The Weibel instability in relativistic plasmas II. Nonlinear theory and stabilization mechanism

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.