Boyang Ding, Junwei Zhao, Ruizhu Chen, Matthias Waidele, Sushant S. Mahajan, Oana Vesa
W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085
Recently, we have characterized the Sun’s high-latitude m=1 inertial mode through analyzing its frequency- and time-dependent power distributions in both polar regions. First, we investigated their relationship with local magnetic fields, monthly sunspot numbers, and zonal flow. Second, we measured their phase speeds in both the zonal and meridional directions and examined the phase relations between the modes in two polar regions. This science nugget will only focus on the first part. Please refer to the full paper[1] for the second part.
Video 1: Longitudinal-flow map after the filter is applied to keep only m=1 mode near its dominant frequencies, projected around the north pole (left) the south pole (right) separately. This animation shows all 147 flow maps consecutively to cover the full analysis period. It dynamically illustrates the evolution of the characteristic m=1 yin-yang pattern.
In this study, we used the subsurface vφ(θ, φ), obtained from the SDO/HMI time-distance helioseismic analysis pipeline[2] for the depth of 0 − 1 Mm. These flow maps, with a spatial sampling rate of 0.12° pixel−1, are generated every 8 hours using SDO/HMI Dopplergrams and span approximately ±80.4° latitude near the central meridian. We used the data from 2010 May 19 to 2024 August 23, spanning a total of 14 years and 3 months and covering Carrington Rotation (CR) 2097 – CR2288. To cover one full rotation in the region, as well as to keep the dominant frequencies of low-m modes near zero, we track the daily subsurface flow maps and construct new synoptic flow maps using the rotation rate at latitude 65°, which is 365.4 nHz following Ref [3].
Running averages of vφ along each latitude over five neighboring rotations are removed to suppress differential rotation and its fluctuations that would otherwise leak into the m=0 mode. A progressive Gaussian smoothing is applied to eliminate small-scale structures. Spherical harmonic decomposition is performed separately on each synoptic flow map, yielding A(l, m, t), and Fourier transform is applied next to get A(l, m, ν). In the spherical harmonic domain, the m = 1 and 0≤l≤20 signals are kept. The combination of the frequency and spherical-harmonic filters will keep only m = 1 mode for further analysis. See Video 1 for an animation of the filtered m =1 mode data over the full period of analysis.
Figure 2. Comparisons of mode power with magnetic field, zonal flow, and sunspot number for the northern hemisphere (left) and southern hemisphere (right). (a) Power of m = 1 mode, integrated across all longitudes of each rotation for each latitude and displayed as a function of time and latitude. (b) Magnetic butterfly diagram during the interested time-period and covering only the high-latitude regions in the northern hemisphere. (c) Comparison of integrated mode power and magnetic flux, both smoothed with a 5-rotation running window. Arrows associated with a number indicate the peaks in mode power and magnetic flux that correspond to each other in the timing of occurrence. The shaded region indicates the activity minimum period. (d) Comparison of integrated mode power, mean zonal flow velocity, and monthly sunspot number, all of which are smoothed with a 13-month running window.
Using the filtered data, we integrate the m = 1 flow power across all longitudes for each rotation and obtain the total power of the mode for all latitudes (Figures 2a and 3a). In each polar region, the power fluctuates over time and tends to break into distinct mode packets. These power breakings coincide with where phase jumps occur, marking the end of an old mode and the start of a new one.
We compare the temporal evolution of the mode power with the evolution of the magnetic field, see Figures 2b and 3b. During the activity minimum period, approximately between the year 2016.5 and 2022.0, the polar magnetic field remains stable in one polarity, and the mode power is stronger than at other times. During the magnetically active periods, magnetic fluxes are observed to be transported from mid-latitude to high-latitude regions. There is a tendency for mode breakings to occur when magnetic fluxes are transported poleward and pass the latitudes around 55° − 60°, while the mode power strengthens when the magnetic field enhances around latitude 70°.
Figure 3. Same as Figure 2 but for the southern hemisphere.
To further quantify this relationship, Figures 2c and 3c compare the mode power (integrated over 60° − 70°) with the magnetic flux (integration of the unsigned magnetic field over 65° − 75°), both of which are smoothed with a 5-rotation running window. During the activity minimum period, three local peaks of mode power and unsigned magnetic flux are marked in both hemispheres, highlighting their good correspondence in the timing of occurrence. This correspondence in timing, though not in magnitude, suggests that the magnetic field significantly influences the m = 1 mode power, which gets boosted as the local magnetic field strengthens.
Figures 2d and 3d present a comparison between the total mode power (integrated over 50° − 80°) and the monthly sunspot number, both smoothed using a 13-month window, as done in Ref [4]. As shown in both polar regions, the mode power exhibits a clear anti-correlation with the sunspot number, weakening during solar maximum and strengthening during minimum, consistent with the recent finding of Ref [5].
We also investigated the relationship between the mode power and the zonal flow speed. As shown in Figures 2d and 3d, the zonal flow velocities, averaged over 55° −65° and smoothed over a 13-month period for consistency with the mode power curve, clearly slow down during activity minimum years, when the mode power substantially enhances. Overall, the mode power tends to be anti-correlated with the zonal flow speed.
References
[1] Ding, B., Zhao, J., Chen, R., et al. 2025, ApJ,
[2] Zhao, J., Couvidat, S., Bogart, R. S., et al. 2012, Solar Phys, 275, 375
[3] Komm, R. W., Howard, R. F., & Harvey, J. W. 1993, Solar Phys, 145, 1
[4] Waidele, M., & Zhao, J. 2023, ApJ Lett, 954, L26
[5] Liang, Z.-C., & Gizon, L. 2025, A&A, 695, A67