SOFIE: Solar Wind & Energetic Particles Model

• SOFIE is a physics-based model that simulates solar energetic particle events by integrating MHD solar wind dynamics, CME eruption modeling, and turbulence-aware field-line geometry.
• It employs data-driven boundary conditions and advanced numerical techniques such as adaptive mesh refinement and focused transport to deliver high-fidelity, operationally relevant predictions.
• The model has been validated against historical SEP events and shows improved forecast metrics, providing actionable insights for both heliophysics research and space weather forecasting.

The SOlar wind with FIeld lines and Energetic particles (SOFIE) model is a comprehensive, physics-based framework for simulating and forecasting solar energetic particle (SEP) events by integrating the underlying solar-wind magnetohydrodynamics (MHD), coronal mass ejection (CME) eruption dynamics, and energetic particle acceleration and transport. SOFIE leverages data-driven boundary conditions and turbulence-aware field-line geometry to enable high-fidelity, operationally relevant SEP predictions, supporting both basic heliophysics research and applied space weather forecasting.

1. Theoretical Basis: MHD, Turbulence, and Field-Line Geometry

The SOFIE approach is rooted in first-principles MHD modeling, using the Real-time Alfvén Wave Solar atmosphere Model (AWSoM-R) to evolve the background solar wind structure, incorporating observed radial magnetic field from GONG synoptic magnetograms at the inner boundary and solving the MHD system with explicit Alfvén-wave transport equations. This establishes the dynamic “stream-aligned” solar wind and coronal magnetic field configuration required for realistic CME eruption and SEP propagation (Zhao et al., 2023, Liu et al., 12 Nov 2025).

Turbulence in the interplanetary medium is represented with a two-component spectrum (slab and 2D), which controls magnetic field-line random walk (FLRW), pitch-angle scattering, and the focused transport regime of SEPs. The field-line connectivity and turbulent scattering coefficients (e.g., parallel mean free path λ∥, cross-field diffusion κ⊥) are parametrized via in situ measurement or derived from the modeled magnetic power spectra (Subashchandar et al., 12 Sep 2025, Laitinen et al., 2018, Sonsrettee et al., 23 Apr 2024).

Field-line topology—including the distinction between closed and open coronal loops—strongly affects both elemental abundance (the “FIP effect”) and the initial SEP acceleration region, linking the Alfvén-wave ponderomotive force and the FIP-dependent fractionation of the source plasma (Reames, 2018).

2. Model Architecture and Numerical Implementations

SOFIE comprises a tightly coupled triplet:

1. AWSoM-R (Alfvén Wave Solar atmosphere Model–Realtime): Solves the ideal single-fluid MHD equations with Alfvén-wave turbulence driving and heating, using adaptive mesh refinement for efficient domain coverage from the low corona (1.05 R⊙) out to ≳2.5 AU. Boundary conditions are set by near-real-time solar magnetograms; Poynting flux and transverse correlation-length parameters are tuned to match steady-state speed, density, and field observations at 1 AU (Zhao et al., 2023).
2. EEGGL (Eruptive Event Generator using Gibson-Low): Triggers CME eruptions by inserting an imbalanced, analytic flux rope atop the parent active region. The rope geometry and twist are scaled to observed CME speed and polarity-inversion-line orientation, ensuring event-specific realism (Zhao et al., 2023).
3. M-FLAMPA (Multiple Field-Line Advection Model for Particle Acceleration): Solves the focused transport (Parker) equation along dynamically evolving field lines, tracing SEP acceleration in the CME-driven shock and subsequent propagation. The core equation is:

$$\frac{\partial f}{\partial t} + \mu v \frac{\partial f}{\partial s} + (1-\mu^2)\frac{v}{2L}\frac{\partial f}{\partial \mu} - \frac{p}{3}\frac{\partial u}{\partial s}\frac{\partial f}{\partial p} = \frac{\partial}{\partial \mu}\left[D_{\mu\mu}\frac{\partial f}{\partial \mu}\right] + Q_{\rm inj}(s,p)\delta(s - s_{\rm shock}).$$

Here, $f$ is the gyrotropic phase-space density, $\mu$ the pitch-cosine, $v$ the particle speed, $L$ the focusing length, $D_{\mu\mu}$ the pitch-angle diffusion coefficient, and $Q_{\rm inj}$ the shock injection term.

Coupling between the modules allows field lines and shock structure from AWSoM-R/EEGGL to provide time-dependent boundary conditions for M-FLAMPA. Spatial coverage is achieved by sampling 648 field lines over a 2D array of footpoints (𝜃, 𝜙) at r=2.5 R⊙ (Zhao et al., 2023).

3. Turbulence, Field-Line Random Walk, and SEP Transport Physics

Turbulence strength and geometry—quantified by the slab fraction $f_s$ and the total amplitude $b/B_0$—dictate both early-time non-diffusive transport, governed by field-line meandering, and late-time cross-field diffusion. The parallel mean free path ($\lambda_{\parallel}$) and field-line diffusion coefficient ($D_{FL}$) are linked by: $$D_{FL} \propto \frac{\delta B}{B}, \qquad \lambda_{\parallel} \propto \frac{B^2}{\delta B^2},$$ resulting in the scaling

$$\sigma_{\phi} \propto (1/\lambda_{\parallel})^{1/4}$$

for the longitudinal SEP event extent in the early meandering phase (Laitinen et al., 2018).

At early times ($t \lesssim \lambda_{\parallel}/v$), SEPs are effectively confined to individual meandering lines:

$\langle \Delta x_\perp^2 \rangle \simeq 2vD_{FL}t.$

At late times, diffusive cross-field transport takes over: $$\langle \Delta x_\perp^2 \rangle \simeq 2\kappa_\perp t.$$ SEP time profiles are therefore convolutions of the field-line path length probability distribution $P(s)$ with the injection function $Q(t)$ and the focused-transport kernel (Sonsrettee et al., 23 Apr 2024).

4. Validation, Performance, and Operational Forecasting

The SOFIE framework has been benchmarked against historical SEP events, including the nine 2012–2017 SHINE challenge events (Zhao et al., 2023) and two notable SEP events (10 Sep 2017 and 4 Nov 2001) in operational tests at NOAA/SWPC (Liu et al., 12 Nov 2025). Key forecast skill metrics include:

• Spearman rank coefficients: $\rho_{sp}(>10\,{\rm MeV}) \approx 0.84$, $\rho_{sp}(>100\,{\rm MeV}) \approx 0.56$ with coarse background mesh, improving to $>0.9$ with high-resolution setup.
• Fraction of forecast points within one order of magnitude: $92.1\!-\!92.7\%$
• Real-time operational feasibility: $4$-day forecast completed in $<5$ hours with 1,000 CPU cores; high-resolution setup in $13\!-\!21$ hr (Liu et al., 12 Nov 2025).

Dynamic adaptive mesh refinement (block-AMR) combines a coarsened background with high-resolution cones along the CME/shock path and at the Earth’s longitude, optimizing trade-offs between speed and CME/SEP fidelity. Forecaster feedback prompted operational improvements: coarse/quick background runs for early alert, followed by detailed forecasts for quantitative assessment (Liu et al., 12 Nov 2025).

5. Observational Synthesis and Physical Coupling

SOFIE unites several key physical processes:

• Fractionation and Source Attribution: FIP-dependent ponderomotive fractionation sets the elemental abundance in SEPs (closed-loop dominated, FIP “crossover” at $\sim$10 eV) and in slow solar wind (open-field, $\sim$14 eV), with shock acceleration (CME or CIR) largely preserving these chromospheric signatures in the energetic population (Reames, 2018).
• Field-Line Mapping and PADs: Accurate mapping of observed field lines back to solar source regions, using Runge-Kutta or “B-step” algorithms and boundary-fitted MHD fields, supports the prediction and classification of suprathermal electron pitch-angle distributions (PADs) and magnetic connectivity (Tasnim et al., 2019).
• Transport Theory Realism: Diffusion coefficients are computed via SOQLT for $\kappa_\parallel$ (resolving the “90-degree problem”) and UNLT for $\kappa_\perp$, using in situ measured or modeled turbulence spectra. Near the Sun ($r \lesssim 0.3$ AU), $\kappa_\perp/\kappa_\parallel \sim 10^{-3}$–$10^{-4}$, confirming the dominance of streaming along field lines and emphasizing the geometric rather than diffusive origin of early SEP event width (Subashchandar et al., 12 Sep 2025, Wang et al., 2023).
• Field-Particle Thermodynamics: Simultaneous, high-cadence measurements of magnetic field intermittency ($\kappa_{\Delta B}$) and particle kappa ($\kappa_{\rm EP}$) demonstrate entropy and degree-of-freedom (d_eff) transfer in CME-driven shocks, with an in situ anti-correlation $\kappa_{\Delta B} \approx -0.15\,\kappa_{\rm EP}$ during ICME passage (Cuesta et al., 14 Apr 2025).

6. Limitations, Uncertainties, and Future Enhancements

Current SOFIE implementations tune a per-event shock injection coefficient $c_i$ to match observed >10 MeV flux, as a placeholder for unresolved suprathermal “seed” populations. Perpendicular diffusion is not yet included in all runs; physically realistic, turbulence-driven spatially varying $\lambda(s,p)$ and $D_{\mu\mu}$ will further improve decay-phase time profiles and poorly connected events (Zhao et al., 2023).

Additional model developments in progress:

• Multi-CME and ICME coupling for sequential event “twin CME” scenarios
• 3D white-light and multi-point coronagraph validation for CME geometry
• Pipeline integration of real-time or near-real-time turbulence power and spectral measurements to set transport parameters on the fly
• Automated magnetic field mapping and PAD-based event classification

7. Broader Physical Context: Unified SOFIE Paradigm

SOFIE provides a unified model for the generation, evolution, and observation of SEPs by integrating:

• (A) Large-scale MHD field-line geometry from realistic solar magnetograms,
• (B) Turbulence-aware field-line wandering and pitch-angle scattering diagnostics,
• (C) Shock acceleration physics respecting the local coronal and heliospheric conditions,
• (D) Real-time or retrospective synthesis of observed CME, solar wind, and in situ energetic particle data.

This framework quantitatively links coronal source physics (field topology, FIP fractionation, CME field-line geometry) to in situ and remote SEP observations, enabling operational space weather forecasts with both physical transparency and validated predictive skill. For researchers, SOFIE offers a tractable, extensible platform in which new physics modules (cross-field transport, non-Maxwellian injection, entropy tracking) can be systematically evaluated in the SEP context.