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Weibel Filament Decay and Thermalization in Collisionless Shocks and Gamma-Ray Burst Afterglows

Models for the synchrotron emission of gamma-ray burst afterglows suggest that the magnetic field is generated in the shock wave that forms as relativistic ejecta plow through the circumburst medium. Transverse Weibel instability efficiently generates magnetic fields near equipartition with the postshock energy density. The detailed saturated state of the instability, as seen in particle-in-cell simulations, consists of magnetically self-pinched current filaments. The filaments are parallel to the direction of propagation of the shock and are about a plasma skin depth in radius, forming a quasi-two-dimensional structure. We argue that the Weibel filaments are susceptible to pressure-driven instabilities and use a rudimentary analytical model to illustrate the development of a particular, kinklike unstable mode. The instabilities destroy the quasi-two-dimensional structure of the Weibel filaments. For wavelengths longer than the skin depth, the kinklike mode grows at the rate equal to the speed of light divided by the wavelength. We calculate the transport of collisionless test particles in the filaments experiencing the instability and show that the particles diffuse in energy. This diffusion marks the beginning of thermalization in the shock transition layer and causes initial magnetic field decay as particles escape from the filaments. We discuss the implications of these results for the structure of the shock and the polarization of the afterglow.