# Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model

## Abstract

We investigate analytically and numerically the transport of cosmic rays following their escape from a shock or another localized acceleration site. Observed cosmic-ray distributions in the vicinity of heliospheric and astrophysical shocks imply that anomalous, superdiffusive transport plays a role in the evolution of the energetic particles. Several authors have quantitatively described the anomalous diffusion scalings, implied by the data, by solutions of a formal transport equation with fractional derivatives. Yet the physical basis of the fractional diffusion model remains uncertain. We explore an alternative model of the cosmic-ray transport: a nonlinear diffusion equation that follows from a self-consistent treatment of the resonantly interacting cosmic-ray particles and their self-generated turbulence. The nonlinear model naturally leads to superdiffusive scalings. In the presence of convection, the model yields a power-law dependence of the particle density on the distance upstream of the shock. Although the results do not refute the use of a fractional advection–diffusion equation, they indicate a viable alternative to explain the anomalous diffusion scalings of cosmic-ray particles.

- Authors:

- Department of Mathematics, University of Waikato, P. B. 3105, Hamilton 3240 (New Zealand)
- Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum (Germany)

- Publication Date:

- OSTI Identifier:
- 22663570

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 841; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCELERATION; ASTROPHYSICS; COMPUTERIZED SIMULATION; CONVECTION; COSMIC RADIATION; DENSITY; DIFFUSION; DIFFUSION EQUATIONS; DISTRIBUTION; EVOLUTION; NONLINEAR PROBLEMS; TRANSPORT THEORY; TURBULENCE

### Citation Formats

```
Litvinenko, Yuri E., Fichtner, Horst, and Walter, Dominik.
```*Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model*. United States: N. p., 2017.
Web. doi:10.3847/1538-4357/AA71BA.

```
Litvinenko, Yuri E., Fichtner, Horst, & Walter, Dominik.
```*Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model*. United States. doi:10.3847/1538-4357/AA71BA.

```
Litvinenko, Yuri E., Fichtner, Horst, and Walter, Dominik. Sat .
"Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model". United States.
doi:10.3847/1538-4357/AA71BA.
```

```
@article{osti_22663570,
```

title = {Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model},

author = {Litvinenko, Yuri E. and Fichtner, Horst and Walter, Dominik},

abstractNote = {We investigate analytically and numerically the transport of cosmic rays following their escape from a shock or another localized acceleration site. Observed cosmic-ray distributions in the vicinity of heliospheric and astrophysical shocks imply that anomalous, superdiffusive transport plays a role in the evolution of the energetic particles. Several authors have quantitatively described the anomalous diffusion scalings, implied by the data, by solutions of a formal transport equation with fractional derivatives. Yet the physical basis of the fractional diffusion model remains uncertain. We explore an alternative model of the cosmic-ray transport: a nonlinear diffusion equation that follows from a self-consistent treatment of the resonantly interacting cosmic-ray particles and their self-generated turbulence. The nonlinear model naturally leads to superdiffusive scalings. In the presence of convection, the model yields a power-law dependence of the particle density on the distance upstream of the shock. Although the results do not refute the use of a fractional advection–diffusion equation, they indicate a viable alternative to explain the anomalous diffusion scalings of cosmic-ray particles.},

doi = {10.3847/1538-4357/AA71BA},

journal = {Astrophysical Journal},

number = 1,

volume = 841,

place = {United States},

year = {Sat May 20 00:00:00 EDT 2017},

month = {Sat May 20 00:00:00 EDT 2017}

}