Coconut Model: MHD Coronal Simulation Framework

Updated 24 January 2026

The Coconut Model is a collection of discipline-specific models, notably a 3D MHD simulation framework that captures solar corona dynamics and supports space weather forecasting.
It employs advanced numerical schemes, including implicit time integration and unstructured geodesic meshes, to ensure high fidelity and stability in simulating coronal physics.
Its integration with observational data and operational forecasting systems highlights its practical impact on bridging simulation outputs with real-world solar wind predictions.

The term "Coconut Model" encompasses several distinct, domain-specific models across astrophysics, planetary science, data management, and computer science. This article presents a comprehensive technical overview and precise delineation of the most prominent academic usages of the Coconut Model, concentrating on the global magnetohydrodynamic (MHD) coronal simulation framework (COCONUT) that underpins modern physics-based space weather forecasting, as established and extended in recent literature (Baratashvili et al., 9 Nov 2025, Wang et al., 28 Aug 2025, Wang et al., 2024, Linan et al., 2024, Guo et al., 2023, Brchnelova et al., 2023, Wang et al., 15 Jan 2026, Perri et al., 2022, Baratashvili et al., 2024, Wang et al., 17 May 2025). As context, variations of the Coconut Model terminology have also been featured for time series data indexing (Kondylakis et al., 2020), agent-based search models (Banisch et al., 2016), deep learning (Daga et al., 15 May 2025), and latent-space reasoning for LLMs (Hao et al., 2024), but the discussion herein focuses on the physical COCONUT framework for solar and heliospheric MHD.

1. Conceptual Basis and Scope of the COCONUT Coronal Model

The COolfluid COroNal UnsTructured (COCONUT) model is a state-of-the-art, fully three-dimensional, time-evolving global MHD simulation code for the solar corona (ranging from 1.01–30 $R_⨀$) designed for high-fidelity, data-driven, and computationally efficient modeling of solar coronal dynamics and solar wind. It solves the conservative or quasi-conservative form of the single-fluid (with optional two-fluid extensions) ideal MHD equations including gravity, heating/cooling terms, anisotropic thermal conduction, optically thin radiative losses, and magnetic divergence cleaning. COCONUT has been validated in both polytropic closure and full thermodynamic MHD, tested against eclipse, EUV, and in situ data, and integrated as the coronal driver for operational Sun-to-Earth forecasting chains (notably with EUHFORIA and ICARUS) (Baratashvili et al., 9 Nov 2025, Baratashvili et al., 2024, Linan et al., 2024).

The code is developed on a cell-centered, unstructured, geodesic polyhedron mesh, supporting both implicit time integration (critical for solar maximum and high-resolution runs) and various boundary condition regimes, from static (quasi-steady) to fully time-varying, magnetogram-driven flows (Wang et al., 28 Aug 2025, Wang et al., 2024).

2. Governing Equations, Physical Processes, and Numerical Schemes

2.1 Fundamental MHD System

The model advances $U = [\rho, \rho \mathbf{v}, \mathbf{B}, E]^\top$ where:

Mass conservation: $\partial_t \rho + \nabla \cdot (\rho \mathbf{v}) = 0$
Momentum: $\partial_t (\rho \mathbf{v}) + \nabla \cdot [\rho \mathbf{v} \mathbf{v} + \mathrm{I}(p + B^2/2) - \mathbf{B B}] = \rho \mathbf{g}$
Induction (GLM cleaning): $\partial_t \mathbf{B} + \nabla \cdot [\mathbf{v} \mathbf{B} - \mathbf{B} \mathbf{v} + \mathrm{I} \phi] = 0$ , $\partial_t \phi + V_{\mathrm{ref}}^2 \nabla \cdot \mathbf{B}=0$
Total energy: $\partial_t E + \nabla \cdot [(E + p + B^2/2) \mathbf{v} - \mathbf{B} (\mathbf{v} \cdot \mathbf{B}) + \mathbf{q}] = \rho \mathbf{g} \cdot \mathbf{v} + S_{\mathrm{heat}}$

with $E = \frac{p}{\gamma - 1} + \frac{1}{2} \rho v^2 + \frac{1}{2} B^2$ ; $\gamma = 5/3$ for full thermodynamic MHD, $\gamma = 1.05$ for polytropic approximation in legacy runs (Baratashvili et al., 9 Nov 2025, Baratashvili et al., 2024, Perri et al., 2022).

2.2 Non-Ideal and Source Terms

Gravity: $g(r) = -GM_⨀/r^2 \hat r$
Thermal conduction: Hybrid Spitzer-Härm (below 10 $R_⨀$) and collisionless (above 10 $R_⨀$); $\mathbf{q}_{\rm Spitzer} = -\xi T^{5/2} (\hat{\mathbf{b}} \cdot \nabla T) \hat{\mathbf{b}}$
Optically thin radiative cooling: $Q_{\rm rad} = -n_e n_p \Lambda(T)$ (CHIANTI tables)
Empirical heating: $Q_H = H_0|B|\exp(-[r-R_⨀]/\lambda)$ or $Q_H \propto |B|(r/R_⨀)\exp(-[r-R_⨀]/\lambda)$, with $H_0$ and $\lambda$ tuned to solar observations.

2.3 Energy Decomposition and Positivity Preservation

To address numerical instability in low- $\beta$ regions (particularly at solar maximum), COCONUT supports an energy decomposition scheme evolving $E' = E - B^2/2\mu_0$ , such that pressure $p$ is recovered without catastrophic subtractions. This eliminates negative-pressure artifacts and allows accurate runs with boundary $|B| > 100$ G (Wang et al., 28 Aug 2025).

Additional limiters (smooth tanh “flatteners”) ensure positivity of $[\rho, p]$ at both boundaries and during iterative solves, capping Alfvén speeds and avoiding unphysical states (Wang et al., 17 May 2025).

2.4 Spatial and Temporal Discretization

Grid: Unstructured, hierarchical geodesic meshes (sixth-level subdivision, $\sim$ 1.5–6M cells), with 73–74 stretched radial layers from $1.01\,R_⨀$ to $25–30\,R_⨀$.
Finite volume: 2nd-order Godunov with HLL or AUSM+up Riemann solvers, Venkatakrishnan limiters on $\rho, p, E'$ , preserving sharp discontinuities at streamers and CME sheaths (Brchnelova et al., 2023).
Time integration: Fully implicit backward Euler or BDF2, “matrix-free” Newton/Gmres solvers, supporting $\Delta t \sim 2–10$ min; up to CFL $\sim10^4$ in steady-state (Wang et al., 2024, Wang et al., 17 May 2025).

3. Boundary Conditions, Input Data, and Coupling

3.1 Coronal Inner Boundary

$B_r$ : Imposed from processed GONG/HMI synoptic magnetograms; low/high- $\ell_{\max}$ spherical harmonic filtering (commonly $\ell_{\max}=10$ or $50$) to balance small-scale resolution and noise.
Velocity: Enforced parallel to $\mathbf{B}$ ; solar rotation imposed.
Density/Temperature: Characteristic base values (e.g., $n_0 = 2\times10^8$ cm $^{-3}$ , $T_0 = 1.8$ MK).
Temporal driving: Quasi-steady (daily snapshots, rotated) or fully dynamic (hourly, interpolated via cubic Hermite), supporting accurate real-time reconstructions of emerging/cancelling flux (Baratashvili et al., 9 Nov 2025).

3.2 Outer Boundary and Heliophysical Coupling

Coronal/Heliospheric Hand-off: At $r=21.5\,R_⨀$ (or $0.1$ AU), COCONUT outputs all primary variables to an interface grid; files are temporally interpolated for seamless handover to EUHFORIA/Icarus. This enables continuous propagation of eruptive disturbances (e.g., CMEs) with shape and amplitude preservation (Linan et al., 2024, Baratashvili et al., 2024).

3.3 CME Initiation (Flux Rope Models)

Titov-Démoulin (TDm): Toroidal current ring, net current parameter $\zeta>1$ drives instability/eruption (Linan et al., 2024).
Regularized Biot-Savart Law (RBSL): Arbitrary axis, complex twist profiles, support for S-shaped (“sigmoid”) rope topologies. Key: forces only $\mathbf{B}$ at insertion, then self-consistent eruption tracks thermodynamic evolution (Guo et al., 2023).

4. Validation, Benchmarking, and Predictive Success

Quantitative validation against ground-based and space-borne observations is a central focus:

Validation Metric	Method/Instrument	Summary Result
White-light streamers	Eclipse/COR2, STEREO-A	COCONUT simulates streamer positions, angular misalignments in $\Delta\theta$ below $0.2\,R_⨀$ (Baratashvili et al., 9 Nov 2025, Perri et al., 2022)
pB curves	COR2 (synthetic vs observed)	Normalized $pB_{\rm norm}$ amplitude/shape matches; best performance in dynamic runs (Baratashvili et al., 9 Nov 2025)
Speed, density, arrival	WIND/OMNI @ 1 AU	Major wind interval arrival-time error $<6$ h, speed errors $<30$ km/s (Wang et al., 17 May 2025, Baratashvili et al., 2024)
CME magnetic/thermo profile	Virtual probes (L1, etc.)	B amplitude, expansion and topology preserved; R $^{-1.5}$ scaling observed (Linan et al., 2024)
Performance	VSC Tier-2 cluster	Full CR in $\leq$ 9 h (1.5M grid cells, 1080 CPU cores); $>48\times$ real-time (Wang et al., 2024, Wang et al., 17 May 2025)

5. Scientific Applications and Operational Impact

COCONUT is used for:

Space Weather Forecasting: End-to-end operational chains (COCONUT+EUHFORIA/Icarus) now replace empirical (e.g., WSA) models, providing data-driven, fully physical Sun-to-Earth wind and CME forecasts (Linan et al., 2024, Baratashvili et al., 2024).
Solar Eclipse/White-Light Reconstruction: Prospective eclipse coronal structure can be predicted $>$ 2 weeks in advance, validated by post-event observations (Baratashvili et al., 9 Nov 2025).
Solar Maximum and Open Flux: Explicit treatment of open field line emergence/cancellation and high-frequency boundary driving, crucial for resolving the “open flux problem”—i.e., reconciling near-surface and 1 AU unsigned flux (Wang et al., 15 Jan 2026).
CME Modelling: Implementation of modern flux rope CME models allows analysis of eruption onset, coronal evolution, and heliospheric propagation with faithful preservation of magnetic and thermodynamic signatures (Linan et al., 2024, Guo et al., 2023).

6. Limitations, Model Developments, and Frontiers

Key documented challenges and ongoing improvements include:

Grid Resolution: Current meshes (1.8° angular, static) insufficient for small-scale features near streamer cusps, current sheets, or polarity inversion boundaries; adaptive mesh refinement (AMR) is under active development (Baratashvili et al., 9 Nov 2025, Wang et al., 15 Jan 2026).
Coronal Heating Physics: Empirical $Q_H$ remains an approximation; ongoing work aims to replace with physics-based turbulence or Alfvén-wave heating (AWSoM, etc.) (Wang et al., 28 Aug 2025, Baratashvili et al., 2024).
Boundary Data Assimilation: Incorporation of far-side flux (helioseismic, Solar Orbiter PHI) and higher- $\ell$ magnetograms to capture emerging AR, especially for highly dynamic configurations (Baratashvili et al., 9 Nov 2025).
Two-Fluid/Chromosphere Coupling: COCONUT-MF demonstrates that inclusion of ion-neutral coupling affects flow properties at the 5–10% level in localized regions; full lower-atmosphere extension is in progress (Brchnelova et al., 2023).
Energy Conservation and Numerical Stability: The energy decomposition and positivity-preserving algorithms have essentially eliminated low- $\beta$ pathologies, but require further robustness at solar maximum (Wang et al., 28 Aug 2025, Wang et al., 17 May 2025).
Real-Time Operations and Scalability: Implicit time integration enables $\gg$ CFL time steps, but further efficiency is sought for “operational” ( $<1$ h/full Sun) runs on hybrid HPC systems (Wang et al., 2024).
Comparative and Synthetic Validation: Systematic forward generation of synthetic WL, pB, and EUV images, direct observational cross-validation, and machine learning–based inversion for density/field reconstruction are under consideration (Baratashvili et al., 9 Nov 2025).

The term "Coconut Model" also denotes structures and algorithms in computational and AI applications:

Data Series Indexing: Coconut is a bottom-up, z-order–based index for bulk-loaded, disk-based timeseries, supporting scalable $k$ -NN and range queries (Kondylakis et al., 2020).
Agent-Based Search Model: Coconut (Diamond's search equilibrium) in economics models production/trading subject to expectations and learning (Banisch et al., 2016).
Deep Learning and Computer Vision: DeepSeqCoco is a convolutional classifier for disease detection in Cocos nucifera images (Daga et al., 15 May 2025).
Latent Reasoning in LLMs: Coconut (Chain of Continuous Thought) modifies transformer inference by propagating hidden states as latent reasoning steps (Hao et al., 2024).

These models are entirely distinct from the coronal/space weather COCONUT—care should be taken with domain context.

In summary, the COCONUT model framework, as implemented in recent MHD coronal simulation studies, defines a high-fidelity, fully implicit, unstructured-grid, thermodynamic MHD code that accommodates observational driving, advanced CME initiation, and real-time forecasting. Its extensions to two-fluid physics, adaptive boundary treatment, and coupled heliospheric flows mark a comprehensive advance in predictive heliophysics and quantitative solar wind modeling at solar minimum and maximum. Remaining challenges are primarily in adaptive mesh refinement, empirical term replacement by physical models, and robust assimilation of real-time magnetogram data for operational reliability (Baratashvili et al., 9 Nov 2025, Wang et al., 28 Aug 2025, Wang et al., 2024, Wang et al., 15 Jan 2026, Baratashvili et al., 2024, Linan et al., 2024, Guo et al., 2023, Brchnelova et al., 2023, Wang et al., 17 May 2025, Perri et al., 2022).