In the present study, the Parker (1965) cosmic-ray TPE is solved stochastically, modifying the model presented by Engelbrecht & Burger (2015b). This equation, neglecting possible sources of CRs such as the Jovian magnetosphere, is given by
1





where p and r denote particle momentum and position, 





 is the omnidirectional cosmic-ray distribution function, such that the cosmic-ray differential intensity is 





 (see, e.g., Moraal 2013), and 





 and K denote, respectively, the solar wind velocity and heliospheric diffusion tensor. Various processes modulating an initial, boundary CR differential intensity jB are described by this TPE, including CR drift due to gradients and curvatures of the heliospheric magnetic field and along the heliospheric current sheet, diffusion (both described by the first term on the right-hand side of Equation (1)), the outward convection of cosmic rays by the solar wind (second term on the right-hand side of Equation (1)), and adiabatic cooling (third term on the right-hand side of Equation (1)). Equation (1) can be written in terms of a set of equivalent Itō SDEs (see, e.g., Strauss & Effenberger 2017, and references therein) given by
2





with subscripts 





 referring to heliocentric spherical polar coordinates 





 and CR kinetic energy E, xi(t) are Itō processes (see, e.g., Gardiner 2004), and dWi satisfy a Weiner process, such that (Strauss & Effenberger 2017)
3





where η (t) represents a pseudo-random, Gaussian distributed number between zero and one. Pseudo-random numbers are here generated using the Mersenne Twister algorithm developed by Matsumoto & Nishimura (1998). Equation (2) is solved in the time-backward manner using the Euler–Maruyama approximation (Maruyama 1955). Therefore, the evolution of a sufficiently large number of pseudo-particles N at a particular energy/rigidity, and at a pre-specified point in phase-space and time 





 (e.g., at Earth), are followed to their exit positions, times, and energies at a pre-specified boundary 





. Then the CR intensity at the initial point can be calculated using (Strauss & Effenberger 2017)
4





using the assumed boundary CR intensity. Note that N = 10,000 is used throughout this study. The tensor B and vector A are related to the CR transport coefficients, and are given by, e.g., Pei et al. (2010) and Engelbrecht & Burger (2015b), such that, for a fully 3D HMF,
5





where κ denotes one of the elements of the diffusion tensor K in Equation (1) in heliocentric spherical coordinates. The present study employs the diffusion tensor transformation proposed by Burger et al. (2008) to convert the diffusion tensor in HMF-aligned coordinates K′, given by
6





where parallel and perpendicular subscripts denote diffusion coefficients parallel and perpendicular to the assumed HMF, and κA the drift coefficient (see, e.g., Forman et al. 1974), to a diffusion tensor in spherical coordinates. Note that elements of the above tensor are related to the mean free paths (MFPs) discussed below by κ = vλ/3, with v the particle speed. The vector A is given by (Engelbrecht & Burger 2015b)
7





with Eo the CR rest-mass energy. Note that the signs of AE, as well as of the solar wind and drift speeds Vsw and Vd, are chosen so as to render explicitly the time-backward nature of the approach taken to solving Equation (2), and thus Equation (1), in this study.
Citation: Strauss & Effenberger 2017
Citation function: Uses; Uses
Citation: Strauss & Effenberger 2017
Citation function: Uses; Uses
Citation: Strauss & Effenberger 2017
Citation function: Uses; Uses