Tempered Superdiffusive Shock Acceleration at a Perpendicular Shock

Abstract A theory for steady-state tempered superdiffusive shock acceleration of energetic particles at a plane perpendicular shock is presented that involves solving analytically a tempered fractional Parker transport equation for perpendicular transport in quasi-2D turbulence. The main predictions of the theory are as follows: (i) Tempered perpendicular superdiffusion (tempered Lévy flights) upstream reduces the effectiveness of classical perpendicular superdiffusion (standard Lévy flights) through a transition toward normal perpendicular diffusion when the particle transport distance upstream surpasses a critical spatial scale. Consequently, the accelerated particle distribution upstream decays spatially as a stretched exponential close to the shock that becomes a power law farther upstream under some conditions (superdiffusion), followed by an exponential rollover (transition to normal diffusion); (ii) Downstream, the accelerated particle distribution for tempered superdiffusion converges to a uniform distribution closer to the shock because of a stronger decay close to the shock compared to classical superdiffusion, and a shift of the peak in the distribution at the shock to just downstream of the shock occurs under some conditions; (iii) The accelerated power-law momentum spectrum at the shock is modulated upstream by the incoming solar wind flow to be a hardened power law instead of featuring an exponential rollover at lower particle momenta as for classical diffusive shock acceleration; and (iv) Tempered superdiffusive shock acceleration tends to be more efficient compared to standard superdiffusive shock acceleration.