More than mass proportional heating of heavy ions by supercritical collisionless shocks in the solar corona

We propose a new model for explaining the observations of more than mass proportional heating of heavy ions in the polar solar corona. We point out that a large number of small scale intermittent shock waves can be present in the solar corona. The energization mechanism is, essentially, the ion reflection off supercritical quasi-perpendicular collisionless shocks in the corona and the subsequent acceleration by the motional electric field ${\bf E} = - (1/c) {\bf V} \times {\bf B}$. The acceleration due to ${\bf E}$ is perpendicular to the magnetic field, in agreement with observations, and is more than mass proportional with respect to protons, because the heavy ion orbit is mostly upstream of the quasi-perpendicular shock foot. The observed temperature ratios between O$^{5+}$ ions and protons in the polar corona, and between $\alpha$ particles and protons in the solar wind are easily recovered.Comment: 11 pages, 2 figure

Self-similar transport processes in a two-dimensional realization of multiscale magnetic field turbulence

We present the results of a numerical investigation of charged-particle transport across a synthesized magnetic configuration composed of a constant homogeneous background field and a multiscale perturbation component simulating an effect of turbulence on the microscopic particle dynamics. Our main goal is to analyze the dispersion of ideal test particles faced to diverse conditions in the turbulent domain. Depending on the amplitude of the background field and the input test particle velocity, we observe distinct transport regimes ranging from subdiffusion of guiding centers in the limit of Hamiltonian dynamics to random walks on a percolating fractal array and further to nearly diffusive behavior of the mean-square particle displacement versus time. In all cases, we find complex microscopic structure of the particle motion revealing long-time rests and trapping phenomena, sporadically interrupted by the phases of active cross-field propagation reminiscent of Levy-walk statistics. These complex features persist even when the particle dispersion is diffusive. An interpretation of the results obtained is proposed in connection with the fractional kinetics paradigm extending the microscopic properties of transport far beyond the conventional picture of a Brownian random motion. A calculation of the transport exponent for random walks on a fractal lattice is advocated from topological arguments. An intriguing indication of the topological approach is a gap in the transport exponent separating Hamiltonian-like and fractal random walk-like dynamics, supported through the simulation.Comment: 10 pages (including cover page), 7 figures, improved content, accepted for publication in Physica Script

Parameter estimation of superdiffusive motion of energetic particles upstream of heliospheric shocks

In-situ spacecraft observations recently suggested that the transport of energetic particles accelerated at heliospheric shocks can be anomalous, i.e. the mean square displacement can grow non-linearly in time. In particular, a new analysis technique has permitted the study of particle transport properties from energetic particle time profiles upstream of interplanetary shocks. Indeed, the time/spatial power laws of the differential intensity upstream of several shocks are indicative of superdiffusion. A complete determination of the key parameters of superdiffusive transport comprises the power-law index, the superdiffusion coefficient, the related transition scale at which the energetic particle profiles turn to decay as power laws, and the energy spectral index of the shock accelerated particles. Assuming large-scale spatial homogeneity of the background plasma, the power-law behaviour can been derived from both a (microscopic) propagator formalism and a (macroscopic) fractional transport equation. We compare the two approaches and find a relation between the diffusion coefficients used in the two formalisms. Based on the assumption of superdiffusive transport, we quantitatively derive these parameters by studying energetic particle profiles observed by the Ulysses and Voyager 2 spacecraft upstream of shocks in the heliosphere, for which a superdiffusive particle transport has previously been observed. Further, we have jointly studied the electron energy spectra, comparing the values of the spectral indices observed with those predicted by the standard diffusive shock acceleration theory and by a model based on superdiffusive transport. For a number of interplanetary shocks and for the solar wind termination shock, for the first time we obtain the anomalous diffusion constants and the scale at which the probability of particle free paths changes to a power-law...Comment: 5 Figure

Wave-particle interactions with parallel whistler waves: nonlinear and time-dependent effects revealed by Particle-in-Cell simulations

We present a self-consistent Particle-in-Cell simulation of the resonant interactions between anisotropic energetic electrons and a population of whistler waves, with parameters relevant to the Earths radiation belt. By tracking PIC particles, and comparing with test-particle simulations we emphasize the importance of including nonlinear effects and time evolution in the modeling of wave-particle interactions, which are excluded in the resonant limit of quasi- linear theory routinely used in radiation belt studies. In particular we show that pitch angle diffusion is enhanced during the linear growth phase, and it rapidly saturates well before a single bounce period. This calls into question the widely used bounce average performed in most radiation belt diffusion calculations. Furthermore we discuss how the saturation is related to the fact that the domain in which the,particles pitch angle diffuse is bounded, and to the well-known problem of $90^\circ$ diffusion barrier diffusion barrier.Comment: to appear on Physics of Plasmas 8 pages, 12 figure

Collisionless Shocks as a Diagnostic Tool for Understanding Energetic Particle Transport in Space Plasmas

We study the transport of energetic particles accelerated at three different shock events observed in the solar wind by the ACE spacecraft. We consider particle propagation for a quasi-parallel, an oblique, and a quasi-perpendicular shock. The transport regime is deduced from the shape of the energetic particle profiles upstream of the shock, and for these events the profiles are well-fitted by power-laws with slope β. This corresponds to a superdiffusive transport with the anomalous diffusion exponent α = 2−β when β <1, and to normal diffusion when β≥1. We checked the resonant turbulence level upstream of the shocks, finding that this is statistically constant, so that the transport regime is not expected to change with the shock distance. For the three shocks under study, particle transport upstream of the shock is mostly superdiffusive, although the superdiffusive character appears to diminish with the increase of the shock normal angle θBn. We discuss possible interpretations of these results

Superdiffusive and Subdiffusive Transport of Energetic Particles in Solar Wind Anisotropic Magnetic Turbulence

The transport of energetic particles in a mean magnetic field and the presence of anisotropic magnetic turbulence are studied numerically, for parameter values relevant to the solar wind. A numerical realization of magnetic turbulence is set up in which we can vary the type of anisotropy by changing the correlation lengths lx, ly, lz. We find that for lx, ly lz, transport can be non-Gaussian, with superdiffusion along the average magnetic field and subdiffusion perpendicular to it. Decreasing the lx/lz ratio down to 0.3, Gaussian diffusion is obtained, showing that the transport regime depends on the turbulence anisotropy. Implications for energetic particle propagation in the solar wind and for diffusive shock acceleration are discussed

Superdiffusive Shock Acceleration

The theory of diffusive shock acceleration is extended to the case of superdiffusive transport, i.e., when the mean square deviation grows proportionally to t α, with α > 1. Superdiffusion can be described by a statistical process called Levy random walk, in which the propagator is not a Gaussian but it exhibits power-law tails. By using the propagator appropriate for Levy random walk, it is found that the indices of energy spectra of particles are harder than those obtained where a normal diffusion is envisaged, with the spectral index decreasing with the increase of α. A new scaling for the acceleration time is also found, allowing substantially shorter times than in the case of normal diffusion. Within this framework we can explain a number of observations of flat spectra in various astrophysical and heliospheric contexts, for instance, for the Crab Nebula and the termination shock of the solar wind

From Lévy walks to superdiffusive shock acceleration

In this paper, we present a general scenario for nondiffusive transport and we investigate the influence of anomalous, superdiffusive transport on Fermi acceleration processes at shocks. We explain why energetic particle superdiffusion can be described within the Levy walk framework, which is based on a power-law distribution of free path lengths and on a coupling between free path length and free path duration. A self-contained derivation of the particle mean square displacement, which grows as (Δx {sup 2}) = 2D {sub α} t {sup α} with α > 1, and the particle propagator, is presented for Levy walks, making use of a generalized version of the Montroll-Weiss equation. We also derive for the first time an explicit expression for the anomalous diffusion coefficient D {sub α} and we discuss how to obtain these quantities from energetic particle observations in space. The results are applied to the case of particle acceleration at an infinite planar shock front. Using the scaling properties of the Levy walk propagator, the energy spectral indices are found to have values smaller than the ones predicted by the diffusive shock acceleration theory. Furthermore, when applying the results to ions with energies of a few MeV accelerated atmore » the solar wind termination shock, the estimation of the anomalous diffusion coefficient associated with the superdiffusive motion gives acceleration times much smaller than the ones related to normal diffusion.« les

Ion Superdiffusion at the Solar Wind Termination Shock

We investigate the propagation of 0.54-3.5 MeV ions accelerated at the termination shock of the solar wind. Data are from Voyager 2 and refer to a time interval about one year long, just before the Voyager 2 termination shock crossing at the end of 2007 August, at roughly 83.7 AU. A recently developed technique, which allows to unravel the transport properties from an analysis of the energetic particle time profiles, is used. The ion time profiles exhibit a power-law decay from a few days to 200 days before the shock front, so that transport is found to be superdiffusive, with a mean square deviation growing like Δx 2 ∝ t α, with α ~ 1.3. This means that ion propagation in the heliosphere can be intermediate between normal diffusion and ballistic motion. The implication of ion superdiffusion on particle acceleration mechanisms at the termination shock is discussed, as well as some observational evidence coming from both Voyager 1 and Voyager 2, which questions diffusive shock acceleration