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We treat the problem of adiabatic losses and stochastic particle acceleration in gamma-ray burst (GRB) blast waves that decelerate by sweeping up matter from an external medium. The shocked fluid is assumed to be represented by a homogeneous expanding shell. The energy lost by nonthermal particles through adiabatic expansion is converted to the bulk kinetic energy of the outflow, permitting the evolution of the bulk Lorentz factor Γ of the blast wave to be selfconsistently calculated. The behavior of the system is shown to reproduce the hydrodynamic self-similar solutions in the relativistic and nonrelativistic limits, and the formalism is applicable to scenarios that are intermediate between the adiabatic and fully radiative regimes.Nonthermal particle energization through stochastic gyroresonant acceleration with magnetic turbulence in the blast wave is treated by employing energy-gain rates and diffusive escape timescales based upon expressions derived in the quasilinear regime. If the magnetic field in the shocked fluid approaches its equipartition value, this process can accelerate escaping particles to 10 20 eV energies, consistent with the hypothesis that ultra-high energy cosmic rays (UHECRs) are accelerated by GRB blast waves. Due to particle trapping by the magnetic turbulence, only the highest energy particles can escape during the prompt and afterglow phases of a GRB for acceleration by a Kolmogorov spectrum of MHD turbulence. Lower energy particles begin to escape as the blast wave becomes nonrelativistic and shock Fermi acceleration becomes more important.

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Long's equation describes two dimensional stratified flow over terrain. Its numerical solutions under various approximations were investigated by many authors under the assumption that the base flow field is without shear. Special attention was paid to the properties of the gravity waves that are predicted to be generated as a result. In this paper we address, analytically, the nature and properties of these solutions when shear is present and derive some constraints on the possible generation of gravity waves under these circumstances.