# Connection between Sub-surface Layers and Surface Magnetic Activity
over Multiple Solar Cycles using GONG Observations

Mackenzie A. Baird Villanova University, 800 Lancaster Ave, Villanova, PA
19085, USA  National Solar Observatory, REU Program, 3665 Discovery Dr.,
Boulder, CO 80303, USA  Sushanta C. Tripathy National Solar Observatory, 3665
Discovery Dr., Boulder, CO 80303, USA  Kiran Jain National Solar Observatory,
3665 Discovery Dr., Boulder, CO 80303, USA

###### Abstract

We investigate the spatio-temporal evolution of high-degree acoustic mode
frequencies of the Sun and surface magnetic activity over the course of
multiple solar cycles, to improve our understanding of the connection between
the solar interior and atmosphere. We focus on high-degree p-modes due to
their ability to characterize conditions in the shear layer just below the
solar surface and analyze 22 years of oscillation frequencies obtained from
the Global Oscillation Network Group (GONG). Considering 10.7 cm radio flux
measurements, the sunspot number, and the local magnetic activity index as
solar activity proxies, we find strong correlation between the mode
frequencies and each activity index. We further investigate the hemispheric
asymmetry associated with oscillation frequencies and magnetic activity
proxies, and find that both were dominant in the southern hemisphere during
the descending phase of cycle 23, while in cycle 24 these quantities
fluctuated between northern and southern hemispheres. Analyzing the
frequencies at different latitudes with the progression of solar cycles, we
observe that the variations at mid-latitudes were dominant in the southern
hemisphere during the maximum activity period of cycle 24 but the values
overlap as the cycle advances towards the minimum phase. The mode frequencies
at the beginning of cycle 25 are found to be dominant in the southern
hemisphere following the pattern of magnetic activity. The analysis provides
added evidence that the variability in oscillation frequencies are caused by
both strong and weak magnetic fields.

Helioseismology (709) — Solar interior (1500) — Solar oscillations (1515) —
Solar activity (1475)

††facilities: GONG, SoHO/MDI††software: Interactive Data Language
(IDL)555https://www.nv5geospatialsoftware.com/docs/home.html.

##  1 Introduction

The acoustic oscillations at the surface of the Sun are known to reveal
characteristics of the solar interior. Without the ability to physically see
the interior, we rely on seismic waves to comprehend the dynamics below the
solar atmosphere mainly through the use of p-mode frequencies. These
frequencies are known to vary with the level of surface magnetic activity and
synchronize with the solar magnetic cycle. The change of oscillation
frequencies in phase with the progression of solar cycle were first reported
by Libbrecht & Woodard (1990) and have been confirmed by a large number of
studies covering low-degree (Chaplin et al., 2004; Basu et al., 2012; Salabert
et al., 2015; Jain et al., 2018), intermediate-degree (Woodard et al., 1991;
Tripathy et al., 2007; Jain et al., 2013; Korzennik, 2023) and high-degree
global mode frequencies (Rabello-Soares et al., 2008; Rhodes et al., 2011;
Tripathy et al., 2015). A few studies, however, have alluded that the
correlation is weak during the low activity phase, for example, during the
solar minima preceding cycle 24 (Salabert et al., 2009; Tripathy et al.,
2010b; Jain et al., 2011), and cycle 25 (Jain et al., 2022). These two minima
were unusually long and deep. This anomaly was present in both the length and
timing of the minima, as determined by helioseismic frequencies, and apparent
in the surface magnetic activity proxies. In addition to temporal variation,
the frequencies of high degree-modes from local helioseismic techniques have
also shown high correlation with the spatial distribution of surface magnetic
fields (Hindman et al., 2000; Howe et al., 2004; Tripathy et al., 2010a,
2015). Since there are numerous evidence that point to a strong linear
relationship between the solar activity and the change in oscillation
frequencies, it has been argued that the helioseismic frequency shifts can be
considered as another proxy of the Sun’s activity cycle (Broomhall &
Nakariakov, 2015; Salabert et al., 2015).

Furthermore, studies with improved and longer data sets indicate that the
relationship between the frequency shifts and solar activity is not always
simple (see review by Basu, 2016). Using intermediate-degree ($7\leq\ell\leq
188$) global mode frequencies, Jain et al. (2009) have shown that the linear
correlation between the change in frequencies and the magnetic activity
differs from phase to phase in the activity cycle. Similar results were
obtained for low-degree ($\ell\leq 3$) mode frequencies computed from Sun-as-
a-Star observations (Broomhall & Nakariakov, 2015).

The early study of frequency variations with solar cycle demonstrated that the
frequency changes are predominantly a function of frequency and independent of
harmonic degree. This led to the perspective that the solar-cycle dependent
frequency changes are confined to a thin layer close to the surface and are
caused by the magnetic field at the tachocline evolving with solar cycle
(Roberts & Campbell, 1986). Other mechanisms such as structural changes in the
sub-surface layers due to temperature (Kuhn, 1988; Foullon & Nakariakov,
2010), modification of the acoustic cavity (Dziembowski & Goode, 2005),
influence of magnetic atmosphere (Jain & Roberts, 1996) etc. have also been
considered (see review by Howe, 2008).

Recent studies further suggest that the magnetic field inside the Sun
influences the oscillation frequencies in different ways depending on its
location and geometry. Analyzing individual multiplet frequencies (frequencies
that are functions of $n$, $\ell$, and $m$) from Global Oscillation Network
Group (GONG) over solar cycles 23 and 24, Jain et al. (2022) found that the
minimum period associated with different acoustic modes occur at different
epochs e.g., occurring a year earlier in deeper layers as compared to the
surface. This led the authors to conclude that the magnetic fields located at
three different locations, i.e., core, tachocline and near-surface shear
layers may play important roles in modifying the oscillation frequencies.
Based on theoretical models where a quasi-degenerate perturbation theory was
used to calculate the effect of the toroidal field on the solar oscillation
frequencies, Kiefer & Roth (2018) suggested that the magnitude of the observed
frequency shifts over the entire solar cycle cannot be explained by the
presence of strong field produced by the dynamo located in the tachocline
region alone and conjectured that part of the frequency shifts may be
explained by the magnetic fields near the surface. Analysing the GONG global
frequencies, Broomhall (2017) has compared the depth of the minimum preceding
and following solar cycle 23 and advocated that the near-surface magnetic
perturbation is responsible for the changes in the frequencies.

Considering the dominant physical mechanism to be sub-surface magnetic
perturbation, a better knowledge of the near-surface layers is required to
advance our understanding. This can be achieved through a detailed analysis of
accurate and precise frequencies of high-degree modes since the lower turning
points of these modes occur very close to the surface. However, determination
of high-degree modes using the techniques of global helioseismology is a
challenging task since individual peaks are blended into ridges. In spite of
this inherent difficulty, there have been several attempts to compute high-
degree oscillation frequencies for different epochs (Korzennik et al., 2013;
Reiter et al., 2015, and references therein).

In addition to global helioseismology, the local helioseismic technique of
ring diagram (Hill, 1988) also produces frequencies of high-degree modes and
are available since mid-2001 from GONG111https://gong.nso.edu. In this paper
we analyze these high-degree acoustic mode frequencies as a tool to understand
the characteristics of subsurface layers and their connection with the surface
magnetic field. This analysis extends our previous study (Tripathy et al.,
2015, hereafter paper I) where we studied a part of solar cycle 23 and the
following minimum. With the data available for another solar cycle, here we
expand the investigation to include cycle 24 and the minimum between cycles 24
and 25. This allows us to extend the results reported in Paper I to another
solar cycle that is weaker compared to cycle 23. The rest of the paper is
organized as follows: We describe the data in Section 2 and present our
results in Section 3. We summarize our findings in Section 4.

##  2 Data Analysis

The high-degree mode frequencies used in this study are computed from merged
GONG 1-min full-disk Dopplergrams (Harvey et al., 1996, 1998) and cover the
period from 2001 July 26 to 2023 March 22. This time range includes
approximately the last seven years of cycle 23, complete cycle 24, and the
first three years of cycle 25. The mode frequencies are determined through the
standard GONG ring-diagram pipeline (Corbard et al., 2003). A good description
of the ring-diagram technique can be found in Hill (1988) but for sake of
completeness we present a brief outline of the procedure used in this work.
The solar surface in each Dopplergram is divided into 189 overlapping regions
(tiles hereafter) covering $\pm 52.5^{\circ}$ in latitude and central meridian
distance (CMD), spaced by $7.5^{\circ}$ in either direction. Each tile covers
an area of 15∘ $\times$ 15∘ in heliographic longitude and latitude after
apodization, and is tracked for 1664 minutes (hereafter ‘ring-day’) using
Snodgrass rotation rate (Snodgrass, 1984). The collection of 189 data cubes
centered on different latitudes and longitudes is commonly referred as a
dense-pack mosaic (Haber et al., 2000). Each data cube is passed through a
three-dimensional Fourier transform to create a three-dimensional power
spectrum with two dimensions in space and one in time. The resulting power
spectrum is fitted with a Lorentzian profile described by

$P\left(k_{x},k_{y},\nu\right)=\frac{A\Gamma}{\left(\nu-\nu_{0}+k_{x}U_{x}+k_{y}U_{y}\right)^{2}+\Gamma^{2}}+\frac{b}{k^{3}}\;,$
(1)

where $P$ is the oscillation power for a wave with a temporal frequency $\nu$.
The horizontal wave number, $k$, is defined as a sum of
$k_{x}\,\hat{x}+k_{y}\,\hat{y}$ where $\hat{x}$ and $\hat{y}$ point in the
prograde and northward directions, respectively. The other six parameters that
are fitted consist of the zonal, $U_{x}$, and meridional, $U_{y}$, velocities,
the mode’s unperturbed central frequency in the rest frame of the Sun,
$\nu_{0}$, the mode amplitude, $A$, the mode width, $\Gamma$, and the
background power, $b$. The fitting algorithm uses the maximum-likelihood
procedure described in Anderson et al. (1990) and provides about 200 modes for
radial order, $n$, between 0 – 6, and degree, $\ell$, between 180 – 1200 in
the frequency range of 1500 – 5200 $\mu$Hz (see Figure 17 in Appendix A to
visualize the modes fitted by this technique). Throughout this study, we will
refer to the centers of the tiles when describing latitude or longitude
values.

Figure 1: Distribution of duty cycle as a function of time. The data below 70%
duty cycle (horizontal line) are not used in this study. The vertical dashed
lines demarcate the solar cycles 23, 24 and 25 based on 13-month smoothed
monthly sunspot number (see Section 2.1 for details).

GONG is a ground-based instrument and experiences gaps in observations due to
weather conditions and maintenance of instruments (Jain et al., 2021). Figure
1 shows the distribution of the duty cycle for the analysis period, where the
duty cycle is defined as the percentage of good observations out of 1664
minutes. It is clear that the duty cycle varies widely with a mean and median
value of 88% and 92%, respectively. Since the mode parameters corresponding to
low duty cycle have large uncertainty in fitted values, we exclude the days
where the duty cycle is less than $70\%$. This yielded 6375 usable ring days
(92%) out of 6960 ring-days. We further split the entire data into individual
cycles consisting of 2107, 3314, and 954 ring-days for cycles 23, 24, and 25,
respectively. Figure 2 shows the number of days as a function of the duty
cycle for individual solar cycles in the form of histograms. We note that
cycle 24 has the maximum days with good observations compared to other cycles.

Figure 2: Number of observed ring-days, expressed in percentage, as a function
of duty cycle for different solar cycles.

###  2.1 Solar Activity

Since the mode frequencies are known to vary with solar activity cycle, we
compute a quantity known as “magnetic activity index” (MAI) as a measure of
solar activity. For each tile, a data cube is generated by tracking and re-
mapping line-of-sight magnetograms in the same fashion as Dopplergrams. The
absolute value of each pixel higher than a specified threshold in the data
cube are then averaged to compute the MAI. It may be noted that the merged
GONG magnetograms are not available prior to 2007 April, hence we use
96-minute magnetograms from the Michelson Doppler Imager (MDI) up to 2007
April 9 and GONG magnetograms sampled every 32 minutes thereafter. The
threshold value for MDI magnetograms is set to 50 G (Basu et al., 2004) and
8.8 G for GONG data based on the estimated noise of the instruments. Since the
measured magnetic strength between different instruments differ, we scaled the
MDI values by 0.45 to generate an uniform set of MAI values. The conversion
factor is based on the calculation of Riley et al. (2014) where pixel values
were compared between the line-of-sight magnetograms. We further calculate the
mean magnetic activity index (MMAI), which represents the spatial average of
the MAI over the dense-pack mosaic. Since MAIs do not have measured
uncertainties, we use the standard error (the standard deviation of MAI of the
dense-pack mosaic divided by the square-root of number of contributing tiles)
as the uncertainty in the measurement of MMAI.

We further use three additional surface activity proxies to characterize the
solar activity; total and hemispheric sunspot number (TSN, HSN, version 2
provided by the WDC-SILSO222https://www.sidc.be/silso/datafiles, Clette et
al., 2014) and the 10.7 cm radio flux,
($F_{10.7},$)333https://www.spaceweather.gc.ca/forecast-prevision/solar-
solaire/solarflux/sx-5-en.php. The latter is a measurement of the integrated
emissions at a wavelength of 10.7 cm over the entire solar-disk (Covington,
1969; Ermolli et al., 2014) and is expressed in solar flux units (1 sfu =
10${}^{-22}\,$W m-2 Hz-1). The values of all activity indices are interpolated
to the same temporal grid to have the same length as the frequency data
points. The activity minimum period is chosen on the basis of the 13-month
smoothed monthly sunspot number and occurred in December 2008 and December
2019 for cycle 24 and 25
respectively444https://www.sidc.be/silso/cyclesminmax. Accordingly, we
consider 2008 December 31 and 2019 December 31 as the days to distinguish
between solar cycles 23 and 24 and 24 and 25, respectively. The two vertical
dashed lines in Figures 1, 5, 7-8, 10-15, and 18 indicate the separation
between the cycles.

##  3 Results and Discussion

###  3.1 Spatial Variability in Frequencies

One of the advantages of using ring-diagram technique is that it allows us to
investigate the variation of frequencies at different locations over the solar
disk and to find their relation with the local magnetic field on a shorter
time scale e.g. one ring-day. Comparing dense-pack mosaics of frequency shifts
and co-temporal magnetograms for a few days, Hindman et al. (2000)
demonstrated that the active regions appear as locations of large frequency
shifts. Subsequent analysis using data over several years suggested that this
conclusion is valid only during periods of high activity but breaks down
during the solar minimum (Tripathy et al., 2010b, 2015). These findings are
similar to those obtained by Jain et al. (2009) for global modes. Here we
extend the analysis to compare the spatial variability between different solar
cycles by computing the daily changes in frequencies, $\delta\nu_{s}$, which
is defined as the weighted sum of the frequency differences of each mode in a
given tile where the weights correspond to square of the fitting uncertainties
$\sigma_{n,l}$ following the relation

$\delta\nu_{s}=\Sigma_{n,l}\frac{\delta\nu_{n,l}}{\sigma_{n,l}^{2}}\Bigm{/}\Sigma_{n,l}\frac{1}{\sigma_{n,l}^{2}}\;,$
(2)

where the frequency differences of each mode ($\delta\nu_{n,l}$) is calculated
with respect to the error-weighted spatial average of the same mode present in
the dense-pack mosaic of each ring-day. Thus, $\delta\nu_{s}$ represents the
frequency shift of individual tiles.

Figure 3: Spatial distribution of frequency shifts over 189 tiles (top) and
coeval MAI (bottom) during high activity period in 2014 as indicated on the
top of the panels. The Pearson’s linear correlation coefficients between
frequency shifts and MAI (from left to right) are 0.94, 0.95, 0.94. Figure 4:
Spatial distribution of average frequency shifts of 189 dense pack tiles (top)
and coeval MAI (bottom) during low activity period in 2018 as indicated on the
top of the panels. The Pearson’s linear correlation coefficients between
frequency shifts and MAI (from left to right) are 0.73, 0.77, 0.78.

In Figures 3 and 4, we display examples of spatial distribution of $\delta\nu$
and the corresponding MAI values for three consecutive days during the high-
and low-activity periods in cycle 24, respectively. Each small square in these
panels corresponds to a 15∘ $\times$ 15∘ tile. It can be easily seen that the
tiles with higher $\delta\nu$ values have good correspondence with those with
high MAI values or vice versa. This is quantitatively checked by calculating
the Pearson’s linear correlation coefficients, $r_{p}$, and these are found to
be 0.94, 0.95 and 0.94 for the high-activity period and 0.73, 0.77 and 0.78
for the low-activity period. The decrease in correlation with low magnetic
activity is consistent with the studies mentioned earlier and confirms that
the correlation indeed varies during the two different periods of solar
activity. Figure 5 displays the temporal variation of $r_{p}$ computed between
frequency shift and MAI over the dense-pack mosaic of each ring-day. One can
easily notice that $r_{p}$ values remain high during the high activity period
but low during the rising and declining phases of the solar cycle. We also
obtain low or negative correlation around the cycle minimum, which is similar
to the results obtained with global-mode analysis for the minimum between
cycles 23 and 24 (Tripathy et al., 2010b).

Figure 5: Temporal variation of Pearson’s linear correlation coefficients
between frequency shifts, $\delta\nu_{s}$, and magnetic activity index, MAI,
for each ring-day. The vertical dashed lines have the same meaning as in
Figure 1.  Figure 6: Pearson’s linear correlation coefficients, $r_{p}$,
between $\delta\nu_{s}$ and mean magnetic activity index, MMAI, for each ring-
day as a function of MMAI. Solar cycle 23 values are shown in the top panel;
the olive and blue dots depict the MMAI values computed from MDI and GONG
magnetograms, respectively. The middle and bottom panel shows the coefficients
for cycles 24 and 25, respectively.

Figure 6 illustrates the variation in $r_{p}$ as a function of MMAI. This
reveals that for MMAI higher than 2 G, the correlation coefficients tend to
remain between 0.8 and 1.0. As substantiated in Paper I, the correlation
coefficients increase with the higher MMAI values and decrease as the solar
activity progresses towards the minimum phase characterized by low MMAI
values. Tripathy et al. (2013) using data for four years near the cycle 24
minimum have also reported a similar pattern in correlation coefficients
calculated between the daily values of $\delta\nu_{s}$ and MMAI. Based on the
argument of weak correlation between low-degree modes and activity proxies,
Foullon & Nakariakov (2010) carried out a parametric study and suggested that
the anomalous frequency shifts are best explained by thermal effects in the
upper regions of the convection zone. However as outlined in Section 1, the
analysis of oscillation frequencies during the solar cycle favors a near-
surface magnetic perturbation. Our findings agree with this explanation except
around the minimum phase where we observe weak correlation. We conjecture that
weak turbulent magnetic fields in the solar photosphere outside of active
regions (Faurobert et al., 2001) may be responsible to change the oscillation
frequencies during the minimal-activity phase of the cycle.

As seen in Figure 6, the behaviour is strikingly similar between different
solar cycles except for a large scatter observed during solar cycle 23 (top
panel). This is due to the use of MDI magnetograms (olive dots) for
calculating the MAI values as these magnetograms are derived from a
combination of 1-min and 5-min observations with different noise levels. With
the use of a uniform set of merged GONG magnetograms obtained with 32 min
cadence beyond 2007 April 09, the scatter in $r_{p}$ is significantly reduced.
We also obtain negative values for $r_{p}$ when MMAI is below 1 G. To assess
the significance of the negative correlations, we computed the Spearman’s rank
correlation since it provides the two-sided significance of deviation from
zero. It is found that the negative correlation has a two-sided significance
value greater than 0.01 which lead us to conclude that these correlations are
insignificant.

###  3.2 Temporal Evolution of Mode Frequencies and Activity

In order to investigate the temporal variation of mode frequencies, we adopt a
method which is analogous to frequency shifts calculated in global $p$-mode
analysis (Woodard et al., 1991) i.e.

$\delta\nu=\Sigma_{n,l}\frac{\Sigma_{i=1}^{189}\nu_{n,l,i}-\overline{\nu_{n,\ell}}}{\sigma_{n,l}^{2}}\Bigm{/}\Sigma_{n,l}\frac{1}{\sigma_{n,l}^{2}}\;,$
(3)

where $\delta\nu$ is the frequency shift over one ring-day and is determined
by subtracting the reference frequency, $\overline{\nu_{n,\ell}}$ from the
weighted sum of the frequencies of each ($n$, $\ell$) mode averaged over the
189 tiles. Once again, the weights are chosen as the square of the fitting
uncertainties, $\sigma_{n,l}$. It may be noted that while computing the
weighted sum of $\nu_{n,l}$ over the dense-pack mosaic, we include only those
modes that are present in at least two-third of the tiles since the fitting
procedure does not yield the same number of identical modes in each tile or
even in each ring-day. Thus, we analyze the oscillation data in degree and
frequency range of $200\leq\ell\leq 900$ and $1500\leq\nu\leq 5200~{}\mu$Hz.
The mean frequencies as a function of harmonic degree, commonly known as
$\ell-\nu$ diagram is shown in appendix A (Figure 17).

Although the analysis covers two activity minima, preceding and following
cycle 24, the reference frequencies are computed from mode frequencies during
the quiet activity period following cycle 24. Since the solar minimum
following cycle 24 occurred in December 2019, the reference frequency of each
mode is calculated as an weighted average over the 189 tiles across the period
2019 December 17–31 (12 ring-days). The associated error, $\sigma_{ref}$, is
also determined by propagating the values of the fitted uncertainties over the
same interval. As mentioned earlier, identical modes are not fitted in every
data set and hence we chose 12 ring-days to compute the reference frequencies
to include all possible modes. We further emphasize that the results presented
below are independent of the choice of reference values since the magnitude of
the frequency shifts over the entire data do not change with a different set
of reference frequencies.

The calculation of the errors associated with the frequency shifts consists of
two steps. In the first step, we compute the mean error over the 189 tiles by
using the standard formula for the error of the weighted mean
($\sigma^{2}={1}/{(\Sigma_{n,l}\;{1}/{\sigma_{n,l}^{2}})}$. In the next step,
this mean error and the reference error, $\sigma_{ref}$ are summed through the
error propagation formula to yield the formal uncertainty corresponding to
$\delta\nu$ of each ring-day.

Although earlier studies have shown that the observational gaps do not
significantly bias the identification of mode frequencies (Chaplin et al.,
2004; Keith-Hardy et al., 2019), for the sake of completeness, we analyze the
frequency shifts as a function of the duty cycle. The results reveal no
relation between the frequency shifts and duty cycle. We expand on this in
Appendix A.

Figure 7 shows the frequency shifts (dots) and associated errors as a function
of time. Since the magnitude of the errors are too small to be visible in the
figure, it is multiplied by a factor of 5 before plotting. Also to avoid
cluttering, errors are drawn for every 250th point. The dots in the figure
reveals a large scatter since we are plotting the daily variation of
$\delta\nu$ as opposed to averages over longer time period. Similar scatter is
also seen in activity proxies plotted later. In order to decrease the scatter
and visualize different features, we smooth the data through a boxcar
averaging over 181 points. The smooth profile shown by the thick black line in
the figure clearly displays different phases of the solar cycles (the lines in
relevant figures throughout the rest of the paper will represent the boxcar
averaged smooth profiles). The other key features include a double-peak
structure at the activity maximum of cycle 24, a minimum following cycle 24 as
deep as the minimum following cycle 23, and a smaller peak amplitude in cycle
24 compared to cycle 23.

Figure 7: Dots denote the frequency shifts corresponding to each ring-day. The
thick black line represents the smoothed profile obtained through a boxcar
average of 181 data points. The horizontal thin line indicates the position of
the solar minimum preceding cycle 24. The vertical dashed lines have the same
meaning as in Figure 1. To avoid cluttering, errors are shown for every 250th
point on the smoothed curve and are multiplied by a factor of 5 to be visible
on this plot.  Figure 8: The dots depict the temporal evolution of frequency
shifts, $\delta\nu$, mean magnetic activity index, MMAI, total sunspot number,
TSN, and 10.7 cm radio flux, $F_{10.7}$. All quantities are scaled to have
values between 0 and 1. The colored lines are the smoothed profiles where
smoothing is performed by a boxcar average of 181 data points. The dashed
vertical lines have the same meaning as in Figure 1. To avoid cluttering,
errors are shown for every 250th point on the smoothed curve and are
multiplied by a factor of 5 to be visible on this plot. For a plot depicting
only the smoothed curves, the reader is referred to Figure 18 in Appendix A.

Figure 8 exhibits the temporal evolution of $\delta\nu$, MMAI, $F_{10.7}$ and
TSN where all quantities are scaled to have values between 0 and 1. At a first
glance, it is observed that the activity indices follow the same cyclic trend
as frequency shifts. The smooth profiles also illustrate the discrepancy as to
the timing of the solar minimum; minimum occurring earlier in frequency shifts
compared to the solar activity proxies. However, the timings of solar maxima
appear to coincide. On closer examination, we note that the minimum of cycle
24 seen in activity indices was deeper compared to cycle 25 minimum while, as
discussed earlier, the minimum seen in frequency shifts is comparable during
both the cycles.

A similar discrepancy was already noticed during the minimum of solar cycle 24
(paper I) where it was pointed out that the minimum period observed in
frequencies persisted for a short period compared to activity indices. In this
analysis, we find that during the cycle 25 an early minimum is detected in the
frequencies with a lead time of about one year as compared to the activity
indices. This provides added evidence to the interpretation of paper I that
the large changes in high-degree frequencies are caused by the strong field
located in the base of the convection zone while the small-scale, weak
turbulent magnetic field is responsible for the smaller shifts which occur
mostly during the minimum phases of the solar cycle.

Figure 9: The open circles shows the frequency shifts, $\delta\nu$, as a
function of total sunspot number, TSN (bottom panel), 10.7 cm radio flux,
F10.7 (middle panel), and mean magnetic activity index, MMAI (upper panel).
Different colors represent different cycles and are indicated in the top
panel. The corresponding colored lines denote fits obtained from the linear
regressions. We do not plot the error bars since these are small but their
values are tabulated in Table 1. Note that the red and black lines in the
upper panel are mostly overlapping. For a plot illustrating only the linear
fits, please see Figure 19 in Appendix A.

To study the response of the frequency shifts to changing levels of solar
activity, we perform linear regression analyses of the frequency shifts with
all the three activity indicators. The results of regression are shown in
Figure 9 as a black solid line. We find that the gradients are positive; the
smallest and largest values are obtained for TSN and MMAI, respectively. For
investigating the differences between solar cycles, we perform regressions for
individual solar cycles and the fits are shown in Figure 9 by different
colored lines. The coefficients of the linear regression and the Pearson’s
linear correlation coefficients are provided in Table 1 (We did not calculate
the coefficients for solar cycle 25 since the data spans only for about three
years). Comparing the slopes between different cycles, we further notice that
the MMAI has the largest gradient in cycle 23 while for the other two indices,
the gradients are larger for cycle 24. This is not surprising since the MMAI
represents the local magnetic activity over the same portion of the solar disk
as the frequency shifts while the other two indices represent the activity
levels over the entire disk. Also it may be noted that we do not have data for
the whole cycle 23 and hence it is hard to say if this steep slope is inherent
to the last half of the solar cycle or unique to all of solar cycle 23.

As with most past studies, we find that the correlation coefficients with
different activity indices is greater than 0.75 which increases to 0.95 or
higher when the data was smoothed by a boxcar average of 181 data points. As
anticipated, the correlation is highest between the frequency shifts and
magnetic activity measured locally. We also notice some differences between
the cycles, the correlation decreasing in cycle 24 compared to cycle 23; a
decreases of 3.6% for MMAI, 6.8% for radio flux and 8.2% for TSN. The small
decrease in linear regressions and correlation coefficients between cycles 23
and 24 may be a consequence of not having complete data for solar cycle 23. It
will be of interest to carry out a statistical analysis of cycles 24 and 25
when data is available for complete cycle 25.

Table 1: Gradients and Linear Correlation Coefficients between Frequency
Shifts and Activity Indices

| Cycle 23 |  | Cycle 24 |  | All Data
---|---|---|---|---|---
Activity Index | Gradient | $r_{p}$ |  | Gradient | $r_{p}$ |  | Gradient | $r_{p}$
MMAI | 0.2606 $\pm$ 2.63$\times 10^{-3}$ | 0.91 (0.99) |  | 0.2532 $\pm$ 2.46$\times 10^{-3}$ | 0.87 (0.98) |  | 0.2576 $\pm$ 1.86$\times 10^{-3}$ | 0.87 (0.95)
F10.7 | 0.0108 $\pm$ 1.26$\times 10^{-4}$ | 0.88 (0.99) |  | 0.0125 $\pm$ 1.51$\times 10^{-4}$ | 0.82 (0.97) |  | 0.0115 $\pm$ 9.40$\times 10^{-5}$ | 0.84 (0.97)
TSN | 0.0071 $\pm$ 1.00$\times 10^{-4}$ | 0.85 (0.99) |  | 0.0076 $\pm$ 1.07$\times 10^{-4}$ | 0.78 (0.95) |  | 0.0074 $\pm$ 6.90$\times 10^{-5}$ | 0.80 (0.95)

Note. — Units of the gradients are $\mu$Hz per unit activity. The numbers in
the parenthesis denote the correlation coefficients corresponding to the
smooth profiles.

Figure 10: Smoothed frequency shifts, $\delta\nu$, as a function of time for
different frequency ranges expressed in $\mu$Hz. The smoothing is performed
with a boxcar average of 181 data points. The errors are shown for every 250th
point and are multiplied by a factor of 5 to be visible on this plot. The
dashed vertical lines have the same meaning as in Figure 1.  Figure 11: Same
as Figure 10, but for different ranges of spherical harmonic degrees.

###  3.3 Frequency Shifts as a Function of Frequency and Degree

The analysis of global mode frequencies have shown that solar cycle dependent
frequency changes are predominantly a function of frequency, with the
magnitude of shifts increasing with higher frequencies (Jain et al., 2000;
Broomhall, 2017) but there are mixed inferences for the degree dependence.
Earlier studies have suggested small or no degree dependence if mode-inertia
is taken into account (Howe et al., 2002; Salabert et al., 2004) which implied
that solar cycle related changes are confined to a thin layer near the
surface. But recent studies involving low-degree modes during the deep minimum
period of cycle 24 indicated degree dependence. Particularly, $\ell$ = 0 and 2
modes indicated that the onset of cycle 24 occurred earlier in 2007 while
$\ell$ = 1 modes showed the onset to be in early 2009 (Salabert et al., 2009;
Tripathy et al., 2010c, 2011).

To determine the frequency and degree dependence of high-degree modes, we
calculate the average frequency shifts in four different frequency ranges with
an interval of 1000 $\mu$Hz in $\nu$ and two different degree ranges. The
$\nu$ and $\ell$ ranges are listed in Figures 10 and 11, respectively where we
plot the temporal variation of the smoothed frequency shifts. For consistency,
the shifts for all ranges are calculated with respect to the same set of
reference frequencies corresponding to the 12 ring days during the minimum
period of solar cycle 24 (see Section 3.2). It is evident that the magnitude
of the shifts increase with $\nu$ and $\ell$. To quantify, we calculate the
difference between the minimum and maximum phase of cycle 24. These values are
shown in Table 2 and confirm that the shifts increase with higher frequencies
and degree. Our results are in qualitative agreement with Broomhall (2017)
where frequency shifts between cycle 23 and 24 minima periods were compared as
a function of mode frequency. The figures also reveal negative shifts in some
$\nu$ and $\ell$ ranges implying that the frequencies are lower than the
reference frequencies. Thus the choice of using the reference frequencies
corresponding to the activity minimum of solar cycle 24 rendered negative
frequency shifts in the two lowest $\nu$ bands and the highest $\ell$ band.
Such dependencies of frequency shifts on $\nu$ and $\ell$ have also been
observed in global p-mode analysis and were attributed to the location of
upper tuning points from where the modes reflect back into the interior (Basu
et al., 2012; Jain et al., 2022).

The frequency shifts also demonstrate a clear solar cycle pattern except for
the lowest frequency range of 1500 – 2500 $\mu$Hz. Chaplin et al. (2001) have
also reported a flat sensitivity below 2500 $\mu$Hz. Table 2 shows the
Pearson’s linear correlation coefficients between frequency shifts and
activity proxies. We find that the frequencies are highly correlated with the
solar activity proxies except for the lowest frequency range where a signature
of anti-correlation is seen. However, given that the coefficients are small,
we interpret this as no correlation.

Table 2: Observed Differences in Frequency Shifts, $\delta\nu$, between the
Maximum and Minimum Phase of Cycle 24 and Pearson’s Linear Correlation
Coefficients, rp, between $\delta\nu$ and Activity Proxies.

| Differences in $\delta\nu$ (nHz) | rp
---|---|---
|  | MMAI | F10.7 | TSN
Frequency range ($\mu$Hz) |  |  |  |
1500 $-$ 5200 | 1246.4 $\pm$ 08.8 |  00.95 | 0.96 | 0.95
1500 $-$ 2500 |  0462.7 $\pm$ 20.1 | -0.19 | -0.13 | -0.17
2500 $-$ 3500 |  0743.4 $\pm$ 09.8 |  00.87 | 0.90 | 0.88
3500 $-$ 4500 | 2514.2 $\pm$ 16.1 |  00.98 | 09.8 | 0.97
4500 $-$ 5200 | 5028.3 $\pm$ 69.2 |  00.99 | 0.98 | 0.98
Degree range |  |  |  |
0200 $-$0900 | 1246.4 $\pm$ 08.8 |  00.95 | 0.96 | 0.95
0200 $-$0500 | 986.9 $\pm$ 09.4 |  00.96 | 0.96 | 0.95
0500 $-$0900 | 1719.4 $\pm$ 12.2 |  00.90 | 0.93 | 0.90

Note. — Differences in $\delta\nu$ and $r_{p}$ are based on the smoothed
values obtained from the boxcar average of 181 data points.

###  3.4 Hemispheric Asymmetry

Hemispheric asymmetry observed in sunspots or other solar activity proxies is
an important aspect of solar cycles as it provides important clues for
understanding dynamical processes in the interior of the Sun. However, its
origin is still not well understood. Observations illustrate that the major
changes in hemispheric flux occur around activity maximum in each cycle due to
the cancellation or transport of flux across the equator through the
meridional flow (Petrie, 2023). Simulations reveal that hemispheric asymmetry
could result due to the presence of an equator-symmetric part in the
oscillating magnetic field (Nepomnyashchikh et al., 2019) or due to a
stochastic process in the solar dynamo suggesting that the solar magnetic
field has some memory in the convection zone (Das et al., 2022). Contrary to
this, Veronig et al. (2021) argue that the solar cycle evolves independently
in the two hemispheres and suggest that the solar cycle prediction methods can
be improved by including the evolution of hemispheric sunspot number.

Figure 12: Temporal evolution of frequency shifts, $\delta\nu$, mean magnetic
activity index, MMAI, and hemispheric sunspot number, HSN, over the (a)
Northern and (b) Southern Hemispheres. All quantities are scaled to have
values between 0 and 1. The colored lines depict the smoothed profiles
obtained by applying a boxcar average over 181 data points. The errors for
$\delta\nu$ and MMAI are shown for every 250th point and are multiplied by a
factor of 5 to be visible on these plots. The dashed vertical lines have the
same meaning as in Figure 1.  Figure 13: The difference between south and
north components of frequency shifts, $\delta\nu$, mean magnetic activity
index, MMAI, and hemispheric sunspot number, HSN, normalized by their maximum
values. The colored lines depict the smoothed profiles obtained by applying a
boxcar average over 181 data points. The errors for $\delta\nu$ and MMAI are
shown for every 250th point and are multiplied by a factor of 5 to be visible
on these plots. The dashed vertical lines have the same meaning as in Figure
1.

In this context, we investigate the presence of hemispheric asymmetry,
especially during solar cycle 24, by analyzing the frequency variations over
the two hemispheres. It may be noted that the m-averaged central frequencies
obtained in global analysis are latitudinally averaged and can not be used to
study variations between hemispheres. Figure 12 shows the frequency shifts,
MMAI and HSN for both the hemispheres separately, where each quantity is
scaled to have values between 0 and 1. The frequency shifts are calculated
using equation 3 where we use the same reference period but averaged over each
hemisphere separately. The solid lines representing the smooth profiles
illustrate the asymmetric nature of these quantities. The frequency shifts
closely follow the asymmetry pattern seen in HSN and MMAI, and attain a peak
amplitude in late 2011 in the northern hemisphere, two years earlier than the
south. We also note two distinct and prominent peaks (double peaks) of similar
magnitudes in $\delta\nu$ and MMAI in the southern hemisphere but only one
strong peak in the northern hemisphere. The secondary peaks in the northern
hemispheres have reduced amplitudes as compared to the main peak. As solar
activity decreases with time, the amplitude of frequency shifts is also found
to be decreasing in both hemispheres but rapidly in the southern hemisphere.
These findings indicate that the magnetic activity in each hemispheres evolve
independently and frequency shifts follow this evolution closely. Comparing
the total magnetic energy of the Sun between the two hemispheres, Inceoglu et
al. (2017) had also concluded that the two hemispheres are decoupled. Our
study supports this assertion.

For a better understanding of the behaviour of hemispheric asymmetry, we
compute the difference between the two hemispheres as south$-$north and
present the result in Figure 13. Here positive values imply that the dominant
contribution is from the southern hemisphere and negative values from northern
hemisphere. It is evident that during cycle 24, the frequency shifts vary in
phase with MMAI and HSN attaining the maximum (minimum) amplitude around the
same epoch but some small deviations are seen during cycle 23. Nevertheless,
the Pearson’s correlation coefficient is found to be greater than 0.90 for the
entire period confirming a close linear relationship between the shifts and
activity proxies in each hemisphere. Additionally both the frequency shifts
and level of solar activity are found to be dominant in the southern
hemisphere during the descending phase of cycle 23 while cycle 24 is
characterized by a three-part structure: (i) the northern hemisphere is found
to be dominant during the ascending phase of the cycle (ii) the southern
hemisphere changes to be dominant during the maximum phase, and (iii) the
southern activity declines during the descending phase of the cycle 24 and the
northern hemisphere turns to be dominant until the cycle reaches the minimum
period. With the start of the new cycle 25, the southern hemisphere is
emerging to be dominant in activity proxies and frequency shifts. Remarkably,
these results agree well with the study of Petrie (2023) where line-of-sight
magnetograms were used to detect the large hemispheric flux fluctuations
during cycle 24.

Figure 14: Top panels display the progression of the solar cycles as seen in
frequency shifts, $\delta\nu$, at two different latitude bands for northern
(left) and southern hemispheres (right). The bottom panels illustrate
corresponding variations in the mean magnetic activity index, MMAI. All
quantities are scaled to have values between 0 and 1. We have only shown the
smoothed profiles obtained by applying a boxcar average over 181 data points.
The errors for $\delta\nu$ and MMAI are shown for every 250th point and are
multiplied by a factor of 5 to be visible on these plots. The dashed vertical
lines have the same meaning as in Figure 1.  Figure 15: Top panels display the
progression of solar cycles as seen in the smoothed frequency shifts,
$\delta\nu$, in northern and southern hemispheres at two different latitude
bands; $0^{\circ}\leq\theta\leq 15^{\circ}$ (left) and
$15^{\circ}\leq\theta\leq 30^{\circ}$ (right). The bottom panels illustrate
corresponding variations in the mean magnetic activity index, MMAI. All
quantities are scaled to have values between 0 and 1. The smoothed values are
obtained by applying a boxcar average over 181 data points. The errors for
$\delta\nu$ and MMAI are shown for every 250th point and are multiplied by a
factor of 5 to be visible on these plots. The dashed vertical lines have the
same meaning as in Figure 1.

###  3.5 Progression of the Solar Cycle at Different Latitudes

We aim at characterizing the frequency shifts during the progression of solar
magnetic cycle at different latitudes to explore similarities or differences
between them. Since the cycle starts at mid-latitudes and subsequently
migrates equatorward (or poleward), we chose two latitude bands,
$0^{\circ}\leq\theta\leq 15^{\circ}$ and $15^{\circ}\leq\theta\leq 30^{\circ}$
(we will refer them as low- and mid-latitude bands hereafter) as
representative latitudes to cover both the start of the cycle and its progress
towards the equator. Figure 14 displays the temporal evolution of the
frequency shifts (upper panels) and MMAI (bottom panels) in the northern and
southern hemispheres for these latitudinal bands separately. While in the
northern hemisphere the frequency shifts are almost comparable at the two
selected latitudes (the level of activity is marginally stronger), the shifts
in the southern hemisphere are higher at the maximum phase in the mid-latitude
band but reach similar values in both the bands around 2003 December. Soon
after the extended minimum period, the frequency shifts between the two
latitudes start to diverge and follow different patterns. The start of the
solar cycle 24 is evident in the mid latitude band where the frequencies
increased rapidly compared to values in the low latitude band. This reveals a
considerable delay in the onset of the cycle in both hemispheres at the lower
latitudes. Simoniello et al. (2016) found similar results using both global
and high-degree mode frequencies. With the progression of the solar cycle
towards the maximum phase, we do not find any anomaly as the frequency shifts
reach their maximum values at mid latitudes first. However, we do observe a
delay in the peak values between the two hemispheres probably due to the
hemispheric asymmetry seen in the magnetic activity. Additionally, the shifts
corresponding to the low latitude band also demonstrate delay between the two
hemispheres; the peak values are reached about a year earlier in the northern
hemisphere. With the onset of cycle 25, as expected, the frequency shifts
increased at the mid latitudes in both hemispheres while the shifts remained
low at the lower latitudes manifesting overlapping periods between the two
cycles. Although some differences are observed between the temporal evolution
of MMAI and $\delta\nu$, the general behavior between them remains consistent
throughout the multiple cycles with a Pearson’s linear correlation coefficient
value greater than 0.95.

When we compare the strength of the frequency shifts at the same latitude but
for different hemispheres (Figure 15), we notice that during the descending
phase of cycle 23 the southern hemisphere (shown in blue) shifts are higher
than the northern ones (red curve). This traces the strength of the magnetic
activity. We further note differences in the timing of the minimum phase
between the hemispheres as well as low- and mid-latitudes (also see, Paper I
and Simoniello et al., 2016). During cycle 24 when the activity started at
higher latitudes, the shifts in the southern hemisphere are higher compared to
the northern ones while the MMAI were comparable in both hemispheres. This
corroborates that the shifts and magnetic activity are poorly correlated
during low activity periods. But soon after 2010, the frequency shifts in the
mid latitude band of the northern hemisphere increase rapidly and peaked in
2011 November while the shifts in the southern hemisphere peaked during 2014
February. Thus there is a delay of more than 3 years between the two peaks.
However, the track of the frequency shifts in the low latitude band is
different; the shifts are higher in the northern hemisphere during the
ascending phase until 2013 August. After this epoch, the frequency shifts of
the southern hemisphere became dominant and reached its peak value in 2014
December while the peak value in the northern hemisphere was reached around
2014 October. During the maximum period of the cycle 24 (2014 October-
December), the activity levels and the frequency shifts in the mid latitude
branch of the southern hemisphere increased and continued to be dominant until
2016 February. As the activity levels further reduced, the frequency shifts
grouped together and remained in that configuration till the minimum was
reached. A similar overlapping pattern is also seen in the low latitude branch
starting from 2018 March. Form the figure it is also evident that the low- and
mid-latitude frequency shifts reached the minimum of cycle 25 in two different
epochs. Table 3 lists the timing of the minimum at these two latitude bands in
both southern and northern hemispheres. Comparing the timing of the minimum
between the two hemispheres we find that there is a delay of about 4 to 6
months for both the frequency shifts and magnetic activity which is shorter
compared to the delay observed in cycle 23 as outlined in Paper I. The only
difference is at the mid latitudes where the frequencies of the northern
hemisphere reached the minimum phase earlier than the southern hemispheres
with a lead time of about 3 months. Finally we note that at the beginning of
cycle 25 both the frequency shifts and magnetic activity are found to be
dominant in the southern hemisphere.

Table 3: Epochs of Minimum between Cycles 24 and 25 as seen in $\delta\nu$ and
MMAI in the Two Hemispheres at Low- and Mid-Latitude Bands and Differences
between the Epochs.

Latitude | $\delta\nu$ | Difference | MMAI | Difference
---|---|---|---|---
| South | North | (months) | South | North | (months)
$0^{\circ}\leq\theta\leq 15^{\circ}$ | 08/2020 | 02/2021 | 6 | 06/2020 | 11/2020 | 50
$15^{\circ}\leq\theta\leq 30^{\circ}$ | 03/2019 | 12/2018 | -3 | 11/2018 | 03/2019 | 40

Note. — Positive and negative differences indicate lead and lag time between
the two hemispheres.

##  4 Summary

Using high-degree acoustic mode frequencies from the Global Oscillation
Network Group, we investigated how the frequencies, which carry information
from the sub-surface layers, connect to the observed surface activity. The
data span over 22 years starting from 2001 July and include the maximum and
descending phase of cycle 23, cycle 24 and three initial years of cycle 25. We
find a strong correlation between the change in frequencies and the measures
of surface activity except during the minimum phase when the local magnetic
activity level decreases below 2 G. The correlation remains significantly high
on both spatial and temporal scales.

Investigating the frequency shifts as a function of $\nu$ and $\ell$, we find
that the magnitude of the shifts increase with $\nu$ and $\ell$ and are in
qualitative agreement with Broomhall (2017) where the analysis was carried out
for global modes. We have also analyzed the shifts as a function of
hemispheres separately, and observe that the shifts are dominant in southern
hemisphere during the descending phase of cycle 23, while a three-part
structure is seen in cycle 24; the shifts are dominant in the northern
hemisphere in the initial phase of the cycle, flips to the southern hemisphere
during the maximum period, and to the northern hemisphere during the
descending phase of the cycle 24. For the initial period of cycle 25, we
detect the frequency shifts in southern hemisphere to be dominant. Since the
hemispheric frequency shifts closely follow the behavior of hemispheric
sunspot number and local magnetic activity index, it demonstrates a strong
relationship between the surface activity proxies and the near surface shear
layer where the resonant modes have lower-turning points. Our analysis also
highlights the differences in the progression of solar cycle at two different
latitudinal bands. The frequency shifts in mid-latitudinal band are higher
than the low-latitudinal bands during the rising phase of the solar cycle,
however they become comparable once the maximum phase is reached. This is in
agreement with the appearance of sunspots during the cycle as they emerge
first at higher latitude and then migrate towards the equator. Since the
evolution of the frequency shifts observed between the two hemispheres have
different characteristics, we concur with a few recent assertions that the
magnetic activity in two hemispheres are decoupled and evolve independently.
Based on the analyses presented here, we emphasize that the frequency shifts
observed in high-degree modes are caused by a combination of strong fields
present at the tachocline and weak turbulent fields present in the near
surface shear layer.

##  5 Acknowledgments

We thank the reviewer for several useful comments. MB thanks Boulder Solar
Alliance for providing the REU opportunity (NSF grant 1950911) and the
National Solar Observatory. This work utilises GONG data obtained by the NSO
Integrated Synoptic Program, managed by the National Solar Observatory, which
is operated by the Association of Universities for Research in Astronomy
(AURA), Inc. under a cooperative agreement with the National Science
Foundation and with contribution from the National Oceanic and Atmospheric
Administration. The GONG network of instruments is hosted by the Big Bear
Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory,
Udaipur Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro
Tololo Interamerican Observatory. We also use data from the Solar Oscillations
Investigation/Michelson Doppler Imager on the Solar and Heliospheric
Observatory. SOHO is a mission of international cooperation between ESA and
NASA. The sunspot data is courtesy of WDC-SILSO, Royal Observatory of Belgium,
Brussels.

##  Appendix A Effect of Observational Gaps on Mode Frequencies

It is well known that gaps in observation leads to aliasing in the frequency
domain and create spurious side lobes that lead to ambiguous identification of
real frequencies and other mode parameters. However, analysis of low- and
intermediate-degree mode parameters have indicated that the bias in estimating
the mode frequencies is not significant while it is important for mode
amplitudes and widths (Chaplin et al., 2004; Broomhall & Nakariakov, 2015;
Kiefer & Roth, 2018; Keith-Hardy et al., 2019). However, as a sanity check, we
have studied the frequency shifts as a function of the duty cycle and the
results are shown in Figure 16. As expected, we do not find any relation
between frequency shifts and the duty cycle confirming that the duty cycle
does not introduce any consequential bias in the identification of high-degree
modes.

Figure 16: Frequency shifts, $\delta\nu$, as a function of duty cycle
expressed in percentage.

##  Appendix B supplemental Figures

Figure 17: Distribution of fitted modes in $\ell-\nu$ plane (commonly referred
as $\ell-\nu$ diagram) corresponding to the reference frequency which is
obtained from an average over 12 ring days corresponding to 2019 December
19-31. Figure 18: The temporal evolution of frequency shifts, $\delta\nu$,
mean magnetic activity index, MMAI, total sunspot number, TSN, and the 10.7 cm
radio flux, $F_{10.7}$. All quantities are scaled to have values between 0 and
1. We only show the smoothed values which are obtained by applying a boxcar
average over 181 data points. The errors for $\delta\nu$ and MMAI are shown
for every 250th point and are multiplied by a factor of 5 to be visible on
this plot. The dashed vertical lines have the same meaning as in Figure 1.
Figure 19: The lines show the result of linear regression fits obtained
between $\delta\nu$ and different activity proxies; bottom panel: total
sunspot number, TSN, middle panel: 10.7 cm radio flux, $F_{10.7},$ and upper
panel: mean magnetic activity index, MMAI. The different colors represent
different solar cycles and are indicated in the top panel. We do not plot the
error bars since these are small but their values are tabulated in Table 1.
Note that the red and black lines in the upper panel are mostly overlapping.

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