Authors: Jakobus A. le Roux (University of Alabama in Huntsville), Rubaiya Khondoker (University of Alabama in Huntsville)

The analysis of a number of heliospheric shock events suggested a power-law decay of the accelerated energetic particle flux upstream instead of the exponential decay predicted by standard steady-state diffusive shock acceleration (DSA) theory. This was interpreted to indicate superdiffusive energetic particle transport (Lévy flights) upstream as part of a superdiffusive shock acceleration process (see publications by Zimbardo and coworkers).  To investigate this problem further, a recently derived tempered fractional Parker transport equation for energetic particle interaction with quasi-2D turbulence was solved analytically. It was used to model tempered Lévy flights in the vicinity of a planar perpendicular shock and study its consequence for superdiffusive shock acceleration in the steady-state limit. The following results will be presented that highlight interesting differences compared with the results of standard steady-state DSA theory: (i) A power-law decay of the upstream energetic particle distribution with increasing distance from the shock followed by an exponential rollover as particle transport transitions from standard superdiffusion (Lévy flights) toward normal diffusion beyond a critical transport distance from the shock (tempered Lévy flights). (ii) A downstream energetic particle distribution that decays with increasing distance from the shock and converges to a plateau further downstream instead of just forming a plateau as in standard steady-state DSA theory. The decay is a consequence of a reduced particle escape probability downstream caused by superdiffusion back to the shock (see previous work by Zimbardo and coworkers). (iii) A reduction in the upstream modulation of accelerated particles compared to normal steady-state DSA theory. (iv) An expression for the time scale of tempered superdiffusive shock acceleration suggesting that tempered Lévy flights increase the efficiency of the superdiffusive shock acceleration process.