The effect of drifts on the decay phase of SEP events

Aims. We study the effect of the magnetic gradient and curvature drifts on the pitch-angle dependent transport of solar energetic particles (SEPs) in the heliosphere, focussing on similar to 3-36 MeV protons. By considering observers located at different positions in the heliosphere, we investigate how drifts may alter the measured intensity-time profiles and energy spectra. We focus on the decay phase of solar energetic proton events in which a temporal invariant spectrum and disappearing spatial intensity gradients are often observed; a phenomenon known as the "reservoir effect" or the "SEP flood". We study the effects of drifts by propagating particles both in nominal and non-nominal solar wind conditions.Methods. We used a three-dimensional (3D) particle transport model, solving the focused transport equation extended with the effect of particle drifts in the spatial term. Nominal Parker solar wind configurations of different speeds and a magnetohydrodynamic (MHD) generated solar wind containing a corotating interaction region (CIR) were considered. The latter configuration gives rise to a magnetic bottle structure, with one bottleneck at the Sun and the other at the CIR. We inject protons from a fixed source at 0.1 AU, the inner boundary of the MHD model.Results. When the drift induced particle net-flux is zero, the modelled intensity-time profiles obtained at different radial distances along an IMF line show the same intensity fall-off after the prompt phase of the particle event, which is in accordance with the SEP flood phenomenon. However, observers magnetically connected close to the edges of the particle injection site can experience, as a result of drifts, a sudden drop in the intensities occurring at different times for different energies such that no SEP flood phenomenon is established. In the magnetic bottle structure, this effect is enhanced due to the presence of magnetic field gradients strengthening the nominal particle drifts. Moreover, anisotropies can be large for observers that only receive particles through drifts, illustrating the importance of pitch-angle dependent 3D particle modelling. We observe that interplanetary cross-field diffusion can mitigate the effects of particle drifts.Conclusions. Particle drifts can substantially modify the decay phase of SEP events, especially if the solar wind contains compression regions or shock waves where the drifts are enhanced. This is, for example, the case for our CIR solar wind configuration generated with a 3D MHD model, where the effect of drifts is strong. A similar decay rate in different energy channels and for different observers requires the mitigation of the effect of drifts. One way to accomplish this is through interplanetary cross-field diffusion, suggesting thus a way to determine a minimum value for the cross-field diffusion strength.Peer reviewe

Interplanetary spread of solar energetic protons near a high-speed solar wind stream

Aims. We study how a fast solar wind stream embedded in a slow solar wind influences the spread of solar energetic protons in interplanetary space. In particular, we aim at understanding how the particle intensity and anisotropy vary along interplanetary magnetic field (IMF) lines that encounter changing solar wind conditions such as the shock waves bounding a corotating interaction region (CIR). Moreover, we study how the intensities and anisotropies vary as a function of the longitudinal and latitudinal coordinate, and how the width of the particle intensities evolves with the heliographic radial distance. Furthermore, we study how cross-field diffusion may alter these spatial profiles. Methods. To model the energetic protons, we used a recently developed particle transport code that computes particle distributions in the heliosphere by solving the focused transport equation (RTE) in a stochastic manner. The particles are propagated in a solar wind containing a CIR, which was generated by the heliospheric model, EUHFORIA. We study four cases in which we assume a delta injection of 4 MeV protons spread uniformly over different regions at the inner boundary of the model. These source regions have the same size and shape, yet are shifted in longitude from each other, and are therefore magnetically connected to different solar wind conditions. Results. The intensity and anisotropy profiles along selected IMF lines vary strongly according to the different solar wind conditions encountered along the field line. The IMF lines crossing the shocks bounding the CIR show the formation of accelerated particle populations, with the reverse shock wave being a more efficient accelerator than the forward shock wave. The longitudinal intensity profiles near the CIR are highly asymmetric in contrast to the profiles obtained in a nominal solar wind. For the injection regions that do not cross the transition zone between the fast and slow solar wind, we observe a steep intensity drop of several orders of magnitude near the stream interface (SI) inside the CIR. Moreover, we demonstrate that the longitudinal width of the particle intensity distribution can increase, decrease, or remain constant with heliographic radial distance, reflecting the underlying IMF structure. Finally, we show how the deflection of the IMF at the shock waves and the compression of the IMF in the CIR deforms the three-dimensional shape of the particle distribution in such a way that the original shape of the injection profile is lost.Peer reviewe

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

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 (less than or similar to 1.5AU) 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 greater than or similar to 1.5AU, 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.Peer reviewe

Observation-based modelling of the energetic storm particle event of 14 July 2012

Aims. We model the energetic storm particle (ESP) event of 14 July 2012 using the energetic particle acceleration and transport model named 'PArticle Radiation Asset Directed at Interplanetary Space Exploration' (PARADISE), together with the solar wind and coronal mass ejection (CME) model named 'EUropean Heliospheric FORcasting Information Asset' (EUHFORIA). The simulation results illustrate both the capabilities and limitations of the utilised models. We show that the models capture some essential structural features of the ESP event; however, for some aspects the simulations and observations diverge. We describe and, to some extent, assess the sources of errors in the modelling chain of EUHFORIA and PARADISE and discuss how they may be mitigated in the future. Methods. The PARADISE model computes energetic particle distributions in the heliosphere by solving the focused transport equation in a stochastic manner. This is done using a background solar wind configuration generated by the ideal magnetohydrodynamic module of EUHFORIA. The CME generating the ESP event is simulated by using the spheromak model of EUHFORIA, which approximates the CME's flux rope as a linear force-free spheroidal magnetic field. In addition, a tool was developed to trace CME-driven shock waves in the EUHFORIA simulation domain. This tool is used in PARADISE to (i) inject 50 keV protons continuously at the CME-driven shock and (ii) include a foreshock and a sheath region, in which the energetic particle parallel mean free path, lambda(parallel to), decreases towards the shock wave. The value of lambda(parallel to) at the shock wave is estimated from in situ observations of the ESP event. Results. For energies below similar to 1 MeV, the simulation results agree well with both the upstream and downstream components of the ESP event observed by the Advanced Composition Explorer. This suggests that these low-energy protons are mainly the result of interplanetary particle acceleration. In the downstream region, the sharp drop in the energetic particle intensities is reproduced at the entry into the following magnetic cloud, illustrating the importance of a magnetised CME model.Peer reviewe

Observation-based modelling of the energetic storm particle event of 14 July 2012

Aims. We model the energetic storm particle (ESP) event of 14 July 2012 using the energetic particle acceleration and transport model named 'PArticle Radiation Asset Directed at Interplanetary Space Exploration' (PARADISE), together with the solar wind and coronal mass ejection (CME) model named 'EUropean Heliospheric FORcasting Information Asset' (EUHFORIA). The simulation results illustrate both the capabilities and limitations of the utilised models. We show that the models capture some essential structural features of the ESP event; however, for some aspects the simulations and observations diverge. We describe and, to some extent, assess the sources of errors in the modelling chain of EUHFORIA and PARADISE and discuss how they may be mitigated in the future. Methods. The PARADISE model computes energetic particle distributions in the heliosphere by solving the focused transport equation in a stochastic manner. This is done using a background solar wind configuration generated by the ideal magnetohydrodynamic module of EUHFORIA. The CME generating the ESP event is simulated by using the spheromak model of EUHFORIA, which approximates the CME's flux rope as a linear force-free spheroidal magnetic field. In addition, a tool was developed to trace CME-driven shock waves in the EUHFORIA simulation domain. This tool is used in PARADISE to (i) inject 50 keV protons continuously at the CME-driven shock and (ii) include a foreshock and a sheath region, in which the energetic particle parallel mean free path, lambda(parallel to), decreases towards the shock wave. The value of lambda(parallel to) at the shock wave is estimated from in situ observations of the ESP event. Results. For energies below similar to 1 MeV, the simulation results agree well with both the upstream and downstream components of the ESP event observed by the Advanced Composition Explorer. This suggests that these low-energy protons are mainly the result of interplanetary particle acceleration. In the downstream region, the sharp drop in the energetic particle intensities is reproduced at the entry into the following magnetic cloud, illustrating the importance of a magnetised CME model

Influence of large-scale interplanetary structures on the propagation of solar energetic particles: The Multispacecraft event on 2021 October 9

An intense solar energetic particle (SEP) event was observed on 2021 October 9 by multiple spacecraft distributed near the ecliptic plane at heliocentric radial distances R ≲ 1 au and within a narrow range of heliolongitudes. A stream interaction region (SIR), sequentially observed by Parker Solar Probe (PSP) at R = 0.76 au and 48° east from Earth (ϕ = E48°), STEREO-A (at R = 0.96 au, ϕ = E39°), Solar Orbiter (SolO; at R = 0.68 au, ϕ = E15°), BepiColombo (at R = 0.33 au, ϕ = W02°), and near-Earth spacecraft, regulated the observed intensity-time profiles and the anisotropic character of the SEP event. PSP, STEREO-A, and SolO detected strong anisotropies at the onset of the SEP event, which resulted from the fact that PSP and STEREO-A were in the declining-speed region of the solar wind stream responsible for the SIR and from the passage of a steady magnetic field structure by SolO during the onset of the event. By contrast, the intensity-time profiles observed near Earth displayed a delayed onset at proton energies ≳13 MeV and an accumulation of ≲5 MeV protons between the SIR and the shock driven by the parent coronal mass ejection (CME). Even though BepiColombo, STEREO-A, and SolO were nominally connected to the same region of the Sun, the intensity-time profiles at BepiColombo resemble those observed near Earth, with the bulk of low-energy ions also confined between the SIR and the CME-driven shock. This event exemplifies the impact that intervening large-scale interplanetary structures, such as corotating SIRs, have in shaping the properties of SEP events

Perpendicular diffusion of solar energetic particles: When is the diffusion approximation valid?

Abstract: Multi-spacecraft observations of widespread solar energetic particle (SEP) events indicate that perpendicular (to the mean field) diffusion is an important SEP transport mechanism. However, this is in direct contrast to so-called spike and drop-out events, which indicate very little lateral transport. To better understand these seemingly incongruous observations, we discuss the recent progress made towards understanding and implementing perpendicular diffusion in transport models of SEP electrons. This includes a re-derivation of the relevant focused transport equation, a discussion surrounding the correct form of the pitch-angle dependent perpendicular diffusion coefficient and what turbulence quantities are needed as input, and how models lead to degenerate solutions of the particle intensity. Lastly, we evaluate the validity of a diffusion approach to SEP transport and conclude that it is valid when examining a large number of (an ensemble of) events, but that individual SEP events may exhibit coherent structures related to the magnetic field turbulence at short timescales that cannot be accounted for in this modelling approach

Ground state properties of a Tonks-Girardeau Gas in a periodic potential

In this paper, we investigate the ground-state properties of a bosonic Tonks-Girardeau gas confined in a one-dimensional periodic potential. The single-particle reduced density matrix is computed numerically for systems up to $N=265$ bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein Condensation(BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, -0.5 and 1, respectively, while in the incommensurate case, they are 0.5, -0.5 and 1.5, respectively. These exponents are related to each other in a universal relation.Comment: 9 pages, 10 figure