmas will be described, illustrated with numerical simulations. The limits posed to particle-in-cell simulations of collisionless instabilities are discussed, and different strategies to overtake these limits are proposed. In Sections IV and V, the exploration of magnetic field generation and particle acceleration using lower dimensionality simulations is presented, employing alternative physical configurations. Generation of sub-equipartition magnetic fields is confirmed, along with the two step evolution of the filamentation instability (on the electron time scale, and on the ion time scale), in simulations running for times three orders of magnitude longer than the electron dynamics time scale. Particle acceleration is also observed in scenarios resulting from the collision of relativistic plasma shells, associated with (i) the filament coalescing process [6], (ii) acceleration in the electric field of the filaments generated in the Weibel instability [11], and (iii) acceleration in the relativistic electron plasma wave generated in the interface region between the two colliding shells [12]. Finally, in Section VI, I state the conclusions and I point out the open question to which PIC simulations might contribute in the near future. PARTICLE-IN-CELL SIMULATIONS Particle-in-cell (PIC) simulations are one of the most common numerical tools in plasma physics [13, 14]. Originated in the pioneering work of Oscar Buneman and John Dawson in the 1960s, it has evolved to a mature technique commonly used in many sub-fields of plasma physics. The idea behind particle-in-cell simulations is quite simple. The motion of a set of charged particles is followed under the action of the self-consistent fields generated by the charged particles themselves. Maxwell’s equations are solved on a grid, with the sources for the equations for the field advance (current and/or charge) determined by depositing the relevant quantities from the particles on the grid (Figure 1). FIGURE 1. General scheme of PIC simulations: the electric field and the magnetic field are calculated in two staggered grids, while charged particles move in all regions of space. After advancing the fields in time, the information to determine the fields on each particle’s position is available, the Lorentz force on the charged particles can be determined, the particles can be pushed, and their position and momentum updated to the new values. After the particle advance, the new quantities to advance the fields can be calculated, thus closing the loop (Figure 2). FIGURE 2. General scheme of a PIC loop [16]. To the extent that quantum mechanical effects can be neglected, these codes make no physics approximations and are ideally suited for studying complex systems with many degrees of freedom. The advent of massivelly parallel computing now allows for PIC simulations using more than 109 particles, in systems with (500)3 cells, using 0.5 TByte of RAM, with runtimes from a few hours to several weeks in computing systems with 100s of CPUs, producing data sets that can easily reach the TByte level. For some problems in plasma based accelerators [15], three-dimensional PIC one-to-one simulations with the exact experimental parameters are already performed, complementing the experimental diagnostics, and acting as virtual experiments. The information available from PIC simulations provides an outstanding tool to test new models and new ideas, but the complexity of developing, maintaining, and running these codes, and exploring the data generated in these simulations, requires research teams with diverse skills, and with dimension comparable to those running smallmedium scale experiments. The possibility to include new physics has also been explored, and it is now quite common to find massivelly parallel PIC codes, such as OSIRIS [17], that include impact and tunnel ionization, binary collisions, radiation damping, and that can provide, on post-processing, information about the radiation spectra [18]. Other authors are also attempting to generalize the PIC technique to non flat metrics (Watson et al [19], and [20]), in order to employ PIC codes to model conditions with strong gravitational fields, for instance in the vicinity of black holes. In the astrophysics of relativistic flows, only very recently 3D numerical PIC simulations have become more common [6, 7, 8, 9, 10], even though pioneering work in lower dimensional PIC simulations relevant for relativistic astrophysics was undertaken in the 80s (e.g. [21, 22]. However, and for many years, PIC simulations have been critical to understand the nonlinear evolution of many collisionless plasma processes. COLLISIONLESS PLASMA INSTABILITIES IN UNMAGNETIZED PLASMAS The general theory for collisionless plasma instabilities was developed in the 1960s by Watson, Bludman and Rosenbluth [23]. For a review of the different collisionless instabilities see [24]. In a unmagnetized plasma, and on the electron collective dynamics time scale, the key instability that can operate when different plasma streams collide is the electromagnetic beam-plasma instability. A detailed and global analysis of the electromagnetic beam-plasma instability was undertaken only very recently. Usually, the limiting scenarios described in the litterature deal only with coupling with the longitudinal electrostatic mode (two stream instability) i.e. the collective electrostatic mode (electron plasma wave) whose wave vector is parallel to the direction of propagation of the plasma stream providing the free energy to the instability, or coupling with the purely transverse electromagnetic mode (Weibel instability) [25], with wave vector transverse to the plasma stream propagation direction. For the two-stream instability, and in the most favorable conditions, the growth rate of the instability Γ scales with the ratio between the electron density in the two streams Γ ∝ (nb/n0), where nb is the electron density of the (lower density) stream, and n0 is the electron density of the background stationary plasma. The wave number for maximum growth is k‖ ≃ ωpe0/vb, where ωpe0 = ( 4πen0/me 1/2 is the electron plasma frequency and vb is the velocity of the plasma stream, with e the electron charge, and me the electron mass. The Weibel instability grows with Γ ∝ (nb/n0), with a typical wave number k⊥ ≃ ωpe0/c, for a warm plasma stream, where c is the velocity of light in vacuum. The typical length scales and time scales are determined solely from the electron density, and given by