The QBO-type signals in the subsurface flow fields during solar cycles 23 and 24

The rotation rates are calculated based on regularized least squares (RLS) code using frequencies derived from MDI data. The MDI data consists of 74 non-overlapping 72 day observation periods starting in May 1996. To calculate the rotation rates between May 1996 and February 2011, 74 sets of rotational-splitting coefficients up to $l=300$ from MDI were used. The rotation rate residuals are calculated by removing the temporal mean from the MDI data (for detailed information about the methods, see Howe, 2009; Howe et al., 2018). It must be noted that the inversion data do not distinguish between the northern and southern solar hemispheres, therefore the values are flipped around the equator (Figures 1a, c and e).

To calculate the rotation rates for the period covering solar cycle 24, we again employed the RLS code using frequencies derived from the HMI data, which consists of 53 non-overlapping 72 day observation periods starting in May 2010. To achieve this, we used 53 sets of rotational-splitting coefficients up to $l=300$ from HMI covering the period from May 2010 to August 2020. The rotation rate residuals, similar to those from MDI, are calculated by removing the temporal mean from those inferred from the HMI data set ( Figures 1b, d, and f).

We decided not to splice the data sets from HMI and MDI as they only have an overlapping period of one year where the solar cycle is at its minimum. Using one year overlapping period over the solar cycle minimum might introduce systematic errors.

Interestingly, although the rotation rate residuals show similar features until 2 years before each solar cycle end, the rotation rate residuals at each depth 0.90$R_{\odot}$, 0.95$R_{\odot}$, and 0.99$R_{\odot}$ display opposite behavior after this time. Between 2006 and 2009, a pole-ward slower-than-average flow band forms around 50^∘ latitude (Figure 1a, c, and e). Between May 2010 and September 2020 during solar cycle 24, on the other hand, the slower-than-average flow band forms around 60^∘ latitude, and then it propagates equator-ward (Figure 1b, d, and f). The physical mechanism which might lead to the observed differences in rotation rate residuals is to be studied further and not within the scope of this study. The acceleration and deceleration compared to the average background flow during solar cycle 24 are slightly weaker compared with those in solar cycle 23.

The rotation rate residuals at each depth show higher frequency fluctuations than the 11-year cycle throughout solar cycle 23 at the low and mid latitudinal bands (Figures 2a and b). The rotation rate residuals for the high latitudinal band, on the other hand, exhibits variations in phase with the solar cycle. The acceleration compared to the mean background flow trend is seen at the onset of solar cycle 23, which then starts to gradually decelerate after around 2001 until 2006 (Figure 2c). Another interesting feature is that the amplitudes of higher frequency variations in the rotation rate residuals increase with increasing depth (Figure 2).

Similar to solar cycle 23, solar cycle 24 shows high-frequency variations (Figures 2). However, different than solar cycle 23, the high latitudinal band in solar cycle 24 does not exhibit in phase variations with the solar cycle. The rotation rate residuals show a very different behavior; the rotation rate residuals fluctuate around 1 nHz. After around 2017, the rotation rate residuals show slower-than-average values followed by a recovery trend by 2020 (Figure 2c). This behavior is related to the equator-ward slower-than-average flow band observed in the higher latitudes in solar cycle 24 (the right panel of Figure 1). Rotation rate residuals, similar to solar cycle 23, show increasingly higher amplitude variations as we go deeper layers into the Sun (Figure 2).

To investigate the presence of the QBOs in low, mid, and high latitudinal bands in the subsurface flows, we use continuous wavelet transformation (CWT) following Torrence & Compo (1998). We must note that lag-1 autocorrelation coefficients, which are used to calculate the red noise spectrum, could not be calculated for some latitudinal bands due to numerical reasons such as insufficient length of data or strong trends in it. Therefore, the significances of the CWTs are calculated against white noise at significance levels of 0.1, 0.5, and 0.01.

During solar cycle 23, even though the rotation rate residuals exhibit QBO-like signals in general, only some of these signals are statistically significant at the 0.1 level. The rotation rate residuals at the target depth of 0.90$R_{\odot}$ at the low and mid latitudinal bands exhibit QBO-like signals during the rising phase of solar cycle 23 (the left panel of Figure 3). At the depth of 0.95$R_{\odot}$, the low latitudinal band shows a statistically significant QBO-like signal between 2000 and 2006, while the mid latitudinal band at the same depth shows a QBO-like signal only on the onset of the solar cycle (the middle panel of Figure 3). There are not any statistically significant QBO-like signals detected at the depth of 0.99$R_{\odot}$ (the right panel of Figure 3).

As for solar cycle 24, the rotation rate residuals show a few QBO-like signals that are statistically significant at the 0.1 level. At the depth of 0.90$R_{\odot}$ and at the low latitudinal band the rotation rate residuals show QBO-like signals between 2014 and $\sim$2019 (the left panel of Figure 4). At the depth of 0.95$R_{\odot}$ a QBO-like signal in the mid latitudinal band can be observed after around 2015 (the middle panel of Figure 4).

The results generally show that both data sets display marginally significant QBO-like signals at the 0.1 level. We, therefore, decided to study the existence of the QBO-like signals further.

The common features observed in the temporal variations of the periodicities throughout solar cycles 23 and 24 are that there are strong indications for the presence of QBO-like signals, and these signals might be suppressed by the presence of longer-term ($\geq\sim$8 years) and higher amplitude variations (Figures 3 and  4). It is difficult to find and analyze the QBOs in data sets without any prior data preparation methods, such as filtering the signals using discrete or fast Fourier transform, spherical harmonic decomposition, passband filtering, and Empirical Mode Decomposition (EMD) analysis (Bazilevskaya et al., 2014).

To remove the possible effects of these higher amplitude variations and to investigate the spatiotemporal behavior of the QBO-like signals even further, we decomposed our data sets using the EMD method, which decomposes the time series data into a finite number of Intrinsic Mode Functions (IMF) and they represent a set of basis functions (Huang et al., 1998). This method has successfully been applied to various time-series data, including solar neutrino flux rates, solar and galactic cosmic rays measured by neutron monitors, surface magnetograms, and polar faculae data, previously (Vecchio et al., 2010, 2012; Deng et al., 2020).

We decompose the rotation rate residuals at each depth and latitudinal band through solar cycles 23 and 24. It must be noted that we did not include latitudes above 70^∘ N and S as there are some data gaps above these latitudes because of the tilt of the rotation axis of the Sun. The data is decomposed until a QBO-like oscillation with a period ranging from $\sim$1 to $\sim$4 years is found in the data. This corresponded to the second intrinsic mode function (IMF2) in the rotation rate residuals that display QBO-like signals.

The IMF2 of the rotation rate residuals during solar cycles 23 and 24 show at each depth, there are faster-than-average and slower-than-average flow bands and their amplitudes decrease as we get closer to the surface (Figure 5). At the depth of 0.90$R_{\odot}$, the faster-than-average and slower-than-average flow bands form around 40^∘ latitudes and they propagate both equator-ward and pole-ward during solar cycle 23 (Figure 5a). During solar cycle 24, on the other hand, this pattern is completely different. The faster-than-average and slower-than-average flow bands form at lower latitudes and propagate pole-ward to higher latitudes (Figure 5b). Another interesting feature during the rising phase of solar cycle 23 is switching between faster-than-average to slower-than-average flow bands in the higher latitudes between 50 – 70^∘. The switching pattern is more pronounced and clear at the 0.90$R_{\odot}$ and 0.95$R_{\odot}$, while it is more convoluted and not as pronounced at the 0.99$R_{\odot}$. This flow pattern in the higher latitudes almost vanishes during the declining phase after the maximum of the cycle at the depth of 0.99$R_{\odot}$. For solar cycle 24, however, these patches persist throughout the solar cycle (the bottom panel of Figure 5). The IMF2 of the rotation rate residuals in solar cycle 23 display a different pattern at every depth than those in solar cycle 24 (Figure 5).

Following that, we averaged the IMF2 of the rotation rate residuals into low, mid, and high latitudinal bands (Figure 6). The IMF2s of the rotation rate residuals calculated for each latitudinal band show QBO-like oscillations (the left panel of Figure 6). The amplitude of these oscillations in each latitudinal band increases with increasing depth, except for that calculated using HMI data in 0.95$R_{\odot}$ (the right panel of Figure 6). The IMF2 of the rotation rate residuals show variations with a maximum amplitude of $\sim$2 nHz during solar cycle 23 around 2002.

To investigate the spatiotemporal behavior of the QBOs-like signals in the IMF2 of the rotation rate residuals, we calculate the CWT of the data during solar cycles 23 and 24.

There are QBO-like signals that are statistically significant at the 0.01 level at the target depth of 0.90$R_{\odot}$ at each latitudinal band. In the low latitudinal band, the QBO-like signal lasts until 2004, while it lasts throughout solar cycle 23 in the mid latitudinal band. The high latitudinal band shows a QBO-like signal, the period of which is 1.5 years between 2000 and $\sim$2004. This signal then becomes shorter in time dropping down to a period of 1 year around 2004, lasting until 2006 (the left panel of Figure 7). However, one must be careful when evaluating periodicities close to 1 year due to the $\beta$-angle of the solar rotation axis. The target depth of 0.95$R_{\odot}$ also shows QBO-like signals significant at the 0.01 level. In the low latitudinal band, a statistically significant QBO-like signal can be observed between 2000 and 2006. In the mid-latitudinal band, a QBO-like signal is only present during the rising phase of solar cycle 23, whereas the high-latitudinal band exhibits a QBO-like signal only during the declining phase of the cycle (the middle panel of Figure 7). At the target depth of 0.99$R_{\odot}$ at the low latitudinal band does not show any statistically significant QBO-like signals. At the middle and high latitudinal bands, on the other hand, the QBO-like signals are statistically significant at the 0.01 level and are observed between 2002 and 2006 at the mid latitudinal band, and 1998 and 2002 at the high latitudinal band (the right panel of Figure 7).

During solar cycle 24, statistically significant QBO-like signals are observed intermittently at each depth and latitudinal band (Figure 8). At the target depth of 0.90$R_{\odot}$ at the low latitudinal band, a statistically significant QBO-like signal is present between 2017 and 2020. At the mid latitudinal band, on the other hand, the QBO-like signal is significant until around 2014. At the high latitudinal band at the same depth, the QBO-like signal has a shorter period of around 1.2 years and lasts between 2014 and 2016 (the left panel of Figure 8). However, similar to the signal observed for the same depth and latitudinal band in solar cycle 23, one must be careful evaluating signals that have periods close to 1 year, which might stem from the $\beta$-angle of the solar rotation axis. A statistically significant QBO-like signal at the depth of 0.95$R_{\odot}$ at the low latitudinal band extends from 2013 to 2018. The mid latitudinal band at the same depth shows QBO-like signals that are significant on the rising and declining phases of the cycle, while the high latitudinal band exhibits a QBO-like signal with a period of $\sim$2.5 years that is statistically significant throughout the cycle. Additionally, there is a statistically significant signal with a period of $\sim$1.2 year, that drops to a little below 1 year with time (the middle panel of Figure 8). The depth of 0.99$R_{\odot}$ shows statistically significant QBO-like signals during solar cycle 24 at the low and high latitudinal bands, while the mid latitudinal band does not exhibit any significant QBO-like signal (the right panel of Figure 8).