David Korda1,2, Michal Švanda2,3 and Thierry Roudier4
1 Department of Geosciences and Geography, Faculty of Science, University of Helsinki, Gustaf Hällströmin katu 2, 00014 Helsinki, Finland
2 Astronomical Institute of Charles University, Faculty of Mathematics and Physics, V Holešovičkách 2, 18000 Prague 8, Czech Republic
3 Astronomical Institute of Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic
4 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, UPS, CNES 14 avenue Edouard Belin, 31400 Toulouse, France
In the last decades, helioseismology provided us valuable information about the overall structure of the Sun and plasma properties in subsurface layers of the quiet Sun. Helioseismology is used to study the Sun via acoustic waves. The dispersion relation of these waves is affected by plasma properties along the wave trajectories. It is believed that the waves are generated randomly via convection in the upper convection zone. In regions of strong magnetic fields, acoustic waves are converted to magneto-acoustic waves and helioseismology loses part of its predictive power. This contrasts with the fact that knowledge of plasma properties in the vicinity of active regions gives us important constraints on the theories of the solar dynamo, and the formation, evolution, and stability of active regions, which are of high importance.
In the time-distance method, authors determine inhomogeneities of plasma properties via studying perturbations of wave travel times. The time-distance method is mostly used to determine plasma flows and sound-speed perturbations. In our recent work, we aimed at horizontal flows. Travel times measured between two points are sensitive to flows in the direction defined by the two points. In order to decrease noise level, acoustic travel times are averaged over an annulus, and the travel time is measured as a mean travel time between the point in the center of the annulus and all points on the annulus. Such a travel-time measurement is very useful in regions of the quiet Sun, where the gradient of plasma properties is small. Close to an active region, the same annulus averaging mixes quiet and active regions together and is unable to determine plasma properties properly close to the active region.
Figure 1| Travel-time measurements at the position of the black dot in the vicinity of an active region delimited by the black circle. The annulus averaging scheme averages quiet and active regions in a single travel time. Arc averaging scheme can correctly denoise travel time up to the edge of the penumbra.
This property of the annulus averaging is due to its omnidirectionality. In this work, we introduced a new averaging scheme. This scheme uses one-direction arc averaging and can correctly average travel times up to the edge of the penumbra (Figure 1). Using a set of arc averaging geometries, we successfully reconstructed standard annulus travel times and showed that both averaging geometries return comparable results in quiet-Sun regions.
Figure 2| Horizontal components of plasma flows around the active region NOAA 11084 on 1 July 2010 (intensitygram as a background image). Left: annulus-geometry model. Middle: arc-geometry model. Right: CST model. The reference arrows in the left bottom corners correspond to 250 m/s, and are all of the same length. Green contours roughly delimit penumbra and umbra. In quiet-Sun regions, all models return comparable flows. At distances about 45 Mm and closer to the sunspot, the annulus model deviates from the other two.
We demonstrated the advantages of the new arc geometries in combination with non-linear travel-time approximation (GB02) on the active region NOAA 11084 observed on 1 July 2010 at Carrington coordinates of 144.5° longitude and -19.1° latitude. As a ground-truth model, we chose Coherent Structure Tracking (CST)[3, 4]. CST model is based on tracking granules, therefore does not suffer from magnetic fields if the granules are still observable. Unlike helioseismology models, CST is sensitive strictly to surface flows.
Figure 3| Azimuthally averaged horizontal flows around the sunspot. The model based on arc travel times predicts the correct extension of the moat flow and has similar properties as the CST model up to the edge of the penumbra (green dashed line).
The helioseismic models were based on annulus (PtA) and arc (PtArc) travel times, and the CST model agree well in regions of quiet Sun (Figure 2). At the distance of about 45 Mm from the sunspot, the annulus model started to deviate from the others, while the arc model had similar properties as the CST model up to the edge of penumbra (Figure 2). Slightly different amplitudes between the arc and CST models were probably caused by different vertical sensitivity and an increase in the magnitude of flows around a sunspot. Because the active region was almost circular, we could compare azimuthally averaged flows (Figure 3). In this plot, the difference between the annulus and other models in quiet-Sun regions not distant from an active region is clearly seen. The annulus model predicted moat flow up to about 37 Mm from the sunspot center. The arc and CST models predicted the extension of the moat flow of only to 28 Mm.
The arc and CST models had similar properties up to the edge of the penumbra. Therefore, we suggest applying the arc averaging scheme with a combination of GB02 travel-time approximation in order to learn about the depth structure of plasma flows around active regions.
 Duvall, Jr., T. L., Jefferies, S. M., Harvey, J. W., & Pomerantz, M. A. 1993, Nature, 362, 430
 Korda, D., Švanda, M., & Roudier, T. 2021, A&A, 654, A84
 Gizon, L., & Birch, A. C. 2002, ApJ, 571, 966
 Roudier, T., Rieutord, M., Malherbe, J. M., & Vigneau, J. 1999, A&A, 349, 301
 Rempel, M. 2011, ApJ, 740, 15