Stochastic approach to the numerical solution of the non-stationary Parker's transport equation

A Wawrzynczak,R Modzelewska,A Gil

doi:10.1088/1742-6596/574/1/012078

Abstract

We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy/rigidity it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parker's transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The advantages and disadvantages of the forward and the backward solution of the PTE are discussed. The obtained stochastic model of the Forbush decrease of the GCR intensity is in an agreement with the experimental data.

Highlights

Many problems in physics, finance or biology can be represented as models of the diffusive transport processes described by the Fokker-Planck type equations (FPE)
Based on the stochastic approach we model the short time variation of the galactic cosmic ray (GCR) intensity, called the Forbush decrease (Fd) [9]
In this paper we present the model of the recurrent Fd taking place due to established corotating heliolongitudinal disturbances in the interplanetary space

Summary

Introduction

Finance or biology can be represented as models of the diffusive transport processes described by the Fokker-Planck type equations (FPE). To ensure the scheme stability and convergence the density of numerical grid must be improved, increasing the computational complexity To overcome this problem the stochastic methods can be applied (e.g.[1],[2]). Modulation of the GCR is a result of action of four main processes: convection by the solar wind, diffusion on irregularities of HMF, particles drifts in the non-uniform magnetic field and adiabatic cooling Forward and backward cases, first we start from some initial position in space and time and integrate along the pseudoparticles trajectories until they reach the boundary. In the backward integration particles are initialized at the Earth orbit and traced backward in time until they reach the heliosphere’s boundary (in this paper assumed at 100 AU, Fig. 1).

A5 2A1

Model of the short time variation of the GCR intensity