Perpendicular Diffusion Coefficient of Comic Rays: The Presence of Weak Adiabatic Focusing

Abstract

The influence of adiabatic focusing on particle diffusion is an important topic in astrophysics and plasma physics. In the past several authors have explored the influence of along-field adiabatic focusing on parallel diffusion of charged energetic particles. In this paper by using the Unified NonLinear Transport (UNLT) theory developed by Shalchi (SH2010) and the method of He and Schlickeiser (HS2014) we derive a new nonlinear perpendicular diffusion coefficient for nonuniform background magnetic field. This formula demonstrates that particle perpendicular diffusion coefficient is modified by along-field adiabatic focusing.For isotropic pitch-angle scattering and weak adiabatic focusing limit the derived perpendicular diffusion coefficient is independent of the sign of adiabatic focusing characteristic length. For two-component model we simplify the perpendicular diffusion coefficient up to second order of the power series of adiabatic focusing characteristic quantity. We find that the first order modifying factor is equal to zero and the sign of the second one is determined by the energy of particles.

On the Pitch-angle-dependent Perpendicular Diffusion Coefficients of Solar Energetic Protons in the Inner Heliosphere

“…where subscript i denotes an axis of some Cartesian coordinate system, v is the component of the particle velocity along said axis, and the angle brackets an ensemble-averaging process. For an alternative approach of incorporating the effects of adiabatic focusing, see Wang et al (2017). In this study, a coordinate system with the z-axis along the uniform component Ideally, an exact expression for the velocity components should be employed in Equation (1).…”

“…Due to the significance of pitch-angle scattering in the transport of SEPs, many studies focus on the pitch-angle diffusion coefficient D μμ and its role in their diffusion parallel to some mean field (e.g., Karimabadi et al 1992;Reames 1999;He & Wan 2012;Agueda & Vainio 2013). To date, however, fewer studies have focused on the pitch-angledependent Fokker-Planck perpendicular diffusion coefficient D ⊥ (see, however, Qin & Shalchi 2014;Wang et al 2017), which enters directly into the Skilling (1971) equation solved in computational studies of SEP transport and, thus, could potentially play a crucial role in studies of space weather (e.g., Schwadron et al 2014;Laitinen et al 2018). These particle transport simulations, however, have shown that calculated SEP intensities and anisotropies are sensitive to the magnitude as well as the pitch-angle dependence of this quantity (Zhang et al 2009;Dresing et al 2012;Laitinen et al 2013Laitinen et al , 2018Strauss & Fichtner 2015;Strauss et al 2017).…”

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

“…By introducing the linear phase space density f (z, µ, t) = f 0 (z, µ, t)/B 0 , from Equation (2), the modified Fokker-Planck equation for the distribution function of energetic charged particles can be obtained (Kunstmann 1979;He & Schlickeiser 2014;Wang et al 2017b;Wang & Qin 2018)…”

“…With strong pitch-angle scattering, the gyro-tropic cosmic-ray phase space density f ( x, µ, t) can be split into the dominant isotropic part F( x, t) and the subordinate anisotropic part g( x, µ, t) (see, e.g., Schlickeiser et al 2007;Schlickeiser & Shalchi 2008;He & Schlickeiser 2014;Wang et al 2017b;…”

“…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.…”