A nonlinear theory of the perpendicular diffusion of charged particles is presented, including the influence of parallel scattering and dynamical turbulence. The theory shows encouraging agreement with numerical simulations.

https://iopscience.iop.org/article/10.1086/376613/pdf

Nonlinear collisionless perpendicular diffusion of charged particles

The computation of charged particle orbits in model turbulent magnetic fields is used to investigate the properties of particle transport in the directions perpendicular to the large-scale magnetic field. Recent results by Qin, Matthaeus, & Bieber demonstrate that parallel scattering suppresses perpendicular diffusion to a subdiffusive level when the turbulence lacks transverse structure. Here numerical computations are used to show that in turbulence in which there is substantial transverse structure, a second regime of diffusive transport can be established. In both the subdiffusion regime and this second diffusion regime, perpendicular transport is intrinsically nonlinear. The regime of second diffusion persists for long times and may therefore be of interest in astrophysical transport problems such as the scattering and solar modulation of cosmic rays.

Perpendicular Transport of Charged Particles in Composite Model Turbulence: Recovery of Diffusion

[1] The force field approximation to the transport equation which describes cosmic ray modulation in the heliosphere is a widely used tool. It is popular because it provides an easy to use, quasi-analytical method to describe the level of modulation with a single parameter. A simple numerical solution of the one-dimensional cosmic ray transport equation is used to show that this is a good approximation for galactic cosmic rays in the inner heliosphere but that its accuracy decreases toward the outer heliosphere. On the other hand, the even simpler convection-diffusion approximation improves with radial distance. The reason for the complementary behavior of these two approximations is that energy losses are relatively important in the inner heliosphere but not in the outer heliosphere. The force field approximation is worse for anomalous cosmic rays at all radial distances due to the exponential cutoff of these spectra at high energies. The ranges of validity are quantified. Since both approximations have their limitations, a simple numerical solution of the one-dimensional transport equation is provided for general use.

Limitations of the force field equation to describe cosmic ray modulation

[1] Transport of charged particles across the mean magnetic field due to broad band, powerlaw-distributed magnetic fluctuations is examined by direct computation of a large number of charged test particle trajectories. Fluctuations consist of a one dimensional slab spectrum with an additional 0.01 percent level of two dimensional fluctuations; the net field has no ignorable directions. We find that, at long times (t), the mean square displacements subdiffusively scale as ∼t1/2. This confirms that perpendicular diffusion is suppressed when turbulence lacks strong three dimensional structure, in agreement with theories of Urch and of Kota and Jokipii.

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Subdiffusive transport of charged particles perpendicular to the large scale magnetic field

We present a new derivation of perpendicular diffusion and drift coefficients, based on the Taylor-Green-Kubo equation and physical arguments as to the form of the particle velocity correlation function. In contrast to previous derivations, which are restricted to particle rigidities greater than 4 GV for typical conditions at 1 AU, our derivation is valid for intermediate-energy, as well as high-energy, cosmic rays. Our results provide a natural explanation for several hitherto puzzling observations, including (1) the form of the perpendicular diffusion coefficient derived from measurements of anomalous cosmic rays, (2) the fact that drift models often describe observations better when drifts are reduced, and (3) the "CR-B" relation, which associates cosmic-ray modulation with enhanced magnetic fields in merged interaction regions.

https://iopscience.iop.org/article/10.1086/304464/pdf

Perpendicular Diffusion and Drift at Intermediate Cosmic-Ray Energies

A new development is given of the solution of the equation of the force-field approximation for the propagation of galactic cosmic rays in the interplanetary region. It leads to simpler methods for determining the force-field parameters. A method is given for determining the separable diffusion coefficient from observations of galactic electron spectrum and near-Earth electron spectra; it is shown that this diffusion coefficient is not unique but may have a periodic-like dependence upon rigidity; and the method is used to obtain diffusion coefficients for 1965 and 1968. Approximate formulae relating small changes in intensity and diffusion coefficient are developed and some applications of these noted; in one it is shown that the form of, and changes in, diffusion coefficient deduced previously for a neutron monitor event during June–September 1969 are unnecessarily constrained and therefore probably not correct.

A study of the force-field equation for the propagation of galactic cosmic rays

The diffusion of charged particles in a static turbulent magnetic field, which is superimposed on a constant magnetic fieldB0k, is considered. Previous calculations of the particle flux in a direction perpendicular tok have related the fluxS⊥ to the particle number densityf byS⊥ = − κ⊥(▽f)⊥ where κ⊥ is found from the power spectrum of the turbulent magnetic field. It is shown that this formula is inconsistent with the notion, developed by Jokipii and Parker (1969), that the perpendicular particle flux primarily arises because of random-walking of magnetic field lines across the directionk. For a simple example of a turbulent magnetic field it is shown that the above expression forS⊥ is incorrect; the particle fluxS⊥ is recalculated and a new relationship betweenS⊥ andf is found. This new expression forS⊥ is shown to be consistent with particle diffusion across the directionk being due to random-walking of the magnetic field lines.

Charged Particle Transport in Turbulent Magnetic Fields: The Perpendicular Diffusion Coefficient

The diffusion of charged particles through a weak stochastic electro-magnetic field which is superimposed on a constant background magnetic field\(B_o \hat k\) is considered. The stochastic electromagnetic fields are assumed to consist of unpolarized Alfven waves propagating at arbitrary angles to the direction\(\hat k\). When the Alfven waves are propagating in directions other than\(\hat k\) and the particle gyro-radius,rg, is sufficiently large (but may be smaller than the correlation length of the stochastic fields) it is shown that the particle flux perpendicular to the direction\(\hat k\) is\(S_ \bot = - \kappa _{_ \bot } (\nabla f)_{_{_ \bot } } + \tfrac{1}{3}\upsilon r_g \hat k \times \nabla f\), wherev is the particle speed andf the particle density. The expression forK⊥ differs from those calculated by previous authors. For small particle gyro-radii the flux S⊥ has a different functional form and is identical to that found by Urch (1977) to describe particle diffusion when the Alfven waves only propagate in the direction\(\hat k\).

Charged Particle Diffusion in the Presence of Alfvén Waves: The Perpendicular Particle Flux

The diffusion of charged particles in a turbulent magnetic field, but with no constant back-ground electromagnetic fields, is discussed and expressions for the particle fluxes calculated.

Charged Particle Diffusion in a Turbulent Magnetic Field

The Fokker-Planck equation which describes the motion of charged particles in a random electromagnetic field is derived from the Liouville equation by a new method. The size of the perturbing magnetic field, for the Fokker-Planck equation to be valid, is calculated in a regime appropriate for cosmic-ray diffusion.

I. H. Urch

Papers

A study of the force-field equation for the propagation of galactic cosmic rays

Charged Particle Transport in Turbulent Magnetic Fields: The Perpendicular Diffusion Coefficient

Charged Particle Diffusion in the Presence of Alfvén Waves: The Perpendicular Particle Flux

Charged Particle Diffusion in a Turbulent Magnetic Field

The Fokker-Planck equation for charged particles moving in random electromagnetic fields