Bibhuti Kumar Jha1,2, Bidya Binay Karak3, Sudip Mandal4 & Dipnakar Banerjee1,2
1. Indian Institute of Astrophysics, Bangalore, India
2. Aryabhatta Research Institute of Observational Sciences, Nainital, India
3. Indian Institute of Technology (BHU), Varanasi, India
4. Max Planck Institute for Solar System Research, Göttingen, Germany
Sunspots are essentially members of a more general class, known as bipolar magnetic regions (BMRs). These BMRs, as identified through magnetogram are found to exhibit a tilt (with respect to solar equator) that increases with latitude. This tilt versus latitude relation is often referred as the Joy’s law. It is believed that flux tubes of coherent magnetic fluxes rise from the base of the convection zone due to magnetic buoyancy and form BMRs at the solar surface. During their rise, they experience Coriolis force, which possibly leads to the systematic tilts in BMRs. The tilt is crucial for the generation of poloidal field through the decay and dispersal of BMRs on the surface, a mechanism popularly known as Babcock–Leighton process. Dynamo models based on this process have been reasonably successful in reproducing various features of solar cycles. However, most of these models are kinematic in nature and the saturation of magnetic field growth in these models has been a major concern. To overcome this issue, these models use a simple nonlinear quenching factor 1/(1+(B/B0)2) in the Babcock–Leighton source term[2,3]. Therefore, the fundamental question we ask here is, ‘what is the process that could lead to this type of nonlinear quenching in the Sun?’ One such possible candidate is the quenching in BMR tilt. In the above theory of tilt formation, the tilt depends on the rise time of the flux tube, which in turn strongly depends on the strength of the magnetic field in the tube. BMRs having strong field strengths are expected to show less tilts and vice versa. In our recent work. we have attempted to check this theoretical idea.
Figure 1| Representative magnetograms of (c) MDI and (d) HMI (saturated to ±1.5 kG) with BMRWS in red box and BMRNS in blue box (a, b, e, and f) showing IC counterparts.
In our work, we have identified the BMRs (Figure 1) from LOS magnetogram observed by MDI during 1996 – 2011 and by HMI during 2010 – 2018.
Figure 2| (a-b) Distributions of Bmax in the BMRs from MDI (left panel) and HMI (right). Red and blue respectively show Bmax distributions of BMRWS and BMRNS. (c) Time-latitude distribution of BMRWS (red) and BMRNS (blue).
Interestingly, we find that the distribution of maximum magnetic field Bmax in detected BMRs are not monotonous. Instead, it is bimodal with peaks near 600 G and 2100 G (Figure 2). This is true for both data sets, which cover two different solar cycles. To understand this bimodality, we looked for near-simultaneous white light images (intensity continuum; IC). After careful examination, we find that not all BMRs have associated sunspots in IC (Figure 1a, b, e, f). Hence, we segregated the BMRs based on these criteria and overplotted the distribution of Bmax in these two classes (BMRWS and BMRNS), and surprisingly these two peaks get well resolved. The time-latitude distributions of these two BMRs classes do not show any differences (Figure 2c). This suggests that these two classes are possibly produced from the same large-scale dynamo.
Figure 3| Magnetic field (Bmax) dependence of: (a) Joy’s law slope γ0 and (c) the tilt scatter σ. In (a) dotted line shows 1/(1+(B/B0)n) with B0 = 2.9 ± 0.1 kG and n = 5.8 ± 0.8 dependence, and the dashed-dotted line shows Joy’s law slope as predicted by the thin flux tube model. (b) and (d) are the same as left panels but as functions of flux.
Finally, to understand the behavior of tilt with the strength of magnetic field and the flux in BMR, we plot Joy’s law slope γ0 (Joy’s law: γ = γ0 sin (latitude)) with magnetic field in BMR (Bmax) and flux in Figure 3a,b. We clearly see that γ0 indeed depends on Bmax. Above 2 kG of magnetic field, the tilt has a decreasing trend in both data sets. This decrease of tilt with the increase of magnetic field is in agreement with the thin flux tube model of BMR formation. Hence, it gives us a hint about the tilt quenching used in dynamo models.
Joy’s law is actually a statistical law; hence it is also important to look at the variation of scatter σ with Bmax. We see a monotonic decrease in the σ with Bmax and flux (Figure 3c, d). This behavior is pretty easy to understand based on the fact that flux tube with stronger field or flux will be less buffeted by convection and also get less time to rise.
In summary, our study reveals that there is some nonlinear quenching in the BMR tilt for Bmax >2 kG. However, evidence of tilt quenching in our study exists only in a narrow range of magnetic fields. Therefore, we believe that our result needs to be investigated with a more extensive data set, especially with stronger cycles.
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