Diffusion of Relativistic Charged Particles and Field Lines in Isotropic Turbulence. I. Numerical Simulations

Context. The dependence of the parallel and perpendicular mean free paths on the particle rigidity is an important topic in the studies of the diffusion and propagation of charged energetic particles in a large-scale turbulent magnetic field.Aims. In this work, we investigate the dependence of the parallel and perpendicular mean free paths on the rigidity of solar energetic particles (SEPs) by means of both the theoretical model and spacecraft observations with regard to several typical SEP events.Methods. A direct method developed by previous studies and derived from the focused transport equation and Taylor-Green-Kubo (TGK) formulation is applied to explicitly determine the parallel and perpendicular mean free paths of SEPs in a turbulent and spatially varying magnetic field.Results. We find that the parallel and perpendicular mean free paths, λ ∥ and λ ⊥ , of energetic protons monotonically decrease with increasing particle rigidity, and the ratio λ ⊥ /λ ∥ monotonically increases with particle rigidity, when the magnetic turbulence is weak. Taking a series of typical SEP events together, it can also be seen that the ratio λ ⊥ /λ ∥ of the perpendicular to the parallel mean free paths remains in the range of 0.001−0.2.

We present direct numerical simulations of charged-particle transport in a turbulent magnetic field. The magnetic field model used in the simulations consists of a composite of statistically homogeneous slab and two-dimensional turbulence representative of solar wind conditions at Earth. This turbulent magnetic field is then added to a uniform background magnetic field. We find that the parallel and perpendicular mean free paths are well described by power laws as a function of rigidity at different turbulence levels. At a low level of turbulence we find that quasi-linear theory and the field line random walk theory for the parallel and perpendicular mean free paths, respectively, provide predictions that are in good agreement with the simulated mean free paths. At intermediate turbulence levels the simulated parallel and perpendicular mean free paths are best accounted for by recently proposed nonlinear theories, while quasi-linear theory and the field line random walk theory overestimate the simulated mean free paths. At high turbulence levels neither quasi-linear theory and the field line random walk theory nor the nonlinear theories provide predictions that are in good agreement with the simulated parallel and perpendicular mean free paths.

Recently, several non-linear theories for cosmic ray transport have been proposed to achieve agreement with numerical test particle simulations. Some of these theories, which can be seen as an improvement of quasi-linear theory, agree with simulations. Because recent reports have shown that the perpendicular mean free path could be larger than the parallel mean free path, the assumption of strong turbulence could be reasonable. Therefore, a comparison between different theories and simulations done in the strong turbulence parameter regime for slab/2D composite geometry is presented. It is shown that transport theories and simulations can explain that the perpendicular mean free path can become larger than the parallel mean free path. Further, it is demonstrated that theories cannot reproduce simulations for strong turbulence with high accuracy.

During 2010 August a series of solar particle events was observed by the two STEREO spacecraft as well as near-Earth spacecraft. The events, occurring on August 7, 14, and 18, originated from active regions 11093 and 11099. We combine in situ and remote-sensing observations with predictions from our model of three-dimensional anisotropic particle propagation in order to investigate the physical processes that caused the large angular spreads of energetic electrons during these events. In particular, we address the effects of the lateral transport of the electrons in the solar corona that is due to diffusion perpendicular to the average magnetic field in the interplanetary medium. We also study the influence of two coronal mass ejections and associated shock waves on the electron propagation, and a possible time variation of the transport conditions during the above period. For the August 18 event we also utilize electron observations from the MESSENGER spacecraft at a distance of 0.31 au from the Sun for an attempt to separate between radial and longitudinal dependencies in the transport process. Our modelings show that the parallel and perpendicular diffusion mean free paths of electrons can vary significantly not only as a function of the radial distance, but also of the heliospheric longitude. Normalized to a distance of 1 au, we derive values of λ ∥ in the range of 0.15–0.6 au, and values of λ ⊥ in the range of 0.005–0.01 au. We discuss how our results relate to various theoretical models for perpendicular diffusion, and whether there might be a functional relationship between the perpendicular and the parallel mean free path.

We employ a recently developed test-particle simulation code to explore the Bohm diffusion and the scattering anisotropy of charged particles interacting with magnetic turbulence. By computing parallel and perpendicular mean free paths, it is shown that a dominant mean magnetic field and a large particle mean free path along the mean magnetic field break the scattering symmetry. In such cases, a strong scattering anisotropy is found, which is in good agreement with recent analytical calculations. It is also shown that the assumption of the Bohm diffusion in isotropic turbulence is only valid for some special cases.

A nonlinear guiding center (NLGC) theory for diffusion of charged particles perpendicular to the mean magnetic field was recently proposed. Here, we draw attention to a number of attractive features of this theory: (1) The theory provides a natural mechanism to connect the perpendicular mean free path with the parallel mean free path. In fact, the parallel mean free path is the only particle property required to determine uniquely the perpendicular mean free path. (2) Under a broad range of conditions, the theory predicts that the perpendicular mean free path will be of order one percent or a few percent of the parallel mean free path, in agreement with numerical simulations of particle transport. (3) For conditions representative of the inner heliosphere, the theory predicts values of the perpendicular mean free path in agreement with observational determinations from Jovian electrons and from modeling Ulysses observations of Galactic protons.

We have solved the two‐dimensional time‐dependent diffusion‐convection equation numerically to obtain the distribution and anisotropy of cosmic rays in the heliosphere. We have assumed that the parallel and perpendicular mean free paths are proportional to the particle Larmor radius, and we have treated each proportionality constant (a, b) as a parameter. We have found that the set (a, b) = (4, 2) gives the steady state solution compatible with observations on the intensity and the solar diurnal anisotropy of cosmic rays in 0.5‐ to 10‐GeV range as obtained at the earth. This set of (a, b) corresponds to the ratio of the diffusion coefficients D∥/D⊥ = 10. In our solution the intensity for the (pre‐1980) interplanetary magnetic field (IMF) state where the solar magnetic dipole and the angular velocity vector are parallel is higher than for the (post‐1980) state where they are antiparallel, while the phase of the diurnal anisotropy is about 15 hours for the parallel state and about 18 hours for the antiparallel state. We have also reproduced the observed small radial gradient for each IMF state. We discuss the nature of the solution in order to understand the effect of the density gradient drift motion on the cosmic ray distribution.

A detailed study of solar wind turbulence throughout the heliosphere in both the upwind and downwind directions is presented. We use an incompressible magnetohydrodynamic (MHD) turbulence model that includes the effects of electrons, the separation of turbulence energy into proton and electron heating, the electron heat flux, and Coulomb collisions between protons and electrons. We derive expressions for the turbulence cascade rate corresponding to the energy in forward and backward propagating modes, the fluctuating kinetic and magnetic energy, the normalized cross-helicity, and the normalized residual energy, and calculate the turbulence cascade rate from 0.17 to 75 au in the upwind and downwind directions. Finally, we use the turbulence transport models to derive cosmic ray (CR) parallel and perpendicular mean free paths (mfps) in the upwind and downwind heliocentric directions. We find that turbulence in the upwind and downwind directions is different, in part because of the asymmetric distribution of new born pickup ions in the two directions, which results in the CR mfps being different in the two directions. This is important for models that describe the modulation of cosmic rays by the solar wind.

A key problem of cosmic ray astrophysics is the explanation of measured parallel and perpendicular mean free paths in the heliosphere. Previous approaches used quasilinear theory in combination with simple turbulence models to reproduce heliospheric observations. Because of recent progress in transport and turbulence theory we present linear and nonlinear diffusion coefficients within an improved dynamical turbulence model to demonstrate that the observed mean free paths can indeed be reproduced theoretically.

Novel insights into the behavior of the diffusion coefficients of charged particles in the inner heliosphere are of great importance to any study of the transport of these particles and are especially relevant with regard to the transport of low-energy electrons. The present study undertakes an exhaustive investigation into the diffusion parameters needed to reproduce low-energy electron intensities as observed at Earth, using a state-of-the-art 3D cosmic ray transport code. To this end, the transport of Jovian electrons is considered, as Jupiter represents the predominant source of these particles in the inner heliosphere, and because a careful comparison of model results with observations taken during periods of good and poor magnetic connectivity between Earth and Jupiter allows for conclusions to be drawn as to both parallel and perpendicular diffusion coefficients. This study then compares these results with the predictions made by various scattering theories. Best-fit parameters for parallel and perpendicular mean free paths at 1 au fall reasonably well within the span of observational values reported by previous studies, but best-fit radial and rigidity dependences vary widely. However, a large number of diffusion parameters lead to reasonable to-good fits to observations, and it is argued that considerable caution must be exercised when comparing theoretical results for diffusion coefficients with diffusion parameters calculated from particle transport studies.

view Abstract Citations (20) References (26) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Acceleration and transport of galactic cosmic rays. Jokipii, J. R. ; Higdon, J. C. Abstract A recently proposed cosmic-ray acceleration mechanism is combined with a model of transport and loss from the Galaxy to construct a model for galactic cosmic rays. Various aspects of the cosmic-ray spectrum, including the secondary-to-primary ratio as a function of energy, are found to fit naturally into the overall picture. A solution to the dynamical-halo model with energy-dependent diffusion coefficient (kappa) is presented. This model is consistent with the observed spectra if kappa is proportional to the square root of the kinetic energy. The observed flattening of the secondary-to-primary ratio below about 1-2 GeV per nucleon is shown to be a natural consequence of the model. Similarly, the primary electron spectrum is predicted to flatten below about 1-2 GeV, which is consistent with deconvolution of the local nonthermal radio spectrum, although the flattening is not quite as much as indicated by the observations. Publication: The Astrophysical Journal Pub Date: February 1979 DOI: 10.1086/156845 Bibcode: 1979ApJ...228..293J Keywords: Acceleration (Physics); Astronomical Models; Cosmic Rays; Galactic Radiation; Shock Wave Propagation; Transport Properties; Astronomical Spectroscopy; Dynamic Models; Electron Energy; Energy Spectra; Halos; Nonthermal Radiation; Space Radiation; Cosmic Rays:Acceleration; Cosmic Rays:Galaxy; Cosmic Rays:Propagation full text sources ADS |

Observations of cosmic rays and other energetic charged particles in the heliosphere over the past decade have created new challenges to the standard theoretical paradigms for energetic‐particle transport. Certainly, some of these will be resolved using the standard Parker (diffusive) transport equation applied to increasingly sophisticated models of heliospheric phenomena. For example, we can apparently understand the modulation of galactic cosmic rays and the acceleration and transport of galactic cosmic rays in terms of this paradigm. Cosmic‐ray reaction back on the plasma can also fit into this paradigm. However, it is also becoming increasingly clear that in some situations the diffusion approximation is not strictly valid. The scattering mean free paths may be large or there are significant anisotropies. For example, observations of solar energetic‐particle events show non‐diffusive effects, particularly in the early phases. Recently observed enhancements on Voyager 1, attributed to the proximity of...

The problem of particle transport perpendicular to a magnetic background field is well known in cosmic-ray astrophysics. Whereas it is widely accepted that quasi-linear theory (QLT) of particle transport does not provide the correct results for perpendicular diffusion, it was assumed for a long time that QLT is the correct theory for parallel diffusion. In the current paper we demonstrate that QLT is in general also incorrect for parallel particle transport if we consider composite turbulence geometry. Motivated through the recent success of the so-called nonlinear guiding center theory of perpendicular diffusion, we present a new theory for parallel and perpendicular diffusion of cosmic rays. This new theory is a nonlinear extension of QLT and provides us with a coupled system of nonlinear Fokker-Planck coefficients. By solving the resulting system of integral equations we obtain new results for the pitch-angle Fokker-Planck coefficient and the Fokker-Planck coefficient of perpendicular diffusion. By integrating over pitch angle we calculate the parallel and perpendicular mean free path. To our knowledge the new theory is the first that can deal with both parallel and perpendicular diffusion in agreement with simulations.

Determining transport coefficients for galactic cosmic ray (GCR) propagation in the turbulent interplanetary magnetic field (IMF) poses a fundamental challenge in modeling cosmic ray modulation processes. GCR scattering in the solar wind involves wave‐particle interaction, the waves being Alfven waves which propagate along the ambient field (B). Empirical values at 1 AU are determined for the components of the diffusion tensor for GCR propagation in the heliosphere using neutron monitor (NM) data. At high rigidities, particle density gradients and mean free paths at 1 AU in B can only be computed from the solar diurnal anisotropy (SDA) represented by a vector A (components Ar, Aϕ, and Aθ) in a heliospherical polar coordinate system. Long‐term changes in SDA components of NMs (with long track record and the median rigidity of response Rm ~ 20 GV) are used to compute yearly values of the transport coefficients for 1963–2013. We confirm the previously reported result that the product of the parallel (to B) mean free path (λ||) and radial density gradient (Gr) computed from NM data exhibits a weak Schwabe cycle (11y) but strong Hale magnetic cycle (22y) dependence. Its value is most depressed in solar activity minima for positive (p) polarity intervals (solar magnetic field in the Northern Hemisphere points outward from the Sun) when GCRs drift from the polar regions toward the helioequatorial plane and out along the heliospheric current sheet (HCS), setting up a symmetric gradient Gθs pointing away from HCS. Gr drives all SDA components and λ|| Gr contributes to the diffusive component (Ad) of the ecliptic plane anisotropy (A). GCR transport is commonly discussed in terms of an isotropic hard sphere scattering (also known as billiard‐ball scattering) in the solar wind plasma. We use it with a flat HCS model and the Ahluwalia‐Dorman master equations to compute the coefficients α (=λ⊥/λ∥) and ωτ (a measure of turbulence in the solar wind) and transport parameters λ||, λ⊥, Gr, Gθs, and an asymmetric gradient Gθa normal to the ecliptic plane. We study their dependence on rigidity (R), p/n intervals, sunspot numbers (SSNs), and solar wind parameters at 1 AU. λ|| exhibits a strong 22y dependence but Gr does not, explaining solar polarity dependence of λ|| Gr. The computed Gr values are an order of magnitude greater than those reported by our colleagues making an ad hoc assumption that α is low (0.01). At high rigidities, the drift contribution at 1 AU is small and unsteady. A new methodology is outlined to compute yearly GCR north‐south anisotropy (Aθ) from the data for a single detector sorted for p/n intervals. We show that Gθa is the main contributor to Aθ in the steady state, and Gθa is shown not correlated with the north‐south excess SSNs.

Context. Cosmic rays, both solar and Galactic, have an ionising effect on the Earth’s atmosphere and are thought to be important in the production of prebiotic molecules. In particular, the H2-dominated atmosphere that follows an ocean-vaporising impact is considered favourable to prebiotic molecule formation. As a first step in determining the role that cosmic rays might have played in the origin of life, we need to understand the significance of their ionising effect. Aims. We model the transport of solar and Galactic cosmic rays through a post-impact early Earth atmosphere at 200 Myr. We aim to identify the differences in the resulting ionisation rates – particularly at the Earth’s surface during a period when the Sun was very active. Methods. We used a Monte Carlo model for describing cosmic ray transport through the early Earth atmosphere, giving the cosmic ray spectra as a function of atmospheric height. Using these spectra, we calculated the ionisation and ion-pair production rates as a function of height due to Galactic and solar cosmic rays. The Galactic and solar cosmic ray spectra are both affected by the Sun’s rotation rate, Ω, because the solar wind velocity and magnetic field strength both depend on Ω and influence cosmic ray transport. We considered a range of input spectra resulting from the range of possible rotation rates of the young Sun – from 3.5–15Ω⊙. To account for the possibility that the Galactic cosmic ray spectrum outside the Solar System is not constant over gigayear timescales, we compared the ionisation rate at the top of the Earth’s atmosphere resulting from two different scenarios. We also considered the suppression of the cosmic ray spectra by a planetary magnetic field. Results. We find that the ionisation and ion-pair production rates due to cosmic rays are dominated by solar cosmic rays in the early Earth atmosphere for most cases. The corresponding ionisation rate at the surface of the early Earth ranges from 5 × 10−21 s−1 for Ω = 3.5Ω⊙ to 1 × 10−16 s−1 for Ω = 15Ω⊙. Thus if the young Sun was a fast rotator (Ω = 15Ω⊙), it is likely that solar cosmic rays had a significant effect on the chemistry at the Earth’s surface at the time when life is likely to have formed. Conclusions. Cosmic rays, particularly solar cosmic rays, are a source of ionisation that should be taken into account in chemical modelling of the post-impact early Earth atmosphere. Modelling of cosmic ray transport and effects on chemistry will also be of interest for the characterisation of H2-dominated exoplanet atmospheres.