We study the long term evolution of magnetic fields generated by an initially unmagnetized collisionless rela tivistic e~+e~- shock. Our two-dimensional particle-in-cell numerical simulations show that downstream of such a Weibel-mediated shock, particle distributions are approximately isotropic, relativistic Maxwellians, and the magnetic turbulence is highly intermittent spatially. The nonpropagating magnetic fields decay in amplitude and do not merge. The fields start with magnetic energy density ～ 0.1 -0.2 of equipartition, but rapid downstream decay drives the fields to much smaller values, below ～10~(-3) of equipartition after ～10~3 skin depths. To construct a theory to follow field decay to these smaller values, we hypothesize that the observed damping is a variant of Landau damping. The model is based on the small value of the downstream magnetic energy density, which only weakly perturbs particle orbits, for homogeneous turbulence. Using linear kinetic theory, we find a simple analytic form for the damping rates for small amplitude, subluminous electromagnetic fields. Our theory predicts that overall magnetic energy decays as (ω_pt)~(-q) with q ～ 1, which compares with simulations. However, our theory predicts overly rapid damping of short-wavelength modes. Magnetic trapping of particles within the highly spatially intermittent downstream magnetic structures may be the origin of this discrepancy and may allow for some of this initial magnetic energy to persist. Absent additional physical processes that create longer wavelength, more persistent fields, we conclude that initially unmagnetized relativistic shocks in electron-positron plasmas are unable to form persistent downstream magnetic fields. These results put interesting constraints on synchrotron models for the prompt and afterglow emission from GRBs. We also comment on the relevance of these results for relativistic electron-ion shocks.
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