A Hitch-hiker’s Guide to Stochastic Differential Equations

Strauss, R. D.; Effenberger, Frederic

doi:10.1007/s11214-017-0351-y

Cited by 79 publications

(55 citation statements)

References 139 publications

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“…Ruffolo 1995;Lario et al 1998;Schwadron et al 2010;le Roux & Webb 2012;Wang et al 2012) or by taking a stochastic approach (see, e.g. Agueda et al 2008;Zhang et al 2009;Dröge et al 2010;Strauss & Effenberger 2017). We take the latter approach and solve the FTE by means of a time-forward Monte Carlo simulation.…”

Section: Particle Transportmentioning

confidence: 99%

Modelling three-dimensional transport of solar energetic protons in a corotating interaction region generated with EUHFORIA

Wijsen

Aran

Pomoell

et al. 2019

A&A

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Aims. We introduce a new solar energetic particle (SEP) transport code that aims at studying the effects of different background solar wind configurations on SEP events. In this work, we focus on the influence of varying solar wind velocities on the adiabatic energy changes of SEPs and study how a non-Parker background solar wind can trap particles temporarily at small heliocentric radial distances ( 1.5 AU) thereby influencing the cross-field diffusion of SEPs in the interplanetary space. Methods. Our particle transport code computes particle distributions in the heliosphere by solving the focused transport equation (FTE) in a stochastic manner. Particles are propagated in a solar wind generated by the newly developed data-driven heliospheric model, EUHFORIA. In this work, we solve the FTE, including all solar wind effects, cross-field diffusion, and magnetic-field gradient and curvature drifts. As initial conditions, we assume a delta injection of 4 MeV protons, spread uniformly over a selected region at the inner boundary of the model. To verify the model, we first propagate particles in nominal undisturbed fast and slow solar winds. Thereafter, we simulate and analyse the propagation of particles in a solar wind containing a corotating interaction region (CIR). We study the particle intensities and anisotropies measured by a fleet of virtual observers located at different positions in the heliosphere, as well as the global distribution of particles in interplanetary space. Results. The differential intensity-time profiles obtained in the simulations using the nominal Parker solar wind solutions illustrate the considerable adiabatic deceleration undergone by SEPs, especially when propagating in a fast solar wind. In the case of the solar wind containing a CIR, we observe that particles adiabatically accelerate when propagating in the compression waves bounding the CIR at small radial distances. In addition, for r 1.5 AU, there are particles accelerated by the reverse shock as indicated by, for example, the anisotropies and pitch-angle distributions of the particles. Moreover, a decrease in high-energy particles at the stream interface (SI) inside the CIR is observed. The compression/shock waves and the magnetic configuration near the SI may also act as a magnetic mirror, producing long-lasting high intensities at small radial distances. We also illustrate how the efficiency of the cross-field diffusion in spreading particles in the heliosphere is enhanced due to compressed magnetic fields. Finally, the inclusion of cross-field diffusion enables some particles to cross both the forward compression wave at small radial distances and the forward shock at larger radial distances. This results in the formation of an accelerated particle population centred on the forward shock, despite the lack of magnetic connection between the particle injection region and this shock wave. Particles injected in the fast solar wind stream cannot reach the forward shock since the SI acts as a diffusion barrier.

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Section: Particle Transportmentioning

confidence: 99%

Modelling three-dimensional transport of solar energetic protons in a corotating interaction region generated with EUHFORIA

Wijsen

Aran

Pomoell

et al. 2019

A&A

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“…The stochastically meandering path is described as fieldline diffusion using a stochastic differential equation (SDE, Gardiner, 2009;Strauss and Effenberger, 2017), with the displacement dr ⊥ across the Parker field direction given as…”

Section: Modelsmentioning

confidence: 99%

“…The particle propagation along meandering field lines, the step B in the FP þ FLRW scheme, is performed using an SDE formulation of the Fokker-Planck equation (Roelof, 1969;Skilling, 1971;Isenberg, 1997;Zhang et al, 2009;Strauss and Effenberger, 2017). The Fokker-Planck equation is given as…”

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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“…In the Monte Carlo model, particles are advanced integrating a Langevin type equation in the form of a stochastic differential equation (e.g. Strauss & Effenberger 2017):…”

Section: Appendix A: Testing Tamsd With Monte Carlo Modelsmentioning

confidence: 99%

Particle transport in hybrid PIC shock simulations: A comparison of diagnostics

Trotta

Burgess

Prete

et al. 2019

Monthly Notices of the Royal Astronomical Society

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Recent in-situ and remote observations suggest that the transport regime associated with shock accelerated particles may be anomalous i.e., the Mean Square Displacement (MSD) of such particles scales non-linearly with time. We use self-consistent, hybrid PIC plasma simulations to simulate a quasi-parallel shock with parameters compatible with heliospheric shocks, and gain insights about the particle transport in such a system. For suprathermal particles interacting with the shock we compute the MSD separately in the upstream and downstream regions. Tracking suprathermal particles for sufficiently long times up and/or downstream of the shock poses problems in particle plasma simulations, such as statistically poor particle ensembles and trajectory fragments of variable length in time. Therefore, we introduce the use of time-averaged mean square displacement (TAMSD), which is based on single particle trajectories, as an additional technique to address the transport regime for the upstream and downstream regions. MSD and TAMSD are in agreement for the upstream energetic particle population, and both give a strong indication of superdiffusive transport, consistent with interplanetary shock observations. MSD and TAMSD are also in reasonable agreement downstream, where indications of anomalous transport are also found. TAMSD shows evidence of heterogeneity in the diffusion properties of the downstream particle population, ranging from subdiffusive behaviour of particles trapped in the strong magnetic field fluctuations generated at the shock, to superdiffusive behaviour of particles transmitted and moving away from the shock.

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A Hitch-hiker’s Guide to Stochastic Differential Equations

Cited by 79 publications

References 139 publications

Modelling three-dimensional transport of solar energetic protons in a corotating interaction region generated with EUHFORIA

Modelling three-dimensional transport of solar energetic protons in a corotating interaction region generated with EUHFORIA

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

Particle transport in hybrid PIC shock simulations: A comparison of diagnostics

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