Aiying Duan^{1}, Chaowei Jiang^{2}, Shin Toriumi^{3}, and Petros Syntelis^{4}

1. Planetary Environmental and Astrobiological Research Laboratory (PEARL), School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai 519000, People’s Republic of China

2. Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen 518055, People’s Republic of China

3. Institute of Space and Astronautical Science (ISAS)/Japan Aerospace Exploration Agency (JAXA), 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan

4. School of Mathematics and Statistics, St. Andrews University, St. Andrews, KY16 9SS, UK

It is well known that sunspots and active regions (ARs) are formed by toroidal magnetic flux that is generated by the solar dynamo action and emerges into the atmosphere. Theories and simulations of flux emergence suggest that the magnetic flux can rise buoyantly through the convection zone; however, at the photosphere, it is trapped because of the strongly sub-adiabatic stratification (i.e., a much smaller temperature gradient than an adiabatic stratification) there, and its subsequent rising into the atmosphere resorts to the buoyancy instability, which is a kind of magnetic Rayleigh-Taylor instability. To trigger the buoyancy instability, the Lorentz force must be built up as large as the gas pressure gradient to hold up an extra amount of mass against gravity, and further rise of the magnetic flux depends mainly on liberation of the gravity potential energy of this extra amount of mass^{[1]}. Considering that the plasma β is on the order of unity in the photosphere, this naturally results in a strongly non-force-free photosphere, particularly during the formation of ARs.

To quantify the global extent of this force, one can make use of the Gauss’s law by expressing the net volumetric Lorentz force above the photosphere in a surface integral of Maxwell stress tensor of the photospheric magnetic field, which are observable and thus the results can be used to compare with and constrain theories and simulations. We defined normalized metrics for the force (namely f_{x}, f_{y}, and f_{z}) and torque (t_{x}, t_{y}, and t_{z}), which are all much less than unity if the photospheric field is close to a force-free state, while any of them being close to unity indicates a strongly non-force-free field.

Figure 1|Comparison of different parameters from two observed flux-emerging ARs and numerical simulations. From top to bottom are, respectively, evolution of the total unsigned flux, flux emergence rate, normalized Lorentz force and torque. From left to right are results for AR 11158, AR 12673, simulations of Ref. [5] and Ref. [4], respectively. For the simulations we show the results for three different heights, which are approximately (z_{0}, z_{1}, z_{2}) = (0.1, 0.3, 0.4) Mm. The horizontal axes show the time for different events. Note that in the plots of normalized forces for the two ARs, the results from the HMI cgem.Lorentz data set, i.e., ε_{x}, ε_{y}, and ε_{z}, are also shown for a double-checking of our calculations.

With this, we find a noticeable difference between the observed photospheric field and that obtained from typical MHD simulations of flux emergence^{[2]}. For instance, in Figure 1, we compare the results of two well-studied ARs, NOAA 11158 and 12673, with two independently developed flux emergence MHD simulations ^{[5,4]}, which are aimed for reproducing AR formation with significant nonpotentiality from a twisted flux tube that is initially placed in the convection zone and emerges into the corona. As can be seen, for the observed ARs, the normalized forces and torques are mostly less than 0.1 for the full duration of flux injection, suggesting that the field is very close to force-free in the emergence process; in the simulations, the forces and torques in the flux injection phase rise to significantly larger values (~ 0.8 for f_{z}) and torques (~ 0.6 for t_{x} andt_{y}), attaining the order of unity. This means that in the simulated cases, the balance between the tension and magnetic pressure gradient is strongly destroyed, i.e., the field is in an extremely non-force-free state.

Figure 2|Magnetic flux evolution for all the 51 flux emerging ARs in our statistic sample. The horizontal axis shows the longitude of the AR’s center with respect to the solar disk center, thus it also indicates the evolution time. Note that the events are selected with longitude between −45° and 45°. Different events are plotted in different colors, which denote the AR’s number.

We further perform a systematic survey of the global Lorentz force and torque focusing on ARs in their full emergence phase using a large sample of SDO/HMI vector magnetograms^{2}. Figure 2 shows the evolutions of total unsigned magnetic fluxes of all the flux-emerging ARs we have studied, which includes 3536 magnetograms (with time cadence of 1 hour). The normalized forces and torques for all the magnetograms have an averages and standard deviations of, respectively, f_{x} = 0.15±0.10, f_{y} = 0.13±0.08, and f_{z} = 0.13±0.08, and t_{x} = 0.05±0.03, t_{y} = 0.11±0.07, and t_{z} = 0.12 ± 0.09. Thus, all of them are close to 0.1, at which the field can be regarded as force-free^{3}. Furthermore, as shown by Figure 3, the small extents of forces and torques seem not to be influenced by the emerging AR’s size, the emergence rate, or the non-potentiality of the field. Interestingly, the last column of Figure 3 shows that the closer to the disk center, the smaller the force (and torque) is. Thus, the increase of the forces (and torques) is most likely due to the errors in the observations, and with a better quality of observed data, presumably the force and torque will be smaller, meaning that the emerging field is actually even closer to force-free than the results shown here.

Figure 3|Distributions of mean magnetic force (eforce=(f_{x}+f_{y}+f_{z})/3) and magnetic torque (etorque = (t_{x}+t_{y}+t_{z})/3) with different parameters. The top panels are for magnetic force, and the bottom ones are for magnetic torque. The horizontal axes from the left to right columns represent, respectively, the total unsigned flux Φ_{u}, the flux changing rate dΦ_{u}/dt, the mean twist parameter α_{tot}, and distance of the ARs from the solar disk center (in unit of solar radius R_{Sun}). The colors indicate ratios of events in a specific bin. Here the bins for magnetic force and magnetic torque are both 0.02; and for Φ_{u}, dΦ_{u}/dt, α_{tot}and distance, they are 0.5 × 10^{22}Mx, 1 × 10^{22}Mx h^{-1}, 0.01 Mm^{-1}, and 0.02 R_{Sun}, respectively.

In summary, our study shows that the photosphere field is very close to force-free during the emergence process, which is not consistent with theories as well as idealized simulations of flux emergence. This result puts an important constraint on future development of theories and simulations of flux emergence.

### References

[1] Cheung, M. C. M., & Isobe, H. 2014, *LRSP*, **11**, 3

[2] Duan, A. Y., Jiang, C. W., Toriumi, S., & Syntelis, P. 2020, *ApJL*, **896**, L9

[3] Metcalf, T. R., Jiao, L., McClymont, A. N., Canfield, R. C., & Uitenbroek, H. 1995, *ApJ*, **439**, 474

[4] Syntelis, P., Archontis, V., & Tsinganos, K. 2017, *ApJ*, **850**, 95

[5] Toriumi, S., & Takasao, S. 2017, *ApJ*, **850**, 39