COCONUT MHD Solar Corona Model

• COCONUT coronal model is a comprehensive 3D, time-dependent MHD solver that simulates solar corona dynamics, including CME initiation and propagation.
• It employs unstructured finite-volume grids, robust implicit solvers, and advanced divergence control to achieve high efficiency and accuracy.
• By coupling with kinetic particle transport models, the framework enables first-principles studies of energetic particle dynamics in evolving solar wind conditions.

The COCONUT (COolfluid COroNa UnsTructured) coronal model constitutes a fully three-dimensional, time-dependent, magnetohydrodynamic (MHD) solver for the solar corona, designed to address large-scale structure, CME initiation and propagation, and coupled energetic particle transport using modern numerical and data-driven techniques. Its architecture is based on the COOLFluiD platform, employing unstructured finite-volume grids, robust implicit solvers, and flexible boundary treatments, and is utilized in both research and operational space weather forecasting contexts.

1. Mathematical Framework and Physical Assumptions

COCONUT solves the conservative form of the ideal MHD equations, with gravity, on an unstructured grid. The governing equations are:

$$\begin{aligned} \frac{\partial \rho}{\partial t} &+ \nabla \cdot (\rho \mathbf{v}) = 0 \ \frac{\partial (\rho \mathbf{v})}{\partial t} &+ \nabla \cdot \left[\rho \mathbf{v} \mathbf{v} + \mathbb{I}(P + \tfrac{1}{2} |\mathbf{B}|^2) - \mathbf{B} \mathbf{B}\right] = \rho \mathbf{g} \ \frac{\partial E}{\partial t} &+ \nabla \cdot \left[(E + P + \tfrac{1}{2} |\mathbf{B}|^2) \mathbf{v} - \mathbf{B} (\mathbf{v} \cdot \mathbf{B})\right] = \rho \mathbf{g} \cdot \mathbf{v} \ \frac{\partial \mathbf{B}}{\partial t} &+ \nabla \cdot (\mathbf{v} \mathbf{B} - \mathbf{B} \mathbf{v}) + \nabla \psi = 0\ \frac{\partial \psi}{\partial t} &+ V_{\rm ref}^2 \nabla \cdot \mathbf{B} = 0 \ \end{aligned}$$

Where $\rho$ is mass density, $\mathbf{v}$ velocity, $\mathbf{B}$ magnetic field, $P$ thermal pressure, $E$ total energy density ($$E = \rho|\mathbf{v}|^2/2 + \rho \epsilon + |\mathbf{B}|^2/2$$), $\psi$ the divergence-cleaning field (GLM method), and $g(r) = -GM_\odot / r^2\, \hat{\mathbf{e}}_r$ gravitational acceleration. The closure is polytropic, $P = (\gamma - 1)\rho \epsilon$, with $\gamma = 1.05$ for nearly isothermal closure, emulating coronal heating in the absence of explicit terms. For more advanced simulations, thermal conduction, radiative losses, and empirical heating functions are included:

$S = -\nabla \cdot \mathbf{q} + Q_{\rm rad} + Q_H,$

where $\mathbf{q}$ is the anisotropic conductive flux (Spitzer–Härm inside $\sim10 R_\odot$, Hollweg collisionless beyond), $Q_{\rm rad}$ optically thin radiative losses, and $Q_H$ an empirical, field-weighted heating.

COCONUT models the domain from $r = 1\,R_\odot$ up to $25$–$30\,R_\odot$ with no polar singularity, capturing the outer solar corona up to the typical heliospheric model interface. Both steady-state and fully time-dependent regimes are supported, with the latter driven by time-sequenced, processed synoptic magnetograms.

2. Numerical Schemes, Grid, and Boundary Treatments

COCONUT employs an unstructured finite-volume mesh, typically realized as a $6^{\rm th}$-level subdivided geodesic polyhedron extruded radially, yielding $1.5$–$2$ million prismatic cells (angular resolution $\sim$0.8°). Temporal integration is conducted using fully implicit backward-Euler (steady-state) or BDF2 (time-accurate), enabling very large timesteps (few seconds to minutes), far surpassing the explicit CFL limitation. Newton–Krylov (GMRES) solvers (with Additive Schwarz preconditioners) solve the resulting nonlinear systems, ensuring rapid convergence even at high resolution.

Magnetic divergence control is enforced via the Dedner GLM hyperbolic/parabolic cleaning scheme, with the $\psi$ field evolved alongside $\mathbf{B}$ to suppress $\nabla\cdot\mathbf{B}$ to machine precision.

Boundary and Initial Conditions

• Inner boundary ($1\,R_\odot$): Radial magnetic field $B_r$ applied from SDO/HMI magnetograms; ghost-cell values for $\rho$, $P$, $\mathbf{v}$ are set to hydrostatic or polytropic profiles with a 1 MK base and low inflow ($v_r \approx 1$ km/s).
• Outer boundary ($25\,R_\odot$): Zero-gradient outflow for all variables.
• Initial condition: Typically a PFSS (Potential Field Source Surface) extrapolation from the input magnetogram, then relaxation to MHD steady-state.

Inner boundary modifications include local density scaling to cap the Alfvén speed in active regions, or smooth tanh-based blending strategies to constrain minimum plasma $\beta$ and maximum $v_A$, ensuring both numerical stability and physical plausibility during solar maxima (Brchnelova et al., 2024, Brchnelova et al., 2023).

3. CME and Flux Rope Modeling

Coronal Mass Ejection (CME) initiation is achieved by instantaneous superposition of an analytical, force-unbalanced flux rope onto the background steady-state corona:

• Titov–Démoulin model (TDFR/TDm): A toroidal flux rope specified by analytical vector potential with major radius $a=0.3\,R_\odot$, minor radius $b=0.1\,R_\odot$, footpoints separated by $0.15\,R_\odot$. The current parameter $\zeta = I/I_S > 1$ ensures eruption.
• Regularized Biot–Savart Laws (RBSL): Allows arbitrary, S-shaped axis geometries via regularized integral kernels, matching observed sigmoid morphologies (Guo et al., 2023). The axial current $I$ and flux $F$ are specified to exceed loss-of-equilibrium thresholds.

Only the magnetic field is perturbed upon rope injection; thermodynamic variables are unchanged. The resulting Lorentz force imbalance induces eruption, driving the flux rope and associated CME radially outward, forming a sheath and ejecta structure consistent with in-situ and white-light observations (Linan et al., 2023).

The model allows parametric study of initial core field $B_0$, eruption speed, and sheath/ejecta properties. Observed relationships include

$$V_{\max} \approx 21.9\,B_0 + 277.1\ \mathrm{km/s} \ \text{(solar minimum)}, \quad B_{\max} \approx 28.3\,B_0 + 163.4\ \mathrm{nT},$$

with similar scaling at solar maximum.

4. Particle Transport Coupling: COCONUT+PARADISE

The COCONUT configuration, including the evolving CME and flux rope, serves as the background for kinetic models such as PARADISE, which simulates test-particle trajectories and distribution functions.

Key features:

• COCONUT outputs 3D snapshots every $\sim289$ s, including all macroscopic plasma quantities.
• PARADISE applies barycentric interpolation to recover local $\rho$, $\mathbf{v}$, $\mathbf{B}$, and their derivatives at each particle position.
• The focused transport equation (FTE), including pitch-angle diffusion $D_{\mu\mu}$ and cross-field spatial diffusion $\kappa_\perp$, is solved stochastically:
  • $\kappa_\perp$ supports both constant perpendicular mean free path (MFP) and Larmor-radius dependent scaling,

  $$\lambda_\perp = (\pi/12)\,\alpha\,(r_L/r_{L0})\,\lambda_\parallel, \quad r_L = m v / (|q| B).$$

• Physical cases demonstrate that absence of cross-field diffusion leads to particle trapping within the flux rope, whereas even small $\lambda_\perp$ values enable escape from the rope's periphery, with core-trapped populations persisting (Husidic et al., 2024).

This coupled MHD–kinetic framework allows first-principles studies of SEP transport, flux rope confinement, and cross-field escape, with significant implications for particle event forecasting.

5. Practical and Operational Validation

COCONUT has undergone extensive benchmarking and validation using both synthetic and observational metrics (Perri et al., 2022, Perri et al., 2022, Baratashvili et al., 9 Nov 2025):

• Streamers, Coronal Holes, HCS: Comparisons of simulated streamer envelopes, coronal-hole boundaries (from EUV), and current sheet positions (from coronagraph synoptic maps) demonstrate high fidelity, with best-case streamer edge matches of up to 85% area overlap and mean HCS deviations of $\sim4^\circ$ using HMI-based driving (Perri et al., 2022).
• Dynamic vs. Quasi-steady Plasmas: Time-evolving, magnetogram-driven runs capture streamer and hole dynamics more accurately, with instantaneous errors in wind speed and density at $20R_s$ exceeding 15–25% when comparing time-evolving to quasi-steady approaches (Wang et al., 2024, Wang et al., 17 May 2025).
• Resource Efficiency: Implicit time-integration yields convergence rates up to $30\times$ faster than explicit solvers for realistic domains, with full-cron simulations achieving wall-times of $\sim9$ hours per Carrington Rotation on $1000+$ cores (Wang et al., 2024).
• Observational Forecasts: Forward-modeled white-light and pB images from synthetic viewpoints reliably match COR2/STEREO and eclipse campaigns, including predictive validation for the April 8, 2024, eclipse (Baratashvili et al., 9 Nov 2025).

6. Model Extensions and Couplings

COCONUT supports multiple extensions for enhanced physics and coupling along the Sun-Earth chain:

• Full-MHD with Realistic Thermodynamics: Thermal conduction (Spitzer/Hollweg), radiative losses (CHIANTI or Rosner et al. 1978), field-weighted or composite heating functions, and realistic $\gamma=5/3$ closure are implemented for accurate wind acceleration and bi-modal structure (Baratashvili et al., 2024).
• Two-fluid Ion-Neutral (COCONUT-MF): Separate evolution of ion and neutral fluids, including collisional, chemical, and charge-exchange source terms, enables analysis of partial ionization effects (typically local 5–10% differences in temperature and flow in the corona) (Brchnelova et al., 2023).
• Energy Decomposition Strategy: To prevent numerical pathologies in low-$\beta$ regions under high magnetic field (e.g., solar maximum), COCONUT advances $E_1 = E - |\mathbf{B}|^2/2$ instead of $E$, avoiding catastrophic cancellation in pressure recovery, and enabling robust, unfiltered modeling up to $|B| \sim 100$ G (Wang et al., 28 Aug 2025).
• Coupling to ICARUS and EUHFORIA: Standardized boundary file output (typically at $21.5\,R_\odot$) for $B_r$ and bulk plasma parameters, facilitates direct interfacing with ICARUS (for MHD heliosphere simulations) and EUHFORIA (for advanced CME and space‑weather forecasting) (Baratashvili et al., 2024, Linan et al., 2024).

7. Limitations, Uncertainties, and Future Development

Physical limitations of COCONUT arise mainly from the employed closure and parametrizations:

• The polytropic model, while efficient, omits explicit coronal heating, radiation, and thermal conduction;
• Empirical or simplistic heating laws do not capture all observed wind-density correlations or fine structure, particularly in the low corona;
• The initial field is based on PFSS, not NLFFF, and thus excludes pre-eruptive non-potentiality;
• MHD is treated as single-fluid unless COCONUT-MF is activated; partial ionization, wave-particle effects, and multi-fluid kinetic terms are, as of 2025, the subject of active research.

Ongoing advances include the implementation of physically motivated heating functions (AWSoM-style), inner-boundary vector-magnetogram assimilation, non-potential initializations, local adaptive mesh refinement for active region resolution, and statistical/AI-based data-assimilation for operational forecasting pipelines.

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The COCONUT coronal model, with its modular, implicit, and unstructured approach, represents a robust, extensible framework for current and next-generation global coronal and heliospheric MHD modeling, advanced particle-transport studies, and operational space weather prediction (Husidic et al., 2024, Brchnelova et al., 2024, Wang et al., 28 Aug 2025, Linan et al., 2023, Baratashvili et al., 2024).