Self-similar collisionless shocks

Observations of gamma-ray burst afterglows suggest that the correlation length of magnetic field fluctuations downstream of relativistic nonmagnetized collisionless shocks grows with distance from the shock to scales much larger than the plasma skin depth. We argue that this indicates that the plasma properties are described by a self-similar solution and derive constraints on the scaling properties of the solution. For example, we find that the scaling of the characteristic magnetic field amplitude with distance from the shock is B proportional to D-SB, with -1 < s(B) <= 0; that the spectrum of accelerated particles is dn/dE proportional to E-2/(sB+1); and that the scaling of themagnetic correlation function is < B-i(x)B-j(x + Delta x)> proportional to Delta x(2sB) (for Delta x >> D). We show that the plasma may be approximated as a combination of two self-similar components: a kinetic component of energetic particles and anMHD-like component representing thermal'' particles. We argue that the latter may be considered as infinitely conducting, in which case s(B) = 0 and the scalings are completely determined (e. g., dn/dE proportional to E-2 and B proportional to D-0, with possible logarithmic corrections). Similar claims apply to nonrelativistic shocks such as in supernova remnants, if the upstream/magnetic field can be neglected. Self-similarity has important implications for any model of particle acceleration and/or field generation. For example, we show that the diffusion function in the angle mu of momentum p in diffusive shock acceleration models must satisfy D-mu mu(p; D) = D-1 (D) over tilde (mu mu)(p/D) (where p is the particle momentum) and that a previously suggested model for the generation of large-scale magnetic fields through a hierarchical merger of current filaments should be generalized. A numerical experiment testing our analysis is outlined.