5. SUMMARY AND CONCLUSIONS
Overall, the present study supports the basic idea that the dropout phenomenon, in which the observed SEP flux from impulsive solar flares rises and drops suddenly and nondispersively, is associated with filamentation of magnetic field line connection to the source (Mazur et al. 2000; Giacalone et al. 2000). The temporal features correspond to particle-rich magnetic filaments convecting past the spacecraft. The impulsive solar flare injects particles over a limited spatial region at the Sun (Reames 1992), and field lines connected to that region have a highly nonuniform distribution at 1 AU. In our view (Ruffolo et al. 2003), at 1 AU some field lines are still trapped in filamentary structures while others have escaped to travel far in the lateral directions. Here, using numerical experiments with model fields, we have shown that this is a natural consequence of anisotropic turbulence, which, for solar wind parameters, implies that the lateral motion of field lines cannot be viewed as uniformly diffusive over a scale of 1 AU. In the 2D+slab model of magnetic turbulence, thought to be a relevant idealization of solar wind fluctuations, the average diffusioncan be quite strong, asis indeedinferred from observations of the lateral transport of SEPs (Ruffolo et al. 2003; McKibben 2005). However, the contribution to the ensemble average diffusion coefficient due to the 2D fluctuations is dominant for these parameters, and the implied diffusive rate of lateral spread, while accurate for the ensemble as a whole, does not apply initially to field lines that start near an O-point of the 2D turbulence. Thus
TABLE 2 Comparison of Processes of Trapping and Diffusion of Magnetic Field Lines for a Uniform Mean Field plus Two-dimensional Field plus Slab Turbulence
Process Gaussian 2D Islanda 2D Turbulenceb
Trapping........................................................................................ Within of the O-point Within local trapping boundaries Ensemble average diffusion.......................................................... Slab diffusion Combination of 2D and slab diffusion, 2D-dominated for solar wind conditionsc Motion within the trap.................................................................. 2D motion plus suppressed slab diffusion 2D motion plus slab diffusion Escape from the trap..................................................................... Suppressed slab diffusion Suppressed slab diffusion Requirements for suppression of diffusive escape....................... Strong 2D field Moderate 2D field plus irregular 2D orbit
a The 2D field is defined by eqs. (12) and (13); see Fig. 1. b See Fig. 2. c Matthaeus et al. 1995.
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the small-scale topology of 2D turbulence plays a role in restricting the lateral motion of field lines, with some subset of field lines (those ‘‘trapped’’) approaching the full diffusive limit much more slowly than the ensemble average behavior. Furthermore, during the trapping phase the contribution of slab fluctuations to the field-line random walk is suppressed by a strong 2D field (Chuychai et al. 2005), an effect that further delays lateral spread.Thepresentworkshowsthattheingredientsof topological trapping and suppressed diffusive escape do apply to the 2D+slab model of turbulence with parameters suitable to describe the solar wind, and we can both qualitatively explain the sharp filamentationpatternsandquantitativelyexplaintheirpersistencebeyond a distance of 1 AU. We note that the 2D+slab model is an idealizedapproximation,andweviewtheseparatecomponentsas representing (1) the structure that causes trapping (2D part) and (2) the turbulence that induces escape (slab part). In reality the wavevector distribution should be broader, and for example, the 2D part might be generalized as a flux tube that varies weakly in the parallel direction (as in reduced MHD [Montgomery 1982; Zank & Matthaeus 1992] or the GS model [Goldreich & Sridhar 1995]). The slab component might generalize in a variety of ways to a broadband incoherent MHD wave spectrum. To quantify the phenomena of trapping and diffusive escape of field lines from topological traps associated with transverse complexity of turbulence, we have introduced several concepts. For a specified 2D island, whether regularly shaped (Gaussian) or irregularly shaped (turbulence), we find that the advance of the mean squared lateral displacement toward its asymptotic untrapped limit is delayed by trapping, and the effect is enhanced when field lines start more deeply inside a 2D island. ‘‘Deeper’’ here means that the field line is insulated from the distant outside region by larger transverse (poloidal or 2D) magnetic flux. Quantifyingthiseffectinasimplewayleadstothenotionsofmaximum trapping length, which occurs for the most deeply trapped field lines, and a critical radius (or value of the potential) beyond which field lines are too weakly trapped to see any delay in lateral transport at all. While some insight derives from this perspective, there are limitations:toquantifytrappinginthis way, one needs to look at one specific magnetic island, and furthermore, one needs to examine conditional statistics of many field lines. Topartiallyalleviatethesedifficulties,weintroducedthenotion of local trapping boundaries (LTBs). All closed 2D field lines might be viewed as potential trapping regions, but it turns out that islands with more flux contained in them, or more properly, more
flux per unit radius, seem to provide better traps. This means that field lines with strong average 2D magnetic field strength (flux density) are good candidates to trap field lines effectively. These are the LTBs, which can be calculated from the 2D magnetic field alone,withnoreferencetothestatisticsof fieldlines.Nevertheless our numerical experiments show that the LTBs provide a good estimate of where trapping will occur. We find from the above discussion that suppressed diffusion is an important process that is expected to contribute to the dropout features of field lines over a distance of 1 AU. The assumption is that the interplanetary magnetic field is highly structured in the direction transverse to the mean field, in the sense of 2D turbulence. Therefore, near injection regions field lines may be near either near O- or X-points, and for some span of distances, these lead to different rates of lateral spread. The field lines near the O-points experience suppressed diffusion and diffuse in the directionperpendiculartothemeanfieldmoreslowlythanthefield lines nearthe X-points.Therefore, thesedifferent ratesofspreading lead to inhomogeneous features and sharp gradients of the field-line density in the direction perpendicular to the mean field. If these field lines represent the guiding centers of the SEPs injected from a localized source near the Sun, the particles then follow those field lines. We find a dropout-like distribution of the field lines (surrogates for particles) at 1 AU if the size of the island is about 0.03 AU. From this study, we can say that the fieldlinescanbetrappedfordistanceslongerorshorterthan1AU, and also for trapping-island sizes that can be larger or smaller than 0.03 AU, depending on the magnetic field parameters as describedabove.Conversely,inprincipleourformulationofthetrapping length might be used to analyze observed dropout features in conjunction with magnetic field parameters of the solar wind, using other methods to model or constrain the topology of the nearbymagneticfield(Hu&Sonnerup2003).Inthiswaythepresent analysis may lead toward a physical explanation of the persistence, sharpness, and intermediate filling factor of dropouts of SEPs at Earth orbit, as found both in observations and in various independent simulation models.