Study of Bipolar Magnetic Regions using AutoTAB: Support of Thin Flux Tube Model?

Anu Sreedevi¹, Bibhuti Kumar Jha^2,3, Bidya Binay Karak¹, & Dipankar Banerjee^3,4

1. Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India
2. Southwest Research Institute, Boulder, CO 80302, USA
3. Aryabhatta Research Institute of Observational Sciences, Nainital 263002, Uttarakhand, India
4. Indian Institute of Astrophysics, Koramangala, Bangalore 560034, India
5. Center of Excellence in Space Sciences India, IISER Kolkata, Mohanpur 741246, West Bengal, India

One intriguing aspect of the Sun’s magnetism is the emergence of sunspots, which are the regions of intense magnetic fields on the Sun’s surface. The number of sunspots rises and falls periodically over an 11-year cycle. They usually appear in pairs with opposite polarity, forming Bipolar Magnetic Regions (BMRs). These BMRs emerge tilted relative to the equator, with the tilt increasing with latitude, known as Joy’s law.

A popular explanation for sunspot emergence is the thin-flux tube model. This model posits that a thin, untwisted flux tube, anchored deep in the convection zone (CZ), rises toward the photosphere due to strong magnetic buoyancy. As the flux tube pierces the solar surface, it forms sunspots observed in white-light images. The Coriolis force acts on the rising flux tube, creating the tilt seen in emerging BMRs^[1].

This theory, being very popular, observational support is limited and does not involve the tracked information of BMRs^[2]. The consequence of such studies is a bias towards long-living BMRs. Hence, our study evaluates the validity of the thin-flux tube model by examining the evolving properties of BMRs^[3]. We automatically detect and track BMRs in line-of-sight (LOS) magnetograms using our in-house developed Automatic Tracking Algorithm for BMRs (AutoTAB) for the period of 1996-2024. Initially, BMRs are detected from magnetograms, and the information is stored as binary files. AutoTAB uses these binary maps and the technique of feature association in successive binary ﬁles to track the BMR in future instances. It has to be noted that the detection and tracking algorithms operate independently. Therefore, AutoTAB allows users to choose different detection methods for BMRs, and AutoTAB can be modified to track other features^[4].

Next, we analyze the tracked BMRs to search for evidence behind the operation of the thin-flux tube model. Specifically, we look for the following signs:

1. Separation of Polarities: As the flux tubes of a BMR rise, the polarities are expected to separate over their lifetime. Our analysis shows that the footpoint separation (D) rapidly increases early in the BMR’s lifetime, then rises slowly and finally saturates. This rapid rise in D is linked to a sudden increase in longitudinal separation (Δφ), suggesting a connection with footpoint tethering from the convection zone.

Figure 1| Joy’s law plot at the first detection of BMR emerging between 45° E-W. (Gaussian) Mean tilt in each 5° latitude bin as a function of the latitude. Blue dashed lines represent Joy’s law fit ( $\gamma = \gamma_{0}\sin\lambda + b$ ) with parameters and $\chi^2$ value for the fit mentioned in the panel. Numbers appearing below the points mark the total number of BMRs in the associated bins.

2. Tilt Consistent with Joy’s Law: If the Coriolis force causes the tilt in BMRs, they should emerge with a tilt that aligns with Joy’s law. Consistent with the thin-flux tube model, we observe that Joy’s law is evident from the first detection of BMRs, which evolve completely on the nearside (Figure 1). This suggests that BMRs statistically emerge with tilt, and some part of the tilt observed in the BMRs is contributed by the Coriolis force. A point to note here is that AutoTAB tracks the BMRs only when they satisfy the flux balance conditions. Hence, we went back in time for selected BMRs to see how their initial signatures emerged. This revealed a nuanced scenario, where the signatures of some BMRs emerge with zero tilt and develop at later phases, while the rest exhibit a signiﬁcant tilt from the beginning. A thorough study is required to verify this further.

Figure 2| (a) Mean $\gamma_0$ is calculated in each flux bin ( $\Phi_m$ ) of length 1 × 10²² Mx and is plotted as a function of $\Phi_m$ . The dashed blue line represents the straight line fit ( $\gamma_0 = a \Phi_m + b$ ), excluding the last two points with parameters mentioned on top of the panel. Here, the error bars represent the standard error in each flux bin, which is bigger in the high flux bins because of less number of BMRs. (b) Mean $\gamma_0$ is calculated in each $B_\mathrm{max}$ bin of length 0.5 kG and is plotted against $B_\mathrm{max}$ . Dashed black vertical line represents $B_\mathrm{max}$ at 1.75 kG.

3. Flux-Tilt Relationship: According to thin-flux tube model simulations, flux tubes with higher flux should rise more slowly, allowing the Coriolis force more time to produce tilt. Consequently, higher flux BMRs should exhibit a greater tilt^[1]. Our results show a linear increase in the averaged Joy’s law slope ( $\overline{\gamma_0}$ ), with latitude dependence removed) with $\Phi_m$ , indicating a clear tilt dependence on magnetic flux (Figure 2). This is further supported by the higher Joy’s law slope observed in higher flux regions. It’s important to note that $\Phi_m$ represents the average flux of the whole region, not the flux inside the flux tube. A similar trend is seen with $B_\mathrm{max}$ : $\overline{\gamma_0}$ initially increases for lower $B_\mathrm{max}$ regions and decreases after 1.75 kG. Here, we suppose a dynamic interplay between drag force and magnetic buoyancy. Higher flux regions, dominated by magnetic buoyancy, emerge with less tilt, while lower flux regions, dominated by the drag force, emerge with higher tilt.

In conclusion, our analysis supports the thin-flux tube model and suggests that the Coriolis force contributes to at least some part of the tilt observed in BMRs. However, more rigorous analysis with richer data from stronger solar cycles is necessary to strengthen these findings and draw definitive conclusions.

References:

[1] Fan, Y., Fisher, G. H., Deluca, E. E., 1993, ApJ, 405, 390
[2] Jha, B. K., Karak, B. B., Mandal, S., Banerjee, D., 2020, ApJL, 889, L19
[3] Sreedevi, A., Jha, B. K., Karak, B. B., Banerjee, D., 2024, ApJ, 966, 112
[4] Sreedevi, A., Jha, B. K., Karak, B. B., Banerjee, D., 2023, ApJS, 268, 58

HMI Science Nuggets

201. Study of Bipolar Magnetic Regions using AutoTAB: Support of Thin Flux Tube Model?

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