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 cosmicray 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 cosmicray transport: a nonlinear diffusion equation that follows from a selfconsistent treatment of the resonantly interacting cosmicray particles and their selfgenerated turbulence. The nonlinear model naturally leads to superdiffusive scalings. In the presence of convection, the model yields a powerlaw 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 cosmicray particles.
 Authors:
 Department of Mathematics, University of Waikato, P. B. 3105, Hamilton 3240 (New Zealand)
 Institut für Theoretische Physik IV, RuhrUniversität Bochum, Universitätsstrasse 150, D44780 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:10.3847/15384357/AA71BA.
Litvinenko, Yuri E., Fichtner, Horst, & Walter, Dominik. Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model. United States. doi:10.3847/15384357/AA71BA.
Litvinenko, Yuri E., Fichtner, Horst, and Walter, Dominik. Sat .
"Anomalous Transport of Cosmic Rays in a Nonlinear Diffusion Model". United States.
doi:10.3847/15384357/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 cosmicray 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 cosmicray transport: a nonlinear diffusion equation that follows from a selfconsistent treatment of the resonantly interacting cosmicray particles and their selfgenerated turbulence. The nonlinear model naturally leads to superdiffusive scalings. In the presence of convection, the model yields a powerlaw 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 cosmicray particles.},
doi = {10.3847/15384357/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}
}

The original nonlinear guiding center (NLGC) theory by Matthaeus et al. was a breakthrough in establishing a theory that promised to reproduce for the first time the anomalous perpendicular diffusion results from test particle trajectory calculations in prescribed static magnetic field turbulence dominated by a twodimensional component. The assumptions used in this approach guaranteed anomalous diffusion to be a classical process (the variance ({Delta}x {sup 2}) {proportional_to} ({Delta}t){sup {alpha}} with {alpha} = 1). However, Shalchi and Kourakis showed that similar calculations can be even better reproduced within the context of a generalized compound diffusion model for anomalous perpendicular diffusion wherebymore »

Effects of drift on the transport of cosmic rays. VI. A threedimensional model including diffusion
We present the first results from a series of computer simulations of the solar modulation of galactic cosmic rays using a full threedimensional model which incorporates all known important effects on particle transport, particle drifts, convection with the solar wind, energy loss, and anisotropic diffusion. The model is timeindependent in the coordinate frame rotating with the Sun, so corotating effects can be studied. We consider modulation in an interplanetary magnetic field model in which the current sheet separating the northerm and southern solar hemispheres is warped and corotating with the Sun. The amplitude of the warp is varied to simulatemore » 
ANOMALOUS TRANSPORT OF HIGHENERGY COSMIC RAYS IN GALACTIC SUPERBUBBLES. I. NUMERICAL SIMULATIONS
We present a simple continuoustime randomwalk model for the transport of energetic particles accelerated by a collection of supernova explosions in a galactic superbubble, developed to simulate and highlight signatures of anomalous transport on the particles' evolution and their spectra in a multishock context. We assume standard diffusive shock acceleration (DSA) theory for each shock encounter. The superbubble (an OB stars association) is idealized as a heterogeneous region of particle sources and sinks bounded by a random surface. The model is based on two coupled stochastic differential equations and is applied for protons and alpha particles. Using characteristic values formore » 
Anomalous component of lowenergy cosmic rays: A comparison of observed spectra with model calculations
Using current modulation theory for the transport of lowenergy cosmic ray particles and the model of Fisk (1976a) for their acceleration by transit time damping largescale field variations in the outer solar system, we construct a set of parameters which reproduce the quiet time spectra of He, O, N, and Ne as observed at 1 AU during the time period 19731975. With an analytic approximation for the acceleration and a numerical solution for the steady state spherical symmetric transport equation, both the observed spectral shapes and the relative intensities for He, O, N, and Ne can be fitted simultaneously remarkablymore »