Influence of the transport regime on the energetic particle density profiles upstream and downstream of interplanetary shocks

The spatial distribution of energetic particles accelerated at shocks depends on their transport properties. Analyses of energetic particle fluxes measured by spacecraft upstream of interplanetary shock waves have pointed out the presence of spatial profiles different from those expected for normal diffusion. We propose that anomalous, superdiffusive transport can help to interpret the observed energetic particle profiles both upstream and downstream of shock waves. We set up a numerical model to compute the energetic particle profiles on both sides of the shock: particles are injected at the shock and then propagate according to a Gaussian random walk in the case of normal diffusion and according to a Lévy random walk in the case of superdiffusion. The latter is characterized by a nonlinear growth of the mean square displacement of particles and by a power law distribution of free path lengths. A Langevin type equation is solved numerically, and energetic particle spatial profiles are obtained for a steady state configuration. A number of numerical solutions are obtained: in the case of normal diffusion, the well known exponential profile upstream of the shock is recovered. In the case of superdiffusion, varying the power exponent and the scale time characterizing the power law distribution of free path times, it is found that power law upstream profiles and spatially nonconstant downstream profiles are obtained. A preliminary comparison between the obtained numerical results and interplanetary shock observations by the ACE spacecraft is carried out, and good agreement between the energetic particle profiles is found. This shows that the present superdiffusive model can be helpful for interpreting the overall time/space trends of particles accelerated at interplanetary shock waves.