A Tempered Fractional Kinetic Transport Theory for Energetic Particle Interaction with Quasi-two-dimensional Turbulence in the Large-scale Solar Wind

Abstract

Observational evidence is accumulating that turbulence in the solar wind is intermittent (non-Gaussian) because of the strong presence of a quasi-two-dimensional (quasi-2D), low-frequency turbulence component containing nonpropagating, closed, small-scale magnetic flux ropes with open meandering field lines in between. le Roux & Zank showed how one can derive fractional focused and Parker-type transport equations that model large-scale anomalous transport in the solar wind as the outcome of energetic particle interaction with quasi-2D turbulence. In this follow-up paper this theory is developed further to address certain limitations. (i) The second moment of the Lévy probability distribution function (PDF) specified in the theory for the particle step size is infinite, indicating unphysical transport. (ii) The expected transition of energetic particle transport from anomalous to normal diffusion beyond a certain critical transport distance was not included. (iii) The competition between anomalous diffusion and advection is not properly sustained at late times. Shortcomings (i) and (ii) are addressed by introducing an exponentially truncated Lévy PDF for the energetic particle step size in the theory, resulting in revised tempered fractional focused and Parker-type transport equations featuring tempered fractional derivatives that enable modeling of tempered Lévy flights. Furthermore, these equations are cast in a tempered fractional telegrapher form to investigate whether the fractional wave equation part of the equation can restore causality in unscattered particle transport during early times and in Lévy flights during intermediate times (Lévy walks). They are also transformed into a tempered fractional Fokker–Planck form to overcome limitation (iii).