Superdiffusion of energetic particles at shocks: A fractional diffusion and Lévy flight model of spatial transport

Context. The observed power laws in space and time profiles of energetic particles in the heliosphere can be the result of an underlying superdiffusive transport behavior. Such anomalous, non-Gaussian transport regimes can arise, for example, as a consequence of intermittent structures in the solar wind. Non-diffusive transport regimes may also play a critical role in other astrophysical environments such as supernova remnant shocks. Aims. To clarify the role of superdiffusion in the transport of particles near shocks, we study the solutions of a fractional diffusion-advection equation to investigate this issue. A fractional generalization of the Laplace operator, the Riesz derivative, provides a model of superdiffusive propagation. Methods. We obtained numerical solutions to the fractional transport equation by means of pseudo-particle trajectories solving the associated stochastic differential equation driven by a symmetric, stable Lévy motion. Results. The expected power law profiles of particles upstream of the plasma shock, where particles are injected, can be reproduced with this approach. The method provides a full, time-dependent solution of the fractional diffusion-advection equation. Conclusions. The developed models enable a quantitative comparison to energetic particle properties based on a comprehensive, superdiffusive transport equation and allow for an application in a number of scenarios in astrophysics and space science.