Effect of Adiabatic Focusing on the Diffusion of Energetic Charged Particles

Abstract

An approximate analytic description of a diffusion coefficient, including the effect of adiabatic focusing, has been developed. This description is formulated with the aid of stochastic differential equations and the steady perturbation solution of the Fokker-Plank transport equation. The analytical formula is based on three important assumptions. First, the pitch-angle diffusion coefficient is set to be separable from the spatial coordinate and the pitch-angle cosine. Second, the spatial dependence of the ratio between the mean-free path and focusing length is assumed to be weak. Third, the pitch-angle distribution relaxes quickly to a steady state. The new analytic formula could be applied to calculate the spatial diffusion coefficient in the interplanetary and interstellar space.

The Diffusion Coefficient with Displacement Variance of Energetic Particles Caused by Adiabatic Focusing

“…which is usually used in previous research (Schlickeiser et al 2007;Shalchi 2011;Litvinenko & Schlickeiser 2013;Effenberger & Litvinenko 2014;Malkov & Sagdeev 2015;Wang & Qin 2016). Here f 0 is the isotropic distribution function, t is time, z is the distance along the background magnetic field, µ = v z /v is the pitch-angle cosine with particle speed v and its z-component v z , D µµ (µ) is the pitch-angle diffusion coefficient, L(z) = −B 0 (z)/[dB 0 (z)/dz] is the adiabatic focusing characteristic length of the large-scale mag-netic field B 0 (z).…”

“…However, it is clear that the mean solar wind magnetic field is not constant in reality, especially when particles are close to the Sun. It is found that the spatially varying background solar wind magnetic fields lead to the adiabatic focusing effect of charged energetic particle transport and introduce correction to the particle diffusion coefficients (see, e.g., Roelof 1969;Earl 1976;Kunstmann 1979;Beeck & Wibberenz 1986;Bieber & Burger 1990;Kóta 2000;Schlickeiser & Shalchi 2008;Shalchi 2009bShalchi , 2011Litvinenko 2012a,b;Shalchi & Danos 2013;Wang & Qin 2016;Wang et al 2017b;Wang & Qin 2018). The adiabatic focusing effect causes a convection term in the energetic particles transport equation of the isotropic distribution function, so DCDV might be modified.…”

The Effect of Solar Wind on Charged Particles’ Diffusion Coefficients

“…Parallel and perpendicular diffusion are very important transport processes of energetic charged particles, so they are widely studied in plasma physics (Schlickeiser 2002;Qin 2007;Shalchi 2009;Qin & Shalchi 2014;Shalchi 2020b). In addition, the impacts of the along-field adiabatic focusing effect on parallel and perpendicular diffusion have been extensively studied (Roelof 1969;Earl 1976;Kunstmann 1979;Beeck & Wibberenz 1986;Bieber & Burger 1990;Kota 2000;Schlickeiser & Shalchi 2008;Shalchi 2011;Litvinenko 2012aLitvinenko , 2012bShalchi & Danos 2013;He & Schlickeiser 2014;Wang & Qin 2016;Wang et al 2017;Wang & Qin 2018, 2019.…”

Parallel and Perpendicular Diffusion Coefficients of Energetic Charged Particles with Adiabatic Focusing

“…One can show that the spatially varying background magnetic fields lead to the adiabatic focusing effect of charged energetic particle transport and introduces correction to the particle diffusion coefficients (see, e.g., Roelof 1969;Earl 1976;Kunstmann 1979;Beeck & Wibberenz 1986;Bieber & Burger 1990;Kóta 2000;Schlickeiser & Shalchi 2008;Shalchi 2009bShalchi , 2011Litvinenko 2012a,b;Shalchi & Danos 2013;Wang & Qin 2016;Wang et al 2017b). To explore the influence of adiabatic focusing on particle transport, perturbation method is frequently used (see, e.g., Beeck & Wibberenz 1986;Bieber & Burger 1990;Schlickeiser & Shalchi 2008;Schlickeiser & Jenko 2010;Litvinenko & Schlickeiser 2013;He & Schlickeiser 2014).…”