The Diffusion Approximation Versus the Telegraph Equation for Modeling Solar Energetic Particle Transport With Adiabatic Focusing. I. Isotropic Pitch-Angle Scattering

Abstract

The diffusion approximation to the Fokker-Planck equation is commonly used to model the transport of solar energetic particles in interplanetary space. In this study, we present exact analytical predictions of a higher order telegraph approximation for particle transport and compare them with the corresponding predictions of the diffusion approximation and numerical solutions of the full Fokker-Planck equation. We specifically investigate the role of the adiabatic focusing effect of a spatially varying magnetic field on an evolving particle distribution. Comparison of the analytical and numerical results shows that the telegraph approximation reproduces the particle intensity profiles much more accurately than does the diffusion approximation, especially when the focusing is strong. However, the telegraph approximation appears to offer no significant advantage over the diffusion approximation for calculating the particle anisotropy. The telegraph approximation can be a useful tool for describing both diffusive and wavelike aspects of the cosmic-ray transport.

Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport

“…The adiabatic focusing effect is in some sense analogous to an initial anisotropic distribution in which it introduces an asymmetry into the transport problem, causing the maximum of the particle density profile to shift from the injection location. Effenberger and Litvinenko 17 compared the diffusion and telegraph analytical solutions with the numerical solution of the Fokker-Planck equation for focused particle transport. Figures 2 and 3 by Effenberger and Litvinenko, 17 for instance, show that the telegraph model reproduces the shape of an evolving density pulse much better than does the diffusion model, especially when focusing is strong, even for times significantly exceeding the scattering time.…”

“…For simple initial conditions, the inverse transform in terms of Bessel functions is well known for the telegraph equation. 17,19 Malkov and Sagdeev 26 pointed out that the fundamental solution G 0 for a generalized telegraph model, derived by Litvinenko and Schlickeiser, 25 does not conserve the total particle number. Of course, this is a well-known property of the fundamental solution, 37 and the solution of an initial-value problem, based on G 0 , does conserve the particle number.…”

“…Of course, this is a well-known property of the fundamental solution, 37 and the solution of an initial-value problem, based on G 0 , does conserve the particle number. 17 Although the inverse transform of the solution of the hyperdiffusion equation probably can be expressed in terms of standard special functions, we are only aware of an asymptotic analytical expression, given in Equation (41) …”

“…8,[10][11][12][13][14] Numerical studies both confirmed the accuracy of the approximation in a relevant parameter range 15 and illustrated its breakdown in the case of strong adiabatic focusing, caused by the mirror force in a nonuniform magnetic field. 16,17 Because the signal propagation speed is infinite in the diffusion limit, attempts to obtain a more accurate description of cosmic-ray evolution are often made use of the telegraph equation and related hyperbolic partial differential equations, characterized by a finite signal propagation speed. Alternative derivations of a modified telegraph equation from the Fokker-Planck equation had been given, which employed different expansion schemes and explored parameter regimes appropriate in concrete applications.…”

“…These features, ultimately caused by a finite particle speed, cannot be reproduced in the diffusion approximation but are captured in the telegraph approximation. 16,17 The hyperbolic nature of the telegraph equation denotes that the telegraph approximation typically yields solutions that contain singular components or sharp spikes at propagating diffusion fronts. 27 Although the d-functional singularities of the analytical solutions cannot be resolved numerically, the solutions of boundary value problems are also predicted to contain discontinuities that are absent in the solutions of the underlying Fokker-Planck equation.…”