Linear Force-Free Spheromak Model

• The linear force-free spheromak model is an analytical solution defining a magnetically dominated flux rope with constant twist, widely used for initializing CME simulations.
• It employs spherical geometry with axisymmetric Bessel function solutions that confine magnetic field lines within a finite domain, ensuring self-contained energy and helicity.
• Numerical implementations in MHD codes demonstrate that this model improves CME arrival time and magnetic field profile predictions compared to traditional hydrodynamic cone models.

The linear force-free spheromak model is a physically-motivated analytical solution representing a compact, self-contained, magnetically dominated flux rope characterized by a constant-α force-free field. This configuration—frequently applied to simulation of Coronal Mass Ejections (CMEs) in heliospheric magnetohydrodynamics (MHD)—captures essential internal magnetic structure, energy content, and helicity, and has enabled significant advances in the predictive modeling of CME propagation and geoeffectiveness, particularly in operational frameworks such as EUHFORIA and Icarus (Scolini et al., 2019, Baratashvili et al., 9 Nov 2025, Singh et al., 2020, Singh et al., 2020, Prasad et al., 2014).

1. Mathematical Foundations and Analytic Structure

The linear force-free spheromak is an exact solution to the MHD equations within a spherical domain of radius $R$, satisfying: $$\nabla \cdot \mathbf{B} = 0, \qquad \nabla \times \mathbf{B} = \alpha \mathbf{B}$$ where $\alpha$ is a spatial constant; it quantifies the twist per unit length along the flux rope axis.

Axisymmetric solutions in spherical coordinates $(r, \theta, \phi)$ make use of separation of variables. For the lowest order (dipolar, $\ell=1$) mode: $$\psi(r, \theta) = A_1\, j_1(\alpha r)\, \frac{\cos \theta}{r}$$ with $j_1(x)$ the spherical Bessel function of order one. The full field is then: $$\begin{aligned} B_r &= 2 B_0 \frac{j_1(\alpha r)}{\alpha r} \cos\theta,\ B_\theta &= -B_0 \frac{1}{\alpha r} \frac{d}{dr}(r j_1(\alpha r)) \sin\theta, \ B_\phi &= H B_0 j_1(\alpha r) \sin\theta, \end{aligned}$$ where $B_0$ is a normalization (central) field, and $H = \pm 1$ sets the chirality.

The boundary condition for a magnetically self-contained spheromak is imposed by requiring the normal field to vanish at $r=R$; that is, $j_1(\alpha R)=0$. The lowest root is $\alpha_1 R \approx 4.4934$ (Baratashvili et al., 9 Nov 2025, Scolini et al., 2019, Prasad et al., 2014). This confines all field lines and currents to the sphere.

2. Parameterization and Initialization from Observations

Practical application of the spheromak model to CME initialization involves determining the critical parameters from remote sensing and/or solar disk magnetic proxies (Scolini et al., 2019, Baratashvili et al., 9 Nov 2025):

Parameter	Description	Physical/Observational Source
$R$	Spheromak radius at inner boundary (e.g., 0.1 AU)	Derived from measured CME half-angular width
$v_{\rm CME}$	Initial speed (preferably radial)	Measured apex speed, separated into radial/exp
$H$	Chirality ($\pm 1$)	EUV sigmoids, filament barbs, magnetic tongues
$\tau$	Tilt angle (orientation in heliocentric coords)	Projected PIL orientation
$\Phi_p$	Poloidal magnetic flux	Inferred from PEA area, reconnected flux
$\Phi_t$	Toroidal (axial) magnetic flux	Dimming regions, analytical integrals
$B_0$	Field normalization	Inverted from $\Phi_p$

The initialization workflow in models like EUHFORIA and Icarus involves:

• Fitting GCS or similar 3D geometric models to coronagraph data to determine launch geometry, apex speed ($v_{\rm 3D}$), and width (yielding $R$).
• Separating the measured apex speed into $v_{\rm rad}$ (radial translational speed) and $v_{\rm exp}$ (expansion speed), and using $v_{\rm rad}$ for spheromak insertion to avoid double-counting Lorentz-driven expansion (Scolini et al., 2019).
• Deriving $\Phi_p$ from post-eruption arcade magnetograms; computing $B_0$ from poloidal flux using the analytic spheromak relation.
• Assigning chirality from solar disk proxies; fixing orientation and tilt from pre-eruptive PIL or tongue analysis.
• Optionally, matching the toroidal flux to observed dimming or loop structure, and adjusting further via synthetic in-situ B profiles at multiple virtual/real spacecraft (Baratashvili et al., 9 Nov 2025, Singh et al., 2020).

3. Model Behavior and Comparison with Hydrodynamic Cone CMEs

The linear force-free spheromak and the non-magnetized cone CME are fundamentally distinct in both field topology and expansion dynamics:

• The spheromak configuration is low-$\beta$, with its expansion dominated by internal magnetic pressure and tension. At the CME-wind interface, an initially force-free spheromak develops a pressure jump driving Lorentz-force-driven expansion normal to its boundary (Scolini et al., 2019, Baratashvili et al., 9 Nov 2025).
• The cone model, being hydrodynamic, lacks internal magnetic structure; its expansion and shock properties rely on thermal overpressure or geometric prescription (Baratashvili et al., 9 Nov 2025).
• Equatorial cross-sections show that the cone tends to retain a roughly constant longitude ("cone angle") and contracts radially due to solar-wind drag, whereas the spheromak expands radially but can contract in longitude due to magnetic tension (Baratashvili et al., 9 Nov 2025).

In practice, launching both models with the same initial apex speed leads to premature arrival of the spheromak at 1 AU by up to $\sim 14$ h. Correcting the initial velocity for the expansion component aligns arrival times to within $1$--$3$ h between the models and observation (Scolini et al., 2019).

4. Analytical Properties: Flux, Helicity, and Energy

The spheromak solution allows closed-form evaluation of the following global quantities (Prasad et al., 2014, Singh et al., 2020, Singh et al., 2020):

• Poloidal Flux: $$\Phi_p = 2\pi\, B_0 R^2 / \alpha \cdot j_1(\alpha R)/R$$ (evaluated at a meridional cross-section).
• Toroidal (Axial) Flux: $$\Phi_t = 2\pi\, \alpha\, \int_{0}^{\pi/2} A(R,\theta)\ d\theta$$ (integral of $B_\phi$ around the core).
• Relative Magnetic Helicity: $H = 2/\alpha \cdot \Phi_p \Phi_t$ (gauge-invariant for axisymmetric fields).
• Magnetic Energy: $E = [\alpha \Phi_t^2 + \Phi_p^2/\alpha]/(2\mu_0)$ for the basic linear force-free spheromak; modified forms retain this structure.

Parametric studies indicate the CME's eruptive speed scales nearly linearly with $\Phi_p$, while $\Phi_t$ sets the axial field amplitude and geo-effectiveness but only weakly impacts propagation (Singh et al., 2020). The sign of helicity (set by $H$ or $\delta$) inverts $B_\phi$ and $B_\theta$ but leaves Lorentz acceleration unaffected.

5. Numerical Implementation in MHD Codes

The insertion and evolution of the spheromak model in global heliospheric MHD codes follows a strict protocol (Baratashvili et al., 9 Nov 2025, Scolini et al., 2019, Singh et al., 2020):

• The analytic solution is imposed within a sphere of radius $R$ at the simulation inner boundary, replacing the pre-existing field, density, and temperature.
• The chosen radius, fluxes, and orientation are tailored to observations.
• Simulation evolves the full set of MHD equations (typically with TVDLF, van Leer, or Rusanov solvers and divergence-cleaning for magnetic field), allowing the spheromak to expand and interact self-consistently with the ambient solar wind (steady or dynamic regime).
• Empirical arrival time and field strength at in-situ and virtual spacecraft are compared to observations for validation and parameter fine-tuning.

A salient modeling limitation is that the fixed constant-$\alpha$, spherical geometry cannot capture dynamic changes in twist or non-sphericity due to CME–solar-wind interactions (e.g., pancaking/flattening, leg extension). Simulation results show that spheromaks fail to generate wide "flanks," underpredicting field signatures in spacecraft located well off the CME nose (Baratashvili et al., 9 Nov 2025, Scolini et al., 2019).

6. Predictive Performance and Empirical Results

Simulation studies consistently show the linear force-free spheromak model provides improved fidelity in predicting CME magnetic clouds compared to cone models:

• For two well-studied events, the spheromak (with radial-only launch speed) reproduced arrival times at Earth to within $1$–$3$ h of observed shocks, matching cone model accuracy (Scolini et al., 2019).
• Key magnetic field metrics for the spheromak were |B| $\sim$ 60% and $B_z$ $\sim$ 40% of observed values (vs. $<$10% for the cone) at Earth; including virtual spacecraft, |B| and $B_z$ recoveries rose as high as 78% (Scolini et al., 2019).
• At four spacecraft (BepiColombo, Solar Orbiter, Parker Solar Probe, STEREO A), spheromak simulations reproduced the sheath + magnetic cloud B double-peak, with magnitude within $10$–$20$% of observed peaks except for cases where $B_0$ was overestimated (Baratashvili et al., 9 Nov 2025).
• Spheromak expansion and shape deformation under dynamic solar wind background yielded more realistic deceleration and spatial profiles than in the steady wind case (Baratashvili et al., 9 Nov 2025).

Empirical methods such as self-similar expansion (SSE) and in-situ SSE underpredicted shock arrival times by $12$–$15$ h, emphasizing the necessity of MHD modeling (Scolini et al., 2019).

7. Limitations and Extensions

The linear force-free spheromak model is limited by several factors:

• Assumption of constant-$\alpha$ (global, spectral approach) excludes radial variation of twist and self-consistent relaxation or untwisting during propagation (Singh et al., 2020, Baratashvili et al., 9 Nov 2025).
• The imposed sharp boundary at $r=R$ leads to numerical artifacts in discretized $\nabla \cdot \mathbf{B}$, requiring explicit divergence-cleaning (Singh et al., 2020, Baratashvili et al., 9 Nov 2025).
• Axisymmetry and fixed orientation limit the ability to model CME deformations, leg structure, and pancaking, especially for off-axis events (Scolini et al., 2019, Baratashvili et al., 9 Nov 2025).
• Modeling the eruption phase, reconnection, or non-force-free boundary layers requires further extensions not present in the analytic spheromak framework (Singh et al., 2020).
• Efforts to allow independent control of $\Phi_p$ and $\Phi_t$, and explicit helicity sign control, increase flexibility at potential cost to force balance and require retuning of pressure balance to match observations (Singh et al., 2020, Singh et al., 2020).

This suggests that while the model's primary value is as an analytically controlled, physically consistent, and easily observationally-constrained initialization within 3D MHD CME simulations, more sophisticated treatments (e.g., time-dependent $\alpha$, ellipsoidal/torus geometries) will be required for events exhibiting significant asphericity or evolutionary change within the heliosphere.