Shock formation in magnetised electron–positron plasmas: mechanism and timing

The shock formation process in electron–positron pair plasmas is investigated in the presence of an ambient perpendicular magnetic field. In initially unmagnetised plasmas, which are dominated by the Weibel or filamentation instability, the shock formation time is a multiple of the saturation time of the linear instability. While in weakly magnetised plasmas the mechanism is still the same, higher magnetisations induce synchrotron maser modes such that the shock formation is dominated by magnetic reflection. As a consequence the formation times are reduced. The focus is on the detailed picture of the particle kinetics, in which the transition between Weibel and magneto-hydrodynamic shocks can be clearly identified.


Introduction
Collisionless shocks can provide efficient particle acceleration and are thus important in the context of cosmic rays. The level of non-thermal acceleration is a critical parameter, thus non-relativistic shocks are better particle accelerators than relativistic shocks (Sironi et al 2013). Moreover, the level of non-thermal acceleration depends on the strength of the ambient, initial magnetisation, as well as on the angle of the field with respect to the particle flow. The main difference regarding particle acceleration between non-relativistic and relativistic shocks is that the latter become poor accelerators in the presence of too strong and too oblique magnetic fields. The reason for this is that in such a situation, particles flowing along the field lines cannot go fast enough to escape upstream, be scattered back to the shock front, and close the Fermi acceleration cycles (see Sironi et al 2015 and references therein). In contrast, non-relativistic shock can easily have particles outrun their front (Guo et al 2014, Park et al 2015. Shocks mediated by the Weibel or filamentation instability (Fried 1959, Weibel 1959) are common in astrophysics and are strongest in initially unmagnetised plasmas with a large anisotropy in the momentum distribution, given by a directed bulk flow or temperature anisotropy. Electromagnetic modes are seeded from noise level and amplified to a large-scale turbulent magnetic field structure. The charged plasma particles are scattered in the magnetic turbulence, which finally leads to an isotropisation of the particle flow.
The shock formation process in initially unmagnetised plasmas has been studied in detail (Bret et al 2013 and it was found that the steady-state formation is associated with the saturation time of the magnetic instability in the plasma, t s . The time to form a steady-state shock in an electron-positron pair plasma is given by where d is the dimension parameter, which is d=2 in 2D and d=3 in a real 3D setup. In electron-ion plasmas, the shock formation time is enhanced due to an extra merging time of the filaments (Stockem Novo et al 2015).
When we speak of a steady-state shock, we refer to the compression ratio, which has saturated to a constant value , where e is the electron charge, m e the mass of an electron and n the electron density. Long-term studies of shock formation in magnetised plasmas show that the dominant mediating process changes to magnetic reflection at high magnetisation (Sironi et al 2013). How the ambient magnetic field ¹ B 0 0 actually influences the particle kinetics and the shock formation time is not clear yet.
In a series of 1D simulations, highly magnetised shocks have been investigated by Gallant et al (1992), which we will call magnetised shocks in the following. Synchrotron maser modes lead to a coherent reflection of the particles in the shock front. The thermalisation in the downstream was found to be faster than the gyration time scale.
The manuscript is organised as follows. The theory of filamentation modes in a plasma with perpendicular magnetic field and the jump conditions in magnetised shocks are summarised in section 2. Particle-in-cell (PIC) simulations are presented and discussed in section 3. The findings are summarised in section 4.
2. Theory of shock formation in magnetised plasmas 2.1. Shock formation in Weibel shocks 2.1.1. The filamentation instability in magnetised plasmas Consider two counter-streaming pair beams of identical density and Lorentz factor g b = -1 1 0 2 , with b = v c 0 . Both beams are cold, and embedded in a magnetic fieldB v 0 0 . In order to investigate the Weibel instability, we look at the growth of perturbations withk v 0 . As evidenced on figure 1, in 3D, the perpendicular magnetic field breaks the axial symmetry, and one has to investigate every possible orientation of the wave-vector in the ( ) x x , 2 3 plane. Since both beams are cold, we can analyse the system through a four-fluids cold model. Electrons and positrons moving to the left constitute for two components and those moving to the right constitute for the other two. Initially, the system is charge and current neutral since both beams are individually so. Similar calculations have been performed for counter-streaming electron beams with a flow-aligned field (Godfrey et al 1975), or even an oblique one (Bret et al 2006, Bret and Dieckmann 2008, Bret 2014). In the fieldfree case, the present system is equivalent to counter-streaming electron beams, because the linear response varies with q 2 . An external magnetic field introduces cyclotron frequencies depending on the sign of the charges. Therefore, we expect to recover previous results in the limit = B 0 0 . The derivation of the dispersion equation is standard. We start writing the matter and momentum conservation equations , stand for the densities and velocities of the 4 species involved. All equilibrium quantities are then perturbed by a small amount, proportional to Figure 1. Sketch of the system considered. In 3D, the perpendicular magnetic field breaks the axial symmetry, and one has to investigate every possible orientations of the wave-vector.
Once linearised, equation (2) allow to express the first order density perturbation n i1 in terms of v i1 . Inserting the latter into the linearised equation (3) gives an expression for v i1 , hence of the first order current J 1 , in terms of E 1 . The dispersion equation is finally obtained from a combination of Maxwell-Ampère's and Faraday's equations 0, which is the final dispersion equation. This tensor has been computed analytically using the Mathematica Notebook described in Bret (2007). The tensor elements are reported in appendix, in terms of the dimensionless variables The dispersion equation for the Weibel modes stems from , 2 3 plane, the numerical study displayed in figure 2 shows that as is the case for a flow-aligned field (Godfrey et al 1975, Bret 2014, the growth-rate saturates at large = | | Z Z . The asymptotic value of the growth-rate can be derived in the following way. This dispersion equation is a polynomial in Z that we shall write which can be solved exactly, giving the growth-rate In order to find out if the instability can be suppressed completely, we have to address the growth-rate. Since it is always smaller than d ¥ , cancelling the Weibel instability means cancelling d ¥ . This in turn, implies having b g q W = cos 0 , that is, no magnetic field. In such a case, D < 0 and d b g = ¥ 2 0 which is the field-free result (Bret et al 2006). We therefore find that in the non-trivial circumstances where b W ¹ , 0 B , the instability can only be cancelled if D > 0, and only for two discrete orientations of the wave-vector, q p =  2 k . As a consequence, such a perpendicular field cannot cancel the Weibel instability, since all modes withk v 0 will be excited during the interaction. Let us now turn to the largest growth-rate in terms of q k . We see from equation (7) that it is reached for (7) gives ,max 2 0 which is the growth-rate in the absence of magnetic field. Among all the Weibel modes which grow, those with q p = [ ] 0 k grow as if there were no magnetic field. As a consequence, the whole system is eventually as unstable as the one with no field, consistently with the forthcoming PIC simulations.
This stands in stark contrast with the case of a flow-aligned field, where the Weibel instability can be completely suppressed (Godfrey et al 1975, Bret et al 2006. A similar effect has been found for the case of two counter-streaming electron beams (Bret 2014). It arises from the fact that the Weibel instability has the particles move essentially sideways. In the presence of a perpendicular magnetic field, such motions may simply be found parallel to the field, cancelling the Lorentz force, and the field effect with it.

The saturation phase in magnetised plasmas
The filamentation instability grows exponentially only in the early stage and enters a saturation phase. At that time, the magnetic field has been amplified from its fluctuation value, which is discussed in (Bret et al 2013), up to equipartition with the gyro-kinetic motion of the electrons and positrons. Thus, the saturation time of the filamentation instability is given by where Π is the number of e-foldings of the instability and dwpe 1 its growth rate. By this time, the field has grown to nearly equipartition with the saturation magnetic field (Medvedev and Loeb 1999, Silva et  for shock formation. The downstream density is still only twice the upstream density at this time, thus, the density compression of 3 according to the Rankine-Hugoniot jump conditions for the 2D case needs another t s since the overlapping region no longer expands . In the 3D case, the expected density jump is »4 so the time t 2 s is required to bring enough material in the central region. The shock formation time in Weibel where d is the dimensionality of the system ( = ( ) d 2 3 for a 2D (3D) setup). Formally, equation (14) is the same as for unmagnetised plasmas with a modified growth rate δ in the case of magnetised plasma.

Magnetised shocks
The analysis of strongly magnetised shocks showed that they are parametrised purely by the upstream magnetisation  where a perpendicular magnetic field and a cold upstream were assumed. In the limit of highly relativistic beams g 1 0  the above equation (17) agrees with the approximation given in Gallant et al (1992) In our 2D geometry the ideal adiabatic gas constant is G = 3 2 ad . Since we are interested in weakly magnetised flows (s 1  ), the shock velocity b sh and compression ratio n n 2 1 can be approximated to pe . In order to form a shock, the particles need to be gyrating several times in this peak field. We estimate a number of Y = 20, which will give an estimate of the shock formation time due to magnetic reflection t g w = -( ) 5 . 2 3 f M , 0 pe 1

PIC simulations of shock formation in magnetised plasmas
In order to compare with the theory, we performed PIC simulations of shock formation in initially magnetised plasmas. The simulations are 2D in space and 3D in momentum space. In the beginning, the 2D simulation box is homogeneously filled with a quasi-neutral plasma of electrons and positrons. We use the reflecting wall setup in order to save computation time, see In order to estimate the shock formation time from the simulations, we average the 2D density over the x 2 direction. We plot the averaged density versus the shock propagation direction . The strength of this precursor is on the order of the estimate of the overshoot, equation (22), but slightly reduced because of the 2D geometry. A 2D perpendicular magnetic field has been self-consistently generated during the shock formation (see figure 6). In agreement with previous findings (Sironi and Spitkovsky 2009), the shock front width decreases when the magnetisation is increased from s = -10 e 5 to s = -10 e 3 but still shows the typical filamentary structure. For higher magnetisations this filamentary structure in the shock front disappears.
In order to identify the filamentation instability as the mediator for Weibel shocks and to investigate the influence of the ambient magnetic field, we analyse the magnetic energy density during the early stage of interaction in a narrow slab close to the reflecting wall in the interval    figure 7 and table 1). For s = -10 e 2 a short quasi-linear growth time of the magnetic field exists with a reduced growth rate, while for s = -10 e 1 no linear behaviour at all can be found anymore. The phase space plots reveal interesting information about the interplay of the filamentation instability and magnetic reflection (see figure 8). Typical for the filamentation instability is a rather slow development of the instability. The positron phase space is not perturbed for a long period while the modes continue to grow and finally saturate. Only after saturation of the linear phase of the magnetic field amplification, the particles are scattered and isotropised in the magnetic turbulence (Stockem Novo et al 2015). At this stage, the phase space is widened and the two beams, the incoming and the beam reflected from the wall, starts to narrow and overlap until the shock downstream region forms. This narrowing of the phase space does not happen at the leading edge of the reflected particle beam but a few tens of plasma skin depths behind, which guarantees a sufficient time of interacting counterflows.
For a typical Weibel shock this behaviour is observed in the particle phase spaces (see figure 8 for s = -10 e 5 ). In an magnetised shock a strong overshoot of the magnetic field is build up at the interface of the counterstreaming beams which immediately reflects almost all particles. The particle mean free path is

Conclusions
We investigated collisionless shock formation of an electron-positron plasma flow with ambient magnetic field. The initial magnetisation was increased in order to analyse the impact of the magnetic field on the shock formation time. In analogy to shock formation in initially unmagnetised plasmas, we derived a theoretical model for the shock formation time. Since the growth rate of the filamentation instability depends only weakly on a perpendicular magnetic field, the shock formation time for very low magnetisations σ is almost the same as for unmagnetised plasmas. The shock formation time for high magnetisations, which is dominated by magnetic reflection, was also estimated.
2D PIC simulations have been performed in order to investigate the transition between both types of shocks. It has been found that the estimated shock formation times in the limiting cases of Weibel and magnetised shocks fit well. For intermediate magnetisations, a mix of both mechanisms was observed. The shock formation time in Weibel shocks is smaller for magnetised shocks, since magnetic reflection is a very quick process. In between, we observed the reflection of a small fraction of particles at the leading edge of the shock, which propagated ahead of the shock as a precursor and finally faded out. In the meanwhile, the filamentation modes had time to evolve and to form a shock of Weibel type. However, the shock formation time was reduced compared to a pure Weibel shock, while the theoretical estimate predicted a constant shock formation time since the filamentation growth rate is constant for higher magnetisations.

Acknowledgments
This work was supported by the European Research Council (ERC-2010-AdG Grant 267841), grant ENE2013-45661-C2-1-P from the Ministerio de Educación y Ciencia, Spain and grant PEII-2014-008-P from the Junta de Comunidades de Castilla-La Mancha. The authors acknowledge the Gauss Centre for Supercomputing (GCS) for providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS share of the supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC).

Appendix. Tensor elements
The tensor  defined by equation (4)  where the first column stand for the x 3 direction, the second column for x 2 , and the third for x 1 , the flow axis. The tensor elements read,