The Weibel instability in relativistic plasmas

Abstract

Aims. We discuss the linear theory of the Weibel instability in a relativistic plasma driven by ultra-relativistic beams, describing the physics of the generation of magnetic fields in the ultra-relativistic shocks associated with Gamma Ray Bursts (GRBs). We perform a detailed analysis of the linear dispersion relation for the benefit of non-linear calculations that we discuss in the companion paper. Methods. We use a covariant approach, where the linear response of the beam-plasma system is determined from the polarization tensor. This tensor relates the four-current density to the four-potential of the electromagnetic field. Showing that two approaches, one based on a fluid model and one on a kinetic description that uses a waterbag distribution for the phase-space density of the beam particles, yield essentially the same result, we compare our results to those obtained by other approaches. We mainly consider the symmetric case of two counterstreaming (but otherwise identical) beams. Results. We show that the effect of an asymmetry in the beam densities is small for typical parameters, and briefly discuss the effect of an ambient magnetic field. The dispersion relation of the Weibel instability driven by ultra-relativistic beams is rather insensitive to the model used to describe the plasma. The properties of the instability, such as the growth rate and the range of unstable wavelengths, are governed by only two parameters: the ratio of the plasma frequency squared of the beam and hot background plasma, and a "Mach number", which is essentially the ratio of the beam momentum and the momentum associated with thermal velocity (∼sound speed) in the beam plasma. We also show that, at least for the parameters associated with the ultra-relativistic shocks in GRBs, the influence of the magnetic field is small, and the results for an unmagnetized plasma can be used. Conclusions.

Multidimensional electron beam-plasma instabilities in the relativistic regime

“…This mathematical complication, which evidently holds for any kind of coordinate system, has restricted many studies of kinetic plasma instabilities in the relativistic regime to peculiar, and often blatantly unrealistic, distribution functions allowing for a simplified handling of the Lorentz factor, and/or regimes characterized by weak ͑i.e., nonrelativistic͒ thermal spreads. 15,16,52,[104][105][106][107][108][109] Let us emphasize that no assumption whatsoever is made in Eq. ͑7͒ about the respective orientations of k and E 1 .…”

Electron Heating in a Relativistic, Weibel-Unstable Plasma

“…In particlular, the Weibel instability (Weibel 1959), which causes fast growth of a strong magnetic field at small scales in the anisotropic plasma flow, has received much attention as the main isotropization mechanism leading to the shock transition in free-streaming ejecta from violent astrophysical events (Medvedev & Loeb 1999;Wiersma & Achterberg 2004;Lyubarsky & Eichler 2006;Achterberg & Wiersma 2007;Bret 2009;Yalinewich & Gedalin 2010;Shaisultanov et al 2012;Shukla et al 2012). As the dominant modes of the Weibel instability are less than the ion gyroradius, which renders them inefficient scatterers of ions, the long standing question concerning its role in collisionless shocks has been how well it competes with other mechanisms (e.g., Galeev et al 1964;Blandford & Eichler 1987;Lyubarsky & Eichler 2006).…”

Weibel instability in a plasma with nonzero external magnetic field

“…In this respect it becomes similar to the ordinary two-stream instability (Schlickeiser and Shukla, 2003). The instability gives rise to the exponential growth of electromagnetic fields which help to restore plasma isotropy and is often considered as one of the most important mechanisms for the generation of magnetic fields in space (Medvedev and Loeb, 1999;Achtenberg and Wiersma, 2007). When a plasma is either free from or possesses negligibly small external magnetic fields, the Weibel wave behaves as a purely zero-frequency non propagating mode.…”