Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

A recently developed time-dependent fractional Parker transport equation is solved to investigate the parallel and momentum superdiffusion of energetic charged particles in an inner heliospheric region containing dynamic small-scale flux ropes (SMFRs). Both types of superdiffusive transport are investigated with fractional transport terms containing a fractional time integral combined with normal spatial or momentum derivatives. Just as for normal diffusion, accelerated particles form spatial peaks with a maximum amplification factor that increases with particle energy. Instead of growth of the spatial peaks until a steady state is reached as for normal diffusion, parallel superdiffusion causes the peaks to dissipate into plateaus followed by a rollover at late times. The peaks dissipate at a faster rate when parallel transport is more superdiffusive. Furthermore, the accelerated particle spectral distribution function inevitably becomes an f 0 ∝ p −3 spectrum at late times in the test particle limit near the particle source despite the potential for spectral steepening from other transport terms. All this is a product of the growing domination of parallel spatial and especially momentum superdiffusion over other transport terms with time. Such extreme late time effects can be avoided by a transition to a normal diffusive state. Finally, fitting spatial peaks observed during SMFR acceleration events with the solution of the fractional Parker transport equation can potentially be used as a diagnostic for estimating the level of spatial and momentum superdiffusion in these events and how the levels of superdiffusion vary with distance from the Sun.


Introduction
It has often been discussed that suprathermal particle distributions in the large-scale, quiet solar wind away from shocks are observed to form power-law spectra.The observations indicate that, averaged over suitably long timescales, the directionaveraged distribution function in the solar wind frame is a power law f 0 (p) ∝ p −5 (e.g., Fisk & Gloeckler 2006, 2008).These spectral slopes have been explained in terms of stochastic acceleration caused by self-consistent interaction of energetic particles with compressive wave turbulence that settles into a quasi-stationary state characterized by a universal f 0 (p) ∝ p −5 power-law spectrum (Fisk & Gloeckler 2006, 2008).In a different approach, Zhang (2010) found that energetic particle interaction with a train of compressive waves inevitably produces an accelerated particle distribution f 0 (p) ∝ p −3 in the test particle limit.Competition from large-scale adiabatic cooling by the radially expanding solar wind can produce a steeper accelerated spectrum that does not necessarily produce an f 0 (p) ∝ p −5 spectrum.However, if acceleration with the train of compressive waves is treated self-consistently, there develops a balance between the acceleration by compressive wave and adiabatic cooling that can produce, on average, an f 0 (p) ∝ p −5 spectrum according to Zhang (2010).Nonetheless, some observations also show that often significant deviations from an f 0 (p) ∝ p −5 spectrum occur (see discussion by Schwadron et al. 2010).Furthermore, Schwadron et al. (2010) also make the argument that, over the long timescale needed to observe an f 0 (p) ∝ p −5 spectrum for suprathermal particles, this power law might be a consequence of an average over many separate stochastic acceleration processes, which individually produce exponential or Gaussian particle spectra with different characteristics that pass over the spacecraft during this long time period.It has been shown before that energetic particle gyroresonant interaction with parallel propagating Alfvén waves produces exponentially accelerated particle spectra (Isenberg 1987), which then implies that multiple adjacent Alfvén wave acceleration regions in the solar wind with different characteristics will produce distinct exponential spectra that, after averaging over a long enough timescale, will yield an f 0 (p) ∝ p −5 spectrum following the Schwadron et al. (2010) approach.
Whereas the above discussion features some recent developments in a long history of trying to explain the formation of suprathermal particles in the large-scale solar wind within the framework of particle acceleration by low-frequency propagating magnetohydrodynamic (MHD) waves, recent work suggests that there is reason to attack this problem also from the viewpoint of energetic particle-particle interaction with and acceleration by dynamic, non-propagating coherent magnetic field structures, such as small-scale magnetic flux ropes (SMFRs).Analysis of recent spacecraft observations near 1 au indicated for the first time a potential connection between multiple interacting (merging) SMFRs, produced by turbulent magnetic reconnection in the vicinity of large-scale reconnecting current sheets in the inner heliospheric solar wind, and efficient particle acceleration (Khabarova & Zank 2017).Previously it was not realized that magnetic reconnection-related processes can result in efficient energetic particle acceleration in the large-scale solar wind (see discussions by Khabarova & Zank 2017;Adhikari et al. 2019).Comprehensive kinetic transport theories were developed to model the energetic particle acceleration by numerous dynamic SMFRs in the large-scale solar wind (e.g., Zank et al. 2014;le Roux et al. 2015).SMFR acceleration mechanisms in these theories include, e.g., first-order Fermi acceleration by the mean compression rate experienced by SMFRs, and stochastic acceleration (second-order Fermi acceleration) by fluctuating turbulent motional electric fields in the guide/background magnetic field direction generated in contracting and merging SMFRs.Solutions of these transport equations were successfully used to reproduce the main observed energetic particle features of identified SMFR acceleration events in the inner heliosphere (e.g., Zhao et al. 2018;Van Eck et al. 2021).
SMFRs are defined as quasi-helical, coherent, non-propagating magnetic field structures advected with the solar wind flow.They are composed of a guide magnetic field component in the axial direction and a 2D twist or magnetic island component perpendicular to the guide field component (Cartwright & Moldwin 2010).SMFRs near Earth typically have cross sections <0.1-0.01 astronomical units (au), which overlap largely with the inertial-range scales of solar wind magnetic turbulence.These structures are different from Alfvén vortex structures that propagate at the Alfvén speed (Hu et al. 2018).There is a longstanding debate as to whether SMFRs mainly originate near the Sun (e.g., Borovsky 2008) or whether they are generated mainly locally in the solar wind.Evidence is increasing that in a magnetically turbulent space plasma with a significant guide/ background magnetic field like the solar wind, SMFRs naturally form as part of a locally generated quasi-two-dimensional (quasi-2D) MHD turbulence component that is thought to dominate other MHD wave turbulence modes, such as shear Alfvén waves.This is supported by observations in the slow solar wind near 1 au (e.g., Matthaeus et al. 1990;Bieber et al. 1996;Greco et al. 2009;Zheng & Hu 2018), MHD simulations (e.g., Shebalin et al. 1983;Dmitruk et al. 2004) and nearly incompressible MHD turbulence transport theory (Zank & Matthaeus 1992, 1993;Zank et al. 2017Zank et al. , 2018Zank et al. , 2020)).Furthermore, observational evidence is accumulating that additional SMFRs are generated locally in the solar wind near the heliospheric current sheet, large-scale current sheets associated with interplanetary coronal mass ejections, and corotating interaction regions through turbulent magnetic reconnection in these current sheets, and that neighboring SMFRs undergo merging through magnetic reconnection at small-scale current sheets in between (Khabarova et al. 2015(Khabarova et al. , 2016;;Hu et al. 2018;Zheng & Hu 2018;Chen et al. 2019;Chen & Hu 2020).Unprecedented numbers (a few hundred per month) of SMFRs have been identified with the powerful Grad-Shafranov method in the inner heliosphere from ∼0.3 to 7 au using Helios, Advanced Composition Explorer (ACE), Wind, Ulysses, and Voyager data (e.g., Hu et al. 2018;Zheng & Hu 2018;Chen et al. 2019;Zhao et al. 2019;Chen & Hu 2020), thus providing support for their expected widespread presence in the inner heliosphere as discussed above.
Some observational evidence exists that energetic particles can be temporarily trapped in SMFRs in the solar wind.Mazur et al. (2000) and Chollet & Giacalone (2008) reported large abrupt drops in the counting rate of solar-flare-generated solar energetic ions (SEPs) at 1 au over small spatial scales of the order of the solar wind turbulence correlation length, suggesting that these sudden drops are connected to the properties of low-frequency MHD turbulence.Ruffolo et al. (2003) offered an explanation for these dropout events that involves SEPs from solar flare events populating two classes of magnetic flux tubes.SEPs that propagate to 1 au while being quasi-trapped in closed SMFR structures will retain a relatively high counting rate, while those SEPs propagating along adjacent open magnetic field lines that spread efficiently in latitude and longitude will arrive at 1 au with a strongly reduced counting rate.More recently, the connection of dropout events with SEP trapping in SMFRs at 1 au was confirmed by Trenchi et al. (2013) who used Grad-Shafranov reconstruction to identify SMFRs in ACE spacecraft data coinciding with SEP counting rate peaks during a series of dropout events of a solar flare related SEP event in 1999.Also, SMFRs were identified in Parker Solar Probe data closer to the Sun by Pecore (2021) employing a real-space evaluation of magnetic helicity, and their potential boundaries using the partial variance of increments method.The authors found that energetic particles are either confined within or localized outside of SMFRs.Energetic particle trapping in magnetic islands produced by primary current sheet magnetic reconnection has been reported in many simulations (e.g., Drake et al. 2006;Guidoni et al. 2016;Xia & Zharkova 2018).In addition, particle simulations, such as those by Drake et al. (2006Drake et al. ( , 2010)), exhibit systematic acceleration of particles temporarily trapped in a contracting magnetic island.
To this can be added that analysis of the statistics of solar wind magnetic field component increments determined from ACE spacecraft data at 1 au yielded a non-Gaussian probability distribution function (PDF) with strong power-law tails due to intermittent strong jumps in the magnetic field component increments (Greco et al. 2009).This PDF has been successfully reproduced by calculating the corresponding PDF for a 2D MHD turbulence simulation containing a strong presence of SMFRs.It appears that strong tails in the latter PDF were generated by large intermittent jumps in the magnetic field component increments at the boundaries of SMFRs.Using Wind data, Zheng & Hu (2018) identified numerous SMFRs at 1 au with the Grad-Shafranov method.From the SMFR data, the PDF of the out-of-plane current density was constructed, which again yielded a power-law PDF consistent with the PDF derived from the 2D MHD turbulence simulation of Greco et al. (2009).Thus, a strong case was presented of an inner heliospheric solar wind filled with numerous coherent SMFR structures that generate intermittent strong magnetic field component increments best modeled statistically as a non-Gaussian power law with prominent tails.It stands to reason that when suprathermal particles follow the magnetic field lines of numerous SMFRs in the solar wind, their response will be intermittently strongly disturbed guiding center trajectories that statistically yield a non-Gaussian power-law PDF.
On the basis of the evidence presented above, it is highly likely that energetic particle transport through an SMFR field should yield anomalous diffusion in ordinary and momentum space.Evidence in favor of such anomalous transport can be found in 3D compressible MHD simulations of a strongly turbulent plasma with large out-of-plane coherent small-scale current densities, and small-scale current sheets between coherent magnetic field filament structures that turbulently reconnect to produce turbulent motional electric field fluctuations.Analysis of simulations of energetic test particle trajectories through the motional electric fields of these 3D MHD simulations show strong superdiffusion in particle energy space (Isliker et al. 2017a(Isliker et al. , 2017b;;Nakanotani et al. 2022).Further work along these lines that focuses on simulating eruptions and jets in the solar atmosphere that result in turbulent reconnecting small-scale current sheets suggest that superdiffusive energetic particle transport occurs both in ordinary and in energy space (Isliker et al. 2019).

Motivation
In le Roux & Zank (2021) we presented a derivation from first principles of a new pitch-angle dependent, nonlinear fractional diffusion-advection equation for energetic particle interaction with dynamic coherent SMFR structures with sizes coinciding with magnetic turbulence inertial-range scales in the large-scale solar wind on the basis of the standard focused transport equation (Skilling 1975).The idea was to capture how this interaction generates coherently disturbed guiding center trajectories, trajectories in momentum space, and in pitch-angle space inside SMFRs due to temporary trapping of energetic particles in SMFRs that statistically results in anomalous spatial, momentum, and pitch-angle diffusion on larger spatial and timescales as the particles scatter when crossing abrupt SMFR boundaries.Following the pioneering approach of Sanchez et al. (2006), this was accomplished partly by specifying a power-law PDF for disturbed guiding center trajectories, disturbances in the rate of momentum change, and variations in the rate of pitch-angle change in the derivation that represent the large-scale asymptotic of a Lévy stable distribution of these disturbances instead of a Gaussian PDF (see le Roux & Zank 2021).Each of these three power-law PDFs was separated into independent PDFs for the particle step size and the waiting time between scattering events generated by crossing of sharp SMFR boundaries, reminiscent of classical continuous-time-random-walk theories with independent PDFs for the step size and the waiting time.Another key element in the theoretical development was retaining time, spatial, momentum, and pitch-angle lags in the average particle distribution that cannot be ignored (as can be done in quasilinear theory for particle interaction with weak wave turbulence) since the distribution function varies significantly on the scale over which the turbulence correlation functions in the transport terms decay in response to the intermittently strong SMFR fields.This approach naturally yielded a transport equation with anomalous diffusion terms featuring fractional derivatives (Sanchez et al. 2006;le Roux & Zank 2021).
Energetic particles are expected to interact numerous times with SMFR structures on large spatial and long timescales in the solar wind.These structures, characterized by large magnetic field component increment variations at their boundaries (e.g., Greco et al. 2009), should strongly scatter particles every time when particles cross their boundaries.Therefore, sufficient particle scattering should occur to maintain nearly isotropic pitch-angle distributions on large spatial and timescales (e.g., Effenberger 2012; Zimbardo et al. 2017).On this basis, le Roux (2022) demonstrated in follow-up work how the pitch-angle dependent fractional diffusion-advection equation can be converted into fractional Parker and Gleeson-Axford transport equations.
An important prediction of the new fractional Parker transport equation is that parallel transport through an SMFR field can be superdiffusive, while energetic particle advection processes remain normal.The transport theory naturally yields that parallel superdiffusion is controlled by a fractional partial time derivative with a fractional index 2 − β ∥ instead of by fractional spatial derivatives because the latter are normal derivatives.Thus, the variance of the particle parallel , where 1 < 2 − β ∥ < 2 if β ∥ < 1, to cover the standard superdiffusive range.It is interesting to note that such a parallel superdiffusive result was also obtained based on solar wind observations that energetic particle gyroresonant interaction with Alfvén waves can be characterized by a power-law distribution of pitch-angle scattering times instead of a single value (Zimbardo & Perri 2020).The benefit of the fractional time derivative approach to superdiffusion is that the problem of the infinite variance of the particle step size (Lévy flights), inherent in specifying a power-law step-size PDF without a cutoff or rollover at large spatial intervals, is avoided (Zimbardo et al. 2021).However, the consequence of modeling parallel superdiffusion through SMFRs with a fractional partial time derivative is that energetic particle advection transport processes are increasingly dominated by parallel superdiffusion with time, which needs further exploration.As a first application, le Roux (2022) investigated the consequences of the new fractional Parker equation for superdiffusive shock acceleration at a parallel heliospheric shock when dynamic SMFRs have a strong presence in its vicinity.Surprising consequences of a fractional time derivative approach to parallel superdiffusion at a parallel shock that needs further reflection are (1) the timescale for superdiffusive shock acceleration is inherently fractional; (2) as superdiffusion increasingly dominates solar wind advection, the accelerated particle distribution upstream converges to a plateau at very late times; and (3) the upstream distribution solution has the characteristics of a combined time-dependent diffusion and wave solution.
However, in that study, the acceleration of energetic particles by dynamic coherent SMFRs was not addressed.The purpose of this paper is to broaden the investigation of the interaction of energetic particles with dynamic SMFRs in the large-scale solar wind following the suggestion from simulations discussed above that this interaction results in both parallel spatial transport, and transport in momentum space, to be superdiffusive.As in le Roux (2022), this study will be conducted assuming that both superdiffusive transport processes can be modeled with fractional time derivatives while keeping the spatial and momentum partial derivatives normal.The repercussions and feasibility of such an approach will be analyzed.

The Fractional Parker Transport Equation for Energetic Particle Acceleration by SMFRs and Its Solution
To model energetic particle acceleration by SMFRs propagating superdiffusively along the large-scale magnetic field, we simplify the fractional Parker transport equation by allowing for anomalous spatial diffusion to occur only along the mean magnetic field where it is predicted to be superdiffusive.This can be justified by our assumption that temporary particle trapping in SMFRs results in inefficient perpendicular subdiffusion so that perpendicular anomalous diffusion to a first approximation can be neglected.Further simplification of the fractional Parker equation can be achieved by assuming that suprathermal particles are predominantly energized by statistical fluctuations in SMFR fields rather than by average SMFR fields.Some evidence in favor of this approach can be found in full particle simulations of particle acceleration during force-free magnetic reconnection (Che & Zank 2020;Che et al. 2021), and in fits of a solution of our normal Parker transport equation for energetic particle acceleration by SMFRs to observations of SMFR acceleration in the inner heliosphere (Van Eck et al. 2021).The simplified fractional Parker transport equation is given by where f 0 (x, p, t) is the energetic particle direction-averaged distribution function depending on position x, particle momentum magnitude p, and time t.In Equation (1), U 0 is the large-scale solar wind flow velocity, b 0 is the unit vector along the large-scale solar wind magnetic field,  , where 1 < 2 − β ∥ < 2 for 0 < β ∥ < 1, so that anomalous parallel diffusion falls in the standard superdiffusive range.The last term on the right-hand side of Equation (1) models suprathermal particle acceleration by the fluctuating component of SMFR fields as a more general anomalous diffusion process that includes both a fractional time integral because its fractional index 1 − β p < 0 due to the restriction 1 < β p < 2, and fractional momentum derivatives with a fractional index α − 1 restricted by 1 < α < 2. After multiplying Equation (1) with the fractional derivative with 1 < 2β p /α < 4 so that anomalous momentum diffusion falls in a momentum superdiffusion range that extends beyond the normal range of 1-2.
To facilitate a time-dependent analytical solution, we consider one-dimensional spatial transport parallel to the solar wind flow from the Sun in the ecliptic plane.For this purpose, we introduce a Cartesian coordinate system in which the large-scale solar wind flows with a constant speed along the x-axis U 0 = U 0 e x , and the interplanetary magnetic field B 0 lies in the x-z-plane (ecliptic plane) with an angle ψ between it and the solar wind flow direction.The effect of adiabatic cooling due to spherical solar wind expansion on energetic particles is accounted for by specifying the divergence of the solar wind flow in the ecliptic plane as ∇ • U 0 = 2U 0 /x.The competition between solar wind advection and superdiffusion in the xdirection is modeled by projecting the parallel superdiffusion coefficient in the x-direction according to the expression . The momentum superdiffusion term is adjusted by letting the fractional momentum derivative index α → 2 whereby . Thus, the momentum derivatives become normal and áD ñ µ D b p t 2 p with 1 < β p < 2 so that anomalous momentum diffusion falls in the standard superdiffusion range.With this simplification both parallel and momentum superdiffusion by SMFRs are the result of fractional time integrals only (or equivalently fractional time derivatives as discussed above), thus making it easier to find time-dependent analytical solutions.The equation to be solved for f 0 (x, p, t) now reads as where The coefficients with a bar on top in Equations (2) and (3) indicate that they are constant because they have been averaged over a selected spatial distance interval Δx and over a selected particle speed interval Δv.Therefore, the coefficient k is the path length along a magnetic field line through a helical SMFR structure, and that is the Alfvén ratio of SMFRs.Furthermore, 2 is the ratio of the average magnetic energy density of the twist/island component δB I over the magnetic energy density of the guide/ background field B 0 component inside SMFRs (SMFRs are considered to be quasi-2D magnetic structures where δB I is modeled to be in the 2D plane perpendicular to B 0 ), δU I is the SMFR flow velocity due to SMFR contracting and merging activity, also defined to be in the 2D plane perpendicular to B 0 , V A0 is the Alfvén speed associated with the B 0 component, v 0 is the minimum energetic particle speed, and v m is the maximum energetic particle speed bounding the speed interval Δv over which the parallel and momentum superdiffusion coefficients are averaged, L I is the average width and L I|| the average length of SMFRs, and d i is the ion inertial length scale.The expression of the escape time t esc can be found in Section 3.2 (further below).The derivation of the expression of ), and a steady-state particle point source to inject particles at a fixed rate N  at position x 0 and momentum p 0 into the SMFR acceleration region given by the final version of the equation to be solved for f 0 (x, z, t) becomes By defining the double Laplace transform , , , 6 Equation (5) can be expressed as an evolution equation for h f x s , , 0 ( )according to which can be interpreted as an ordinary second-order differential equation with respect to x for solution purposes.
The solution of Equation ( 7) is where 1 2 4 2 3 3 1 .9 The solution requires that l > Re 0 1 ( ) and l < Re 0 2 ( ) be combined with the boundary conditions . The next step is to perform the inverse Laplace transform ) of solution (8) to recover the momentum dependence of the solution ( = z p p ln 0 ( )).After some manipulation, the inverse Laplace transform can be expressed in terms of a cosine transform so that the solution becomes The cosine transform is evaluated using transform 27 on page 17 of the table of integral transforms by Erdelyi et al. (1954) to yield the solution 11 where . 12

Time-dependent Solutions at Late Times
Time-dependent analytical solutions can be found by assuming the same level of superdiffusiveness for anomalous parallel and momentum diffusion.This means requiring that 2 − β ∥ = β p so that 2 − (β ∥ + β p ) = 0.In combination with assuming a late time solution by letting s → 0 in expression b (s), the time-dependent solution at late times is where K 0 is the modified Bessel function of the second kind containing in its argument the simplified expression for b(s) given by 14

A Solution for an Intermediate Level of Superdiffusiveness
To evaluate the implications of superparallel and momentum diffusion for energetic particle transport and acceleration in a dynamic SMFR region in the large-scale solar wind, two specific levels of superdiffusion are analyzed.The simplest solution follows when specifying the fractional time index related to the level of parallel superdiffusion to be β ∥ = 1/2 in the general late time solution (13).Since it was assumed that 2 − β ∥ = β p , it follows that for parallel superdiffusion projected in the , so that both superdiffusion processes exhibit the same intermediate levels of superdiffusion.The evaluation is based on taking two limits, the high-particle-momentum limit , or the far-from-thesource limit k -D pp x x ln . Consider first the high-momentum limit.The argument of the modified Bessel function in Equation ( 13) is large so that it can be approximated with the expression , and the inverse Laplace in Equation ( 13) can be evaluated.This results in the solution where having introduced the characteristic timescales for momentum superdiffusion , and adiabatic cooling t = x U 3 2 Similarly, in the far-from-the-source limit use can be made of the large-argument approximation of K 0 to evaluate the inverse Laplace transform in Equation (13).Consequently, where which includes the effective fractional superdiffusion time- . After simplifying k 2 with the aid of the far-from-the-source limit, it follows that 21 Consider the high-momentum solution limit (17).The steady-state portion of the accelerated spectrum is a power law f 0 (p) ∝ p −3 with a weak rollover with increasing momentum (see line one of Equation ( 17)).The time-dependent part of the solution indicates that the exponential rollover of the accelerated spectrum is stronger at earlier times, which is a natural consequence of the evolution of an accelerated spectrum with time (see line two of Equation ( 17)).Assume the Sun to be at x = 0 and that x > 0. For x > x 0 (distances in the anti-sunward direction beyond the particle source located at x = x 0 ), the distribution function f 0 increases with increasing separation distance x − x 0 > 0. This increase, which occurs as parallel superdiffusion projected in the x direction beyond the source is in the same direction as particle advection by the outflowing solar wind (U 0 > 0), becomes less pronounced with time until at very late times the distribution becomes a plateau (see line three of Equation (17)).For x < x 0 , on the other hand, (at sunward distances inside the particle source), the distribution function f 0 decreases in the sunward direction with the increasing separation distance x 0 − x > 0. The decrease, as a consequence of particle superdiffusion inside the source in the sunward direction being in the opposite direction of outward solar wind advection (modulation), also dies out with time to form a plateau at very late times.Note also that the respective spatial increases and decreases in f 0 relative to the source are stronger at higher energies.Furthermore, particle escape strengthens the spectral rollover because of its presence in line two, and accentuates the sunward decrease in f 0 inside the particle source and increase of f 0 away from the Sun outside the source since it also appears in line three.However, at late times, these escape effects on both the spectral and spatial profiles evaporate as the exponential functions in lines two and three containing the escape time converge to one.
Spatial plateau formation on both sides of the particle source at late times is a manifestation of spatial superdiffusion growing in importance relative to the solar wind advection term.This can be seen by applying dimensional analysis to the fractional time integrals in transport Equation (2) whereby having assumed β p = 2 − β ∥ as in our analytical solution.
Keeping in mind that 0 < β ∥ < 1, inspection of the evolution of all the transport terms with the characteristic timescale (T) indicates that, at late times, the spatial and momentum superdiffusion terms will dominate all other transport terms so that solar wind advection cannot compete against spatial superdiffusion any longer.Thus, the decrease in f 0 in the sunward direction of the particle source, and the increase of f 0 away from the Sun beyond the particle source disappear as time-dependent superdiffusive particle spreading on both sides of the source continues unopposed.Note also that the transformation of f 0 into a plateau will be accelerated for higher levels of superdiffusion (smaller values for β ∥ closer to zero) as superdiffusion more quickly dominates solar wind advection.This accelerated trend toward a plateau will be verified further below by considering a solution for a lower level of superdiffusiveness for comparison.Inspection of the far-from-the-source limit (20), indicates a power law f 0 (p) ∝ p −3/2 , and that f 0 both beyond the particle source decreases in the direction away from the Sun and inside the particle source decreases toward the Sun (see line two).This spatial decrease in f 0 is stronger during earlier times as the exponential function in line three indicates.In comparison, the opposite solution limit discussed above exhibits a softer powerlaw spectrum ( f 0 (p) ∝ p −3 ) while f 0 in contrast increases beyond the source in the direction away from the Sun.
Considering both solution limits together, one can envision that energetic particles form spatial peaks in f 0 with an increasing amplitude at higher energies beyond the particle source that dissipates at late times into a plateau followed by a decrease, and that inside the particle source, the sunward decrease in f 0 transforms into a plateau followed by a rollover at late times as superdiffusion increasingly dominates spatial advection.The accelerated particle momentum spectrum becomes increasingly hard further away from the source at lower particle energies.This manifests far from the source as a spectral rollover at lower energies toward a p −3/2 spectrum because at high energies the spectrum converges to the softer p −3 power law as described by the high-energy solution limit (17).This spectral rollover at lower particle momenta away from the particle source is simply a consequence of the shift of the accelerated particle spectrum to higher momenta with time as particles propagate further away from the source position while continuing to be accelerated by SMFRs.A cutoff at lower momenta is avoided by the superdiffusive spreading of the particles during acceleration that generates the rollover.
Equation ( 22) is also useful to help us understand why the spectrum inevitably becomes an f 0 (p) ∝ p −3 close to the particle source at late times in the case of superdiffusion.At late times the transport terms for spatial and momentum superdiffusion dominate other transport terms as discussed above.However, because spatial superdiffusive spreading dominates solar wind advection, the spatial gradient converges to zero at late times in the vicinity of the particle source.Consequently, momentum superdiffusion becomes the sole dominant transport process at late times.The solution of Equation ( 22) then inevitably yields an f 0 (p) ∝ p −3 power-law spectrum in the test particle limit (if the particle pressure becomes high enough to generate a backreaction on the SMFR field, the efficiency of SMFR acceleration and thus the spectral power law can be modified).This is different from the case of normal spatial and momentum diffusion (β ∥ → 1 in Equation ( 22)) because then other transport terms, if strong enough, can compete with momentum diffusion at any time whereby the accelerated spectrum will deviate from an f 0 (p) ∝ p −3 power law at late times.For example, terms that model adiabatic cooling by the expanding solar wind and particle escape from the SMFR acceleration region will steepen the accelerated spectrum at late times.

A Solution for a Low Level of Superdiffusiveness
In this case, the fractional time index related to the level of parallel superdiffusion is specified to be β ∥ = 2/3 in the general late time solution (13).Since 2 − β ∥ = β p for parallel superdiffusion projected in the x direction, the mean-square particle displacement is áD ñ µ D b -= x t 2 2 43  , and for momentum superdiffusion the mean-square particle momentum displacement is áD ñ µ D b = p t 2 4 3 p so that both superdiffusion processes have the same low levels of superdiffusion.
First, the solution is presented for the high-particlemomentum limit k -D pp x x ln . As before, the large-argument limit of the modified Bessel function K 0 in solution (13) is implemented, enabling evaluation of the inverse Laplace transform in the solution.The result is that ´- where t k 3 , and K 1/3 is a modified Bessel function of the second kind.Note, that in this case the characteristic timescale for momentum superdiffusion is redefined as t = D 1 the expression for k in Equation (24).To capture the timedependent peak in the particle distribution beyond the particle source, we relate K 1/3 first to a U function according to the expression p = 56 ,53 ,2 ), then approximates the U function in the small argument limit (the argument of K 1/3 in solution ( 23) is small at late times) as U(x) ≈ Γ(2/3)/Γ(5/6)/(2x) 2/3 to get 25 After simplifying k 3/2 for the high-particle momentum limit, the solution is To find the solution for the far-from-the-source limit k -D pp x x ln , the large-argument approximation of K 0 in solution (13) is again implemented before evaluating the inverse Laplace transform in this solution specifically for β ∥ = 2/3.The outcome is The expression for k includes a definition of the fractional timescale for effective spatial superdiffusion t . Following the same approach in determining the expression for K 1/3 in the small argument (late time) limit, solution (27) becomes By applying the large-distance limit to simplify the expression for k 3/2 in Equation (24), the final result is where Comparison of the solutions for low and intermediate levels of superdiffusion shows that the solutions predict qualitatively the same behavior for energetic particles undergoing spatial and momentum superdiffusion in the dynamic SFMR field.In both cases, one sees time-dependent spatial peak formation in the accelerated distribution function beyond the particle source in the downwind direction that flattens to a plateau followed by a decay, while inside the particle source there occurs a timedependent decay in the distribution function in the upwind direction that evolves to a plateau followed by a rollover at very late times.The spectral hardening from an f 0 (p) ∝ p −3 spectrum close to the particle source to an f 0 (p) ∝ p −3/2 spectrum further away at lower particle momenta also occurs for both levels of superdiffusion.Quantitatively, the spatial peaks for different levels of superdiffusion in the solutions decay at different rates.In the case of the low level of superdiffusion, the decay is determined by the factor t −1/3 (see line three in solution (30)) whereas in the case of an intermediate level of superdiffusion the decay is determined by t −1/2 (see the third line of solution (21)).Therefore, the stronger the superdiffusion, the more quickly the spatial peaks will decay with time.It is unlikely that spatial peaks will exist in the case of strong superdiffusion.This issue will be investigated further below with the aid of more general seminumerical solutions.

The Diffusive Solution
A time-dependent diffusive solution can be obtained by letting the time-fractional indices β ∥ → 1 and β p → 1 in Equation (11) and evaluating the inverse Laplace transform.The solution is where By letting t → ∞ in the time integral in line two of Equation (32), it can be evaluated to yield a steady-state solution where α and β are specified in Equation (33), and K 0 is a modified Bessel function of the second kind.This solution can be more directly compared to the solutions for superdiffusion by taking the same limits.Application of the high-momentum limit k -D p p x x ln ) , combined with specifying strong momentum diffusion by requiring t t  , and When applying the far-from-the-source limit k -D p p x x ln ) , and the strong momentum diffusion limit as above, the solution becomes The diffusive solution contains the same features as the superdiffusive solutions.There is a similar formation of a spatial peak in the distribution function f 0 when x > x 0 .This can be deduced from the increase in f 0 in the high-momentum limit (35), followed by a decrease in f 0 in the far-from-the-source limit (36), when x > x 0 .It is less easy to see the increase in the amplitude of the spatial peak with increasing particle energy in the diffusive solution compared to the superdiffusive solutions discussed above.
The increase in the amplitude becomes clear when realizing that the limit for which the increase in f 0 is predicted is valid for larger distances beyond the source at higher particle momenta.It was discussed above for the superdiffusive solution at late times that the accelerated particle spectra inevitably tend to f 0 (p) ∝ p −3 close to the source, rolling over to p −3/2 spectra far from the source at lower suprathermal particle momenta.Having assumed a strong momentum diffusion limit in the diffusive solution, we find the same rollover trend from an f 0 (p) ∝ p −3 close to the source to p −3/2 spectra with increasing distance beyond the particle source in the lower-energy spectra by considering the diffusive solution limits (35) and (36).However, it should be noted that the expression of the exponent of the momentum power law in Equation (35) suggests that if other transport processes, such as adiabatic cooling (t U 0 ), effective parallel spatial diffusion (t k 0 ), and particle escape (t esc ), are more efficient, the spectrum for accelerated particles in the diffusive case will be steeper than f 0 (p) ∝ p −3 at late times.In the case of superdiffusion these competing transport processes, even if potent early on, will not be competitive with super momentum diffusion a late times so that f 0 (p) ∝ p −3 is the only possible outcome.A good question is why effective parallel diffusion, a competition between parallel diffusion and solar wind advection, can steepen the accelerated particle spectrum.A possible explanation is that the parallel diffusion of particles through SMFRs reduces the contact time with individual SMFRs, resulting in less efficient acceleration.However, solution (35) suggests that increasing the parallel diffusion coefficient (increasing t k 0 ) reduces the steepening effect.This apparent contradiction can be explained by taking into account that a larger diffusion coefficient means that diffusive particle spreading becomes more efficient, thus reducing the spatial gradient in the particle distribution.Consequently, the parallel diffusion term as a whole is reduced, which increases the contact time with individual SMFRs and the efficiency of SMFR acceleration.
A key difference between the diffusive solution and the superdiffusive solutions is that the spatial peak in the accelerated particle distribution function for x > x 0 survives at very late times in the diffusive case as the steady-state solution limits (34) and (35) indicate.As discussed above, in the superdiffusive cases the spatial peak in f 0 dissipates at late times as a consequence of the increasing dominance of superdiffusive particle spreading over solar wind advection as a result of the presence of the growing fractional time integral in the parallel superdiffusion term.The more superdiffusive the transport, the more quickly the decay of the peak sets in.This does not happen in the diffusive case.Thus, for a given acceleration time the amplitude of the peak will decrease with the level of superdiffusiveness assumed.
A number of energetic particle events associated with SMFR acceleration were identified in the inner heliosphere in which the flux enhancements in time profiles are characterized by an amplification factor that increases with particle energy.Steadystate analytical solutions of normal Parker-type transport equations for energetic particle interaction with dynamic SMFRs, such as Equation (34), were used to successfully fit these flux-enhancement profiles in the diffusive limit (Zhao et al. 2018;Adhikari et al. 2019;le Roux et al. 2019;le Roux 2022).In future work, these efforts can be expanded by combining diffusive timedependent solutions, such as Equation (32) (see also the timedependent solution in le Roux et al. 2019), with more general superdiffusive solutions presented further below to infer what level of superdiffusiveness, if any, is needed to explain the observed flux enhancements of SMFR acceleration events constrained by the acceleration time.The acceleration time can be estimated by the solar wind transit time between the Sun and the observed acceleration event since SMFRs are non-propagating structures advected with the solar wind.

A Solution Combining Normal Diffusion with Momentum Superdiffusion
Besides investigating analytical solutions with both parallel and momentum superdiffusion at the same level of superdiffusiveness, one would like to know how the solutions change when parallel and momentum superdiffusion feature different levels of superdiffusiveness.It was realized that analytical solutions are possible in the extreme limits where one of the two transport processes is diffusive and one is superdiffusive.In this section, an analytical solution is presented for the case where normal parallel diffusion is combined with momentum superdiffusion.For this purpose, it was specified that β ∥ = 1 in the term for parallel superdiffusion in transport Equation (5).Then the fractional time integral in the parallel superdiffusion term goes to 1 and normal parallel diffusion is restored.Equation (11), the general solution of Equation ( 5 where 1 < β p < 2 and the expression for b(s) in the argument of the modified Bessel function in Equation ( 37) is given by Equation (12).The expression for b(s) can be simplified in the limit of late times (s → 0) to be By specifying an intermediate level of momentum superdiffusion (β p = 3/2), the high-momentum limit ), and late times (s → 0 or t → ∞ ), the large-argument limit of the modified Bessel function in solution (37), can be taken and the inverse Laplace transform in Equation ( 37 where, as before, ) is the timescale for adiabatic cooling, and t = D ] refers to the timescale for momentum superdiffusion.
In the second line of Equation (39) one can see that effective parallel diffusion, adiabatic cooling, and particle escape can potentially modify the accelerated particle spectrum due to momentum superdiffusion to be steeper than an f 0 (p) ∝ p −3 spectrum at higher particle momenta, but that at late times these steepening effects disappear.Thus, the accelerated spectrum inevitably should become an f 0 (p) ∝ p −3 power law at late times as momentum superdiffusion increasingly dominates other transport effects through the presence of the growing fractional time integral in the momentum superdiffusion term as discussed above.Thus, the late time result for the accelerated spectrum for normal diffusion combined with momentum superdiffusion is predicted to be the same as for parallel and momentum superdiffusion discussed above (see Equation (17) and its discussion above).A subtle difference is that effective parallel superdiffusion is less likely to temporarily steepen the spectrum in the case of parallel and momentum superdiffusion compared to this case.That is why the spectral steepening effect from effective parallel superdiffusion is ignored in solution (17), but the steepening effect from normal parallel diffusion is included in solution (39).This also suggests that when normal parallel diffusion is combined with momentum superdiffusion, it will take longer for spectral steepening effects to become insignificant compared to the case of parallel and momentum superdiffusion as is discussed further below.
Another interesting prediction of the late time solution (39) is that the accelerated particle distribution function increases beyond the particle source (x > x 0 ) to form a peak that does not decay with time.This is in contrast with the solution for parallel and momentum superdiffusion where the spatial peak decays increasingly with time (see solutions (17) and ( 21)).The explanation is that because there is normal parallel diffusion, the parallel diffusion term does not grow relative to the solar wind advection term to suppress the spatial peak with time as happens in the case of parallel superdiffusion.Thus, the combination of normal parallel diffusion with momentum superdiffusion is predicted to have the same spatial peak behavior as in the case of the normal parallel and momentum diffusion solution (see solutions (32), (35), and (36)).However, one would expect that as momentum superdiffusion grows with time, and given the accelerated growth of momentum superdiffusion with time for higher levels of momentum superdiffusiveness (larger values for β p ), that some suppression of peak formation should occur as the advection and normal parallel diffusion terms become relatively less important.Some evidence for this is presented further below.

A Solution Combining Parallel Superdiffusion with Normal Momentum Diffusion
In this section, it is discussed how the characteristics of the analytical solution for energetic particles interacting with dynamic SMFRs change when parallel superdiffusion is combined with normal momentum diffusion.This was accomplished by letting β p = 1 whereby the momentum superdiffusion term in transport Equation (5) becomes a normal momentum diffusion term (fractional time integral becomes one in the momentum superdiffusion term).Equation (11), the general solution of transport Equation (5), then becomes where 0 < β ∥ < 1, and the expression for b(s) in the argument of the modified Bessel function K 0 in Equation ( 40) is given by Equation (12).The expression for b(s) can be simplified in the limit of late times (s → 0) to become Upon neglecting particle escape (τ esc → ∞ ), specifying an intermediate level of parallel superdiffusion β ∥ = 1/2, taking the high-momentum limit k -D p p x x s ln , and imposing the late time limit (s → 0 or t → ∞ ), the largeargument limit of the modified Bessel function K 0 in solution (40) can be taken and the inverse Laplace transform in Equation (40) can be evaluated to produce the solution ´- ´- where ). Inspection of solution (42) reveals that at late times the accelerated energetic particle distribution at higher particle momenta due to momentum diffusion is a steeper power law than f 0 (p) ∝ p −3 because of competition from adiabatic cooling.This steepening effect survives at late times because of normal momentum diffusion that does not grow in time to increasingly dominate adiabatic cooling.This is in contrast to the case of parallel and momentum superdiffusion discussed further above, or the case of normal parallel diffusion combined with momentum superdiffusion (previous section).In these cases, the analytical solutions predict an f 0 (p) ∝ p −3 spectrum at late times because the steepening effect from adiabatic cooling disappears as momentum superdiffusion increasingly dominates adiabatic cooling due to the growth in the factional time integral in the momentum superdiffusion term.The predicted survival of the adiabatic cooling spectral steepening effect late times in this solution is in agreement with the prediction of the solution for normal parallel and momentum diffusion (see Equation (35)) because in both cases there occur normal momentum diffusion.The additional steepening effect due to effective parallel superdiffusion in Equation (42) (see the presence of t k 0 in the exponential term that decays with time in Equation (42)), however, becomes increasingly insignificant at late times as parallel superdiffusion increases with time due to the fractional time integral present in this term.This can be understood in terms of effective superdiffusive particle spreading becoming increasingly efficient with time so that the spatial gradient in the particle distribution becomes increasingly small.Consequently, the parallel superdiffusion term as a whole decreases, thus raising the contact time of energetic particles with individual SMFRs and increasing the effective efficiency of SMFR acceleration.
Solution (42) also predicts that the accelerated particle distribution will increase beyond the particle source (x > x 0 ) to form a spatial peak, but that this peak will decay at late times (see last exponential function in Equation ( 42)).This is a direct consequence of the parallel superdiffusion term growing in time to dominate the solar wind advection term.Thus, the prediction is qualitatively similar to the case of parallel and normal superdiffusion discussed above because both cases have a growing parallel superdiffusion term in common.

More General Semi-numerical Solutions
Given that the analytical solutions presented above for equal levels of parallel and momentum superdiffusiveness only apply to two levels of superdiffusion (β ∥ = 1/2, and β ∥ = 2/3), it was decided to produce more flexible semi-numerical timedependent solutions that apply to any equal level of superdiffusiveness.For this purpose the inverse Laplace transform, as given in solution (13), was calculated numerically with the aid of the Talbot solver (Talbot 1979) available in MATLAB (McClure 2022).The numerical evaluation was done after taking the late time limit (s → 0) in the complete expression for b(s) (see Equation ( 12)) whereby a simplified expression for b (s) was specified in the modified Bessel function K 0 in Equation ( 13) that is given by 43 Note that this expression is a more complete expression for b(s) than the one used for deriving the analytical solutions (see Equation ( 14)).Plots of the solutions were produced with MATLAB and are presented below.The solutions were calculated for solar wind conditions in the vicinity of 1 au by specifying the following reasonable values for parameters in the solutions (for parameter definitions, see the discussion below Equation . The results show that for ´--1.4 10 s 4 1.1 when β ∥ = 0.9.By comparin the values for k b - , one can see that parallel superdiffusion is more effective than momentum superdiffusion by about an order of magnitude.This outcome has some bearing on the question of whether parallel superdiffusion or momentum superdiffusion is mostly responsible for causing the decay of the spatial peaks beyond the particle source which will be discussed again when considering the results shown in Figures 7 and 9 further below. Please note that in the figures below time-dependent solutions are shown after a maximum SMFR acceleration time of ∼50 hr.The idea is to show that, on a timescale shorter than it takes for SMFRs to be advected from the Sun to Earth orbit (∼100 hr), acceleration is efficient enough so that well-developed accelerated power-law spectra can form when inserting reasonable values for energetic particle transport parameters related to interaction with SMFRs.In the discussion above of the analytical time-dependent solutions it has been predicted that at late times the accelerated energetic particle spatial peak profiles beyond the particle source x > x 0 will eventually decay away to form a plateau in the solutions containing superdiffusion.Not enough time elapsed in the time-dependent solutions displayed in the figures below for this to happen.The closest occurrence to a large-scale plateau appears in Figure 2 where the blue curve (the most superdiffusive case) did manage to form a limited plateau followed by a decrease.This plateau will continue to develop as particles propagate further away from the particle source position.
In Figure 1, time-dependent solutions are shown of the spatial peaks that form in the SMFR-accelerated energetic particle distribution f 0 at distances downstream of the particle source x − x 0 > 0 in astronomical units.Particle escape from the SMFR acceleration region has not been included.The figure illustrates the variation in the amplitude of the spatial peak as a function of the level of superdiffusiveness.The dashed curve represents the time-dependent diffusive solution as given by solution (32) (the reference solution) whereas the rest of the curves are all superdiffusive solutions.The black curve is valid for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.1 , the red curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.15 , the green curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.25 , and the blue curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.45 .The curves were normalized to a value of 1 at the particle source (x − x 0 = 0).The results are shown at time t ≈ 20 hr, indicating that normal spatial and momentum diffusion produces a spatial peak in f 0 with the largest amplification factor.The peak amplification factors become progressively smaller with increasing levels of spatial and momentum superdiffusion until it disappears at an intermediate level of superdiffusion 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.45 .Since our analytical solution for a similar level of superdiffusion (see Equations ( 17) and ( 21)) clearly shows that peak formation is possible, it means that in this intermediate case, the peak decayed quickly enough to disappear after ∼20 hr of acceleration.As discussed above, the higher the level of superdiffusiveness, the more quickly the peak decays because then superdiffusive spatial particle spreading dominates the opposition from solar wind advection on a shorter timescale.
In Figure 2, the spatial profiles the same diffusive and superdiffusive cases as in Figure 1 are presented at a later time t ≈ 52 hr.Comparing the results to Figure 1, one can see the opposite trend in the time evolution of the spatial peaks for the diffusive case as compared to the superdiffusive cases.In the diffusive case, the amplitude of the spatial peak increased somewhat with time as the profile converges to a steady state.This supports the diffusive analytical solution (see Equations ( 35) and (36) above) in which the spatial peak is an established feature at late times as the competition of solar wind advection and normal spatial diffusion settles into a steady-state pattern.In the superdiffusive cases, on the other hand, the amplitude of the spatial peaks decreased further as the dominance of spatial superdiffusion over solar wind advection became further entrenched.For the case of the strongest level of superdiffusion (blue curve), the peak disappeared so that what remains is a small plateau followed by a decrease.
Displayed in Figure 3 are the time-dependent solutions for the accelerated particle spectra f 0 (p) at the particle source location (x = x 0 ) as a function of particle momentum p normalized to the source momentum p 0 .The effects of particle escape from the SMFR acceleration regions are not included.The solutions represent different levels of superdiffusion at a time t ≈ 52 hr.The curves are normalized to a value of 1 near the source momentum p/p 0 = 1 to make it easier to detect differences in the spectral slopes.The dashed curve represents the case of normal parallel and momentum diffusion and serves as a reference solution to compare with the superdiffusion solutions (solid curves).The black curve indicates superdiffusion with the statistics 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.15 , the red curve 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.25 , the green curve 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.45 , and the blue curve 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.55 .The green and blue curves (the top curves) are somewhat difficult to distinguish because they almost overlap.Asymptotically, the spectral slopes of the accelerated spectra are power laws f 0 (p) ∝ p − a where a ranges from a ≈ 3.5 for the diffusive solution (dashed curve) to a ≈ 3 for the most superdiffusive solution (blue curve).Thus, the stronger parallel and momentum superdiffusion become, the harder the spectrum becomes.Note also that the spectrum converges to an f 0 (p) ∝ p −3 spectrum (maximum hardness) at intermediate levels of superdiffusion when 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.5 , approximately.The f 0 (p) ∝ p −3 -spectrum is consistent with the analytical solutions at the high-momentum limit that predict f 0 (p) ∝ p −3 at late times (see Equations ( 17) and ( 26)) because the growing momentum superdiffusion term is then the dominant transport process, as argued above.The steeper spectra at lower levels of superdiffusion indicate that the dominance of momentum superdiffusion is less well established at this time so that other transport processes, such as adiabatic cooling, can still steepen the spectrum.This delay in the dominance of parallel momentum diffusion at lower levels of superdiffusion is consistent with the finding discussed above that the stronger the level of superdiffusion, the more quickly the dominance of momentum superdiffusion over other transport processes is established.At lower momenta, the blue curve (top curve) deviates from the f 0 (p) ∝ p −3 power law predicted by the analytical solution by becoming progressively steeper because, although the high-momentum inequality is still fulfilled, the accompanying large-argument limit of the modified Bessel function K 0 in the analytical solution is not fulfilled.
In Figure 4, spectra for the same superdiffusive cases as in Figure 3 are shown at a distance of ∼0.07 au beyond the particle source (x − x 0 ≈ 0.07 au).The normalization that was used in Figure 3 was applied without change in this figure.At larger momenta, the spectra in Figure 4 are increasingly fulfilling the high-momentum inequality, and the accompanying large-argument limit of the modified Bessel function K 0 in the analytical solutions (17) and (26), thus converging toward f 0 (p) ∝ p −3 spectra as predicted by these solutions.At lower particle momenta as shown in Figure 4 the spectra roll over asymptotically toward a harder f 0 (p) ∝ p −3/2 power law as found in the analytical solutions at late times (see Equations ( 18) and ( 27)) because then the far-from-the-source limit becomes more and more applicable.This spectral rollover that occurs at lower particle momenta away from the particle source is simply a consequence of the shift of the accelerated particle spectrum to higher momenta with time as particles propagate further away from the source position and continue to experience momentum diffusion, as discussed above in the case of the analytical solutions.
Particle escape out of the SMFR acceleration region was not considered in Figures 1-4.In Figure 5, it is illustrated how the enhancements in the accelerated distribution beyond the particle source are affected by sideways (perpendicular) particle escape from the SMFR acceleration region.By implication, this suggests that the SMFR acceleration region has a finite extent in the perpendicular direction.Energetic particles that are temporarily trapped in SMFRs when they follow the quasihelical field lines of multiple coherent SMFR structures, can be thought to be undergoing subdiffusive transport perpendicular to the mean magnetic field direction in a statistical average sense as discussed above (see also the discussion in le Roux & Zank 2021).Accordingly, the escape time in the fractional Parker transport equation (see Equation (2)) is modeled as where is the transport coefficient for perpendicular subdiffusion averaged over a spatial interval Δx, and particle speed interval Δv.The expression for the perpendicular subdiffusion coefficient is ) , and β ⊥ is the fractional time index for perpendicular subdiffusion.More information about the derivation of this expression can be found in Figure 1.Time-dependent solutions for energetic protons propagating superdiffusively along the background magnetic field while simultaneously undergoing momentum superdiffusion during interaction with the collection of dynamic SMFRs in the large-scale solar wind.Results are shown at time t ≈ 20 hr.Energetic particle escape from the SFMR acceleration region is neglected.Plotted are enhancements in the direction-averaged particle distribution f 0 (x) as a function of distance beyond the particle source x − x 0 > 0 in astronomical units for different levels of parallel and momentum superdiffusion.The curves are normalized to a value of 1 at the particle source (x − x 0 = 0).The dashed curve denotes a reference solution for normal parallel and momentum diffusion, the black curve is valid for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.1 , the red curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.15 , the green curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.25 , and the blue curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.45 .Figure 3. Time-dependent solutions of SMFR-accelerated energetic proton spectra f 0 (p) as a function of particle momentum normalized to the source momentum p 0 at the particle source position x = x 0 for different levels of parallel and momentum superdiffusion at t ≈ 52 hr.Particle escape is not included.The dashed curve indicates a reference solution for normal parallel and momentum diffusion whereas the solid curves indicate parallel and momentum superdiffusion.The solid black curve is valid for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.15 , the red curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.25 , the green curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.45 , and the blue curve for 〈Δx 2 〉 ∝ 〈Δp 2 〉 ∝ Δt 1.55 .The green and blue curves (the top curves) are somewhat difficult to distinguish because they almost overlap.The curves are normalized to a value of 1 near the injection momentum p/p 0 = 1.le Roux & Zank (2021) and le Roux (2022).If β ⊥ → 1, normal perpendicular diffusion is recovered and the expression is similar to the classical case of field-line random-walk perpendicular diffusion (Jokipii 1966).Since the fractional time index for perpendicular subdiffusion is restricted to β ⊥ < 1 and the fractional spatial index is specified to be α = 2, this implies that the variance of the perpendicular particle displacement is governed by áD ñ µ D b ŷ t 2 , thus covering the standard subdiffusive range.The energetic particle distribution function spatial enhancement results in Figure 5 are shown in exactly the same format as in Figure 2, covering the same cases for parallel spatial and momentum superdiffusion.In Figure 5, the difference is that for each superdiffusion case particle escape is specified at an intermediate subdiffusive level (β ⊥ = 0.5).Both Figures 2 and 5 have the same reference solution for normal parallel and momentum diffusion without particle escape.Comparing the results of Figure 5 with those of Figure 2, the results are complicated.Subdiffusive particle escape results in a reduction in the maximum amplification factor of the spatial peaks in the particle distribution for low levels of superdiffusion (black and red curves).The reduction in the amplification factor becomes progressively smaller as the level of parallel spatial and momentum superdiffusion increases (the value of the fractional time index 2 − β ∥ increases).However, for stronger levels of superdiffusion (green and blue curves), we find the opposite trend where perpendicular subdiffusive escape increases the maximum amplification factor and where the increase in the amplification factor becomes progressively larger with increasing superdiffusiveness.However, close to the particle source (x − x 0 ≈ 0) particle escape has no noticeable effect on the flux amplification factor, except for the blue curve (case of maximum superdiffusion) where the escape results in a noticeable increase in the amplification factor.The analytical solutions close to the source (Equations ( 17) and (26)) confirm that an increase in the amplification factor should occur when particle escape is activated.
It is also investigated how the accelerated particle spectra at the source position are affected by perpendicular subdiffusive particle escape.The effect of particle escape can be evaluated by comparing the results in Figure 6 (with particle escape included) with the results in Figure 3 (no particle escape).The results in Figure 6 are shown in exactly the same format as in Figure 3 covering the same cases for parallel spatial and momentum superdiffusion (solid curves).Both Figures 3 and 6 have the same reference solution for normal parallel and momentum diffusion without particle escape (dashed curves).In Figure 6, particle escape is specified in terms of the same intermediate subdiffusive level (β ⊥ = 0.5) for each parallel and momentum superdiffusive case.Comparing the results of this figure with those in Figure 3, one notices that the spectra in this figure have steeper power laws asymptotically at higher momenta due to particle escape effects for all parallel and momentum superdiffusive cases.The power-law index ranges from a ≈ 4 for the least superdiffusive case (black curve) to a < 3 (barely) for the most superdiffusive case (blue curve), whereas in Figure 3 the index ranged from ≈3.5 to a = 3.This means that the more superdiffusive parallel and momentum transport becomes, the less particle escape is able to steepen the accelerated particle spectrum.That is why the blue curves in Figures 3 and 6 (solutions with the highest levels of superdiffusion) have almost the same slopes.This is consistent with the discussion above that stronger levels of superdiffusion result in more rapid domination of the superdiffusive terms over the other transport terms including the particle escape term, thus rendering the escape term to be less potent (see the discussion related to Equation (22) above).This is supported by comparison of the analytical solution for stronger superdiffusion (line two of Equation ( 17)) with the solution for weaker superdiffusion (line two of Equation ( 26)).The comparison shows that particle escape steepens the particle spectrum in a time-dependent fashion and that the steepening effect will vanish more quickly in the case of a stronger level of superdiffusion.
So far, in all solutions discussed the same level of superdiffusion was specified for both anomalous parallel and anomalous momentum transport.An interesting question is how the formation of spatial peaks in the accelerated particle distribution beyond the particle source is differently affected if the particles experience momentum superdiffusion combined with normal parallel diffusion during interaction with SMFRs in the absence of particle escape.The solutions for this case were computed using solution (37) by evaluating the inverse Laplace transform in Equation (37) numerically without escape effects with the aid of MATLAB as discussed at the beginning of this section.The solutions are presented in Figure 7 following the same format in Figure 2.This includes showing results at the same time t ≈ 52 hr and for the same levels of momentum superdiffusion (solid curves).The dashed curve in this figure indicates the reference solution for normal parallel and momentum diffusion without particle escape also shown in Figure 2. The results in this figure show that, overall, the amplitude of the spatial peaks (solid curves) has decayed with time to become significantly less than for the case of normal  3, but now shown for a distance of x − x 0 ≈ 0.07 au beyond the particle source.The same normalization of the curves as in Figure 3 is followed.The black and red curves (top curves) are somewhat difficult to distinguish because of their overlap.parallel and momentum diffusion (dashed curve) and that the decrease in the amplitude is more pronounced with increasing levels of momentum superdiffusion.This trend is in qualitative agreement with the results in Figure 2 for parallel and momentum superdiffusion despite the effect of normal parallel diffusion.In Figure 2, the decay in the spatial peaks was ascribed to the increasing domination of the parallel superdiffusion term over the solar wind advection term with time, and the acceleration of this domination at higher levels of parallel superdiffusion because of the presence of the growing fractional time integral in the parallel superdiffusion term.In the discussion of the results of Figure 2 the possibility that the growing momentum superdiffusion term also plays a role in the increasing peak decay was not introduced.Since normal parallel diffusion was specified to calculate the results of this figure, the reason for the diminishing spatial peaks must be sought in the momentum superdiffusion term that grows relative to the solar wind advection term as b - T 1 p (1 < β p < 2) because of the presence of the fractional time integral in this term (see Equation (2)).As time increases, the momentum superdiffusion term will increasingly dominate both the solar wind advection and parallel diffusion terms that together are responsible for the spatial peak formation, thus rendering peak formation to be increasingly inefficient.This increasing domination will occur more rapidly for stronger levels of momentum superdiffusion (larger values of β p ) as indicated by the fractional growth timescale.That is why the decay in the peak amplitudes with time in this figure is qualitatively similar to the decay in the case of combined parallel and momentum superdiffusion shown in Figure 2.However, comparison of the spatial peaks in this figure with those in Figure 2 does reveal that when there is normal parallel diffusion (this figure), the amplitude of the spatial peaks is larger (less decayed over the same time period) compared to when there is parallel superdiffusion (Figure 2).This shows that in Figure 2 both growing parallel and momentum superdiffusion terms are contributing to the more effective decay of the spatial peaks with time whereas in this figure less decay occurs because only momentum transport is superdiffusive.The decay of the spatial peaks by momentum superdiffusion with time in this figure is not captured by the simplified analytical solution (39).
In Figure 8, the accelerated particle spectra are displayed at the particle source position x = x 0 after about 52 hr of particle acceleration for the same case as in Figure 7, i.e., normal parallel diffusion combined with superdiffusive momentum diffusion disregarding particle escape effects.This figure follows the format of Figure 3 by retaining the same levels of momentum superdiffusion, and the dashed curve indicates the reference solution for normal parallel and momentum diffusion without particle escape as in Figure 3. Qualitatively, the results are the same as in Figure 3 where both parallel and momentum anomalous transport are superdiffusive and particle escape was also neglected.As in Figure 3, the results indicate that the accelerated spectra become progressively harder as the level of momentum superdiffusion increases.This occurs for the same reason as outlined in the discussion of the results shown in Figure 3 (the stronger the level of momentum superdiffusion, the more quickly the growing momentum superdiffusion term dominates other transport processes that contribute to spectral steepening, such as adiabatic cooling, to converge to an f 0 (p) ∝ p −3 power-law spectrum at late times).However, where the spectra for the strongest levels of momentum superdiffusion (green and blue curves) did converge to the predicted f 0 (p) ∝ p −3 spectrum shown in Figure 5. Spatial profiles of the enhancements in the SMFR-accelerated proton distribution at t ≈ 52 hr following exactly the same format as Figure 2. The only difference in the displayed solutions in this figure compared to those in Figure 2 is that perpendicular subdiffusive particle escape from the SMFR acceleration region, characterized statistically by 〈Δy 2 〉 ∝ Δt 0.5 , is included for each superdiffusive case (solid curves).The reference solution for normal parallel and momentum diffusion (dashed curve) does not include particle escape effects, just like in Figure 2. Figure 6.Same solutions and format as Figure 3, with the differences that perpendicular subdiffusive particle escape from the SMFRs acceleration region, characterized statistically by 〈Δy 2 〉 ∝ Δt 0.5 , is included for all superdiffusive cases (solid curves).The dashed curve, overlapping with the red curve, is the reference solution for normal parallel and momentum diffusion without particle escape, just as in Figure 3.
Figure 3, the corresponding spectra in this figure are somewhat steeper.This difference can be explained by comparing the late time analytical solution (17) for parallel and momentum superdiffusion with the analytical solution (39) for normal parallel diffusion combined with momentum superdiffusion.It is predicted in these solutions that normal effective parallel diffusion contributes to temporary steepening of the accelerated particle spectrum whereas the temporary steepening effect of parallel superdiffusion is negligible.Consequently, it will take longer for temporary steepening effects to become insignificant in the case of normal parallel diffusion where both effective parallel diffusion and adiabatic cooling contribute to spectral steepening, than in the case of parallel superdiffusion where only adiabatic cooling is expected to contribute to spectral steepening.Therefore, the result of steeper spectra in Figure 8 compared to Figure 3.
In Figure 9, calculations are shown to illustrate how the formation of spatial peaks in the accelerated energetic particle distribution beyond the particle source at a time t ≈ 52 hr is affected differently when parallel superdiffusion is combined with normal momentum diffusion in the absence of particle escape.The solutions for this case were computed using solution (40) by evaluating the inverse Laplace transform in Equation (40) numerically without escape effects with the aid of MATLAB as discussed at the beginning of this section.The calculations were done for the same levels of parallel superdiffusion as in Figure 2 and the format of Figure 2 is followed.The results indicate that the amplitude of the spatial peaks for parallel superdiffusion combined with normal momentum diffusion (solid curves) have decayed with time to become significantly less than for the case of normal parallel and momentum diffusion (dashed curve) and that the decrease in the amplitude is more pronounced with increasing levels of parallel superdiffusion.As discussed above, through the presence of the fractional time integral in the parallel superdiffusion term, parallel superdiffusion grows with time to increasingly dominate solar wind advection, thus causing spatial peak formation to dissipate.This effect is accelerated for stronger levels of parallel superdiffusion.The results are qualitatively in agreement with the results shown in Figure 2 where parallel superdiffusion is combined with momentum superdiffusion and both contribute to the decay in the spatial peaks, and with the results in Figure 7 where normal parallel diffusion is combined with momentum superdiffusion and the latter is responsible for the spatial peak decay.A more detailed comparison of the results in this figure with those in Figures 2  and 7 reveals that the amplitudes of the spatial peaks in this figure for the most part fall between those of Figures 7 (largest amplitudes) and 2 (smallest amplitudes).This shows that the parallel superdiffusion term has a larger impact in decreasing the amplitudes of the spatial peaks than the momentum superdiffusion term for a given time period.This is to be expected because it was estimated above that the parallel superdiffusion term is larger than the momentum superdiffusion term in the semi-numerical solutions.
Finally, in Figure 10 solutions of energetic proton spectra accelerated by SMFRs are presented at the particle source position (x = x 0 ) at t ≈ 52 hr for the same case as in Figure 9, i.e., parallel superdiffusion combined with normal momentum diffusion in the absence of particle escape effects.This figure follows the format of Figure 3 by retaining the same levels of parallel superdiffusion, and the dashed curve indicates the reference solution for normal parallel and momentum diffusion without particle escape as in Figure 3.The results indicate that the accelerated spectra become progressively harder as the level of parallel superdiffusion increases, but different from the results in Figure 3, the hardest spectra ended up being softer than f 0 (p) ∝ p −3 , but they are still clearly harder than the solution for normal parallel and momentum diffusion (dashed curve).A clue for these differences can be found by comparing the late-time analytical solutions for parallel and momentum superdiffusion (Equation ( 17)), parallel superdiffusion and normal momentum diffusion (Equation ( 42)), and normal parallel and momentum diffusion (Equation ( 35)) at the source location x = x 0 ignoring particle escape.In the latter case, Equation (35) suggests a power-law spectrum µ ) .The spectrum is steeper than f 0 (p) ∝ p −3 because of the combined steepening effect of adiabatic cooling (the characteristic timescale t = x U 2 3 U 0 0 in the power-law exponent) and effective normal parallel diffusion (the characteristic timescale t in the power-law exponent).For the case of parallel superdiffusion and normal momentum diffusion (this figure) the spectra are also steeper than f 0 (p) ∝ p −3 , but not as much as for normal parallel and momentum superdiffusion.This is because µ ( at late times (see Equation (42)) only adiabatic cooling contributes to spectral steepening.Effective parallel superdiffusion is unable to steepen the accelerated spectra at late times as its steepening ability decays with time the more it grows as explained above (see the second line of Equation (42)).In the case of parallel and momentum superdiffusion, one gets f 0 (p) ∝ p −3 at late times because then both adiabatic cooling and effective parallel superdiffusion are unable to steepen the spectrum.The former's lack of Figure Spatial profiles of the enhancements in the SMFR-accelerated proton distribution at t ≈ 52 hr following exactly the same format as Figure 2. The only difference is that in this figure the solid curves were calculated for normal parallel spatial diffusion.The solutions were calculated for the same levels of momentum superdiffusion and without particle escape as in Figure 2. The dashed curve represents the reference solution of normal parallel and momentum diffusion without particle escape as in Figure 2.
steepening ability at late times can be attributed to the growth in the momentum superdiffusion term that increasingly dominates the adiabatic cooling term with time as discussed above.Considering the progressively steeper spectra in this figure for increasingly lower levels of parallel superdiffusion, this is a consequence of the increasing likelihood of some steepening by effective parallel superdiffusion in the given time as the growth rate of parallel superdiffusion slows down with decreasing superdiffusiveness.

Summary
Evidence is increasing that in a magnetically turbulent, collisionless plasma medium with a significant background magnetic field, such as the solar wind, coherent, nonpropagating SMFR structures naturally form a part of a locally generated quasi-2D MHD turbulence component that dominates other MHD wave turbulence modes in the inner heliosphere (e.g., Matthaeus et al. 1990;Zank et al. 2017).Additional SMFRs are generated locally in the solar wind by turbulent reconnection at large-scale current sheets (e.g., Khabarova et al. 2015Khabarova et al. , 2016)).Energetic particles are found to be temporarily trapped in dynamic SMFRs often separated by turbulently reconnecting small-scale current sheets (Dmitruk et al. 2004;Pecore 2021).This can cause SMFR merging and the production of fluctuating motional electric fields that can efficiently energize suprathermal particles (Dmitruk et al. 2004;Drake et al. 2013).Indications from simulations are that this produces energetic particle spatial transport and energization that are both superdiffusive (Isliker et al. 2017a(Isliker et al. , 2017b(Isliker et al. , 2019;;Nakanotani et al. 2022).
To investigate such superdiffusive behavior in a field of interacting SMFRs theoretically, a new fractional Parker transport equation, which was developed recently from first principles (le Roux & Zank 2021; le Roux 2022) to model energetic particle trapping effects in SMFRs, is solved.Consistent with the simulations, the transport equation predicts that energetic particle transport through dynamic, coherent SMFRs in the background magnetic field direction is superdiffusive when the time-fractional index β ∥ is constrained to be 0 < β ∥ < 1.The reason that this β ∥ range ensures parallel superdiffusion is that the parallel transport process is controlled solely by a fractional time integral (or equivalently by a fractional time derivative from a different perspective) because the spatial derivatives are predicted to be normal by the theory.This results in the variance of the parallel particle displacement being áD ñ µ D b x t 2 2   , which then covers the standard range for superdiffusion.Furthermore, particle scattering in momentum magnitude space, due to interaction with dynamic SMFRs, is determined by an anomalous momentum diffusion process that, more generally, includes both a fractional time integral (or derivative) and fractional momentum derivatives.To ensure a momentum superdiffusion process as predicted by simulations, the fractional time index of this process was restricted to 1 < β p < 2. The chance of finding analytical solutions was enhanced by imposing normal momentum derivatives.Accordingly, áD ñ µ D b p t 2 p yielding the standard superdiffusive range.Thus, both parallel spatial and momentum superdiffusion of energetic particles in a dynamic SMFR field were investigated as fractional time diffusion processes instead of as fractional space or momentum diffusion processes (Lévy flights).This, combined with assuming 2 − β ∥ = β p (parallel and momentum superdiffusion are specified to have the same level of superdiffusiveness), enabled analytical and semianalytical time-dependent solutions as a first step.Accordingly, Figure 8. Solutions of SMFR-accelerated energetic proton spectra at the particle source position x = x 0 at t ≈ 52 hr.The solutions have the same format as Figure 3, with the only difference that in this figure curves were calculated for normal spatial parallel diffusion.The solutions were calculated for the same levels of momentum superdiffusion and without particle escape as in Figure 2. The dashed curve represents the reference solution of normal parallel and momentum diffusion without particle escape as in Figure 2.
Figure 9. Spatial profiles of the enhancements in the SMFR-accelerated proton distribution at t ≈ 52 hr following exactly the same format as Figure 2. The only difference is that in this figure the solid curves were calculated for normal momentum diffusion while retaining the levels of parallel superdiffusion in Figure 2. The dashed curve represents the reference solution of normal parallel and momentum diffusion without particle escape as in Figure 2. a time-fractional approach to parallel spatial and momentum superdiffusion of energetic particles in an SMFR acceleration region in the large-scale solar wind was investigated.
The main results from the solutions are: 1.As was discovered before in the case of normal momentum and spatial diffusion, the accelerated particle distribution function forms spatial peaks beyond the particle point source in the downwind direction (antisunward direction) with a maximum amplification factor that increases with particle energy and simultaneously shifts to larger distances beyond the particle source (Zank et al. 2015;Zhao et al. 2018;Adhikari et al. 2019;le Roux et al. 2019).In the superdiffusive case, however, the spatial peaks dissipate into plateaus followed by a spatial rollover at late times as energetic particle parallel superdiffusion increasingly dominates solar wind advection as a function of time due to the growth in the fractional time integral present in the superdiffusion term.In contrast, in the case of normal diffusion spatial peaks grow in time until a steady state is reached so that the spatial peak formation is an established feature at late times.The growth might be attributed to that diffusive particle spreading becomes less dominant relative to solar wind advection as the spatial gradient in the particle distribution downstream of the particle source decreases (solar wind medium downstream of particle source fills up with source particles).2. The stronger the level of parallel and momentum superdiffusion, the more quickly the spatial peaks downstream of the particle source will decay toward a plateau with a rollover as superdiffusive particle spreading more quickly dominates solar wind advection.Thus, considered over a reasonable finite acceleration time, such as the time it takes for the solar wind to flow from the Sun to Earth (∼100 hr), energetic particle spatial peaks still exist downstream of the particle source for normal diffusion and low to intermediate levels of superdiffusiveness, but for progressively stronger levels of superdiffusion only limited spatial plateaus with decreasing downstream lengths followed by a rollover exist.Given enough time, the spatial plateaus will become more extended as particles propagate further downstream of the particle source.3. The accelerated particle distribution decreases exponentially with distance from the particle source in the upwind direction (sunward direction) because superdiffusion toward the Sun encounters resistance from the radial solar wind outflow from the Sun (superdiffusive modulation).However, as was reported before in le Roux (2022) in the case of superdiffusive particle transport ahead of a parallel shock, at very late times the upwind distribution will eventually relax to a plateau as energetic particle parallel superdiffusive spreading increasingly dominates opposition from solar wind advection.4. The accelerated particle spectra close to the fixed particle point source inevitably converge to an f 0 (p) ∝ p −3 powerlaw spectrum at very late times in the SMFR acceleration region (in the test particle limit).This is a direct consequence of the momentum superdiffusion term becoming the sole dominant transport term at late times.
The simultaneous growing dominance of the parallel spatial superdiffusion term gets interrupted at later times because when it starts dominating solar wind advection, superdiffusive particle spreading toward a spatial plateau profile (zero spatial gradient in the distribution function) renders parallel superdiffusion ineffective compared to momentum superdiffusion.This is different from the case of normal parallel and momentum diffusion where the accelerated spectra at late times will not necessarily converge to an f 0 (p) ∝ p −3 power-law spectrum because other transport terms, if strong enough to compete with momentum diffusion initially, can maintain their competitiveness at late times.5. Far downstream of the particle source, the accelerated particle spectra are largely independent of the level of parallel and momentum superdiffusion at late times for the same reason as for spectra close to the source, namely, the increasing domination of momentum superdiffusion over other transport terms with time.Different from the spectra close to the source, the far downstream spectra develop a rollover at lower particle momenta asymptotically toward an f 0 (p) ∝ p −3/2 power law.This is simply a consequence of the accelerated spectrum as a whole shifting to higher momenta during superdiffusive SMFR acceleration while particles propagate superdiffusively further downstream of the source.The rollover is not very prominent because of the effective superdiffusive spreading out of particles in momentum space.The rollover effect was reported before because it also occurs in solutions for normal parallel and momentum diffusion during SMFR acceleration (e.g., Zhao et al. 2018;le Roux et al. 2019).
Figure 10.Solutions of SMFR-accelerated energetic proton spectra at the particle source position x = x 0 at t ≈ 52 hr.The solutions have the same format as Figure 3, with the only difference being that in this figure the curves were calculated for normal momentum diffusion.The levels of momentum superdiffusion shown in Figure 3 were retained.The dashed curve represents the reference solution of normal parallel and momentum diffusion without particle escape as in Figure 2. The top curve is the blue curve.
6. Assuming that SMFR-accelerated particles in the inner heliosphere can escape out of the SMFR region through perpendicular transport in the heliospheric latitudinal direction (SMFR acceleration region has a finite latitudinal extent), and that the escape is a subdiffusive process (áD ñ µ D b ^x t 2 where β ⊥ < 1) due to temporary particle trapping effects in SMFR structures, the results show that subdiffusive escape does affect the amplitude of spatial peaks downstream of the particle source in a complicated way.For an intermediate level of perpendicular subdiffusion (β ⊥ = 0.5) and low levels of parallel and momentum superdiffusion, the maximum amplification factor of the spatial peaks is reduced and the reduction becomes progressively smaller when increasing the level of superdiffusion (increasing the fractional time index 2 − β ∥ ).The opposite trend occurs at strong levels of superdiffusion where perpendicular subdiffusive escape increases the maximum amplification factor, which progressively grows with increasing superdiffusiveness.7. Perpendicular subdiffusive particle escape at an intermediate level affects the accelerated particle spectra near the particle source in a more straightforward manner.The more superdiffusive parallel and momentum transport becomes, the less particle escape is able to steepen the SMFR-accelerated particle spectrum.This is consistent with the discussion above that the more superdiffusive the momentum transport is, the more rapidly momentum superdiffusion dominates all other transport terms, including the escape term, thus rendering the escape term to be less effective in steepening the spectrum.
To make allowance for the possibility that parallel and momentum superdiffusion will not necessarily have the same level of superdiffusiveness (2 − β ∥ ≠ β p ), the extreme limits of normal parallel diffusion combined with momentum superdiffusion, and parallel superdiffusion combined with normal momentum diffusion, were investigated without particle escape effects.The main results are: 1.In the case of normal parallel diffusion combined with momentum superdiffusion, the spatial peaks in the accelerated particle distribution beyond the particle source decrease in amplitude with an increasing level of momentum superdiffusiveness and with time just as in the case of parallel and momentum superdiffusion.However, the amplitude of the peaks is larger in the former case because only the growth in momentum superdiffusion causes the peaks to decay with time, while in the latter case, the growth of both parallel and momentum superdiffusion contributes to the decay.2. For normal parallel diffusion combined with momentum superdiffusion the accelerated particle spectra at the particle source at higher particle momenta form increasingly hard power-law spectra as the level of momentum superdiffusiveness increases just as in the case of parallel and momentum superdiffusion.Overall, the spectra are steeper in the former case because effective normal parallel diffusion is more likely to contribute to spectral steepening at late times than growing parallel superdiffusion in the latter case.3. Considering the opposite limit of parallel superdiffusion combined with normal momentum diffusion, the amplitude of the spatial peaks decrease with increasing levels of superdiffusiveness and with time as before, but overall the amplitudes fall mostly between those of normal parallel diffusion combined with momentum superdiffusion (largest amplitudes) and the amplitudes of parallel and momentum superdiffusion (smallest amplitudes).This shows that the growing parallel superdiffusion term has a larger impact in decreasing the amplitudes of the spatial peaks than the growing momentum superdiffusion term.This was to be expected because the parallel superdiffusion term was specified to be larger than the momentum superdiffusion term in the illustrated solutions.4. In the limit of parallel superdiffusion combined with normal momentum diffusion, the accelerated particle spectra at the particle source at higher particle momenta develop increasingly hard power-law spectra for increasing levels of momentum superdiffusiveness just as in the case of parallel and momentum superdiffusion.Also in this case the spectra are overall steeper compared to the case of parallel and momentum superdiffusion due to the sustained steepening effect of adiabatic cooling at late times.In the case of parallel and momentum superdiffusion, the growth of the momentum superdiffusion term increasingly suppresses the spectral steepening effect of adiabatic cooling.
Potentially, the maximum amplification factor of the spatial peaks generated in an observed inner heliospheric SMFR acceleration event can be used as a diagnostic for the level of superdiffusiveness in the parallel spatial and momentum diffusion associated with the SMFR event by finding that level in the solution that best reproduces the observed spatial peak profiles.The maximum amplification factor in observed SMFR acceleration events appears to be increasing with radial distance from the Sun (Van Eck et al. 2021).Although that appears to be caused by the decrease in the efficiency of SMFR acceleration with increasing distance from the Sun (Van Eck et al. 2021), partly it might be that the level of superdiffusiveness is also dropping as the parallel transport and acceleration become more diffusive as the SMFR acceleration region is advected radially outward with the solar wind flow.This is consistent with the expectation that in complex dynamic systems, anomalous diffusion processes are intermediate or transitional, crossing over into normal diffusion states for system evolution times longer than a certain critical time (e.g., Ito & Miyazaki 2003).In the solar wind, a crossover from anomalous diffusion to normal diffusion is also expected to occur when energetic particles temporarily trapped in SMFR structures escape and propagate along diffusing magnetic field lines (Ruffolo et al. 2003), or when solar energetic particles escaping nearly scatter free in front of traveling shocks scatter on magnetic turbulence at later times (e.g., Effenberger & Litvinenko 2014).
Perhaps the most controversial aspects of modeling superdiffusion with a fractional time integral (or derivative) and normal spatial or momentum derivatives occur at very late times: (1) Superdiffusive particle spatial spreading then totally dominates solar wind advection so that, in the upwind direction of the particle source, modulation by the opposing solar wind flow ceases and a spatial plateau develops instead of the usual exponential spatial decay characteristic of the diffusive solution.
(2) Then momentum superdiffusion dominates all other transport terms so that an accelerated spectrum steeper Zank (2021).Energetic particle advection by the solar wind flow (second term on the left-hand side), and adiabatic energy changes due to the divergence of the solar wind flow (third term on the left-hand side) are normal advective transport processes.Diffusion processes due to energetic particle interaction with the random fields of numerous dynamic SMFRs, however, are anomalous.The first term on the right-hand side of Equation (1) represents anomalous diffusion along the mean magnetic field which, interestingly, features a fractional time integral because its fractional index β ∥ − 1 < 0 due to the restriction 0 < β ∥ < 1, combined with normal spatial derivatives (see le Roux & Zank (2021) for more details).After multiplying Equation (1) with a fractional time derivative b a left Caputo fractional time derivative (le Roux & Zank 2021), while the parallel anomalous diffusion term only features normal spatial derivatives.Accordingly, the mean parallel particle displacement is áD ñ µ D b - momentum diffusion term just includes fractional momentum derivatives.This implies that áD ñ µ D b a p t 2 2 p

]
field projected in the x-direction (solar wind flow direction) and averaged over the intervals Δx and Δv, while b averaged over Δx and Δv and normalized to p 2 .The expressions of these coefficients in Equation (3) include the parameters b b

Figure 2 .
Figure 2. The same solutions as in Figure 1 but shown at a later time t ≈ 52 hr.

Figure 4 .
Figure4.The same spectral solutions as in Figure3, but now shown for a distance of x − x 0 ≈ 0.07 au beyond the particle source.The same normalization of the curves as in Figure3is followed.The black and red curves (top curves) are somewhat difficult to distinguish because of their overlap.
Upon simplifying the expression for k 2 by applying the high-momentum condition listed above, we finally get