Direct Numerical Comparison of Data Driving Strategies for Solar Flux Emergence in Magnetohydrodynamics: B-driving vs E-driving

The gist: E-driving provides better overall reproduction of the reference evolution than B-driving across various diagnostics in controlled numerical experiments.

Data Driving Context

Data driving is an important technique for modelling realistic magnetic field and plasma evolution in regions of the solar atmosphere where direct observations are either absent or insufficient, using observational data to drive the simulation while the modelled solar atmosphere evolves self-consistently. Data-driven numerical simulations can be thought of as timedependent boundary value problems, where an assumed initial atmospheric state is evolved using dynamic data at the photosphere. This approach allows for realistic physics-grounded coupling between the lower and upper solar atmosphere, which has been successful in reproducing important observed magnetic signatures including the buildup of non-potential magnetic energy and helicity in the corona. The goal of data-driving is the reconstruction and evolution of the magnetic field and plasma in the corona above realistic active regions, which is essential for modelling solar eruptions, coronal heating, and the solar wind.

B-driving vs. E-driving

A common approach for data-driven simulations is B-driving, which uses the observed magnetic field at the surface of the Sun as a boundary condition. However, only prescribing B at the boundary does not generally determine its numerical evolution when the update depends on the reconstructed interface states or fluxes. E-driving provides an alternative approach by instead driving the simulation using the electric field on the boundary, which directly controls the magnetic field update. Across these diagnostics, E-driving gives better overall reproduction of the reference evolution than Bdriving. For configurations considered in an Eulerian CT scheme, E-driving is more accurate than the commonly used ghost-cell B-driving strategies and simpler than constructing an update-consistent B-driving boundary condition.

B-driving Limitations

In the B-driving strategies considered here, magnetic field data are assigned directly to the ghost cells without inverting the numerical boundary-update algorithm, which constrains the driven boundary evolution only approximately. Direct flux prescription can generate spurious MHD states, numerical instability, and violations of the solenoidal constraint ∇ · B = 0. Conventional implementations of B-driving that assign ghost values directly from driving data without accounting for the relevant boundary update algorithm are not correctly controlling the boundary condition numerically and are referred to as “approximate data-driving” strategies.

E-driving Mechanism

E-driving in Athena++ is implemented by directly prescribing the edge-centred EMFs, E, representing the lower z-boundary electric field in the CT update. Specifically, for all i and j across the data-driven boundary (where kl is the first domain cell in z): E vB x,i,j+1/2,kl−1/2 =−(vyBz − vzBy) obs and E vB y,i−1/2,j+1/2,kl−1/2 = (vxBz − vzBx) obs. The E vB values are calculated from the driving-data magnetic field and velocity using the ideal Ohm’s law.

Comparison against Ground Truth

In the comparison of data driving strategies for the ground truth simulation, E1 and E2 perform best in the sense that the quantities in Figures 6 and 7 are closest to the GT time series. The B1 and B2 results deviate significantly from the GT Hr series but all results follow the GT trend. The E-driven strategies E1 and E2 best reproduce the GT structures (see Figures 4 and 5 of S. Toriumi et al. (2020)) and the GT flux-tube height (Figure 7).

Conclusion on Strategy Choice

In general, E-driving is preferable to conventional B-driving in these particular circumstances because the latter applies “approximate” data-driving boundary conditions for the magnetic field evolution, while E-driving specifies these directly. The inclusion of the correction electric field means that the E-driving strategy here is effectively performing data assimilation, which illustrates the practical advantage of using standardized terminology. However, it remains to be seen whether the uncertainty in electric field inversions from observational data compromises this advantage when compared to the B-driving. The treatment of thermodynamic variables in these data-driving strategies may limit their accuracy.

Limiting Factors

The deviation in magnetic energy and magnetic free energy profiles for the GT simulations indicates under-injection of magnetic energy, which occurs despite an otherwise accurate reproduction of the magnetic field. The E-driving cases use identical density and pressure boundary conditions but vary in their velocity boundary condition, and the constant density and pressure inflow boundary conditions used limit accuracy. Future work should investigate more sophisticated treatment of the boundary conditions for density and pressure because they contribute to relevant physical processes like flux-tube buoyancy. The poor reconstruction of the ground-truth (GT) results shows the importance of physically meaningful velocity boundary conditions for B-driving. The E1 result is endorsed as the preferred driving strategy out of those investigated here.

Numerical Consistency Issues

The Athena++ B-driving implementation is a conservative data driving scheme, and if the ghost-zone state used to drive the simulation is not MHD consistent, errors can arise in the physical realisation, but the conservation properties of the scheme are maintained. The LareXd B-driving implemented here is susceptible to similar problems as the Athena++ E-driving implementation and an additional magnetic field inconsistency. Enforcing domain values bypasses the conservative remap and disrupts the coupling between the Lagrangian and remapped states, risking producing an implicit non-conservative source term and an MHD-inconsistent state.

Future Directions

A promising alternative approach to data driving is the characteristic data-driving boundary condition by L. A. Tarr et al. (2026), although this approach evolves a distinct MHD state rather than the target observational/GT state and so differs from our specific objective. A crosscode comparison with this technique and conventional data-driving strategies would be beneficial to the literature. The boundary conditions on the thermodynamic variables at the driven boundary remain a limiting factor, and it is important to improve these because they contribute to relevant physical processes like flux-tube buoyancy.

Acknowledgments

The authors gratefully acknowledge the support for this research through an Australian Research Council Discovery Project DP230101240. We thank Associate Professor Shinsuke Takasao for sharing his dual energy formalism implementation and Boris correction implementation in Athena++ and for providing additional details for comparison to his ground-truth simulations.

Author Contributions

K. Tapinou completed the Athena++ code development and simulations, and was the primary investigator/author of the manuscript. A. Raboonik completed the LareXd data driving implementation and simulation, and contributed to analysis, debugging of the code, and manuscript preparation/editing. D. Pontin advised throughout the work, and contributed to simulation design, analysis, and manuscript preparation/editing. M. Wheatland advised throughout the work, and contributed to simulation design, analysis, and manuscript preparation/editing. M. Cheung advised throughout the work, and contributed to simulation design, analysis, and manuscript preparation/editing. H. Schunker advised throughout the work, and contributed to manuscript preparation/editing.

Appendix A Details

Athena++ is a popular simulation tool in the wider astrophysics community but has not been adopted in solar physics, yet it possesses a constrained-transport (CT) algorithm to ensure a divergence-free magnetic field. The numerical fluxes on the lower z-boundary, which are vital for understanding data driving, are given by the flux vector H in Equation (4).

1. Data-Driven Modeling Capability: The improved system can perform realistic modeling of solar atmosphere evolution by utilizing observational data to drive simulations, moving beyond static techniques such as extrapolation or dataconstrained simulations.

2. Boundary Condition Strategy Selection: The system can quantitatively evaluate B-driving versus E-driving methods in a controlled set of numerical experiments, determining that E-driving gives better overall reproduction of the reference evolution than Bdriving across diagnostics like boundary magnetic flux, relative magnetic helicity, total and free magnetic energy, and magnetic morphology.

3. Numerical Stability Assessment: The system can identify that update-consistent B-driving is not implemented in the validation tests because the ghost-cell values required to impose a prescribed update of the lower z-boundary are non-trivial, indicating that direct flux prescription or overwriting domain values can generate spurious MHD states, numerical instability, and violations of the solenoidal constraint.

4. Data Assimilation Integration: The system can implement a correction electric field calculation, such as "the correction electric field calculated following the method of M. C. Cheung & M. L. DeRosa (2012), which is formally a data assimilation scheme that incorporates the tracked forecast-observation deviation rather than the observation alone."

5. Strategy Preference for Flux Emergence: The system can recommend E-driving as preferable to conventional B-driving in these particular circumstances when reproducing flux emergence, because the update cannot be controlled exactly by specifying ghost-cell values of B and v from a single data slice; in an Eulerian CT algorithm the boundary EMFs govern the update.

Data driving is an important technique for modelling realistic magnetic field and plasma evolution in regions of the solar atmosphere where direct observations are either absent or insufficient. This technique uses observational data to drive the evolution of the simulation, while the modelled solar atmosphere evolves self-consistently. A common approach for data-driven simulations is B-driving, which uses the observed magnetic field at the surface of the Sun as a boundary condition. However, only prescribing B at the boundary does not generally determine its numerical evolution when the update depends on the reconstructed interface states or fluxes. To address this, E-driving provides an alternative approach by instead driving the simulation using the electric field on the boundary, which directly controls the magnetic field update. We systematically quantify the performance of B- and E-driving in a controlled set of numerical experiments. We test both methods in the Athena++ solver using (a) an analytical flux-emergence model (Y. Fan & S. Gibson 2003) and (b) a "ground-truth" reference simulation (S. Toriumi & S. Takasao 2017; S. Toriumi et al. 2020) of active region emergence. For E-driving, the electric field is incorporated into Athena++'s constrained-transport (CT) update, preserving the discrete solenoidal constraint. We assess the reproduction of (i) boundary magnetic flux, (ii) relative magnetic helicity, (iii) total and free magnetic energy, and (iv) magnetic morphology. Across these diagnostics, E-driving gives better overall reproduction of the reference evolution than B-driving. For the configurations considered in an Eulerian CT scheme, E-driving is more accurate than the commonly used ghost-cell B-driving strategies and simpler than constructing an update-consistent B-driving boundary condition.