Two toroidal flux-rope coronal mass ejection (CME) models — one derived from the modified Miller-Turner (mMT) approximately force-free torus and one from the Soloviev Grad-Shafranov equilibrium — were implemented in the EUHFORIA space-weather forecasting tool, with the declared next step of validating both against real observed events. The mMT half of that objective has been completed: the Horseshoe (three-quarters-torus) mMT variant was validated on the 12 July 2012 and 8-10 September 2014 events, matching FRi3D-level magnetic-field prediction accuracy at near-spheromak runtime. The Soloviev model, which uniquely offers an adjustable cross-section (elongation and triangularity) and controllable magnetic helicity but is injected with a uniform pressure that deliberately violates the linear pressure profile of its equilibrium, has never been validated against observations, and the impact of that pressure inconsistency on the simulated evolution remains unquantified. The open problem is the surviving half of the source question 'which model is most suitable for modeling real events': carry out the observation-constrained, event-scale validation of the Soloviev model in EUHFORIA — inferring its torus parameters from remote observations within the closed-contour bounds, simulating real CME events, and benchmarking arrival time, sheath and magnetic-ejecta structure, and the geoeffective BzB_z profile at L1 against measurements and the mMT/Horseshoe, spheromak, and FRi3D baselines.

Background

Coronal mass ejections (CMEs) are large eruptions of magnetized plasma from the solar corona; when their magnetic structures impinge on Earth's magnetosphere they drive geomagnetic storms. The dominant predictor of storm intensity ('geoeffectiveness') is the southward component of the interplanetary magnetic field, conventionally the BzB_z component measured in geocentric solar-ecliptic coordinates, which enables reconnection with Earth's field. Reliable forecasting therefore requires predicting not just the CME shock arrival time but the full in-situ plasma and magnetic-field time series at Earth, measured by monitors near L1 (the first Sun-Earth Lagrange point, about 1% of the Sun-Earth distance upstream of Earth).

EUHFORIA (EUropean Heliospheric FORecasting Information Asset; Pomoell and Poedts 2018) is a 3D magnetohydrodynamic (MHD) simulation tool that models the solar wind from an inner boundary sphere at 0.1 AU (about 21.5 solar radii) out past Earth. At that inner boundary, CMEs are inserted as analytic structures superposed on a data-driven background solar wind, and their subsequent self-consistent MHD evolution is computed. The choice of CME submodel strongly affects both forecast quality and runtime: the cone model inserts plasma without magnetic field; the spheromak model inserts a globally force-free (current everywhere parallel to magnetic field, so the Lorentz force vanishes) spherical magnetic configuration; FRi3D ('Flux Rope in 3D') inserts a flexible, more realistic flux rope — a twisted bundle of magnetic field lines — but at large computational cost (hours per run versus tens of minutes for the spheromak).

Linan et al. (2024) implemented two additional CME submodels in EUHFORIA, both with toroidal (donut-shaped) geometry, motivated by the fact that CME flux ropes are anchored by two legs at the Sun and are locally torus-like rather than spherical. The first is the modified Miller-Turner (mMT) model, based on the approximately force-free toroidal equilibrium of Miller and Turner (1981); it has a circular cross-section and a fixed field-line twist. The second is derived from the Soloviev solution, an exact axisymmetric equilibrium of the Grad-Shafranov equation — the standard nonlinear partial differential equation governing axisymmetric MHD equilibria, relating the poloidal magnetic flux function to the plasma pressure and current. The Soloviev construction assumes a spatially linear pressure profile and yields a torus whose poloidal cross-section (the cross-section cutting through the doughnut ring, as opposed to the toroidal direction around it) is shaped by free parameters: the inverse aspect ratio ϵ\epsilon (minor radius over major radius), the elongation σ\sigma (how stretched the cross-section is vertically), and the triangularity τ\tau (how much its outline departs from an ellipse toward a D-shape); the sign of its magnetic helicity — a scalar measuring the overall twist and linkage of the magnetic field lines — is also controllable. These extra degrees of freedom are absent from the mMT model and are potentially valuable for matching observed CME geometries, but they come with a consistency cost and a bookkeeping requirement made explicit by the implementing paper: (i) for numerical convenience the implementation injects uniform density and temperature — hence uniform pressure — which deliberately contradicts the linear pressure profile required by the Soloviev equilibrium, so only the equilibrium's magnetic field is actually used, and the dynamical consequence of this inconsistency is unknown; and (ii) the torus parameters must stay inside the region of (ϵ,τ,σ)(\epsilon, \tau, \sigma) space where the outermost magnetic flux surface (ψ=1\psi=1, with ψ\psi the normalized flux function) is a closed contour, since open, non-physical contours would be injected otherwise.

The implementing paper demonstrated both toroidal models in synthetic runs against a realistic background wind and declared, as its single stated next step, validation against real observed CME events 'to conclude which model is most suitable for modeling real events.' That validation has so far been carried out only for the mMT model: Maharana et al. (2024) introduced a three-quarters-torus 'Horseshoe' variant of the mMT model (the missing quarter mimics the flux-rope legs), built explicit procedures to constrain the torus geometry and magnetic field from observed CME signatures (size and speed from coronagraph images, which image the faint solar corona by occulting the bright solar disk; magnetic flux from magnetograms, i.e. maps of the solar surface field, and from flare ribbons, the bright emission arcs marking the magnetic-reconnection footpoints of the eruption; plus in-situ context), and validated the model on the isolated 12 July 2012 event and the 8-10 September 2014 CME-CME interaction event, scoring predicted BzB_z time series against observations with dynamic time warping (DTW) — a metric that compares two time series while allowing non-linear warping of the time axis, thus rewarding correct profile shape even with arrival-time offsets — and running ensembles to propagate observational uncertainty in the axial field strength. The Horseshoe model achieved accuracy comparable to FRi3D at runtime close to the spheromak model. The Soloviev model was not part of that validation, and no subsequent study has validated it: as of 2026 the Soloviev model exists in EUHFORIA as an implemented but observationally unvalidated capability, with the effect of its uniform-pressure injection still unquantified.

Problem Statement

Carry out the observation-constrained, event-scale validation of the Soloviev toroidal CME model in EUHFORIA — the still-uncompleted half of the objective declared by Linan et al. (2024) ('to conclude which model is most suitable for modeling real events'), whose mMT half was completed by Maharana et al. (2024). Specifically, determine whether the Soloviev model, injected at the 0.1 AU inner boundary with its analytic equilibrium magnetic field but a uniform (rather than the equilibrium's linear) pressure profile, can reproduce measured in-situ signatures of real CME events at L1 as well as or better than the validated mMT/Horseshoe, spheromak, and FRi3D models. A complete answer must: (1) provide documented procedures to infer the Soloviev torus parameters — inverse aspect ratio ϵ\epsilon, elongation σ\sigma, triangularity τ\tau (within the region of parameter space where the outermost flux surface ψ=1\psi=1 is a closed contour), magnetic flux or axial field strength, helicity sign, initial speed, and density — from remote-sensing and contextual observations of real eruptions; and (2) simulate real CME events in EUHFORIA with these observation-constrained parameters and compare the predicted L1 time series — shock/arrival time, sheath and magnetic-ejecta structure, and in particular the geoeffective BzB_z profile — against the in-situ measurements and against the mMT/Horseshoe, spheromak, and FRi3D models run under matched background winds, using established profile metrics (for example dynamic time warping on BzB_z) and ensemble modelling to propagate observational uncertainty. The deliverable is a defensible, evidence-based assessment — positive or negative — of whether the Soloviev model is suitable for operational real-event geoeffectiveness forecasting relative to the alternative models.

References

  1. L. Linan, A. Maharana, S. Poedts, B. Schmieder, R. Keppens (2024), 'Toroidal Miller-Turner and Soloviev coronal mass ejection models in EUHFORIA - I. Implementation', Astronomy & Astrophysics 681, A103. DOI: 10.1051/0004-6361/202347794; preprint arXiv:2310.17239, https://doi.org/10.1051/0004-6361/202347794
  2. A. Maharana, L. Linan, S. Poedts, J. Magdalenić (2024), 'Toroidal modified Miller-Turner CME model in EUHFORIA: Validation and comparison with flux rope and spheromak', Astronomy & Astrophysics 691, A146. DOI: 10.1051/0004-6361/202450459, https://doi.org/10.1051/0004-6361/202450459
  3. J. Pomoell, S. Poedts (2018), 'EUHFORIA: European heliospheric forecasting information asset', Journal of Space Weather and Space Climate 8, A35. DOI: 10.1051/swsc/2018020, https://doi.org/10.1051/swsc/2018020
  4. G. Miller, L. Turner (1981), 'Force free equilibria in toroidal geometry', The Physics of Fluids 24, 363. DOI: 10.1063/1.863351, https://doi.org/10.1063/1.863351
  5. M. Flossie, A. Maharana, L. Linan, K. Plets, A. Isavnin, S. Poedts (2025), 'Optimization of the FRi3D CME Model in EUHFORIA', Space Weather 23, e2025SW004401. DOI: 10.1029/2025SW004401, https://doi.org/10.1029/2025SW004401
  6. L. Linan, T. Baratashvili, A. Lani, B. Schmieder, M. Brchnelova, J. H. Guo, S. Poedts (2025), 'Coronal mass ejection propagation in the dynamically coupled space weather tool: COCONUT + EUHFORIA', Astronomy & Astrophysics 693, A229. DOI: 10.1051/0004-6361/202451854, https://doi.org/10.1051/0004-6361/202451854
  7. L. Linan, T. Baratashvili, A. Maharana, J. Guo, A. Lani, B. Schmieder, S. Poedts (2025), 'Advanced flux-rope CME models in EUHFORIA and coupling with COCONUT', EGU General Assembly 2025, abstract EGU25-17746. DOI: 10.5194/egusphere-egu25-17746, https://doi.org/10.5194/egusphere-egu25-17746