A Generalized Diffusion Tensor for Fully Anisotropic Diffusion of Energetic Particles in the Heliospheric Magnetic Field

Abstract

The spatial diffusion of cosmic rays in turbulent magnetic fields can, in the most general case, be fully anisotropic, i.e. one has to distinguish three diffusion axes in a local, field-aligned frame. We reexamine the transformation for the diffusion tensor from this local to a global frame, in which the Parker transport equation for energetic particles is usually formulated and solved. Particularly, we generalize the transformation formulas to allow for an explicit choice of two principal local perpendicular diffusion axes. This generalization includes the 'traditional' diffusion tensor in the special case of isotropic perpendicular diffusion. For the local frame, we motivate the choice of the Frenet-Serret trihedron which is related to the intrinsic magnetic field geometry. We directly compare the old and the new tensor elements for two heliospheric magnetic field configurations, namely the hybrid Fisk and the Parker field. Subsequently, we examine the significance of the different formulations for the diffusion tensor in a standard 3D model for the modulation of galactic protons. For this we utilize a numerical code to evaluate a system of stochastic differential equations equivalent to the Parker transport equation and present the resulting modulated spectra. The computed differential fluxes based on the new tensor formulation deviate from those obtained with the 'traditional' one (only valid for isotropic perpendicular diffusion) by up to 60% for energies below a few hundred MeV depending on heliocentric distance.

Modulation of galactic cosmic rays during the unusual solar minimum between cycles 23 and 24

“…Here we assume different values of the parameters a and b for nonaxisymmetric perpendicular diffusion because of nonaxisymmetry of turbulence [e.g., Matthaeus et al, 2003] or the background magnetic field. Note that Effenberger et al [2012a] also discussed the effects of different perpendicular diffusion coefficients.…”

High-rigidity Forbush decreases: due to CMEs or shocks?

“…We note, in particular, that their results not only reproduce the standard results of Giacalone & Jokipii (1999) and Casse et al (2002) but also extend the regime of validity to include strong turbulence and high rigidities. Our approach is similar to that of Effenberger et al (2012), who adopt empirical expressions for the perpendicular diffusion coefficients. It should be mentioned, however, that they allow for the possibility of anisotropic perpendicular diffusion.…”

Anisotropic diffusion of Galactic cosmic ray protons and their steady-state azimuthal distribution

“…If the two perpendicular diffusion coefficients are not equal, this transformation is of particular importance in establishing the appropriate orientation in the calculation of the global diffusion tensor. Recently, Effenberger et al (2012) established a generalized scheme based on the local field geometry to account for this. In the present study, however, both perpendicular diffusion coefficients are set equal to reduce the set of unknown parameters (connected to the unknown detailed turbulence properties in the ISM), i.e.…”