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.

Abstract. We discuss magnetic field generation by the proton Weibel instability in relativistic shocks, a situation that applies to the external shocks in the fireball model for Gamma-ray Bursts, and possibly also to internal shocks. Our analytical estimates show that the linear phase of the instability ends well before it has converted a significant fraction of the energy in the proton beam into magnetic energy: the conversion efficiency is much smaller (of order m e /m p ) in electron-proton plasmas than in pair plasmas. We find this estimate by modelling the plasma in the shock transition zone with a waterbag momentum distribution for the protons and with a background of hot electrons. For ultra-relativistic shocks we find that the wavelength of the most efficient mode for magnetic field generation equals the electron skin depth, that the relevant nonlinear stabilization mechanism is magnetic trapping, and that the presence of the hot electrons limits the typical magnetic field strength generated by this mode so that it does not depend on the energy content of the protons. We conclude that other processes than the linear Weibel instability must convert the free energy of the protons into magnetic fields.

Aims. We discuss the onset of the nonlinear stage of the electromagnetic Weibel instability in a relativistic plasma, and the process of current coalescence that follows this instability. The Weibel instability has been proposed as a possible source of the magnetic fields needed to explain the non-thermal synchrotron emission from gamma ray bursts. Methods. We present two different calculations of the nonlinear saturation of the Weibel instability: one based on a fluid model, and one using kinetic plasma theory. These approaches yield a similar result for the amplitude of the magnetic field at saturation. We then consider the further growth of the magnetic field due to the coalescence of current filaments, a process that has been observed in numerical simulations. Results. These calculations show that the exponential linear stage of the instability is terminated by trapping of the beam particles in the wave. The trapping leaves a magnetic field that acts as the seed field for further amplification through coalescence. Further field amplification is limited to magnetic fields on scales less than the effective plasma skin depth of a background plasma. We show that coalescence of current filaments thicker than a few times the skin depth proceeds at a exponentially slow rate. Conclusions. The amplitude of saturation is determined mostly by the plasma frequency of the hot (shocked) background plasma, which is usually dominated by the electrons. The typical field amplitude at this stage is almost independent of the mass of the beam particles. Further field amplification through current coalescence, a process that follows the exponential Weibel instability, "stalls" once the current filaments reach a size that is comparable to the skin depth of the background plasma. This process concentrates the currents, but the resulting field amplification is small. This implies that the resulting magnetic field energy density never reaches equipartition with the kinetic energy density of the heavy particle species (ions) in the incoming beams.

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