Keiji Hayashi

NorthWest Research Associates, 3380 Mitchell Lane, Boulder, CO 80301, USA

In our recent paper (Ref[1]; hereafter referred as Paper I), an algorithm was presented to determine the electric field vectors that exactly match the temporal variation of observed solar-surface magnetic field vectors. This method allows three-dimensional time-dependent MHD simulations to introduce the arbitrarily varying three-component solar-surface magnetic field vector data as the time-varying boundary values, maintaining the divergence-free condition () in the simulated volume. In the era of advanced observations of the Sun with high spatial and temporal resolution (such as SDO/HMI, Hinode, and DKIST), this method offers a new capability of realistically modeling the nonlinear and highly non-potential features of the solar active region to help enhance our understanding on the dynamics in the solar corona.

Figure 1|Cartoon plot of the electric field to be calculated (white arrows), the given observed temporal variation of magnetic field (dark gray arrows) in the simulation grid system: The three electric field vectors are calculated for three heights, z=-1/2, 0 and +1/2 (the grid size Δz is set equal to 1 for simplicity, and the solar surface is expressed as z=0).

In this method, the observed temporal variation of magnetic field vector is split into three parts, and each of the three parts is solved separately with the two-dimensional Poisson solver (see Figure 1). Here we tentatively call it as SEE3Po (Solar-surface Electric field Estimation with 3 Poisson solvers).

Recently, we conducted an MHD simulation using this model, with higher spatial resolution than in Paper I. Same as in Paper I, we use the SHARP HMI vector magnetic field data^{[2,3]} for active region NOAA 11158. In the simulation, the three electric field vectors are calculated for each of the 12-minute HMI data cadence, and the magnetic field vectors on the bottom boundary z=0 are updated as . Figure 2 compares the three components of the simulated magnetic field (B_{x}, B_{y} and B_{z}) on the bottom boundary surface with the HMI observations at an instant 14 hours after a reference time. The MHD simulation using the SEE3Po traces the temporal evolution of the observed very well, over the 70 intervals of the 12-minute HMI vector magnetic field data.

Figure 2|Scatter plot of the reproduced three components of the bottom-boundary magnetic field and the observed ones, at +14 hours (or 70 intervals of the 12-min cadence HMI vector data). Brown, green, cyan and blue colors are for the number density of computation grid, 10^{3+}, 10^{2}, 10^{1}and 10^{0}, respectively. The divergence from the diagonal is overall smaller than those in Figure 4 of Paper I, because the grid size is smaller (~720km) than in Paper I (~1440km) and hence the discretization error is.

Notice that the electric field vectors appearing at the cell interface ( and at in Figure 1) must be used to calculate the temporal variation of the magnetic field in the computation grids on the second layer next to the bottom boundary surface (at ). Indeed, this horizontal electric field vectors at are the variables that actually drive the simulated magnetic field and hence other MHD variables, maintaining .

Figure 3|A cut-away view of the simulated magnetic field lines, in a simulated active region (NOAA AR 11158), viewed from the north east direction. The blue (red) color on the field line represents the positive (negative) value of the scalar or the degree of magnetic field twist. The colors on the bottom surface show the solar-surface $B_z$ (blue for positive and red for negative polarity).

Figure 3 demonstrates the simulated magnetic field in the selected NOAA AR 11158. The simulation numerically produces a variety of complex, non-potential magnetic features, such as twists of field lines over the polarity inversion lines and around the strong-field sunspot regions. The currents of opposite signs often come from the same sunspot region (at a right negative (red) spot region). The sign of (relative) current direction is rather constant along the field line, which shows that the MHD-simulated magnetic field is near force-free situations as the Alfven time of the simulated system is well smaller than the HMI data cadence and/or typical time scale of the actual solar-surface magnetic field evolutions.

The SEE3Po method determines a unique set of the three electric field vectors from the temporal difference of the two consecutive vector observation data maps. This does not mean that this method is free from the gauge uncertainty: Although the gauge uncertainty (in the total of and ) does not alter at , the other two components B_{x} and B_{y} suffer from it. This is one of many points to solve to improve the algorithm of this kind.

### References

[1] Hayashi, K., Feng, X. S., Xiong, M., & Jiang, C. 2018, *ApJ*, **855**, 11

[2] Hoeksema, J. T., Liu, y., Hayashi, K., et al. 2014, *Solar Phys.*, **289**, 3483

[3] Bobra, M. G., Sun, X., Hoeksema, J. T., et al. 2014, *Solar Phys.*, **289**, 3549