Solar energetic particles, or SEPs, from suprathermal (few keV) up to relativistic (
 $$\sim $$
 few GeV) energies are accelerated near the Sun in at least two ways: (1) by magnetic reconnection-driven processes during solar flares resulting in impulsive SEPs, and (2) at fast coronal-mass-ejection-driven shock waves that produce large gradual SEP events. Large gradual SEP events are of particular interest because the accompanying high-energy (
 $${>}10$$
 s MeV) protons pose serious radiation threats to human explorers living and working beyond low-Earth orbit and to technological assets such as communications and scientific satellites in space. However, a complete understanding of these large SEP events has eluded us primarily because their properties, as observed in Earth orbit, are smeared due to mixing and contributions from many important physical effects. This paper provides a comprehensive review of the current state of knowledge of these important phenomena, and summarizes some of the key questions that will be addressed by two upcoming missions—NASA’s Solar Probe Plus and ESA’s Solar Orbiter. Both of these missions are designed to directly and repeatedly sample the near-Sun environments where interplanetary scattering and transport effects are significantly reduced, allowing us to discriminate between different acceleration sites and mechanisms and to isolate the contributions of numerous physical processes occurring during large SEP events.

Large gradual solar energetic particle events

The Ulysses mission provides an opportunity to study the evolution of magnetohydrodynamic (MHD) turbulence in pure high-speed solar wind streams. The absence at high heliocentric latitudes of the strong shears in solar wind velocity generally present near the heliocentric current sheet allows investigation of how fluctuations in the magnetic field and plasma relax and evolve in the radially expanding solar wind. We report results of an analysis of the radial and latitudinal variation of the turbulence properties of the fluctuations, especially various plasma-field correlations, in high latitude regions. The results constrain current theories of the evolution of MHD turbulence in the solar wind. Compared to similar observations at 0.3 AU by Helios, we find spectra that are similar in having a large frequency band with an f¹ power spectrum in the outward traveling component of the waves, followed at higher frequencies by a steeper spectrum. Ulysses observations establish that at high latitudes the turbulence is less evolved (i.e., has a smaller inertial range) than it is in the ecliptic at the same heliocentric distance, apparently due to the absence of strong velocity shear. Once Ulysses is in the polar coronal hole, properties of the turbulence appear to be determined by the heliocentric distance of the spacecraft rather than by its helio-latitude.

Properties of Magnetohydrodynamic Turbulence in the Solar Wind as Observed

The original nonlinear guiding center (NLGC) theory by Matthaeus et al. was a breakthrough in establishing a theory that promised to reproduce for the first time the anomalous perpendicular diffusion results from test particle trajectory calculations in prescribed static magnetic field turbulence dominated by a two-dimensional component. The assumptions used in this approach guaranteed anomalous diffusion to be a classical process (the variance � Δx 2 �∝ (Δt) α with α = 1). However, Shalchi & Kourakis showed that similar calculations can be even better reproduced within the context of a generalized compound diffusion model for anomalous perpendicular diffusion whereby anomalous diffusion is nonclassical (0 <α< 1 (subdiffusion) or 1 <α< 2 (superdiffusion)). In this paper, it is shown how NLGC theory can be generalized to model compound diffusion conditions consistent with the generalized compound diffusion model in terms of a nonuniform plasma medium. Such a medium is assumed to generate a distribution of cosmic-ray pitch-angle scattering times for cosmic rays interacting resonantly with a minor small-scale slab turbulence component. This suggests that the anomalous perpendicular diffusion from the test particle simulations might be explained best within the framework of anomalous diffusion in a nonuniform plasma medium. It is argued that, during intermediate times when the magnetic field appears to be static in the quiet solar wind near Earth, generalized NLGC theory predicts possibly subdiffusive anomalous perpendicular transport for intermediate-energy (E � 400–500 MeV) cosmic rays because they experience nonuniform scattering conditions. These conditions are suggested to be a product of stochastic wave growth of small-scale slab turbulence resulting intermittently in large patches of intense slab turbulence where scattering times are reduced and cosmic rays are trapped. Classical anomalous diffusion is expected to be restored at high cosmic-ray energies (E � 400–500 MeV) in accordance with original NLGC theory because they encounter more uniform scattering conditions due to weaker stochastic wave growth. Magnetic turbulence observations near Earth during 2003 were used to show that the solar wind can be characterized as a nonuniform medium sustaining a power-law distribution of cosmic-ray pitch-angle scattering times. However, application of generalized NLGC theory to the 2003 observations produced unexpectedly a negative exponent for the variance of the anomalous diffusion ( α< 0). This indicates that particle bunching occurs as cosmic rays are trapped in plasma regions associated with strong particle scattering. Such effects can plausibly be explained by the intermittent presence of strong compressive turbulence downstream of shocks, stream, and interaction regions.

http://iopscience.iop.org/article/10.1088/0004-637X/716/1/671/pdf

A Generalized Nonlinear Guiding Center Theory for the Collisionless Anomalous Perpendicular Diffusion of Cosmic Rays

The Parker spiral model for interplanetary magnetic fields gives a first-order approximation to the magnetic field geometry in the solar wind (SW). How much real magnetic fields deviate from a Parker spiral in the turbulent SW, however, remains a broadly open question from the theory-modeling viewpoint. By generalizing to three-dimensional space a method, recently introduced, that allows for the simulation of realistic turbulent field lines with as many as 1015 turbulent modes, the distributions of the magnetic field orientations relative to the Parker spiral fields or mean fields are computed in quiet slow and fast SW at 0.3 and 1 AU. A comparison of these distributions with distributions determined from Helios 2 in situ data shows excellent agreement, further confirming the validity of the method. The angle distributions in three-dimensional space are relatively broad and peak away from 0°. The locations of these peaks, anywhere between 5° and 65° on timescales of 1.5 minutes to 10 hr in quiet wind within 1 AU from the Sun, are studied as a function of the SW conditions, length of the field segments, and realization of the turbulent field. Analytical predictions for the statistical distributions of field-line orientations are also made and compared to the numerical results.

Distributions of Magnetic Field Orientations in the Turbulent Solar Wind

A quantitative comparison of the magnetic field-line mean cross-field displacements predicted by the generalized quasilinear theory and of those deduced from Helios 2 in situ measurements is presented for the quiet slow solar wind (SW) at 0.3 and 1 AU. While the supradiffusivity of the magnetic field lines, X2 ∝ (Δz)μ with 2 ≥ μ > 1, is confirmed on all field-aligned scales Δz < 0.1 AU, the precise amplitude of the mean cross-field displacement X may not be obtained without some correction of the in situ measured spectra at frequencies <10-4 Hz. The SW cross flow and resulting transverse decorrelation and spectral flattening would cause the global amplitude of X to be underestimated by up to a factor 2. The value of the transport exponent μ below 0.01 AU is little affected by the uncertainty at very low frequencies; it is very close to the value determined by the spectrum around (Δz)-1. By plotting segments of wandering magnetic field lines computed from Helios 2 data as perturbations to Parker spiral fields, a lower limit of the order of 20° is also established for the median angular deviation of the wandering field lines. The cumulative angular deviation of a magnetic field line wandering from the source region of impulsive solar energetic particles (SEPs) out to 1 AU may, in some instances, be a few times larger. It occurs in longitude and latitude and could explain the relatively low magnetic connectivity of impulsive SEPs. The variability of the slow SW turbulence is also demonstrated.

Mean Cross-Field Displacement of Magnetic Field Lines in Slow Solar Wind: A Confirmation of the Supradiffusion Predicted by the Generalized Quasilinear Theory

Charged energetic particles propagating in solar wind magnetic fields span field irregularities down to very short turbulent scales, not described by the original quasi-linear theory for weak magnetic turbulence. This theory only predicts a field line diffusion on the largest scales, well above the correlation length, inverse of the spectral flattening wavenumber. The quasi-linear prediction for the transport and behavior of magnetic field lines is generalized here to all scales and arbitrary three-dimensional turbulence spectra. New analytical expressions are derived for the field line mean square cross-field displacement Δx2, and analytical proof is presented for the anomalous transport of the field lines. We find Δx2 ∝ (Δz)β, where Δz is the elapsed distance along the average field and β, the transport exponent, can take any value between 0 and 2. A decreasing turbulence spectrum results in a field line supradiffusion (β > 1), while an inverted spectrum implies a subdiffusion (β < 1). Simple expressions are derived for the transport exponent and coefficient. A powerful new method is presented to compute magnetic field lines in the quasi-linear regime of turbulence that allows rapid computation of field lines generated from any three-dimensional turbulence spectrum, including some 1015 modes and more. Individual field lines computed with this method show how a spectral steepening results in a smoothing of the field lines and how harder spectra give increasingly more short-scale fluctuations. The field line self-similarity, characteristic of power-law spectra, is demonstrated visually, and the anomalous transport of the field lines is confirmed numerically.

https://iopscience.iop.org/article/10.1086/504506/pdf

B. R. Ragot

Papers

Quasi-linear Magnetic Field Lines: Anomalous Transport and Self-Similarity, from the Analytic to the Numerical