We present a new analysis of the transport of cosmic rays in a turbulent magnetic field that varies in all three spatial dimensions. The analysis utilizes a numerical simulation that integrates the trajectories of an ensemble of test particles from which we obtain diffusion coefficients based on the particle motions. We find that the diffusion coefficient parallel to the mean magnetic field is consistent with values deduced from quasi-linear theory, in agreement with earlier work. The more interesting and less understood transport perpendicular to the average magnetic field is found to be enhanced (above the classical scattering result) by the random walk, or braiding, of the magnetic field. The value of κ⊥ obtained is generally larger than the classical scattering value but smaller than the quasi-linear value. The computed values of κ⊥/κ∥, for a representation of the interplanetary magnetic field, are 0.02-0.04; these values are of the same general magnitude as those assumed in recent numerical simulations of cosmic-ray modulation and transport in the heliosphere, and give reasonable agreement with spacecraft observations of cosmic rays. Some consequences of these results for the interpretation of heliospheric observations are discussed.

https://iopscience.iop.org/article/10.1086/307452/pdf

The Transport of Cosmic Rays across a Turbulent Magnetic Field

Energetic particles are accelerated in rich profusion at sites throughout the heliosphere. They come from solar flares in the low corona, from shock waves driven outward by coronal mass ejections (CMEs), from planetary magnetospheres and bow shocks. They come from corotating interaction regions (CIRs) produced by high-speed streams in the solar wind, and from the heliospheric termination shock at the outer edge of the heliospheric cavity. We sample all these populations near Earth, but can distinguish them readily by their element and isotope abundances, ionization states, energy spectra, angular distributions and time behavior. Remote spacecraft have probed the spatial distributions of the particles and examined new sources in situ. Most acceleration sources can be "seen" only by direct observation of the particles; few photons are produced at these sites. Wave-particle interactions are an essential feature in acceleration sources and, for shock acceleration, new evidence of energetic-proton-generated waves has come from abundance variations and from local cross-field scattering. Element abundances often tell us the physics the source plasma itself, prior to acceleration. By comparing different populations, we learn more about the sources, and about the physics of acceleration and transport, than we can possibly learn from one source alone.

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Particle Acceleration at the Sun and in the Heliosphere

This paper presents a coherent exposition of the modern statistical theory of the transport of fast charged particles (cosmic rays) in the solar wind. Observations are discussed only as they illustrate the phenomena under discussion. A brief introductory section surveys the historical development of the theory. The dominant effect on the motion of cosmic rays in the solar wind is the interplanetary magnetic field, which is irregular and which is therefore best treated statistically, using random functions. The magnetic irregularities scatter the cosmic rays in pitch angle, so that to a good approximation the cosmic rays diffuse through the irregular magnetic field. Using a statistical analysis of the equations of motion, one may relate the diffusion tensor to the power spectrum of the magnetic field, which is in principle measurable. The resulting general transport theory relates the motion of cosmic rays, statistically, to the solar-wind velocity and magnetic field. Application of the theory both to the modulation of galactic cosmic rays by the solar wind and to the propagation of solar cosmic rays is discussed in detail. It is concluded that the present theory explains the principal phenomena quite well. Future theoretical work will probably be devoted to obtaining better solutions of the equations, to obtaining better values of the parameters, and to studying higher-order or more subtle effects.

Propagation of cosmic rays in the solar wind

A theory for the self-consistent configuration of upstream hydromagnetic waves, upstream energetic storm particle (ESP) events, and downstream postshock ion enhancements at interplanetary traveling shocks is presented. The observations of upstream ultralow frequency waves and those ESP events and postshock enhancements which exhibit approximately isotropic ion distributions in the solar wind or shock frame are briefly reviewed. The theory of Lee (1982) for application to interplanetary traveling shocks is modified and analytical solutions for the wave spectrum as a function of wavenumber and z are presented along with the ion omnidirectional distribution functions as functions of energy and z for all ion species. The theory quantitatively explaines the observed features of the shock-associted energetic ions and predicts the configuration of upstream hydromagnetic waves.

Coupled hydromagnetic wave excitation and ion acceleration at interplanetary traveling shocks

We extend the Lee (1982) self-consistent theory of upstream wave excitation and particle energization to address observations by Voyager 2 of sunward propagating MHD waves and diffuse suprathermal particle populations upstream of the Jovian bow shock. Two new ideas are incorporated into the theory. First, the interplanetary seed wave population is taken to be an equal admixture of waves propagating both toward and away from the shock parallel to the interplanetary magnetic field. Second, finite connection times are incorporated approximately into the theory in an effort to understand whether the particle spectra at high energy are limited by particle escape or finite connection time. It is found that finite connection times dominate the particle distribution at energies above 40 keV. In this manner the suprathermal proton distributions can be accounted for by a multiple reflection, shock acceleration theory. We find that the theory can also account for the low-frequency waves observed upstream of the shock in conjunction with the suprathermal ions.

Coupled hydromagnetic wave excitation and ion acceleration upstream of the earth's bow shock

The transport of charged particles diffusing in a random magnetic field parallel to a relatively large guiding field is presented. The same coefficient of diffusion is obtained by three methods. Two corrections must be added to the expression in which the diffusive flux is proportional to the gradient of the density. Explicit expressions are given for a characteristic time and a characteristic length which describe the corrections. The well known divergence of the coefficient of diffusion, which is implied by the quasilinear analysis of pitch angle scattering, does not occur if the scattering rate is finite at 90 deg pitch angle. This effect is illustrated by formulas which give the coefficient of diffusion when the quasilinear expression is perturbed by a variable amount of isotropic scattering.

/pdf/the-diffusive-idealization-of-charged-particle-transport-in-2haju7axpa.pdf

The diffusive idealization of charged particle transport in random magnetic fields

Charged particles propagating along the diverging lines of force of a spatially inhomogeneous guiding field were considered as they are scattered by random fields. Their longitudinal transport is described in terms of the eigenfunctions of a Sturm-Liouville operator incorporating the effect of adiabatic focussing along with that of scattering. The relaxation times and characteristic velocities are graphed and tabulated. The particle density is evaluated as a function of space and time for two different regimes. In the first regime (relatively weak focussing), a diffusive mode of propagation is dominant but coherent modes are also dominant. In the second regime (strong focussing), diffusion does not occur and the propagation is purely coherent. This supercoherent mode corresponds exactly to the so-called scatter-free propagation of kilovolt solar flare electrons. On a larger scale, focussed transport provides an interpretation of many observed characteristics of extragalactic radio sources.

/pdf/the-effect-of-adiabatic-focusing-upon-charged-particle-1rkxzzyuzk.pdf

The effect of adiabatic focusing upon charged particle propagation in random magnetic fields

This paper describes the effects of strong adiabatic focusing in terms of eigenfunctions of an operator which incorporates both scattering by random fields and focusing by a spatially inhomogeneous guiding field. It is found that focused transport differs from rectilinear transport in the following ways: (1) the coherent velocity toward stronger fields is larger than that toward weaker fields; (2) coherent effects are more prominent relative to diffusive effects when focusing is present; and (3) initial anisotropies at injection have a minimal influence upon rectilinear transport but they have a pronounced effect on focused transport. These features explain certain well-known discrepancies between the diffusive picture and the observed profiles of solar particle events. In particular, they explain the so-called scatter-free propagation of kilovolt flare electrons.

The transport equation for cosmic rays scattered by magnetic field irregularities carried in a rarefied conducting fluid has been reexamined. To lowest order in the ratio, U/w, of flow speed to random particle speed, the analysis gives the standard equation first derived by Parker (1965), but additional terms are found in the next order of this ratio. One new term, which reflects viscous damping of fluid motions by the energetic-particle gas, describes both a change in the mean particle momentum and a spreading around the mean. Other new terms, which derive from accelerations of the fluid, describe inertial drift and energy changes. Although these effects are small, they are potentially important at shocks, because they are proportional to the square of velocity derivatives. In addition, they can be significant for the case of pure velocity shear, in which the adiabatic energy change is zero.

J. A. Earl

Papers

The diffusive idealization of charged particle transport in random magnetic fields

The effect of adiabatic focusing upon charged particle propagation in random magnetic fields

The effect of adiabatic focusing upon charged particle propagation in random magnetic fields

Cosmic-ray viscosity

Cloud-Chamber Observations of Primary Cosmic-Ray Electrons