A stochastic differential equation code for multidimensional Fokker–Planck type problems

Kopp, A.; Büsching, I.; Strauss, R. D.; Potgieter, M. S.

doi:10.1016/j.cpc.2011.11.014

Cited by 108 publications

(113 citation statements)

References 20 publications

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“…We only apply the backward method in this study, since it is well-suited to the given problem. For more details on the numerical scheme and especially on determining the root of the diffusion tensor, we refer the reader to Kopp et al (2012) and Strauss et al (2011) where the basis of the code used in this study is discussed in greater detail. Exemples of pseudo-particle trajectories are shown in Fig.…”

Section: The Numerical Solution Methods Based On Stochastic Differentimentioning

confidence: 99%

Anisotropic diffusion of Galactic cosmic ray protons and their steady-state azimuthal distribution

Effenberger

Fichtner

Scherer

et al. 2012

A&A

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Galactic transport models for cosmic rays involve the diffusive motion of these particles in the interstellar medium. Owing to the large-scale structured Galactic magnetic field, this diffusion is anisotropic with respect to the local field direction. We included this transport effect along with continuous loss processes in a quantitative model of Galactic propagation for cosmic ray protons that is based on stochastic differential equations. We calculated energy spectra at different positions along the Sun's Galactic orbit and compared them to the isotropic diffusion case. The results show that a larger amplitude of variation and different spectral shapes are obtained in the introduced anisotropic diffusion scenario, which in turn emphasizes the need for accurate Galactic magnetic field models.

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Section: The Numerical Solution Methods Based On Stochastic Differentimentioning

confidence: 99%

Anisotropic diffusion of Galactic cosmic ray protons and their steady-state azimuthal distribution

Effenberger

Fichtner

Scherer

et al. 2012

A&A

Self Cite

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“…The set of stochastic differential equations (SDE), being equivalent to equation (1), for a pseudo-particle in position (r, , ) and momentum p using spherical coordinate can be written as equation (11) [see also Pei et al, 2010;Strauss et al, 2012]. Kopp et al [2012] and Effenberger et al [2012b] also present a general discussion on the SDE technique for solving Parker transport equation.…”

Section: Methodsmentioning

confidence: 99%

Modulation of galactic cosmic rays during the unusual solar minimum between cycles 23 and 24

et al. 2014

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During the recent solar minimum between cycles 23 and 24 (solar minimum P23/24), the intensity of galactic cosmic rays (GCR) measured at the Earth was the highest ever recorded since space age. It is the purpose of this paper to resolve the most plausible mechanism for this unusually high intensity. A GCR transport model in three‐dimensional heliosphere based on a simulation of Markov stochastic process is used to find the relation of cosmic ray modulation to various transport parameters, including solar wind (SW) speed, distance of heliospheric boundary, magnitude of interplanetary magnetic field (IMF) at the Earth, tilt angle of heliospheric current sheet, and values of parallel and perpendicular diffusion coefficients. We calculate GCR proton energy spectra at the Earth for the last three solar minima P21/22, P22/23, and P23/24, with the transport parameters obtained from observations. Besides weak IMF magnitude and slow SW speed, we find that a possible low magnetic turbulence, which increases the parallel diffusion and reduces the perpendicular diffusion in the polar direction, might be an additional possible mechanism for the high GCR intensity in the solar minimum P23/24.

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“…Dalla et al, 2015). The remaining equation is solved using the SDE code described in further detail in Kopp et al (2012). The particles are propagated along a path that consists of a Parker spiral field superposed with stochastic fluctuations, resulting in particle paths that meander about the Parker spiral.…”

Section: Modelsmentioning

confidence: 99%

The effect of turbulence strength on meandering field lines and Solar Energetic Particle event extents

Laitinen

Effenberger

Kopp

et al. 2018

J. Space Weather Space Clim.

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-Insights into the processes of Solar Energetic Particle (SEP) propagation are essential for understanding how solar eruptions affect the radiation environment of near-Earth space. SEP propagation is influenced by turbulent magnetic fields in the solar wind, resulting in stochastic transport of the particles from their acceleration site to Earth. While the conventional approach for SEP modelling focuses mainly on the transport of particles along the mean Parker spiral magnetic field, multi-spacecraft observations suggest that the cross-field propagation shapes the SEP fluxes at Earth strongly. However, adding cross-field transport of SEPs as spatial diffusion has been shown to be insufficient in modelling the SEP events without use of unrealistically large cross-field diffusion coefficients. Recently, Laitinen et al. [ApJL 773 (2013b); A&A 591 (2016)] demonstrated that the early-time propagation of energetic particles across the mean field direction in turbulent fields is not diffusive, with the particles propagating along meandering field lines. This early-time transport mode results in fast access of the particles across the mean field direction, in agreement with the SEP observations. In this work, we study the propagation of SEPs within the new transport paradigm, and demonstrate the significance of turbulence strength on the evolution of the SEP radiation environment near Earth. We calculate the transport parameters consistently using a turbulence transport model, parametrised by the SEP parallel scattering mean free path at 1 AU, l Ã k , and show that the parallel and cross-field transport are connected, with conditions resulting in slow parallel transport corresponding to wider events. We find a scaling s ';max ∝ð1=l Ã k Þ 1=4 for the Gaussian fitting of the longitudinal distribution of maximum intensities. The longitudes with highest intensities are shifted towards the west for strong scattering conditions. Our results emphasise the importance of understanding both the SEP transport and the interplanetary turbulence conditions for modelling and predicting the SEP radiation environment at Earth.

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A stochastic differential equation code for multidimensional Fokker–Planck type problems

Cited by 108 publications

References 20 publications

Anisotropic diffusion of Galactic cosmic ray protons and their steady-state azimuthal distribution

Anisotropic diffusion of Galactic cosmic ray protons and their steady-state azimuthal distribution

Modulation of galactic cosmic rays during the unusual solar minimum between cycles 23 and 24

The effect of turbulence strength on meandering field lines and Solar Energetic Particle event extents

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