The energy spectrum of energetic particles downstream of turbulent collisionless shocks

Using simple analytic considerations, numerical simulations, and data analysis, we discuss the physics of charged-particle acceleration by turbulent, rippled, collisionless shocks. The standard theory of diffusive shock acceleration predicts that the energetic-particle energy spectrum, in the region of shocked plasma, is a function of the plasma density jump. But because of the interaction of the shock with plasma turbulence, the jump in plasma density varies in time and from place to place on the shock front. Here we show that for reasonable parameters, the shape of the energetic-particle energy spectra downstream of any given shock is nearly independent of location along the shock front, even though the density jump varies. This is because energetic particles are mobile and sample many turbulent fluctuations during their acceleration. This result holds for shocks having smaller scale ripples than the large-scale radius of curvature (D-c) of the shock. Thus, it applies to the interpretation of spacecraft observations of traveling interplanetary shocks provided the spacecraft separation is less than Dc. This result is confirmed with simple analytic considerations and numerical simulations that solve the combined magnetohydrodynamic equations for a plasma and energetic test particles using the well-known Parker transport equation. This conclusion is further supported by our analysis of ACE and Geotail observations of a few interplanetary shocks.