Issue 
J. Space Weather Space Clim.
Volume 8, 2018
Flares, coronal mass ejections and solar energetic particles and their space weather impacts



Article Number  A13  
Number of page(s)  11  
DOI  https://doi.org/10.1051/swsc/2018001  
Published online  16 February 2018 
Research Article
The effect of turbulence strength on meandering field lines and Solar Energetic Particle event extents
^{1}
Jeremiah Horrocks Institute, University of Central Lancashire,
Preston, UK
^{2}
International Space Science Institute,
Bern, Switzerland
^{3}
Bay Area Environmental Research Institute,
Petaluma,
CA, USA
^{4}
Université Libre de Bruxelles, Service de Physique Statistique et des Plasmas,
CP 231,
1050
Brussels, Belgium
^{*} Corresponding author: tlmlaitinen@uclan.ac.uk
Received:
23
May
2017
Accepted:
7
January
2018
Insights into the processes of Solar Energetic Particle (SEP) propagation are essential for understanding how solar eruptions affect the radiation environment of nearEarth 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, multispacecraft observations suggest that the crossfield propagation shapes the SEP fluxes at Earth strongly. However, adding crossfield transport of SEPs as spatial diffusion has been shown to be insufficient in modelling the SEP events without use of unrealistically large crossfield diffusion coefficients. Recently, Laitinen et al. [ApJL 773 (2013b); A&A 591 (2016)] demonstrated that the earlytime propagation of energetic particles across the mean field direction in turbulent fields is not diffusive, with the particles propagating along meandering field lines. This earlytime 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, λ_{∥}^{*}, and show that the parallel and crossfield transport are connected, with conditions resulting in slow parallel transport corresponding to wider events. We find a scaling σ_{φ,max}∝(1/λ_{∥}^{*})^{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.
Key words: Cosmic rays / diffusion / Sun: heliosphere / Sun: particle emission / turbulence
© T. Laitinen et al., Published by EDP Sciences 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Solar Energetic Particles (SEPs), accelerated in solar eruptive events, pose a significant space weather threat to manmade technology and astronauts (Committee on the Evaluation of Radiation Shielding for Space Exploration, 2008). To forecast SEP fluxes near Earth's orbit, it is important to understand how their acceleration is related to flares, coronal mass ejections and other related phenomena during solar eruptions. Furthermore, as the particles propagate through a turbulent solar wind medium, predicting the fluxes and fluences at 1 AU requires understanding of how the solar wind turbulence affects the charged particle motion.
The propagation of SEPs in a turbulent medium is typically modelled as random walk due to the stochastic nature of magnetic field fluctuations, and described as spatial and velocity diffusion using a FokkerPlanck formalism (Parker, 1965; Jokipii, 1966). The propagation along the mean field is usually modelled as either spatial or pitch angle diffusion (Jokipii, 1966). The crossfield transport, on the other hand, is usually described as spatial diffusion due to random walk of the turbulent magnetic field lines (Jokipii, 1966), compounded by the parallel scattering (Matthaeus et al., 2003; Shalchi, 2010; Ruffolo et al., 2012). These approaches have support in fullorbit particle simulations (Giacalone and Jokipii, 1999) and galactic cosmic ray observations (Burger et al., 2000; Potgieter et al., 2014). However, several recent observational studies suggest faster propagation of SEPs across the mean field than predicted by the current theoretical understanding: they often require a ratio of the crossfield diffusion coefficient to the parallel one of order κ_{⊥}/κ_{∥} ∼ 0.1 − 1 (Zhang et al., 2003; Dresing et al., 2012; Dröge et al., 2014)^{1}, whereas values κ_{⊥}/κ_{∥} ≲ 0.01 are more consistent with the interplanetary turbulence conditions at 1 AU (Burlaga and Turner, 1976; Pei et al., 2010; Laitinen et al., 2016).
Recently, Laitinen et al. (2013b, L2013 in the following) demonstrated, using fullorbit particle simulations in turbulent magnetic fields superposed on a constant background magnetic field, that SEPs can propagate rapidly to large crossfield distances along turbulently meandering fieldlines already early in SEP event history. While the concept of fieldline meandering is included in earlier models in the diffusion coefficient, L2013 showed that the initial SEP crossfield transport is nondiffusive, and cannot be modelled using a diffusion approach. As further shown in Laitinen and Dalla (2017), the particles remain on their initial meandering field lines up to tens of hours before decoupling and spreading more freely across the meandering field lines. Thus, the initial evolution of SEP events is dominated by systematic widening of the SEP crossfield distribution, while diffusion dominates the evolution of the SEP crossfield distribution only tens of hours after the SEP injection. L2013 pointed out also that the earlytime nondiffusive SEP propagation across the mean field direction is much faster than the timeasymptotic crossfield diffusion.
Using the novel modelling approach introduced in L2013, Laitinen et al. (2016) developed a particle transport model in the heliospheric Parker Spiral magnetic field configuration. They demonstrated that in moderate turbulence conditions, parametrised by the parallel scattering mean free path λ_{∥} = 0.3 AU, fast spreading of SEPs across the field to a wide range of longitudes, as found by multispacecraft observations of SEP events (e.g. Lario et al., 2006; Dresing et al., 2012; Dresing et al., 2014; Richardson et al., 2014), could be obtained already with a narrow source region.
Laitinen et al. (2016) considered a case study of SEP propagation in magnetic turbulence characterised by the value of the parallel scattering mean free path at 1 AU, whereas in reality the turbulence, and as a consequence the particle transport parameters, can vary considerably from event to event (Burlaga and Turner, 1976; Bavassano et al., 1982; Palmer, 1982; Wanner and Wibberenz, 1993), and even be different during an event at different heliographic longitudes (Dröge et al., 2016). Using fullorbit simulations with a constant background magnetic field, Laitinen and Dalla (2017) and Laitinen et al. (2017) showed that the initial crossfield extent of the SEP distribution depends strongly on the turbulence amplitude. Thus, it is important to evaluate the effect of different levels of turbulence amplitudes on the SEP event width in the Parker Spiral geometry. In this study, we compare longitudinal SEP event extents for different levels of plasma turbulence, as parametrised by parallel mean free paths . We concentrate on 10 MeV protons, which have received less attention in multispacecraftobserved SEP event modelling. While electrons and protons are often considered to be accelerated in different processes and source regions (e.g. Reames, 1999), the recently observed similar heliolongitudinal extents for electrons and protons during different events (e.g. Richardson et al., 2014) may suggest similar processes responsible for the spreading of these particles in interplanetary space, warranting closer analysis of their crossfield transport. We present the employed models in Section 2, the results in Section 3 and discuss and draw our conclusions in Sections 4 and 5.
2 Models
The FokkerPlanck and Field Line Random Walk (FP + FLRW) model used in this study is based on the findings of L2013, who used fullorbit particle simulations in turbulent magnetic fields to show that the initial crossfield propagation of charged particles in turbulent magnetic fields is nondiffusive. The particles tend to follow their field lines, which spread across the mean field direction due to turbulent fluctuations. Until a particle decouples from its field line, its propagation across the mean background magnetic field is deterministic, in the sense that particles which scatter in their pitch angle from one pitch angle hemisphere to the other will just retract their path along the same stochastically meandering path. Thus, the particle crossfield transport behaviour remains nonMarkovian at times shorter than the timescale over which the particle decouples from its original field line. L2013, and subsequently Laitinen and Dalla (2017) and Laitinen et al. (2017) showed that this nonMarkovian propagation can dominate an SEP event for up to tens of hours, depending on the turbulence conditions. The slow onset of the decoupling of particles from the meandering field lines can explain the intensity dropouts observed in some SEP events (e.g. Mazur et al., 2000), as shown for example in simulations by Tooprakai et al. (2016).
While the motion resulting from fieldline meandering and fieldparallel scattering has been described as compound (sub)diffusion by earlier researchers (e.g. Kóta and Jokipii, 2000), L2013 discovered that during early times, the particle propagation cannot be described as diffusion at all, as the particles retain memory of their propagation history.
L2013 introduced the FP + FLRW model, where this process is described as combination of particle propagation along meandering path (supplemented with pitch angle scattering) and particle diffusion across the meandering field. As shown in L2013 and subsequently further investigated in Laitinen et al. (2017), the model agrees well with fullorbit simulations at early times, when the particles are still tied to their field lines, and at the timeasymptotic limit, where the particle crossfield propagation is fully diffusive.
In the fullorbit simulations in L2013, the meandering of field lines was created by using a superposition of Fourier modes corresponding to spectra of slab and 2Dmode waves, which can be cumbersome particularly in scenarios more complex than the constant background field used in that work. For simpler and faster FokkerPlanck description of particle propagation, the FP + FLRW mode considers a description of diffusively meandering particle paths, based on results of, e.g., Matthaeus et al. (1995). In the FP + FLRW approach, rather than calculating the complete fluctuating magnetic field, the effect of diffusive spreading field lines on the propagating particles is estimated by propagating each particle on a separately drawn stochastic meandering path. Thus, schematically the algorithm of the FP + FLRW model for each pseudoparticle in the simulations proceeds as follows:

1.
calculate a diffusively meandering path, unique to the particle being simulated;

2.
propagate a pseudoparticle until end of simulation time using the following scheme:

(a)
step along the meandering path;

(b)
diffuse and adiabatically focus the pitch angle. Here, for the focusing the mean background magnetic field is used;

(c)
take a diffusive crossfield spatial step in direction perpendicular to the meandering path.

(a)
It should be emphasised that in the FP + FLRW model each simulated pseudoparticle has only one meandering field line, and the particle diffuses across this meandering field line: The pseudoparticle does not switch from one meandering path to another. The individual particles propagating each at their unique meandering paths facilitate the initial nondiffusive evolution of the particle distribution seen in L2013 and Laitinen et al. (2017), whereas the particle's spatial crossfield diffusion from this meandering path facilitates the timeasymptotic diffusive particle transport.
An alternative approach, with a particle changing from one meandering path to another, could also be devised. However, such a model would require precise description of the decoupling of particles from their initial fieldlines, and the relation of that decoupling process to the turbulent field line separation, which are not yet well understood. Thus, as the simpler FP + FLRW model was shown by L2013 to reproduce the fullorbit particle simulations well, the singlemeanderingpath approach is welljustified.
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 (1) where dr_{∥} is a step along the local Parker spiral direction, and W_{⊥} a Gaussian random number with zero mean and unit variance. The fieldline diffusion coefficient D_{FL}(r) is calculated based on Matthaeus et al. (1995), using the radiallyevolving 2D component of the turbulence spectrum discussed below.
This method of calculating the meandering field line using a stochastic method is naturally statistic in nature, and does not reproduce patchy spatial particle distributions seen in some fullorbit particle simulations (e.g. Tooprakai et al., 2016), which may explain intensity dropouts observed in SEPs (e.g. Mazur et al., 2000) (see also discussion in Laitinen and Dalla, 2017). It also cannot replicate the coherence of nearby field lines, but, as shown in (Ruffolo et al., 2004), such coherence is lost in 2Ddominated turbulence at scales which are small compared to heliospheric scales.
The particle propagation along meandering field lines, the step B in the FP + FLRW scheme, is performed using an SDE formulation of the FokkerPlanck equation (Roelof, 1969; Skilling, 1971; Isenberg, 1997; Zhang et al., 2009; Strauss and Effenberger, 2017). The FokkerPlanck equation is given as (2) where v and μ are the particle's velocity and pitchangle cosine, respectively, and Q is the particle source function. V_{sw} is the solar wind velocity, b a unit vector along the Parker magnetic field B, and b_{m} a unit vector along the meandering path given by equation (1). The focusing length L = − B/(∂ B/∂ s), with s the arclength along the fieldline, is calculated using the mean Parker spiral field. The particles scatter in pitch angle cosine μ = v_{∥}/v according to the pitchangle diffusion coefficient D_{μμ}, and across the mean field direction according to the spatial crossfield diffusion coefficient κ_{xx} = κ_{yy} ≡ κ_{⊥}, the nonzero elements of the cartesian diffusion tensor . In this study, we ignore the energy changes given by the 5th term in equation (2), as they are expected to be small during the early phase of the SEP event (e.g. 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. The magnitude of the magnetic field is taken as the mean Parker spiral field value, (3) where B_{0} = 5 nT is the magnetic field strength at r_{0} = 1 AU, and , where V_{sw} = 400 km/s, Ω_{⊙} = 2.8631 ⋅ 10^{−6} rad/s is the solar rotation rate and θ the colatitude.
As shown by L2013 and Laitinen and Dalla (2017), at intermediate timescales, of the order of the parallel scattering timescale of the particles, the particles begin to decouple from their field lines and eventually timeasymptotically approach diffusive crossfield propagation. We include the transition to the timeasymptotic crossfield diffusion into our simulations by diffusing the particles across the meandering field. While this approach is not precise, L2013 demonstrated that it is much more accurate than using only crossfield diffusion from the mean field, or only particle propagation along meandering field lines. The crossfield diffusion coefficient κ_{⊥} is calculated using the NonLinear Guiding Centre theory (NLGC, Matthaeus et al., 2003) for the spectrum described below. We do not incorporate the recently suggested dependence of κ_{⊥} on the particle's pitch angle (Dröge et al., 2010; Qin and Shalchi, 2014; Strauss and Fichtner, 2015) since fullorbit results (Laitinen and Dalla, 2017) indicate that it might be more complicated than the suggested proportionality to μ or 1 − μ^{2}.
It should be noted that although both the meandering path of the particle and the crossfield diffusion coefficient are calculated from the same turbulence spectrum, this does not amount to taking the effect of meandering field lines on particles into account twice. As discussed in Laitinen and Dalla (2017), the initial crossfield spreading of the particles is caused by the particles following the meandering field lines. The crossfield diffusion, on the other hand is caused by particles decoupling from the field lines and following new field lines, which meander relative to the original field line (see also Ruffolo et al., 2012). Thus, the two phenomena, while both related to fieldline meandering, are separate and must be both accounted for.
Finally, the particles also scatter as they propagate along the meandering field line. We model this by using a quasilinear pitchangle diffusion coefficient D_{μμ} (e.g. Jokipii, 1966), with additional scattering at μ = 0 to close the resonance gap, as suggested by Beeck and Wibberenz (1986) (see Laitinen et al. (2016) for details).
The particle and field line diffusion coefficients are calculated using a heliospheric turbulence spectrum with slab and 2D components (Gray et al., 1996). The turbulence spectrum is given as (4) where k_{∥} and k_{⊥} = k_{⊥} are the parallel and perpendicular wavenumbers, and S_{∥}(k_{∥}) and S_{⊥}(k_{⊥}) are broken power law spectra as given in Laitinen et al. (2016). It should be noted that our turbulence model differs from the one introduced by Giacalone (2001), in which the turbulence is generated by motion of magnetic field footpoints due to solar supergranulation. The latter does not allow for further turbulence evolution of the magnetic fields in interplanetary space (e.g. Bruno and Carbone, 2013, and references therein), and thus limits the meandering of interplanetary field lines to the angular scale of the supergranular motion.
We model the radial evolution of the turbulence within the heliosphere using the WKB transport approximation (Richter and Olbers, 1974; Tu et al., 1984). For simplicity, we neglect wave refraction, changes in the wave geometry and the modulus of k, as well as nonlinear evolution of the spectral shape (see Laitinen et al., 2016, for discussion). We further consider constant radial solar wind velocity V_{sw,r0}, and electron density . With these assumptions, the radial evolution of the turbulence spectrum can be written as (5) where V_{sw,r0} = 400 km/s is the constant solar wind velocity, and subscript 0 denotes the values at reference distance r_{0}, and v_{a,r0} = 30 km/s is the Alfvén velocity at r_{0} = 1 AU. The resulting ∝r^{3} trend of the wave power is consistent with turbulence observations (e.g. Bavassano et al., 1982).
The spectral power of the slab and 2D components, S_{∥,⊥}(k_{∥,⊥}, r_{0}), is parametrised by the total turbulence amplitude δB^{2} = 2 ∫ S(k)dk, and the energy ratio between the slab and 2D modes, for which we use 20%:80%, as suggested by Bieber et al. (1996). The total turbulence amplitude is parametrised by the parallel mean free path at 1 AU, , as given by the quasilinear theory (Jokipii, 1966) for the slab spectrum S_{∥}(k_{∥}) at 1 AU. It should be emphasised that the parallel mean free path is fixed using only at 1 AU: elsewhere all particle and field line transport parameters are calculated consistently using the turbulence model given by equations (4) and (5). Thus, we constrain the radial evolution of both parallel and perpendicular transport parameters consistently, instead of using an ad hoc parametrisation.
The parallel and crossfield mean free paths for 10 MeV protons are shown as a function of distance from the Sun in the left panel of Figure 1 for the modelled turbulence corresponding to values of 0.1, 0.3 and 1 AU. Close to the Sun, the parallel mean free path is large, indicating nearly scatterfree propagation, and decreases to the parametrised value at 1 AU, after which it increases again. On the other hand, the crossfield mean free path is very short close to the Sun and increases initially faster than ∝r, indicating that the diffusion coefficient ratio κ_{⊥}/κ_{∥} is not constant in the heliosphere, but depends strongly on the radial distance from the Sun. Similar results of the radial dependence of the particle diffusion coefficients have recently been presented in several studies (Pei et al., 2010; Chhiber et al., 2017; Strauss et al., 2017).
In the right panel of Figure 1, we describe how the crossfield particle and field line diffusion coefficients evolve in the heliosphere, by presenting their ratio as a function of radial distance from the Sun. As discussed in L2013, particles propagate initially along meandering field lines that spread diffusively according to diffusion coefficient D_{FL}. Timeasymptotically, the crossfield propagation is diffusive, described by the particle crossfield diffusion coefficient κ_{⊥}, which is much slower than the spreading of particles nondiffusively along the field lines, due to particles scattering along the meandering field lines. As can be seen in the right panel of Figure 1, the spreading of particles across the field due to the early process, at rate vD_{FL}, is 1–2 orders of magnitude faster than the timeasymptotic diffusive crossfield spreading of the particles, and the difference increases as a function of distance from the Sun. The ratio vD_{FL}/κ_{⊥} decreases for weaker turbulence (larger ), and, as discussed in Laitinen et al. (2016), vD_{FL} and κ_{⊥} calculated using the NLGC (Matthaeus et al., 2003) converge to the same value in the limit of no parallel scattering for a particle beam.
Fig. 1 Left panel: Parallel (thick curves) and perpendicular (thin curves) particle mean free paths as function of radial distance from the Sun, for 10 MeV protons and different turbulence strengths parametrised by . Right panel: The ratio of the field line and crossfield particle diffusion coefficients. 
3 Results
We study the effect of turbulence strength on SEP event evolution in time, both along and across the mean field direction. We use a simple injection profile (6) where (r, θ, φ) define the spherical coordinate system, is the solar radius, and E_{0} = 10 MeV the energy of the simulated protons. The coronal magnetic field can be complicated, varying considerably from event to event. However, in this study we are interested in SEP propagation in general, instead of during a particular SEP event. For this reason, we model the coronal magnetic field simply as a Parker spiral starting from the injection height at , reserving case studies that investigate the spatial structure of the source region of particles for future work.
The results of our study can be extended to other injection profiles by simply convolving the impulse response with more complicated injection profiles. It should be noted that Strauss et al. (2017) demonstrated recently that the source size at or near the Sun plays only a minor role in the crossfield extent of an SEP event in cases where crossfield propagation of particles is efficient. Thus, our results can be considered to represent the SEP event evolution as injected from a narrow to an intermediatesize SEP source region.
We first show an overview of the early SEP event extent in Figures 2 and 3, as a snapshot of the spatial distribution of 10 MeV protons in the inner heliosphere, two hours after the injection. The red circle depicts Earth's orbit, and the black spiral curve the Parker spiral connected to the injection location. The SEP distribution is given as a function of heliolongitude and the heliocentric radial distance, integrated over latitudes ±10°.
In Figure 2, we present the SEP distributions for turbulence parametrised with AU. The left panel depicts the model where the fieldline meandering is omitted (the FP model in Laitinen et al., 2016), whereas the right panel is obtained from the model described in Section 2. As discussed in Laitinen et al. (2016), the primary effect of including the field line meandering into the modelling is that the particles spread rapidly across the mean Parker field direction to a wide range of heliolongitudes, as compared to the slow spreading of the SEPs across the mean field depicted in the left panel of Figure 2.
In Figure 3, we show the effect of changing the turbulence strength on the radial and crossfield extent of an SEP event. In the left panel, the turbulence amplitude has been increased to result in stronger parallel scattering conditions, as parametrised by 0.1 AU. The differences with the 0.3 AU case in the right panel of Figure 2 are notable. The strong parallel scattering of the SEPs prevents the particles from propagating as far into the heliosphere as in the 0.3 AU case. On the other hand, the core of the SEP distribution in the left panel of Figure 3 is considerably wider. This is caused by the particles following the meandering field lines which diffuse with D_{FL} ∝ δB/B in 2Ddominated turbulence (Matthaeus et al., 1995). Thus, while particle propagation along the mean field line is inhibited by strong scattering in stronger turbulence, the crossfield transport of the SEP distribution is enhanced by the stronger meandering of the field lines.
For weaker turbulence (the AU case), presented in the right panel of Figure 3, we see that the reduced parallel scattering of the SEPs causes an increased radial extent of the particle distribution: the front of the SEP population has propagated to nearly 2 AU from the Sun, consistent with nearlyscatterfree propagation of 10 MeV protons of ∼1 AU/h. On the other hand, the population is narrower in the crossfield direction, as compared to the cases with = 0.3 AU and 0.1 AU. This is due to the reduced turbulent meandering of the field lines in weaker turbulence.
In Figures 4 and 5, we show the evolution of the SEP event at 1 AU as a function of heliographic longitude and time, for = 0.1 AU and 1 AU, respectively. In both cases, the particles spread rapidly across the mean field direction, with the onset seen at a wide range of longitudes within the first 2 h of the event. In the strong scattering case (Fig. 4), the diffusive nature of the crossfield propagation of the particles after the initial fast spreading results in the parabole shape of the highintensity contours as a function of time and longitude, which suggest the diffusive scaling of the longitudinal variance of the particles as after the initial fast expansion along the meandering field lines.
The low scattering case (Fig. 5) is very different from the stronger scattering case shown in Figure 4. The proton intensity increases rapidly to its maximum value during the first two hours from the injection, and then begins to decay. This is due to the particles being nearly scatterfree and focusing adiabatically outwards when they first arrive to 1 AU. The longitudinal width of the particle distribution is almost completely determined by the diffusive spreading of the field lines: there is no appreciable additional longitudinal widening of the particle distribution after the first two hours during the simulation period. This is most likely due to a combination of two effects: The crossfield particle diffusion coefficient for the case = 1 AU is half of that of the case = 0.1 AU, thus less crossfield spreading of the particles can be expected. In addition, the particles escape from the inner heliosphere very efficiently due to adiabatic focusing and weak parallel scattering. Thus, the widening of the SEP distribution due to crossfield propagation of SEPs is compensated by the escape of particles to the outer heliosphere, resulting in almost constant intensity at longitudes far from the longitude φ =− 62 ° connected to the SEP source along the Parker spiral.
The longitudinal extent of the SEP events, as observed using multiple spacecraft observations, is typically quantified by fitting a Gaussian curve to the observed peak intensities at different longitudes. Several case and statistical studies report the standard deviation σ_{φ,max} of the Gaussian to be in the range of 30° − 50 ° for both electrons and ions at different energies, in both gradual (Dresing et al., 2012; Lario et al., 2013; Richardson et al., 2014; Dresing et al., 2014) and impulsive SEP events (Wiedenbeck et al., 2013; Cohen et al., 2014). We present the longitudinal distribution of the peak intensities during the first 10 hours for our simulation cases in Figure 6, including the observational range σ_{φ,max} = 30° − 50 ° shown with the gray area, and the conventional FokkerPlanck result for 0.3 AU with the black curve.
As can be seen, the longitudinal width of the SEP event at 1 AU depends on the turbulence strength, with strong turbulence resulting in considerably wider SEP events, with σ_{φ,max} = 41° for the = 0.1 AU case, as compared to the narrow σ_{φ,max} = 23° for the = 1 AU case. We demonstrate the dependence of σ_{φ,max} on the turbulence amplitude, as parametrised by , in Figure 7. The blue curve shows the expected trend for SEPs propagating solely along meandering field lines, which implies which, with λ_{∥} ∝ B^{2}/δB^{2} results in . As shown in Figure 7, our model results follow this scaling well. The slight deviation of the expected trend is likely to be caused by more efficient crossfield diffusion of particles from the meandering field lines by the peak time for small , as the peak times will be progressively later for smaller .
Also evident in the longitudinal distribution of the SEPs in Figure 6 is the asymmetry of the distribution with respect to the longitude connected to the injection site, φ =− 62 °. The centres of the Gaussians are shifted to the West, with the longitude of the centre of the Gaussian φ_{max} =− 48 ° for the strong turbulence case with = 0.1 AU. For weaker turbulence cases, φ_{max} approaches the bestconnected longitude, with −57° and −59° for the =0.3 AU and 1 AU cases, respectively. Similar shifts were also found in simulations by Strauss and Fichtner (2015) and Strauss and Fichtner (2015). A shift of the maximum of around 10− 15 ° to the West has been reported in multispacecraft observed SEP events (Lario et al., 2006, 2013; Richardson et al., 2014). It should be noted though that the multispacecraft measurements are typically performed using a maximum of three measurement points, which makes estimation of the exact shape and asymmetries of the longitudinal distributions difficult. In addition, Richardson et al. (2014) reported the centre of the longitudinal peak distribution as 15° ± 35 ° west of the connected longitude, emphasising the large errors associated in both determining the SEP source location and errors in inferring the magnetic connection of the observing spacecraft to the source location.
Fig. 2 Distribution of 10 MeV SEPs integrated between latitudes −10° and 10°, in arbitrary units, 2 h after impulsive injection at (r, φ, θ) = (1 r_{⊙}, 0, π/2), for the FP model (left panel) and the FP + FLRW model (right panel), respectively, with . The red curve depicts 1 AU radial distance, and the thick black spiral curve the Parker field connected to the injection location. 
Fig. 3 Distribution of 10 MeV SEPs integrated between latitudes −10° and 10°, as in Figure 2, with left panel for FP + FLRW with = 0.1 AU, and right panel for FP+FLRW with = 1 AU. 
Fig. 4 10 MeV SEP intensity at 1 AU, as a function of time and heliographic longitude, with magnetic connection along the Parker spiral at φ =− 62 °, for AU. 
Fig. 5 10 MeV SEP intensity at 1 AU, as a function of time and heliographic longitude, with magnetic connection along the Parker spiral at φ =− 62 °, for AU. 
Fig. 6 The 10 MeV SEP maximum intensity at 1 AU during the first 10 h of the event, as a function of longitude. The thin dashed curves show the fitted Gaussian profiles with , 33°, 23° for 0.1 AU (red curve), 0.3 AU (blue curve) and 1 AU (green curve), respectively, and 10° for the reference FP simulation case with 0.3 AU (black curve). The gray area depicts the observational range σ_{φ,max} = 30° − 50 °. 
Fig. 7 The Gaussian σ_{φ,max} fitted to the 10 MeV SEP maximum intensity at 1 AU during the first 10 h of the event, as a function of mean free path (green circles). The blue curve depicts the trend expected for particles propagating along meandering field lines. 
4 Discussion
Our study shows that meandering field lines are able to efficiently spread SEPs across the mean Parker Spiral direction at wide range of heliospheric turbulence conditions, even in weak scattering conditions. The propagation of SEPs along meandering field lines results in longitudinally wide SEP events, with dependence of the longitudinal width scaling with the SEP parallel mean free path as . The evolution of the SEP event after the initial phase is strongly dependent on the amount of turbulence in the heliosphere. In strong scattering environment, the longitudinal extent of the SEPs increases diffusively, (Fig. 4), whereas in the weak turbulence case, after the initial fast expansion, the longitudinal extent of the SEP event remains unchanged for the first 10 h (Fig. 5).
Our results also emphasise the importance of correctly accounting for the link between the interplanetary turbulence conditions and the particle transport coefficients. High turbulence amplitudes result in strong particle scattering along the mean field direction, and hence a short parallel mean free path (e.g. Jokipii, 1966) The particle propagation across the mean field, on the other hand is more efficient in stronger turbulence, as shown in both simulation studies (e.g. Giacalone and Jokipii, 1999; Laitinen and Dalla, 2017; Laitinen et al., 2017 and theoretical work (e.g. Matthaeus et al., 2003; Shalchi, 2010; Ruffolo et al., 2012). This can be clearly seen in Figure 1, where the evolution of the parallel and perpendicular scattering mean free paths are anticorrelated. Using quasilinear theory and the fieldline diffusion coefficient from Matthaeus et al. (1995), D_{FL}, we find , a scaling which is also consistent with Dröge et al. (2016) (their Fig. 17). Also, changing the geometry of the turbulence from the oftenused energy ratio 20%:80% would have an influence on the SEP event evolution: increasing the proportion of the 2D component would result in faster onsets with wider crossfield extents. The overall dependence of shown in Figure 7 would however, likely stay similar for a fixed ratio. Other refinements of turbulence modelling, such as incorporating scaledependence for the k_{⊥}/k_{∥} anisotropy (Goldreich and Sridhar, 1995; Shalchi et al., 2010; Laitinen et al., 2013a) and dynamical evolution of the turbulence (accounted for parallel propagation in Bieber et al., 1994), would naturally affect both early and late crossfield evolution of particle populations. These are left for future studies.
The interdependency between the parallel and crossfield SEP transport parameters, is typically ignored in parametric 3D SEP transport studies (e.g. Zhang et al., 2009; Dröge et al., 2010; He et al., 2011; Giacalone and Jokipii, 2012), and adhoc values are typically used. While the recent studies by Laitinen et al. (2016) and Strauss et al. (2017) did model SEP propagation with consistently modelled SEP transport coefficients or one set of turbulence parameters, our paper is to our knowledge the first to consistently study the effect of varying turbulence strength on both parallel and crossfield propagation when modelling SEP events in 3D.
The turbulence parameters are typically observed using insitu instruments onboard individual spacecraft, providing a singlepoint measurement of the turbulence properties. The SEPs, however, propagate across the mean field, sampling different heliolatitudes and longitudes, and their propagation is affected by the 3D turbulent structure of the heliosphere. Thus, to improve our ability to estimate the radiation environment in nearEarth space, we should consider a 3D picture of turbulence in the heliosphere. The need for longitudinally resolved particle transport conditions was recently highlighted also by Dröge et al. (2016), who found that the SEP intensities observed by the STEREO and ACE spacecraft during a single SEP event may require different diffusion coefficients for fitting the SEP observations at different longitudes. Recent work by Thomas et al. (2017) has shown promise of using solar wind observations at the Lagrangian point L5, 60° behind Earth at Earth's orbit, for forecasting the solar wind conditions at L1. Such forecast would also make it possible to evaluate either the average or longitudinally dependent SEP transport parameters at a wide range of longitudes, from L5 to Earth and beyond. The recently proposed space weather missions to L5 (Akioka et al., 2005; Gopalswamy et al., 2011; Lavraud et al., 2016; Trichas et al., 2015) thus could bring considerable improvement to our ability to model SEP events and improve our knowledge of the radiation environment in nearEarth space.
5 Conclusions
We have studied how the strength of the turbulence in the interplanetary medium affects SEP event evolution within the new paradigm introduced by L2013 that includes the early nondiffusive crossfield transport of SEPs along meandering field lines. We found that

the parallel and crossfield transport of SEPs are inherently linked through the turbulence properties, with high levels of turbulence resulting in diffusively spreading wide, graduallyrising SEP events, and low turbulence in fast SEP events which remain at nearly constant longitudinal extent after the initial rapid crossfield spreading along meandering field lines;

the longitudinal distribution of 10 MeV proton peak intensity follows approximately a Gaussian shape, with the longitudinal width of the distribution scaling as ;

in strong turbulence, the longitudinal distribution of the particles is asymmetric with respect to the longitude connected to the injection site, with the center of the fitted Gaussian distribution shifted by 14° to the west. Weaker turbulence cases are less skewed with respect to the connected longitude.
Our results show that knowledge of the turbulence conditions of the heliospheric plasmas is crucial for modelling the crossfield propagation of the SEPs early in the events. To forecast the particle radiation conditions at Earth due to solar eruptions we must understand the full chain of phenomena including the injection of the particles at Sun, the physics behind their propagation in the interplanetary medium, and state of the interplanetary turbulence during the SEP propagation within the heliosphere.
Acknowledgements
TL and SD acknowledge support from the UK Science and Technology Facilities Council (STFC) (grants ST/J001341/1 and ST/M00760X/1), and FE from NASA grants NNX14AG03G and NNX17AK25G. The contribution of AK benefited from financial support through project He 3279/151, funded by the Deutsche Forschungsgemeinschaft (DFG), at the CAU Kiel, where large parts of this work were carried out. Access to the University of Central Lancashire's High Performance Computing Facility is gratefully acknowledged. The editor thanks R. Du Toit Strauss and an anonymous referee for their assistance in evaluating this paper.
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Cite this article as: Laitinen T, Effenberger F, Kopp A, Dalla S. 2018. The effect of turbulence strength on meandering field lines and Solar Energetic Particle event extents. J. Space Weather Space Clim. 8: A13
All Figures
Fig. 1 Left panel: Parallel (thick curves) and perpendicular (thin curves) particle mean free paths as function of radial distance from the Sun, for 10 MeV protons and different turbulence strengths parametrised by . Right panel: The ratio of the field line and crossfield particle diffusion coefficients. 

In the text 
Fig. 2 Distribution of 10 MeV SEPs integrated between latitudes −10° and 10°, in arbitrary units, 2 h after impulsive injection at (r, φ, θ) = (1 r_{⊙}, 0, π/2), for the FP model (left panel) and the FP + FLRW model (right panel), respectively, with . The red curve depicts 1 AU radial distance, and the thick black spiral curve the Parker field connected to the injection location. 

In the text 
Fig. 3 Distribution of 10 MeV SEPs integrated between latitudes −10° and 10°, as in Figure 2, with left panel for FP + FLRW with = 0.1 AU, and right panel for FP+FLRW with = 1 AU. 

In the text 
Fig. 4 10 MeV SEP intensity at 1 AU, as a function of time and heliographic longitude, with magnetic connection along the Parker spiral at φ =− 62 °, for AU. 

In the text 
Fig. 5 10 MeV SEP intensity at 1 AU, as a function of time and heliographic longitude, with magnetic connection along the Parker spiral at φ =− 62 °, for AU. 

In the text 
Fig. 6 The 10 MeV SEP maximum intensity at 1 AU during the first 10 h of the event, as a function of longitude. The thin dashed curves show the fitted Gaussian profiles with , 33°, 23° for 0.1 AU (red curve), 0.3 AU (blue curve) and 1 AU (green curve), respectively, and 10° for the reference FP simulation case with 0.3 AU (black curve). The gray area depicts the observational range σ_{φ,max} = 30° − 50 °. 

In the text 
Fig. 7 The Gaussian σ_{φ,max} fitted to the 10 MeV SEP maximum intensity at 1 AU during the first 10 h of the event, as a function of mean free path (green circles). The blue curve depicts the trend expected for particles propagating along meandering field lines. 

In the text 
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