Modeling Particle Acceleration and Release from Solar Eruptions

Xiaocan Li¹, Joel Dahlin²,³, Fan Guo¹, Bin Zhuang⁴, Sophie Masson⁵,⁶, and Spiro Antiochos⁷

¹Los Alamos National Laboratory, Los Alamos, NM 87545, USA; xiaocanli@lanl.gov

²Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

³Astronomy Department, University of Maryland, College Park, MD 20742, USA

⁴Institute for the Study of Earth, Oceans, and Space, University of New Hampshire, Durham, NH, USA

⁵Sorbonne Université, École polytechnique, Institut Polytechnique de Paris, Université Paris Saclay, Observatoire de Paris, Université PSL, CNRS, Laboratoire de Physique des Plasmas (LPP), Paris, France

⁶Observatoire Radioastronomique de Nançay, Observatoire de Paris, CNRS, PSL, Université d'Orléans, Nançay, France

⁷Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA

Received 2025 February 26; revised 2025 April 7; accepted 2025 April 8; published 2025 May 19

Abstract

Determining the relative contribution of solar flares versus coronal mass ejections in large solar energetic particle (SEP) events is a long-standing problem. Flare-accelerated particles may travel through complex magnetic fields in the eruption region and escape into interplanetary space, thereby contributing to large SEP events. The process by which flare accelerated particles are released into the heliosphere is poorly understood and yet is critical to advancing our understanding of SEPs. In this work, we address the release problem by solving the focused transport equation in the context of a 2.5D ARMS magnetohydrodynamic simulation of a breakout coronal mass ejection (CME)/flare event. We find that particles accelerated by flare reconnection can be released into interplanetary space through interchange reconnection between closed and open field lines. These particles can contribute directly to SEP events and may become an important seed population for further acceleration by CME-driven shocks. Additionally, we find that the energetic particle fluxes in the inner heliosphere remain elevated for an extended period, allowing them to contribute to SEP acceleration by subsequent CMEs. This study represents the first direct particle modeling of how flare-accelerated particles can contribute to major SEP events.

Unified Astronomy Thesaurus concepts: Interplanetary particle acceleration (826); Solar magnetic reconnection (1504); Solar flares (1496); Solar corona (1483); Space plasmas (1544)

1. Introduction

The role of flare-accelerated particles in large solar energetic particle (SEP) events is a long-standing subject of investigation (H. V. Cane et al. 1986; J. T. Gosling 1993; E. W. Cliver 2006; D. V. Reames 2017). It has been well established that the magnetic reconnection responsible for flare heating can also be highly efficient at accelerating particles, at least, for electron acceleration (e.g., R. P. Lin & H. S. Hudson 1976). Although much more difficult to observe directly, there is indirect observational evidence and compelling theoretical work arguing that ions are also efficiently accelerated by flare reconnection. The escape of these flare-accelerated ions onto open field lines is one of the primary models proposed for the origin of the so-called impulsive SEP events (H. V. Cane et al. 1986; S. Masson et al. 2013), but the large, gradual events are well-known to be due to acceleration by coronal mass ejection (CME)-driven shocks (J. T. Gosling 1993; D. V. Reames 2017).

While CME-driven shocks are the primary accelerator of large SEP events, a preaccelerated suprathermal population may be required to achieve efficient acceleration. These suprathermal particles typically have energies from ~10 keV nuc⁻¹ to ~1 MeV nuc⁻¹ or to even higher energies (G. M. Mason & G. Gloeckler 2012).<fnref n="8" />

These seed particles could originate from solar flares accompanying CMEs (R. A. Mewaldt et al. 2006), previous SEP events (N. Gopalswamy et al. 2004; G. Li et al. 2012; B. Zhuang et al. 2020), corotating interaction regions (R. W. Ebert et al. 2012; G. M. Mason & G. Gloeckler 2012; R. J. Filwett et al. 2017), turbulent magnetic flux ropes (G. P. Zank et al. 2014; J. A. Le Roux et al. 2015; L.-L. Zhao et al. 2018; O. Khabarova et al. 2021; O. Pezzi et al. 2021), compressional turbulence (L. A. Fisk & G. Gloeckler 2006, 2008), or shocks themselves when then move through preexisting turbulence (F. Guo et al. 2021 and references therein). Heavy-ion and ³He measurements have provided unique tracers for distinguishing between the flare component and solar wind sources, as flare-accelerated particles often have enhanced ³He/⁴He ratio (up to ~1–10) and Fe/O ratio (up to ~1) (G. M. Mason et al. 2004). Previous studies have inferred, based on these measurements, that a significant amount of flare-related particles (either concomitant with the CMEs or remnants of earlier events) have contributed to the large-scale CME-related events (G. M. Mason et al. 1999; R. A. Mewaldt et al. 2002, 2006; A. J. Tylka et al. 2005; M. I. Desai et al. 2006). However, the exact mechanisms by which energetic particles propagate out of the flare regions and contribute to large, gradual SEP events, as well as to the impulsive events, remain poorly understood.

When accelerated, most flare particles are expected to remain trapped in the postflare loop and ejected flux rope unless there is a change in magnetic connectivity. This raises doubts about how efficiently particles can be transported onto the open field lines. Therefore, it is important to understand how particles produced in solar flares get released into the heliosphere (H. V. Cane et al. 1986) and possibly be accelerated by CME-driven shocks. The release process is likely facilitated by

<figure><img src="image_1.png" /></figure>

<footnote n="8">⁸ It has been discussed that the particular energy range of the seed populations can be different for SEP events (S. W. Kahler & A. G. Ling 2019), and several authors have also discussed an extended energy up to ~2 MeV nuc⁻¹ (e.g., R. A. Mewaldt et al. 2012; S. W. Kahler & A. Vourlidas 2014; B. Zhuang et al. 2022).</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Schematics of particle acceleration in the solar flare region and transport into the solar eruption region. Left: a zoomed-in diagram of a flare region filled with magnetic islands (large black circles) and turbulent fluctuations (blue circles). The accelerated particles (orange) propagate to the rest of the eruption region. Right: an illustration of the global eruption region. The solid black curves represent the magnetic field lines, and the orange shades indicate the regions with accelerated particles. Particles accelerated in the flare region fill the postflare arcade and the ejected flux ropes. They get released along open field lines (facilitated by interchange reconnection) and propagate in the CME front region and heliosphere.</figcaption></figure>

interchange reconnection, which creates connectivity between closed and open field lines (Figure 1; D. V. Reames 2002). S. Masson et al. (2013) have proposed that the external magnetic reconnection between the erupting and the open magnetic field in a CME breakout model (S. K. Antiochos et al. 1999) can naturally account for the prompt escape of flare-accelerated energetic particles onto open interplanetary magnetic flux tubes. They present detailed 2.5-dimensional (2.5D) magnetohydrodynamic (MHD) simulations of a breakout CME/flare event with a background solar wind. The results show that if the event occurs near a coronal-hole boundary, interchange reconnection between the open field and the CME flux rope can occur, which could be more efficient in 3D (S. Masson et al. 2019). This mechanism allows the flare-accelerated particles injected in the flux rope to access solar wind field lines after the eruption. These particles can reach the CME front by streaming or diffusing along these field lines. Distinct from the background solar wind plasmas, they are “suprathermal” in nature, and depending on the timing of their release, they can be further accelerated by CME-driven shocks, making them an important seed population for CME-driven shock acceleration.

However, a systematic study of this particle transport process in a rigorous, quantitative solar eruption model and their release to the CME front has not yet been conducted. The primary challenge lies in the absence of computational tools capable of evolving energetic particles in the complex regions of solar eruptions captured by advanced MHD models. The recently developed Global Particle Acceleration and Transport (GPAT)

model (X. Li et al. 2018, 2022) addresses this gap by solving the energetic particle transport equations using MHD fields as the background. This model enables the study of particle acceleration during solar flares, the transport of the flare-accelerated particles in solar eruption regions, and their propagation to the CME front. Such a study is essential for learning how the flare-accelerated particles can contribute to the seed populations for large SEP events (D. V. Reames 2002) and how they escape into the interplanetary medium (K. L. Klein et al. 2008; S. Masson et al. 2013) where they can contribute to impulsive SEP events. Additionally, such a study could improve our knowledge of the longitudinal spread of energetic particles (N. Dresing et al. 2012; M. E. Wiedenbeck et al. 2013; C. M. S. Cohen et al. 2017; M. Zhang & L. Zhao 2017) if the flare-accelerated particles can access open field lines across a broad region.

In this paper, we aim to study how flare-accelerated particles contribute to the seed population for CME-driven shock acceleration and how the flare-accelerated particles escape into the interplanetary medium. We reproduce the 2.5D MHD simulation by S. Masson et al. (2013) and solve the focused transport equation using the MHD results. Our study focuses on how flare-accelerated particles are released from the flare region, facilitated by reconnection processes, and contribute to the seed population for the CME-driven shock acceleration. In Section 2, we will describe the numerical simulations and the focused transport equation. In Section 3, we will present the simulation results. In Section 4, we will discuss the results and conclude the paper.
## 2. Numerical Simulations

We reproduce the 2.5D MHD simulation by S. Masson et al. (2013) using the Adaptively Refined Magnetohydrodynamics Solver (ARMS; C. R. DeVore & S. K. Antiochos 2008). ARMS is a well-tested code that has been employed extensively in studies of eruptive flares and CMEs (B. J. Lynch et al. 2008, 2009, 2016; J. T. Karpen et al. 2012; S. Masson et al. 2013; J. T. Dahlin et al. 2019, 2022a, 2022b). The initial conditions are identical to those used by S. Masson et al. (2013). The axisymmetric 2.5D MHD simulation employs a null-point topology susceptible to triggering a breakout CME (S. K. Antiochos et al. 1999), embedded below a helmet streamer formed by a simple isothermal solar wind. The spherical computational domain consists of the volume $r \in [1R_{\odot}, 125R_{\odot}]$, $\theta \in [-\pi/2, \pi/2]$, and $\phi \in [-\pi, \pi]$. The grid is stretched exponentially along $r$ and spaced uniformly along $\theta$. The mesh is adaptively refined up to six levels of refinement upon the current density (as described in detail in the appendix of J. T. Karpen et al. 2012). Both the inner and outer boundaries along $r$ are open to flows of mass, momentum, energy, and magnetic fluxes. The magnetic field is line-tied to the inner boundary. The initial radial flow profile is set using Parker's solar wind model (E. N. Parker 1958), and the initial magnetic field consists of a superposition of a global dipole field and a small-scale magnetic dipole near the northern coronal hole, representing an active region. The initial condition is relaxed for $2.1 \times 10^5$ s to achieve a steady state. Subsequently, the system is driven by photospheric shearing motions across the polarization inversion line of the active region. The shearing motions inject free magnetic energy and magnetic stress into the force-balanced system, leading to four episodes of magnetic reconnection: classical closed/closed null-point reconnection, the interchange closed/open reconnection, flare reconnection, and flux rope reconnection with the ambient interplanetary magnetic field (IMF). For a detailed discussion of the magnetic topology evolution and magnetic reconnection processes, we refer the readers to S. Masson et al. (2013).

We perform particle transport modeling starting from the MHD time $2.55 \times 10^5$ s, which is designated as $t = 0$ for the rest of the discussion. This time lies between the first and second episodes of reconnection. We select the particle modeling domain near the solar surface from 1 to 5 $R_{\odot}$ in the radial direction and from $-7\pi/16$ to $7\pi/16$ in the latitudinal direction. The MHD data are recorded every 25 s until $t = 2000$ s. The magnetic field and velocity field are used to solve the focused transport equation (M. Zhang et al. 2009; G. P. Zank 2014; M. Zhang & L. Zhao 2017; X. Kong et al. 2022). When solving the transport model, the MHD data are resampled to a uniform grid in the spherical coordinate with a grid size of $1024 \times 1024$. Although this resolution is enough to capture the reconnection processes, it is inadequate for a detailed study of particle acceleration in the flare region. Furthermore, it may well be that for truly accurate modeling of the particle acceleration, we would need a fully 3D calculation, which would be computationally prohibitive. Therefore, for the remainder of the discussion, we will focus on particle transport and release from the flare region rather than on particle acceleration in the flare region or by the CME-driven shock.

Figure 2 shows the out-of-plane current density $j_{\phi}$ (along the longitudinal direction) at $t = 0$. The photospheric shearing motions cause the arcade's field lines to bulge, compressing and

$t = 0.0s$

<figure><img src="image_3.png" /><figcaption>Figure 2. The out-of-plane current density $j_{\phi}$ when starting to solve the transport equation using the 2.5D ARMS MHD simulation of a breakout CME/flare event. The black lines are the magnetic field lines. The blue arrow points to the sheared arcade, where the current density starts to intensify to form an extended flare current sheet due to the photospheric shearing motions; the red arrow points to the interchange reconnection region near the null point.</figcaption></figure>

deforming the separatrices and the null point under the helmet streamer, leading to the formation of a current sheet near the null point (indicated by the red arrow). This region is the site of first, the classical closed/closed null-point reconnection as described in the breakout model (S. K. Antiochos et al. 1999) and second, the interchange reconnection between the closed arcades overlaying the sheared field and the open field lines from the northern coronal hole. Due to the shearing motions, the electric current density in the sheared arcade starts to intensify (indicated by the blue arrow), eventually forming an extended and thin flare current sheet. In Section 3, we will study how the flare-accelerated particles are released from the flare region.

We solve the focused transport equation using the GPAT code (X. Li et al. 2018; X. Li & F. Guo 2024). The following form of focused transport equation is used (e.g., M. Zhang et al. 2009; P. Zuo et al. 2013; M. Zhang & L. Zhao 2017; X. Kong et al. 2022):

$$ \frac{\partial f}{\partial t} = \nabla \cdot (\kappa_{\perp} \nabla f) - (v_{\mu}b + \mathbf{V} + \mathbf{V}_{d}) \cdot \nabla f + \frac{\partial}{\partial \mu} \left( D_{\mu\mu} \frac{\partial f}{\partial \mu} \right) - \frac{d\mu}{dt} \frac{\partial f}{\partial \mu} - \frac{dp}{dt} \frac{\partial f}{\partial p}, \quad (1) $$

where the particle distribution function $f(t, \mathbf{r}, \mathbf{p}, \mu)$ depends on time $t$, position $\mathbf{r}$, momentum $\mathbf{p}$, and cosine of particle pitch
angle $\mu$; $v$ is the particle velocity; $\mu \equiv \cos \theta = v \cdot b/v$; and $b \equiv B/B$ is the unit vector along the magnetic field. The terms on the right-hand side are the cross-field spatial diffusion with a tensor $\kappa_{\perp} = \kappa_{\perp}(\vec{I} - bb)$ (M. Zhang et al. 2009), streaming along the ambient or average magnetic field direction $b$; advection with the background plasma flow $V$; guiding-centering drift $V_d$; pitch-angle diffusion with coefficient $D_{\mu\mu}$; focusing $d\mu/dt$; and adiabatic heating/cooling $dp/dt$. In the adiabatic approximation, the drift velocity, focusing rate and cooling rate may be calculated from the ambient magnetic field $B = Bb$ and plasma velocity $V$ through

$$V_d = \frac{cpv}{qB} \left\{ \frac{1 - \mu^2 \mathbf{B} \times \nabla B}{2 B^2} + \mu^2 \frac{\mathbf{B} \times [(\mathbf{B} \cdot \nabla) \mathbf{B}]}{B^3} \right. \\ \left. + \frac{1 - \mu^2 \mathbf{B}(\mathbf{B} \cdot \nabla \times \mathbf{B})}{2 B^3} \right\}, \quad (2)$$

$$\frac{d\mu}{dt} = \frac{1 - \mu^2}{2} \left[ -v\mathbf{b} \cdot \nabla \ln B + \mu \nabla \cdot \mathbf{V} - 3\mu b_i b_j \frac{\partial V_i}{\partial x_j} - \frac{2b_i}{v} \frac{dV_i}{dt} \right], \quad (3)$$

$$\frac{dp}{dt} = -p \left[ \frac{1 - \mu^2}{2} \left( \nabla \cdot \mathbf{V} - b_i b_j \frac{\partial V_i}{\partial x_j} \right) + \mu^2 b_i b_j \frac{\partial V_i}{\partial x_j} + \frac{\mu b_i}{v} \frac{dV_i}{dt} \right], \quad (4)$$

where $V_d$ includes gradient, curvature, and parallel drifts. We ignore $\partial V_i/\partial t$ in $dV_i/dt$, assuming that the flow is not dramatically evolving compared to the rapid particle motion. The focused transport equation can be written in a Fokker-Planck form, which is equivalent to a set of stochastic differential equations that can be solved efficiently using a large number of pseudoparticles (interested readers are encouraged to check out the code documentation for more details in X. Li & F. Guo 2024). We assume a turbulence correlation length of $2 \times 10^5$ km, a slab component of 20% of total turbulent fluctuation $\delta B^2$. We further assume $\delta B^2/B_0^2 = 1$, where $B_0$ is the mean magnetic field, and the turbulence fluctuations develop a Kolmogorov power spectrum with a power-law index of $-5/3$. The turbulence is assumed to be homogenous in the simulation domain. We inject protons with the same energy of 10 keV in the center of the flare current sheet<fnref n="9" /> and let them evolve with the background plasma flow $V$ and magnetic field $B$ from the MHD simulation. When calculating $\kappa_{\perp}$, we assume $\kappa_{\perp}/\kappa_{\parallel} = 0.05$, following earlier test-particle simulations (J. Giacalone & J. R. Jokipii 1999), where $\kappa_{\parallel}$ is the pitch-angle-averaged parallel diffusion coefficient evaluated using $D_{\mu\mu}$ based on an isotropic particle distribution. The simplification is a common practice in this kind of modeling partly because of the significant numerical noise to evaluate $\kappa_{\parallel}$ from $D_{\mu\mu}$ based on anisotropic particle

distributions as the simulation proceeds. We have tried different $\kappa_{\perp}/\kappa_{\parallel}$ and find that the conclusions are insensitive to the chosen $\kappa_{\perp}/\kappa_{\parallel}$. Note that our $\kappa_{\perp}/\kappa_{\parallel}$ is much larger than what is used in a recent study by E. Husidic et al. (2024), where a much smaller $\kappa_{\perp}/\kappa_{\parallel} = 10^{-5}-10^{-3}$ can lead to significant spreading of energetic particles.

### 3. Simulation Results

#### 3.1. MHD Simulation Results

The top panels of Figure 3 show the current density $|j|$ at $t = 0, 650$, and $1550$ s. During the early stages of flare reconnection (panels (a) and (c)), the magnetic fluxes in the sheared arcade undergo internal reconnection. Figures 3(b) and (d) show the zoomed-in view around the flare region, revealing the intensified flare current sheet, the newly formed flare arcade, and the large flux rope. Additionally, one magnetic island is formed in the flare current sheet ($r \sim 1.3 R_\odot$) due to the plasmoid instability. We anticipate that particles accelerated in the flare region in this phase are primarily trapped in the flare region and the escaping CME flux rope. However, some energetic particles might be able to access the open magnetic fluxes near the equatorial region ($\theta < 30^\circ$) due to cross-field spatial diffusion. These particles could contribute to the impulsive SEPs observed in the interplanetary space.

As flare reconnection continues, all the magnetic fluxes in the sheared arcade are reconnected, and the magnetic flux in the large flux rope peaks. The large flux rope begins to rise but remains low near the flare region due to the downward magnetic tension force of the arcade. As it rises, the flux rope pushes on the overlying arcade that first reconnects with the helmet streamer magnetic field through closed/closed reconnection. Once all the closed fluxes from the helmet streamer have been processed, the overlying arcade reconnects with the open field from the northern coronal hole, as shown in Figure 3(c). The zoomed-in view shown in Figure 3(d) highlights the elongated null-point current sheet and the breaking and reconnection of the field lines. Interchange reconnection plays a critical role in the breakout model as it removes the overlying magnetic fluxes, decreasing the downward magnetic tension force (S. K. Antiochos et al. 1999; S. Masson et al. 2013), allowing the large flux rope to be released into the interplanetary space. Additionally, it allows flare-accelerated energetic particles to access the open field lines (indicated by green lines in Figure 3(c)) and be released into interplanetary space. Figure 3(e) shows that after erupting, the flux rope continuously reconnects with the ambient open field lines and therefore, acts like a long-lived source of energetic particles that escape onto open field lines.

The internal reconnection in the sheared arcade leads to the acceleration of particles. Those flare-accelerated particles are injected into the postflare loops and the forming flux rope (see Figure 1). We expect that these particles can be released into the interplanetary space through magnetic reconnection between the flux rope and the ambient open magnetic field, as anticipated by S. Masson et al. (2013).

The zoomed-in view in Figure 3(f) shows the reconnection region between the flux rope and IMF. The flux rope reconnection region is extended and features two X-points and one magnetic island, indicating that the reconnection process is active and ongoing. Particles confined in the flux rope can then be continuously released into the interplanetary space through this reconnection region. Consequently, particles

<footnote n="9">⁹ We avoid injecting particles in other region (e.g., interchange reconnection sites), making sure that our study focuses solely on the release of flare-accelerated particles.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Magnetic evolution in the MHD simulation. Top: current density $|j|$ and magnetic field lines at $t = 0, 650$, and $1550$ s in the northern hemisphere. The colored lines are the magnetic field lines. The orange lines indicate closed field lines. The blue lines indicate open field lines. The green lines indicate reconnected field lines that are open. Bottom: the out-of-plane current density $j_\phi$ in the zoomed-in regions shown in the top panels. (b) and (d) The region around the flare and the interchange reconnection. (f) The region around the reconnection site between the ejected flux rope and the ambient IMF. The black lines are the magnetic field lines.</figcaption></figure>

can access over 40° in the latitude of the heliosphere (indicated by green lines). Although cross-field diffusion enables the particles to escape the ejected flux rope (E. Husidic et al. 2024), without magnetic reconnection to change the magnetic connectivity, particles cannot reach all the green field lines, especially those with relatively low energies (see Section 3.2 for details). Additionally, the energetic particles can also travel back to the Sun along these field lines. These particles can produce emissions and form emission sources, especially γ-ray emissions, away from the flare region, as is often observed in solar eruption events (M. Pesce-Rollins et al. 2015; M. Jin et al. 2018; M. Ajello et al. 2021).

## 3.2. Particle Release from the Flare Region

Using the MHD fields, we solve the focused transport equation using a large number of pseudoparticles. At the start of each MHD output time interval (25 s), about 10 million protons, each with an initial energy of 10 keV, are injected into the flare current sheet. We take this to be the region where the current density $|j| > 100$ and $r < 2 R_\odot$. The normalization of the current is such that this condition corresponds to $j R_\odot/c > 100$ G. Although the simulation is not specifically optimized for studying particle acceleration due to the low spatial resolution, protons can still be energized due to underlying flow compression and shear, as described by Equation (4). Figure 4(a) shows the temporal evolution of the proton energy spectrum in the flare region (the same region shown in Figure 3(d)). Once the flare reconnection begins, the

particles are rapidly accelerated to a few MeV. In the early stages of the simulation ($t < 750$ s), the particles develop a power-law tail that continuously hardens as flare reconnection proceeds. During the later stages, the entire particle population is accelerated to high energies, resulting in a flatter spectrum although not necessarily a power law. The flatter spectrum is likely due to more efficient Fermi acceleration, as the guide field gets weaker during the flare evolution (J. T. Dahlin et al. 2015, 2016, 2022a). Note that during particle modeling, we use a particle-splitting technique to improve the high-energy statistics. When a particle's momentum reaches $8 \times 1.5^{n_{\text{split}}}$ times its initial value $p_0$, it is split into two identical particles with half the original weight before splitting, where $n_{\text{split}}$ is the number of times the particle has been split. These two particles will evolve independently due to the stochastic nature of the transport process. In addition to the energy spectrum, the modeling also tracks the evolution of particle pitch angles. Figure 4(b) shows that protons at different energies exhibit similar anisotropic distributions, predominantly aligned parallel to the magnetic field lines. This anisotropy is likely because the primary acceleration mechanism in magnetic reconnection is the Fermi mechanism (J. F. Drake et al. 2006; J. T. Dahlin et al. 2014, 2017; X. Li et al. 2017, 2019; Q. Zhang et al. 2021, 2024), leading to particle parallel energy gain.

Once the flux rope gets released into the interplanetary space, the flare-accelerated particles can propagate out and be observed in the outer region of the simulation. Figure 4(c) shows the evolution of the particle spectrum in the local sector
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Particle energy and pitch-angle distributions. (a) Time evolution of the proton energy spectrum near the flare region (same region as Figure 3(d)). (b) Pitch-angle distributions of protons with different energies near the flare region at $t = 1000$ s. (c) Time evolution of the proton energy spectrum near the flux rope reconnection region (same region as Figure 3(e)). (d) Pitch-angle distributions of protons at different energies near the flux rope reconnection region at $t = 1500$ s.</figcaption></figure>

shown in Figure 3(e). Due to the time-of-flight effect, the particles show energy dispersion, with higher-energy particles arriving earlier than the lower-energy ones. When the flux rope reaches the region at $t \sim 1500$ s, the spectrum shows a power-law tail similar to what is observed in the flare region at $t \sim 750$ s (Figure 4(a)). Later in the simulation, the spectrum becomes harder as particles that are accelerated later in the flare region reach this area. Figure 4(d) shows that the pitch-angle distributions are similar to what is observed in the flare region (Figure 4(b)), suggesting that the large-scale magnetic field and turbulent scattering during the propagation do not significantly change the pitch-angle distribution of the particles. These results indicate that the flare-accelerated particles can be released from the flare region, and their properties can be retained during the propagation near the Sun. It is important to note that the scale and duration of our simulation are likely insufficient to capture the isotropization typically observed in SEP observations and modelings (J. W. Bieber et al. 1980; T. T. von Rosenvinge et al. 2009; M. Zhang et al. 2009). To confirm that these particles contribute to the seed population for CME-driven shock acceleration, it is necessary to show that they are released from the ejected flux rope and propagate ahead of the CME-driven shock.

To reveal the particle release processes, we accumulate the proton distribution maps in different energy bands and present the results in Figure 5. It shows the proton distribution maps in four energy bands up to a few hundred keV at $t = 1500$ s. At

this point, the front of the CME shock (thick black curve),<fnref n="10" /> driven by the ejection of the large flux rope, has reached $r \sim 4 R_⊙$. An estimate shows the CME-driven shock reaches a speed of $>1500$ km s⁻¹. To catch up with the CME front, protons need to reach a minimum thermal speed of the CME speed, which corresponds to a thermal energy of 11.7 keV. Therefore, unless protons with an energy of $\sim 11.7$ keV can stream freely along the magnetic field lines, they will be unable to catch up with the CME-driven shock. Moreover, these particles undergo adiabatic cooling as they propagate outward, and their spatial diffusion, especially perpendicular to the magnetic field, is not fast enough to allow them to diffuse across the shock. Figure 5(a) shows that these particles are indeed primarily located behind the shock and are trapped in the flux rope. However, once the flux rope field lines are opened up due to the interchange reconnection between the flux rope and IMF, these particles in the energy bin of 10–30 keV can be released into the interplanetary space due to the combined effects of free streaming and spatial diffusion parallel the magnetic field lines, forming a stream of particles beyond the CME-driven shock (black arrow in Figure 5(a)).

To further illustrate the release process, we present the time evolution of the proton maps in the energy band of 10–31 keV in Figures 6(a)–(c). As shown, most particles remain behind the CME-driven shock, and only some of them can be released

<footnote n="10">¹⁰ The front of the CME shock is identified when there is a sudden jump in the flow compression $-\nabla \cdot \mathbf{V}$ along the radial direction.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Proton distribution maps in different energy bands at *t* = 1500 s. The thick black lines indicate the CME-driven shock at the time of the snapshots. The black arrow in (a) indicates the stream of low-energy particles escaped from the flare region through flux rope reconnection.</figcaption></figure>

from the flux rope. The stream of low-energy particles starts to emerge at *t* = 1200 s and becomes prominent after *t* = 1375 s. The stream of particles is primarily located in the region where the flux rope reconnection occurs, indicating that these particles are released through the reconnection region. We expect that the CME-driven shock can further accelerate these particles if they can be partially trapped in the vicinity of the shock. However, these particles are not widely spread in front of the shock and thus cannot explain the wide longitudinal spread of SEPs observed in large SEP events (N. Dresing et al. 2012; M. E. Wiedenbeck et al. 2013; C. M. S. Cohen et al. 2017; M. Zhang & L. Zhao 2017). Most of the other particles stay behind the shock until the CME-driven shock is out of the particle modeling domain (*r* > 5 *R*<sub>☉</sub>). These particles could become a seed population for following CME-driven shocks, as proposed by the twin-CME scenario (G. Li et al. 2012; L. Ding et al. 2013; B. Zhuang et al. 2020), and we will further discuss this in Section 3.3. The particle release process is also evident in the trajectories of the pseudoparticles, as shown in Figure 7. The two particles with the lowest final energies are primarily advected with the background plasma flow driven by the flux rope ejection. These particles, located within the dashed box at *r* > 4 *R*<sub>☉</sub> at *t* = 1500 s, have thus been released from the flux rope by then.

In contrast to the low-energy particles, high-energy particles
spread to a broader region, including the front of the CME-
driven shock, as shown in Figures 5(c) and (d). Their spatial
diffusion is large enough to allow them to diffuse across field
lines and access the open field lines of the streamer near the
equatorial plane (θ = 0°).<fnref n="11" /> These particles can be released into
interplanetary space along the open field lines and contribute to
the impulsive SEP events observed in situ. As the simulation
progresses and more magnetic field lines open due to flare

reconnection and flux rope reconnection processes
(Figure 3(c)), these high-energy particles can stream or diffuse
along the field lines, enabling them to spread ahead of the
CME-driven shock at multiple locations. This explains why
these high-energy particles are broadly distributed along the
CME front, as shown in Figures 5(c) and (d).

To illustrate the release processes of these high-energy
particles, we also plot the time evolution of the proton maps for
the 255–626 keV energy band in Figures 6(d)–(f) and show the
sample trajectories of the high-energy pseudoparticles in
Figure 7. These high-energy particles can propagate ahead of
the CME-driven shock and are spread out in front of the shock
after *t* = 900 s (Figure 6(e)) due to their high speed and large
spatial diffusion coefficients. This behavior is also evident in
their trajectories. Figure 7 shows that these high-energy
particles (with final energies over hundreds of keV) have more
stochastic trajectories, indicating they experience stronger
scattering compared to low-energy particles and more easily
diffuse across the field lines. Since these particles can spread
widely ahead of the CME-driven shock, they can become an
important seed population for the CME-driven shock
acceleration, which is critical for large SEP events.

3.3. Contribution to the Seed Population for Subsequent CMEs

So far, we have focused on the release of flare-accelerated
particles facilitated by magnetic reconnection. These particles
could be further accelerated by the accompanying CME event.
In addition to this process, particles accelerated by an earlier
flare-CME eruption can serve as the seed population of the
subsequent CME-driven shock acceleration, as proposed in the
twin-CME scenario (G. Li et al. 2012; L. Ding et al. 2013;
B. Zhuang et al. 2020). To examine whether the flare-
accelerated particles can contribute to the subsequent CME-
driven shock acceleration, we track the particle fluxes at r = 4
R☉ and show the results in Figure 8. r = 4 R☉ is selected based

<footnote n="11">¹¹ The spreading near the equatorial plane will depend on the $\kappa_{\perp}/\kappa_{\parallel}$, which is unknown in the corona.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Time evolution of the proton distribution maps in the energy band of (a)–(c) 10–31 keV and (d)–(f) 255–626 keV. The thin black lines are the magnetic field lines. The thick black lines indicate the CME-driven shock.</figcaption></figure>

on the observations of type II radio bursts indicating that the shock is generated and starts accelerating particles at or below this distance (e.g., N. Gopalswamy et al. 2009; P. Mäkelä et al. 2015). During the modeling, we turn off particle injection in the flare region at $t = 2000$ s and continue to track the particle evolution until $t = 12,500$ s using a static MHD background at $t = 2000$ s. Because the simulation employs open-boundary conditions, once particles go beyond the simulated domain, they are removed from the simulation loop. Figure 8 shows the particle fluxes quickly rise to the peak, with the high-energy particles rising earlier than low-energy particles due to their higher velocities. Once the particle injection is turned off at $t = 2000$ s, the particle fluxes at $r = 4 R_⊙$ decrease first and then gradually increase to a peak between $t = 4000$ and $t = 6000$ s. The initial decrease is due to leakage at the outer boundary as the flux rope exits the particle modeling domain ($r > 5 R_⊙$), while the subsequent increase occurs because particles in the flare region are continuously accelerated and propagate outward. After that, the particle fluxes exponentially decay over time due to the leakage at the boundary and the spatial diffusion of the particles. The flux decay time is about

1200 s for all the energy bands, so the high-energy particle fluxes will stay at an elevated level (e.g., >1% of the peak flux) over 1.5 hr. Consequently, these particles could become an important seed population for (especially the concurrent) CMEs that erupt within a couple of hours later. We note that the estimated decay time is lower than that in the so-called twin-CME scenario (~8 hr; G. Li et al. 2012). Note that we use a considerably larger perpendicular diffusion coefficient ($\kappa_{\perp}/\kappa_{\|} = 0.05$) than earlier studies (e.g., $\kappa_{\perp}/\kappa_{\|} = 10^{-5}-10^{-3}$ in E. Husidic et al. 2024), which could lead to faster diffusion of the particles out of the domain and a short decaying time. These diffusion coefficients depend critically on the turbulence properties near the Sun, which are not yet well understood. Additionally, our current simulation does not account for shock-accelerated particles or shock-induced turbulence downstream of the shock. However, our simulations estimate how long the flare-generated seed particles can remain in the corona, where the particle acceleration by the shock takes place. We are looking forward to comparing with measurements from Parker Solar Probe (N. J. Fox et al. 2016) during its perihelion in the corona.
<figure><img src="image_8.png" /><figcaption>**Figure 7.** The trajectories of six pseudoparticles with different final energies. (a) The trajectories colored by their kinetic energies. These particles are located in the sectors with dashed boundaries at *t* = 1500 s. (b) Time evolution of the kinetic energies of the particles.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Time evolution of the proton fluxes at *r* = 4 *R*⊙ in seven energy bands binned logarithmically from 10 keV to 5.2 MeV. The dashed lines indicate the exponential decay over time. The vertical black line indicates when the particle injection in the flare region is turned off at *t* = 2000 s.</figcaption></figure>

**4. Discussions and Conclusions**

In this study, we investigated how flare-accelerated particles contribute to the seed population for CME-driven shock acceleration by solving the focused transport equation with 2.5D ARMS MHD simulation of a breakout CME/flare event. We found that these particles can be released from the flare region and the ejected flux rope through reconnection processes between the flux rope and open field lines. Low-energy particles are primarily released through the flux rope reconnection process between the closed field lines of the flux rope and the open IMF, forming a stream of particles ahead of the CME-driven shock. In contrast, high-energy particles (> a few hundred keV), due to their higher speed and large spatial diffusion coefficient, can be released at multiple points and spread widely along the CME front. These high-energy particles could become an important seed population for SEP

acceleration by the accompanying CME. Additionally, we found that the energetic particle fluxes can remain at elevated levels for a couple of hours after the eruption, potentially contributing to acceleration during subsequent CME events.

Results from this study indicate that the low-energy particles are primarily released at a radial distance beyond $4 R_⊙$, forming a stream ahead of the shock and likely acting as seed particle populations. Past studies have shown that the release of efficiently accelerated particles can occur when the shock propagates to $r > 4 R_⊙$ (e.g., E. W. Cliver et al. 2004; D. Lario et al. 2017; B. Zhuang et al. 2020, 2024), in which the evolution of the shock strength (e.g., the shock speed relative to the Alfvén speed or fast magnetosonic speed in the corona) is proposed to be one of the major reasons. Our simulations suggest that an additional mechanism for efficient particle acceleration above $4 R_⊙$ is the release of seed populations primarily at that radial distance. However, due to the limited resolution of the current simulation and the simple turbulence model used, it cannot capture the CME-driven shock acceleration, making it challenging to distinguish these two origins. Higher-resolution simulations are necessary to resolve the shock structure and differentiate between these scenarios accurately.

Particle maps produced by the model show that energetic particles are broadly distributed near the Sun. This result suggests that the rapid changes in magnetic connectivity from solar eruptions can have a significant impact on SEP transport in the inner heliosphere. This effect should be taken into account in future SEP studies as well as the particle transport caused by magnetic field random walk (R. Chhiber et al. 2021).

Although high-energy particles are widely spread along the CME front, the axisymmetric simulation examined in this study cannot capture the important longitudinal spread of SEP. To address the longitudinal spread of SEPs (N. Dresing et al. 2012; M. E. Wiedenbeck et al. 2013; C. M. S. Cohen et al. 2017; M. Zhang & L. Zhao 2017), it is necessary to perform simulations in 3D. However, it is reasonable to expect that particles can spread widely in the longitudinal direction
because the reconnection layers are inherently 3D. Preliminary results from our particle modeling using 3D ARMS MHD simulations (S. Masson et al. 2019; J. T. Dahlin et al. 2022a) suggest that 3D magnetic reconnection between the flux rope and the open field lines occurs over a wide longitudinal range over the CME front and enables energetic particles to spread broadly in the longitudinal direction. We will present these findings in a separate publication.

Another effect that is likely to increase the longitudinal spread of SEPs is the presence of ubiquitous, turbulent-like reconnection in the solar corona driven by the photospheric convective motions, both granular and supergranular. This type of reconnection has been proposed as the origin of a vast range of solar activity. In the closed corona, it underlies the nanoflare reconnection model for coronal heating (e.g., J. A. Klimchuk 2006) and the helicity condensation model for filament channel formation (S. K. Antiochos 2013). In and near open field regions, it is the basis for the so-called component reconnection process that drives the slow wind (A. F. Rappazzo et al. 2012) and the interchange reconnection driving the fast wind (S. D. Bale et al. 2023). Although likely important for particle transport, this turbulent reconnection is highly challenging to calculate rigorously, because it is inherently 3D and requires inclusion of the small-scale photospheric motions that are prohibitive to calculate numerically in a global simulation such as ours. Determining the effect of turbulent reconnection on the SEP longitudinal spread clearly requires future studies.

In our simulations, we found that protons can be quickly accelerated to a few MeV due to the underlying flow compression and shear in the flare current sheet. However, it is worth noting that the moderate resolution (1024 × 1024) of the MHD data might not be enough to study the acceleration processes in the flare current sheet in detail, which requires a much higher resolution typically achieved in local simulation boxes (X. Li et al. 2018; G. Murtas et al. 2024). Additionally, the acceleration processes might not be fully captured by the background MHD magnetic fields and plasma flows, making it necessary to include additional acceleration mechanisms, for example, second-order acceleration caused by plasma turbulence (F. Pecora et al. 2018; L. Comisso & L. Sironi 2022).

Our findings indicate that the properties of flare-accelerated particles, such as their spectral shape and pitch-angle distributions, are largely preserved during their release. These characteristics could be crucial diagnostics for determining whether flare-accelerated particles can contribute to the seed population for large SEPs. However, it is impractical to simultaneously determine these properties in both solar flares and CME shock regions. This is why flux ratios, such as ³He/⁴He and Fe/O (G. M. Mason et al. 2004), are essential. By injecting multiple species into the flare region and tracking the evolution of these flux ratios as the particles propagate outward, one can potentially compare the results with in situ observations. To maintain the focus and clarity of this paper, we will defer such a study to a future publication.

It is worth noting that we assume a constant turbulence correlation length in the simulation. In reality, the turbulence correlation length may increase with radial distance from the Sun (M. E. Ruiz et al. 2014; G. P. Zank et al. 2018; M. E. Cuesta et al. 2022), resulting in a larger spatial diffusion coefficient as particles propagate outward. Consequently, low-energy particles can also diffuse across the shocks like high-energy particles, allowing them to be picked up by the CME-driven shock and

accelerated to higher energies. Additionally, we assume a constant ratio between the perpendicular diffusion coefficient and the pitch-angle-averaged parallel diffusion coefficient. However, since the particle parallel and perpendicular mean-free paths might have different radial dependencies (R. Chhiber et al. 2017), this could result in a radially varying ratio of the diffusion coefficients. Further investigation is needed to determine how this variation affects particle transport near the Sun. Moreover, shock-accelerated energetic particles will likely modify the shock structure or generate additional turbulence near the CME-driven shock (D. Caprioli & A. Spitkovsky 2014), and the propagation of the shock in preexisting turbulence also modifies its structure and affects coherent structures (F. Guo et al. 2012; D. Trotta et al. 2022; G. Prete et al. 2025). These effects will lead to more efficient particle trapping near the shock and stronger particle acceleration.

In conclusion, by solving the focused transport equation using a 2.5D ARMS MHD simulation of a breakout CME/flare event, we found that particles accelerated during solar flares can be released into interplanetary space through reconnection regions between closed and open field lines. These particles could become an important seed population for large SEPs accelerated by CME-driven shocks. Note also that the basic processes invoked in this study, the closed-closed reconnection at the flare current sheet and the interchange reconnection at the breakout sheet, are widely believed to be ubiquitous throughout the corona. Both the nanoflare model for heating the closed corona (e.g., J. A. Klimchuk 2006), and the interchange reconnection model for heating and accelerating the slow and fast winds (e.g., A. F. Rappazzo et al. 2012; S. D. Bale et al. 2023) invoke ubiquitous turbulent-like reconnection involving either closed-closed or closed-open flux. Consequently, the implications of our study may well have applicability not only to flares/CMEs but to a vast range of solar activities.

## Acknowledgments

We thank the support from NASA LWS grant 80NSSC21K1313 and 80HQTR21T0087. X.L. acknowledges the support from NASA through grant NNH240B72A, National Science Foundation grant Nos. AST-2107745 and AGS-2334930, University of California, Berkeley through subcontract No. 00011753, and Smithsonian Astrophysical Observatory through subcontract No. SV1-21012. J.D. acknowledges the support from NASA through grant 80NSSC23K1065. F.G. acknowledges the support from NASA grants 80HQTR21T0104 and NSF grant AST-2109154. S.A. acknowledges support from the NASA LWS and the NSF SHINE Programs. Simulations were performed at the National Energy Research Scientific Computing Center (NERSC).

## ORCID iDs

Xiaocan Li https://orcid.org/0000-0001-5278-8029

Joel Dahlin https://orcid.org/0000-0002-9493-4730

Fan Guo https://orcid.org/0000-0003-4315-3755

Bin Zhuang https://orcid.org/0000-0002-5996-0693

Sophie Masson https://orcid.org/0000-0002-6376-1144

Spiro Antiochos https://orcid.org/0000-0003-0176-4312

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