Differential rotation
Figure 1 shows the solar cycle-related variations of the differential rotation during solar cycle 24 obtained using the magnetic element feature tracking technique. Figure 1a shows the temporal variation of the sunspot numbers as a reference (data are from the Royal Observatory of Belgium). Solar cycle 24 has had the weakest cycle during the last 100 years. During the rising phase of the cycle, from May 2010 to December 2011, as represented by the vertical dashed line in Fig. 1, the number of sunspots increased and reached the first peak, corresponding to the peak number of sunspots in the northern hemisphere. During the maximum phase of the cycle from January 2012 to December 2015, the number of sunspots reached the second peak in approximately the middle of 2014, which corresponds to the peak number of sunspots in the southern hemisphere. During the declining phase of the cycle from January 2015 to March 2020, the numbers of sunspots gradually and monotonically decreased with time, reaching almost zero in 2020. Sunspots for the new cycle, which have emerged at high latitudes, can often be observed these days (not shown here).
The temporal variations of the solar differential rotation speed profiles derived from the entire dataset from May 01 2010 to March 26 2020 are shown in Fig. 1b. The velocities are taken relative to the Carrington frame of reference, which has a sidereal rotation rate of 14.184 deg/day. A faster rotation is indicated by yellow, and a slower rotation is indicated by blue. The latitudinal centroid of the sunspot area in each hemisphere for each rotation is shown in red. We cannot observe any temporal variation in the profile shown in Fig. 1b.
Figure 1c shows the differences between the differential rotation profiles from the average. The average differential rotation profile was estimated by averaging the profiles of the entire dataset. Faster (prograde relative to the average profile)/slower (retrograde) flows are indicated by yellow/blue. During the rising phase of the cycle, we can see the faster/slower flows on the equatorward/poleward sides of the sunspot area. The torsional oscillations (e.g., Howard and Labonte 1980) appear as a faster flow on the equatorward sides of the sunspot area and as a slower flow on the poleward sides. During the maximum phase of the cycle, the slower flow areas move from high latitudes to low latitudes associated with the movement of the sunspot area. Although not clear, faster flow areas that occur close to the pole can be seen during the maximum phase of the cycle. During the declining phase of the cycle, the faster flow areas move toward the equator, and the slower flow areas appear at high latitudes.
Figure 2a shows the average differential rotation speed profile in the northern and southern hemispheres during the entire period of solar cycle 24. The red/blue lines represent the northern/southern hemisphere results. For comparison, we also added the fitted curve of the differential rotation speed profile developed by Hathaway and Rightmire (2011), as shown by the black line. The uncertainties of the velocities can be evaluated from the standard deviations. The standard deviations are a few m \(\text {s}^{-1}\), which is negligibly small (not shown here, see Imada and Fujiyama (2018)). The differential rotation velocity range is + 30/− 180 m \(\text {s}^{-1}\) over a latitude range of 0/60\(^\circ \) in the Carrington rotation frame, respectively. As shown in Fig. 2a, the angular rotation rate is nearly identical to that found by Hathaway and Rightmire (2011) for solar cycle 23 using a different method (a cross-correlation technique). We can see a weak north–south asymmetry in solar cycle 24, which was previously reported by Imada and Fujiyama (2018). At high latitudes (\(\sim 60^\circ \)), the rotational speed was slightly faster in the southern hemisphere than in the northern hemisphere.
Figure 2b–d shows the average differential rotation speed profile during the rising, maximum, and declining phases of solar cycle 24, respectively. We cannot see any north–south asymmetry in Fig. 2b. The flattening of the profile at the equator, which has also been discussed in previous studies (Snodgrass 1983), can be seen during the rising phase of solar cycle 24. During the maximum phase of the cycle, we can clearly see that the rotation speeds in both hemispheres decelerate at mid-latitude (\(\sim 20^\circ \)) and accelerate at high latitudes (\(\sim 60^\circ \)), as shown in Fig. 2c. By contrast, we can see that the rotation speeds in both hemispheres accelerate at mid-latitudes and decelerate at high latitudes during the declining phase of the cycle, as shown in Fig. 2d.
Meridional flow
Figure 3 shows the solar cycle-related variations in the poleward meridional flow profile during solar cycle 24 obtained by the magnetic element feature tracking technique. Figure 3a is the same as Fig. 1a. Figure 3b shows the temporal variations in the meridional flow profile derived from the entire dataset from May 1, 2010 to March 26, 2020. A poleward flow is indicated by yellow, and an equatorward flow is indicated by blue. The latitudinal centroid of the sunspot area in each hemisphere is shown in red. For a meridional flow, we can find two typical types of temporal variation of the profile in Fig. 3b. First, in the declining phase of the cycle, the meridional flow is accelerated in the middle latitude. Second, although not clear, equatorward flows at high latitudes (\(\sim 60^\circ \)) occurred during approximately 2016–2018 (for the southern hemisphere) and 2018–2020 (for the northern hemisphere).
Figure 3c shows the differences of the meridional flow profiles from the average. The average meridional flow profile was estimated by averaging the profiles of the entire dataset. Faster/slower poleward flows are indicated by yellow/blue. Although not as clear as a differential rotation, we can see the faster/slower flows on the equatorward/poleward sides of the sunspot area during the rising phase of the cycle. The faster flow areas that occur close to the pole can also be seen during the maximum phase, although faintly. During the declining phase of the cycle, the faster flow areas move toward the equator, and the slower flow areas appear at high latitudes. The faster area in the low latitudes appears first in the northern hemisphere at the beginning of the declining phase (\(\sim \)2016), and later also appears in the southern hemisphere (\(\sim \)2017). In the northern hemisphere, the slower flow area near the pole was more pronounced than in the southern hemisphere.
Figure 4a shows the bf average meridional flow profile in the northern and southern hemispheres during the entire period of solar cycle 24. The red/blue lines represent the northern/southern hemisphere results. For comparison, we also added the fitted curve of the meridional flow profile developed by Hathaway and Rightmire (2011), as shown in the black line. The uncertainties of the velocities can be evaluated from the standard deviations. The standard deviations are a few m \(\text {s}^{-1}\), which is still enough small. The meridional flow velocity profile peaked at + 15 m \(\text {s}^{-1}\) at 45\(^\circ \). As shown in Fig. 4a, the meridional flow in this study is faster/slower at low/high latitudes than that fitted by Hathaway and Rightmire (2011). We can see a weak north–south asymmetry in solar cycle 24. At high latitude (\(\sim 60^\circ \)), the poleward flow is slightly faster in the southern hemisphere than in the northern hemisphere.
Figure 4b–d shows the average meridional flow speed profile during the rising, maximum, and declining phases of solar cycle 24, respectively. The meridional flow profile during the rising phase is nearly identical to that estimated by Hathaway and Rightmire (2011), although the meridional flow in the southern hemisphere is slightly faster than that in the northern hemisphere. During the maximum phase of the cycle, the flow of the northern hemisphere at high latitudes (\(\sim 50^\circ \)) seems to be accelerated. The meridional flow of the middle latitude during the declining phase of the cycle was accelerated at up to 17 m \(\text {s}^{-1}\) in both hemispheres and decelerated at high latitudes (\(>60^\circ \)).