Nonlinear collisionless damping of Weibel turbulence in relativistic blast waves

Abstract

The Weibel/filamentation instability is known to play a key role in the physics of weakly magnetized collisionless shock waves. From the point of view of high energy astrophysics, this instability also plays a crucial role because its development in the shock precursor populates the downstream with a small-scale magneto-static turbulence which shapes the acceleration and radiative processes of suprathermal particles. The present work discusses the physics of the dissipation of this Weibelgenerated turbulence downstream of relativistic collisionless shock waves. It calculates explicitly the first-order nonlinear terms associated to the diffusive nature of the particle trajectories. These corrections are found to systematically increase the damping rate, assuming that the scattering length remains larger than the coherence length of the magnetic fluctuations. The relevance of such corrections is discussed in a broader astrophysical perspective, in particular regarding the physics of the external relativistic shock wave of a gamma-ray burst. IntroductionThe physics of collisionless shock waves has drawn wide interest, from pure theoretical plasma physics, starting with the pioneering work of Moiseev and Sagdeev (1963), to high energy astrophysics (e.g. Blandford and Eichler 1987), where it plays a key role in explaining most of the observed non-thermal spectra, and more recently, to laboratory high energy density physics, where collisionless shock waves are about to be produced through the interactions of laser beam-generated plasmas (e.g. Drake and Gregori 2012). At low magnetization -meaning that the unshocked plasma carries a magnetic field of small energy density compared to the shock kinetic energy -the physics of these collisionless shock waves is driven by the filamentation instability, also dubbed Weibel instability: this filamentation instability takes place in the shock precursor, where the incoming background plasma -as viewed in the reference frame in which the shock lies at rest -mixes with a population of shock-reflected or supra-thermal particles. This has been demonstrated by ab initio particle-in-cell (PIC) simulations, see e.g. Kato and Takabe (2008) for non-relativistic unmagnetized shock waves and Spitkovsky (2008a) for their relativistic counterparts, of direct interest to the present work. This filamentation instability and its various branches have consequently received a great deal of attention (see e.g. for relativistic shock waves Medvedev and Loeb

Constraining the magnetic field structure in collisionless relativistic shocks with a radio afterglow polarization upper limit in GW 170817

“…Early 2D PIC simulations that showed a gradual power law decay of B prompted the consideration of decaying magnetic fields in the bulk of the shocked region for afterglow modeling (Rossi & Rees 2003;Lemoine et al 2013;Lemoine 2015) and also in models of prompt emission from internal shocks (Pe'er & Zhang 2006;Derishev 2007). Moreover, afterglow modeling of many GRBs in the optical band revealed a wide distribution with a rather low value of the radially averaged magnetic field in the shocked region with 10 −8 B 10 −2 under the explicit assumption that the circumburst density n = 1 cm −3 (Santana et al 2014).…”

Physics and Phenomenology of Weakly Magnetized, Relativistic Astrophysical Shock Waves

“…Figure 10 shows the spatial decay law observed in a PIC simulation of a pair shock with γ sh = 17. Phase mixing erodes the magnetic fluctuations by erasing the small-scale structures first, with a damping rate ω ∼ −|k| 3 c 3 /ω 2 p in terms of (transverse) wavenumber k [64,65]. In the reference frame of the blast, the shock front moves away, with respect to a given plasma element, at velocity c/3 (or c/2 is 2D numerical simulations).…”

Relativistic Shocks: Particle Acceleration and Magnetization

“…In a relativistic plasma, small-scale turbulence is dissipated at a damping rate Chang et al 2008;Lemoine 2015) as a function of the wavenumber k, indicating that small scales are erased early on. Larger modes can survive longer; power on scales exceeding the Larmor radius of the bulk plasma decays on long, ω ∝ k 2 MHD scales (Keshet et al 2009).…”

“…Also shown are the transverse averages (at t 1 , dashed blue, and t 2 , solid red) of (c) the electromagnetic energy EM ≡ [(B 2 + E 2 )/8π]/[(γr − 1)γrn mc 2 ] (with E the electric field amplitude in the downstream frame, included in the definition of EM because in the simulation frame the induced electric field in the upstream medium is E ∼ B) normalized to the upstream kinetic energy, (d) density normalized to the far upstream density nu = γrn , and (e) particle momentum γβ (with β the velocity in c units) in the x-direction averaged over all particles (higher γβx ) and over downstream-headed particles only. Lemoine 2015); the scale-free limit corresponds to a flat magnetic power spectrum (Katz et al 2007).…”

“…As discussed in Section 3.2, one notably expects this turbulence to relax through collisionless damping on hundreds of c/ω p (Chang et al 2008;Keshet et al 2009;Lemoine 2015) while the electrons typically cool on much longer length scales. In GRB external blast waves, the shocked region is typically 7−9 orders of magnitude larger than c/ω p in size, which leaves room for a substantial evolution of B , even if it decreases as a mild power-law in distance from the shock, as suggested by the above studies.…”