Evidence for superdiffusive shock acceleration at interplanetary shock waves

Recent analysis of time profiles of energetic particles accelerated at interplanetary shocks has shown evidence for superdiffusive transport upstream of the shock fronts, namely for a transport characterized by a particle mean square displacement that grows faster than linearly in time. While for normal, diffusive transport exponential particle time profiles are predicted, a large number of interplanetary shock events, included the termination shock of the solar wind, exhibits energetic particle time profiles that upstream decay as power laws. This power law trend has been derived in the framework of particle superdiffusion. The standard theory of diffusive shock acceleration has been further extended to the case of particle superdiffusive transport (superdiffusive shock acceleration), allowing for the derivation of harder energy spectral indices both for relativistic and non-relativistic particles. Here we test this theory for a couple of interplanetary shock waves that accelerate protons and that have been observed by the ACE spacecraft. We show that power law particle time profiles upstream of the shocks are common and clearly indicate superdiffusive transport. The particle energy spectra in some events are in a very good agreement with the superdiffusive shock acceleration prediction.