Krishnendu Mandal, Alexander G. Kosovichev
New Jersey Institute of Technology, Newark, New Jersey, USA

Solar torsional oscillations carry important information about how the solar dynamo operates inside the Sun. At the surface, there is clear evidence that the torsional oscillation pattern closely follows the magnetic butterfly diagram, indicating a strong connection between zonal flow variations and the solar magnetic cycle. This correspondence suggests that by studying torsional oscillations, we can gain in-sight into dynamical processes occurring in the solar interior. Although the magnetic field in the deep interior cannot yet be measured reliably, torsional oscillations can be probed throughout much of the convection zone using helioseismology.

Figure 1 | Left panel: Torsional oscillations at several depths (as indicated in each panel title), overlaid with the magnetic butterfly diagram (black markers), highlighting the phase lag between flows at different depths and the surface magnetic pattern. The results are based on the combined MDI and HMI da-tasets. More about the data products can be found in this work [2]. Right panel: Depth-dependent variations of torsional oscillations at selected latitudes (indicated in each panel title). At low latitudes, the signal originates near the tachocline and propagates upward with a time delay before reaching the surface. In contrast, at high latitudes (around 50°), the signal appears first at the surface. This con-trasting behavior is characteristic of a dynamo wave–like signature in solar torsional oscillations.

Our analysis reveals a dynamo-wave–like signature in the evolution of solar torsional oscillations. The signal appears to originate at intermediate latitudes (approximately 45°–60°), after which one branch migrates equatorward. This equatorward-propagating branch takes time to reach the surface, producing a characteristic slanted structure in radius–time diagrams at low latitudes (see right panel off Figure 1). In contrast, the high-latitude branch propagates more rapidly. A movie illustrating the propagation of the dynamo-wave–like signature in solar torsional oscillations is available in the online version of our recent publication ^[1] (click here to access the movie link). This behavior is qualitatively consistent with the dynamo wave propagation proposed by Parker, whose theoretical work describes how magnetic fields may propagate within the solar interior. These results are based on frequency splitting measurements obtained from data provided by Global Oscillation Network Group (GONG), the Solar Dynamics Observatory/HMI, and the Solar and Heliospheric Observato-ry/MDI instruments.

Figure 2 | Left panel: We calculate the average a-coefficients (shown by black markers, with black error bars) for modes with turning points between 0.7R_☉ and 0.8R_☉ and plot them alongside the sunspot number (light blue line). The variation of the a-coefficients is further smoothed with a Gaussian filter and shown as a red solid line. The plot is based on the combined MDI and HMI 4 × 72-day datasets. Right panel: we plot the jump in radial profile obtained from the inversion of a₃ coefficient between depths of 0.8R_☉ and 0.6R_☉. The dashed vertical line indicates the cycle minimum, while the solid vertical line marks the cycle maxi-mum.

We subsequently examine the a₃ coefficient of the frequency splittings, which carries significant information about the dynamics of the solar tachocline. We analyze all modes whose lower turning point radius lie between 0.6R_☉ and 0.8R_☉ , ensuring substantial sensitivity to the tachocline region. We find that the temporal variation of the a₃ coefficient correlates strongly with the sunspot number observed at the surface. During Solar Cycle 23, the amplitude of the a₃ coefficient was significantly larger than in Cycle 24, and it is increasing again as Cycle 25 approaches its maximum as can be seen in the left panel of Figure 2. This close correspondence in amplitude suggests that the a₃ coefficient may serve as a proxy for the strength of the solar cycle. However, since the variation of a₃ is nearly in phase with the sunspot number, its predictive capability appears limited. Nevertheless, the strong correlation indicates that this coefficient provides valuable insight into how the solar dynamo operates and how the tachocline evolves over the solar cycle. We arrive at the same conclusion from the inversion results of the a₃ coefficient, as shown in the right panel of Figure 2.

Figure 3 | We show ∂Ω/∂θ, with Southern Hemisphere values multiplied by −1 to enforce hemispheric symmetry, as it naturally reverses sign across the equator. The analysis is based on 4 × 72-day GONG datasets. The right panel shows ∂Ω/∂r as a function of the solar cycle at three depths 0.78, 0.75, and 0.6R_☉.

The gradients of solar rotation are expected to be strongest near the near-surface shear layer (NSSL) and the tachocline. We therefore examine two quantities: the radial gradient of rotation and the latitudinal gradient of rotation. After subtracting their temporal mean values, we analyze their variations over the solar cycle. We find that a clear butterfly-like migration pattern in the rotational shear is also present near the tachocline, as shown in Figure 3. This pattern closely correlates with the surface magnetic butterfly diagram. These findings provide strong evidence that the tachocline is also a primary site of solar dynamo action, demonstrating that the dynamo is not confined solely to the NSSL. Instead, our results indicate that the tachocline plays a crucial role in the generation and evolution of the solar magnetic field. For more details, please refer to our recent publication^[3]. A comprehensive description of the theoretical analysis and methods used in this study can be found in [4].

References:

[1]Mandal, K., Kosovichev, A. G., Pipin, V., Korzennik, S., ApJ (2025), DOI: 10.3847/1538-4357/ae061e
[2]Korzennik, S. G., Front. Astron. Space Sci.9, 1031313. DOI: 10.3389/fspas.2022.1031313 (2023).
[3]Mandal, K. & Kosovichev, A. G., Scientific Reports, Volume 16, Issue 1, id.4222, DOI: 10.1038/s41598-025-34336-1
[4]Mandal, K. , Kosovichev, A. G., Pipin, V. , ApJ (2024), Volume 973, Issue 1, id.36, 16 pp. DOI: 10.3847/1538-4357/ad5f2c