# Long-term Variations in Solar Activity: Predictions for Amplitude and
North–South Asymmetry of Solar Cycle 25

J. Javaraiah Bikasipura, Bengaluru-560 111, India
Formerly worked at Indian Institute of Astrophysics, Bengaluru-560 034, India.
corref, email: %**** ̵pasym-corrected.tex ̵Line ̵100<EMAIL_ADDRESS><EMAIL_ADDRESS>email<EMAIL_ADDRESS>

###### keywords:

Sun: Dynamo – Sun: Solar activity – Sun: Sunspots – (Sun:) Space Weather –
(Sun:) Solar–Terrestrial relations

##  1 Introduction

The existence of differences between the strengths of solar activity in the
Sun’s northern and southern hemispheres is well known as the north–south
asymmetry in solar-activity. It exists in most solar activity phenomena
(hath15). Besides this, it is also known that the epochs of maxima of solar
cycles are not the same in the northern and southern hemispheres, that is in
some solar cycles the maximum in one of the hemispheres does not coincide with
the maximum of the total (whole sphere/visible whole disk) activity (e.g. see
Figure 1 in jj20 and also see mci13). In some solar cycles the activity peak
occurs first in the northern hemisphere and in some other cycles, it first
occurs in the southern hemisphere. The existence of some systematic phase
difference between the cycles of activity in the northern and southern
hemispheres is also known (e.g. zolo10; ng10; mura12; mci13; rcj21). There
exist $\approx$12-year, $\approx$55-year, and $\approx$100-year periodicities
in north–south asymmetry (cob93; verma93; jg97; li02; deng16; jj20; non21).
However, the origin of these as well as the known short-term periodicities in
the north–south asymmetry (e.g. knaack04; rj15; chow16; mb16; rcj21) is not
yet clear (nort14; sc18; nep19). Our recent analysis indicates that the
$\approx$12-year and $\approx$51-year periodicities of north–south asymmetry
are manifestations of the differences in strengths of the corresponding
periodicities in northern and southern hemispheres and their origin might be
connected to the configurations of giant planets (jj20). It has been observed
that in several solar cycles there exist some differences in the maximum
epochs of sunspot number and sunspots area cycles (hath15). Recently, we have
analysed the combined sunspot-group data from Greenwich Photoheliographic
Results (GPR) during 1874 – 1976 and Debrecen Photoheliographic Data (DPD)
during 1977 – 2017 and have predicted the north–south asymmetry in the average
sunspot-group area at the maximum epoch of Sunspot Cycle 25 (jj21). Here we
analyse the aforementioned sunspot-group data and predict the north–south
asymmetry at the maximum epochs of solar cycles in the northern and southern
hemispheres. This may help in better understanding the solar dynamo and the
solar-terrestrial relationship. Here our approach is different from that of
Javaraiah (2021), where we have used the same method that was used in our
earlier articles (jj07; 2008; 2015). In those earlier articles the prediction
was made based on the existence of a good correlation between the amplitude of
a solar cycle and the sum of the areas of sunspot groups in the $0^{\circ}$ –
$10^{\circ}$ latitude interval of the southern hemisphere just after around
one year from the maximum of the preceding cycle (hereafter area-sum). Here we
study the cycle-to-cycle variations in the maxima of Solar Cycles 12 – 24 in
the average areas of sunspot groups in northern and southern hemispheres, as
well as in the whole sphere, by determining the best-fit cosine functions to
these data. In principle, from this method it is possible to make predictions
for several upcoming solar cycles. However, due to the considerable
uncertainties in the obtained best-fit cosine functions, here we make cautious
predictions for the amplitude and north–south asymmetry of Solar Cycle 25
only.

In the next section we describe the data analysis. In Section 3 we present the
results, and in Section 4 we present the conclusions and briefly discuss them.

##  2 Data Analysis

Here the data and the analysis are the same as in Javaraiah (2019). In
Javaraiah (2021) and here we have used the values of the amplitudes ([$R_{\rm
M}$]), i.e. the highest value of 13-month smoothed monthly mean sunspot
number, and the maximum epochs of sunspot cycles given by Pesnell (2018).
These were determined by Pesnell (2018) from the time series of 13-month
smoothed monthly mean values of Version 2 international sunspot number (SN)
available at www.sidc.be/silso/datafiles. The updated GPR (1874 – 1976) and
DPD (1977 – 2017) daily sunspot-group data were downloaded from the website
fenyi.solarobs.unideb.hu/pub/DPD/ (for details see gyr11; bara16; gyr17).
These data contain, besides the heliographic positions and other parameters,
the corrected whole-spot area [ msh: millionth of solar hemisphere] of each
sunspot group for its each day observation. First we determined the mean area
of sunspot groups in the Sun’s whole sphere (WSGA), northern hemisphere
(NSGA), and southern hemisphere (SSGA) during each calender month of the years
1874 – 2017, and then we determined the 13-month smoothed monthly mean values
and the corresponding standard errors ($s={\sigma}/{\sqrt{13}}$, where
$\sigma$ is the standard deviation). From the time series of the 13-month
smoothed monthly mean values we obtained the values and the corresponding
epochs of the following parameters during SN, WSGA, NSGA, and SSGA Cycles 12 –
24 (note that for brevity and convenience solar cycle variation of a parameter
is expressed as a cycle of that parameter).
$T_{\rm M}$: epoch of SN cycle maximum,
$T_{\rm W}$: epoch of WSGA cycle maximum,
$T_{\rm N}$: epoch of NSGA cycle maximum,
$T_{\rm S}$: epoch of SSGA cycle maximum,
It should be noted that usually in a solar cycle these epochs are not in an
increasing or decreasing order. However rarely there could be an order, and
even all these four epochs of a solar-cycle could be the same.
RWA: value of WSGA at $T_{\rm M}$,
RNA: value of NSGA at $T_{\rm M}$,
RSA: value of SSGA at $T_{\rm M}$,
WAM: value of maximum of a WSGA Cycle,
WNA: value of NSGA at $T_{\rm W}$,
WSA: value of SSGA at $T_{\rm W}$,
NAM: value of maximum of an NSGA Cycle,
NSA: value of SSGA at $T_{\rm N}$,
NWA: value of WSGA at $T_{\rm N}$,
SAM: value of maximum of an SSGA Cycle,
SNA: value of NSGA at $T_{\rm S}$, and
SWA: value of WSGA at $T_{\rm S}$.

We calculated a best-fit cosine function to the cycle-to-cycle modulation in
each aforementioned parameter and by extrapolating the best-fit cosine curves
we obtained the values of the above parameters for the 25th WSGA, NSGA, and
SSGA cycles. We also calculated the best-fit cosine function to the cycle-to-
cycle modulation of relative north–south asymmetry $(N-S)/(N+S)$, where $N$
and $S$ represent the values of the parameters that correspond to the northern
and southern hemispheres, respectively, at the epochs of maxima of sunspot,
WSGA, NSGA, and SSGA cycle. We predict north–south asymmetry at the
corresponding epochs of the 25th cycles. In the calculations of the best-fit
cosine functions, weights equal to the corresponding standard error of the
parameters are used. All the linear least-squares fits were calculated by
using the Interactive Digital Library (IDL) software FITEXY.PRO, which is
downloaded from the website idlastro.gsfcnasa.gov/ftp/pro/math/. An advantage
of using this software is that the errors in both abscissa and ordinate values
will be taken into account in the calculation of linear fits by the least-
squares method.

##  3 Results and Predictions

In Table 1, we have given the 13-month smoothed monthly mean values of WSGA,
NSGA, and SSGA at the epochs of maxima of SN, WSGA, NSGA, and SSGA of Cycles
12 – 14. Figure 1 shows the values of the parameters NAM, SAM, WAM, and RWA
given in this table versus time (respective epochs). The large (small) error
bars in the values of these parameters of most of the large (small) solar
cycles are due to the strong (weak) fluctuations in the monthly mean sunspot-
group area around the maximum epochs of the large (small) solar cycles. In
Table 2, we give the information obtained from this figure and Table 1 about
dominant hemisphere, phase-leading hemisphere, and the hemisphere coinciding
with the whole sphere. In the same table, the difference between the epochs of
$R_{\rm M}$ and WAM is also given. As we can see in Figure 1 and Table 2 in
some solar cycles North is dominant and in some other cycles South is
dominant. In many solar cycles the maximum in one of the hemispheres does not
coincide with the maximum of the total (whole sphere) activity. In some solar
cycles the activity peak occurs first in the northern hemisphere and in some
other cycles, it occurs first in the southern hemisphere.

Table 1.: The values [ msh] of WSGA, NSGA, and SSGA at the epochs $T_{\rm M}$,
$T_{\rm W}$, $T_{\rm N}$, and $T_{\rm S}$ of the maxima of SN, WSGA, NSGA, and
SSGA cycles ($n=$ 12, 13,…,24), respectively, determined from the
corresponding time series of 13-month smoothed monthly mean areas of the
sunspot groups in whole sphere, northern hemisphere, and southern hemisphere.
The maximum [$R_{\rm M}$] values of SN are also given (all the maximum values
are indicated with bold).

|  |  At maximum $R_{\rm M}$ of SN |  | At maximum WAM of WSGA
---|---|---|---|---
$n$ | $T_{\rm M}$ | $R_{\rm M}$ | RWA | RNA | RSA | $T_{\rm W}$ | WAM | WNA | WSA
12 | 1883.96 | ${\bf 124.4\pm 12.5}$ | $1371\pm 122$ | $414\pm 75$ | $957\pm 105$ | 1883.96 | ${\bf 1371\pm 122}$ | $414\pm 75$ | $957\pm 105$
13 | 1894.04 | ${\bf 146.5\pm 10.8}$ | $1616\pm 110$ | $621\pm 57$ | $995\pm 100$ | 1894.04 | ${\bf 1616\pm 110}$ | $621\pm 57$ | $995\pm 100$
14 | 1906.12 | ${\bf 107.1\pm 9.2}$ | $1044\pm 140$ | $761\pm 139$ | $283\pm 43$ | 1905.45 | ${\bf 1161\pm 158}$ | $745\pm 137$ | $416\pm 90$
15 | 1917.62 | ${\bf 175.7\pm 11.8}$ | $1535\pm 171$ | $829\pm 124$ | $707\pm 89$ | 1917.54 | ${\bf 1554\pm 165}$ | $853\pm 123$ | $701\pm 92$
16 | 1928.29 | ${\bf 130.2\pm 10.2}$ | $1324\pm 123$ | $631\pm 96$ | $693\pm 65$ | 1926.29 | ${\bf 1467\pm 211}$ | $809\pm 181$ | $658\pm 98$
17 | 1937.29 | ${\bf 198.6\pm 12.6}$ | $2120\pm 176$ | $1309\pm 140$ | $811\pm 108$ | 1937.29 | ${\bf 2120\pm 176}$ | $1309\pm 140$ | $811\pm 108$
18 | 1947.37 | ${\bf 218.7\pm 10.3}$ | $2641\pm 210$ | $1051\pm 121$ | $1590\pm 237$ | 1947.37 | ${\bf 2641\pm 210}$ | $1051\pm 121$ | $1590\pm 237$
19 | 1958.20 | ${\bf 285.0\pm 11.3}$ | $3441\pm 208$ | $1749\pm 164$ | $1693\pm 144$ | 1957.96 | ${\bf 3480\pm 239}$ | $1801\pm 153$ | $1679\pm 166$
20 | 1968.87 | ${\bf 156.6\pm 8.4}$ | $1556\pm 82$ | $951\pm 83$ | $605\pm 46$ | 1970.54 | ${\bf 1628\pm 93}$ | $926\pm 70$ | $701\pm 121$
21 | 1979.96 | ${\bf 232.9\pm 10.2}$ | $2121\pm 162$ | $1064\pm 139$ | $1057\pm 142$ | 1981.71 | ${\bf 2338\pm 177}$ | $1065\pm 134$ | $1274\pm 193$
22 | 1989.87 | ${\bf 212.5\pm 12.7}$ | $2269\pm 193$ | $1121\pm 124$ | $1148\pm 118$ | 1989.45 | ${\bf 2591\pm 179}$ | $1401\pm 141$ | $1190\pm 126$
23 | 2001.87 | ${\bf 180.3\pm 10.8}$ | $2157\pm 206$ | $1073\pm 111$ | $1084\pm 152$ | 2002.20 | ${\bf 2334\pm 179}$ | $951\pm 114$ | $1383\pm 141$
24 | 2014.29 | ${\bf 116.4\pm 8.2}$ | $1600\pm 116$ | $420\pm 40$ | $1180\pm 144$ | 2014.45 | ${\bf 1629\pm 114}$ | $418\pm 40$ | $1211\pm 143$
|  | At maximum NAM of NSGA |  | At maximum SAM of SSGA |
| $T_{\rm N}$ | NWA | NAM | NSA | $T_{\rm S}$ | SWA | SNA | SAM |
12 | 1882.37 | $978\pm 145$ | ${\bf 476\pm 72}$ | $501\pm 116$ | 1883.96 | $1371\pm 122$ | $414\pm 75$ | ${\bf 957\pm 105}$ |
13 | 1894.12 | $1585\pm 115$ | ${\bf 649\pm 61}$ | $935\pm 115$ | 1893.96 | $1585\pm 111$ | $577\pm 55$ | ${\bf 1008\pm 99}$ |
14 | 1906.04 | $1109\pm 143$ | ${\bf 822\pm 135}$ | $287\pm 45$ | 1907.45 | $1098\pm 129$ | $512\pm 81$ | ${\bf 585\pm 103}$ |
15 | 1917.71 | $1492\pm 178$ | ${\bf 855\pm 116}$ | $638\pm 84$ | 1917.62 | $1535\pm 171$ | $829\pm 124$ | ${\bf 707\pm 89}$ |
16 | 1926.29 | $1467\pm 211$ | ${\bf 809\pm 181}$ | $658\pm 98$ | 1928.21 | $1338\pm 121$ | $584\pm 102$ | ${\bf 754\pm 57}$ |
17 | 1937.54 | $2088\pm 195$ | ${\bf 1374\pm 158}$ | $714\pm 109$ | 1938.62 | $1855\pm 164$ | $715\pm 91$ | ${\bf 1141\pm 151}$ |
18 | 1949.62 | $2060\pm 196$ | ${\bf 1200\pm 114}$ | $860\pm 153$ | 1947.45 | $2625\pm 215$ | $1011\pm 129$ | ${\bf 1614\pm 232}$ |
19 | 1959.54 | $2847\pm 233$ | ${\bf 2220\pm 227}$ | $627\pm 77$ | 1957.79 | $3428\pm 257$ | $1673\pm 183$ | ${\bf 1755\pm 142}$ |
20 | 1967.62 | $1596\pm 142$ | ${\bf 1108\pm 100}$ | $488\pm 73$ | 1969.87 | $1549\pm 113$ | $767\pm 103$ | ${\bf 783\pm 105}$ |
21 | 1979.37 | $2067\pm 164$ | ${\bf 1211\pm 111}$ | $856\pm 117$ | 1981.96 | $2293\pm 199$ | $976\pm 123$ | ${\bf 1317\pm 179}$ |
22 | 1989.45 | $2591\pm 179$ | ${\bf 1401\pm 141}$ | $1190\pm 126$ | 1991.54 | $2405\pm 167$ | $894\pm 129$ | ${\bf 1511\pm 189}$ |
23 | 2000.96 | $1831\pm 167$ | ${\bf 1122\pm 119}$ | $709\pm 65$ | 2002.20 | $2334\pm 179$ | $951\pm 114$ | ${\bf 1383\pm 141}$ |
24 | 2011.71 | $905\pm 116$ | ${\bf 687\pm 100}$ | $218\pm 40$ | 2014.45 | $1629\pm 114$ | $418\pm 40$ | ${\bf 1211\pm 143}$ |

table1

Figure 1.: Values given in Table 1 versus time (corresponding epochs) of the
parameters: NAM, i.e. the values of maxima of NSGA Cycles 12 – 24; SAM, i.e.
the values of maxima of SSGA Cycles 12 – 24; WAM, i.e. the values of maxima of
WSGA Cycles 12 – 24; and RWA, i.e. the values of the 13-month smoothed monthly
mean sunspot-group areas at the maxima of Sunspot Cycles 12 – 24.f1 Table 2.:
The information obtained by using Table 1 and Figure 1 about dominant
hemisphere (DH), phase-leading hemisphere (LH), and whether the hemisphere
coincides/is close to the whole sphere (CH) in a solar cycle ($n$). The
corresponding differences between the values of parameters and between the
epochs are given within parentheses. In the last column the differences
between $T_{\rm M}$ and $T_{\rm W}$ are also given. In the 2nd column the
“same” implies that there is no phase difference. In the 4th and 5th columns a
zero/small absolute value indicates that the corresponding peaks are
coincident/very close.

$n$ | DH (NAM$-$SAM) | LH ($T_{\rm N}-T_{\rm S}$) | CH ($T_{\rm N}-T_{\rm W})/(T_{\rm S}-T_{\rm W}$) | $T_{\rm M}-T_{\rm W}$
---|---|---|---|---
12 | south ($-481$) | north ($-1.59$) | south ($-1.59$/$0.0$) | $0.0$
13 | south ($-359$) | same ($0.16$) | both ($0.08/-0.08$) | $0.0$
14 | north ($237$) | north ($-1.4$) | north ($0.59/2.0$) | $-0.67$
15 | north ($148$) | same ($0.09$) | both ($0.17/0.08$) | $-0.08$
16 | north ($55$) | north ($-1.92$) | north ($0.0/1.92$) | $-2.0$
17 | north ($233$) | north ($-1.08$) | north ($0.25/1.33$) | $0.0$
18 | south ($-414$) | south ($2.17$) | south ($2.25/0.08$) | $0.0$
19 | north ($465$) | south ($1.75$) | south ($1.58/-0.17$) | $-0.24$
20 | north ($325$) | north ($-2.25$) | south ($-2.92/-0.67$) | $1.67$
21 | south ($-106$) | north ($-2.59$) | south ($-2.34/0.25$) | $1.75$
22 | south ($-110$) | north ($-2.09$) | north ($0.0/2.09$) | $-0.42$
23 | south ($-261$) | north ($-1.24$) | south ($-1.24/0.0$) | $0.33$
24 | south ($-524$) | north ($-2.74$) | south ($-2.74/0.0$) | $0.16$

table2

Figure 2.: (a) WSAn of an even-numbered WSGA cycle versus WSAn+1 of following
odd-numbered WSGA cycle and (b) SNAn of an even-numbered SSGA cycle versus
SNAn+1 of following odd-numbered SSGA cycle. The horizontal and vertical error
bars represent the standard errors of the abscissa and the ordinate,
respectively; both are taken into account in the calculations of linear least-
square fits. The continuous line represents the best-fit linear relationship.
The dotted lines (red) are drawn at one-rms (root-mean-square deviation)
levels. The best-fit linear equation and the values of correlation coefficient
[$r$], $\chi^{2}$ and the corresponding probability (PROB), and rms are also
shown. The filled squares (red) represent the predicted values of WSA and SNA
at the epochs of maxima of WSGA Cycle 25 and SSGA Cycle 25.f2

In the case of sunspot number, the amplitude of Solar Cycle 21 is larger than
that of Solar Cycle 22, whereas, as can be seen in Figure 1, in the case of
sunspot area the behavior is opposite to that of sunspot number, i.e. Solar
Cycle 22 is larger than Solar Cycle 21 (hath15). This difference could be
because the ratio of small to large numbers of sunspots/sunspot groups may be
larger in Solar Cycle 21 than in Solar Cycle 22. The aforementioned opposite
behavior is also found in the cases of whole sphere, northern, and southern
hemispheres’ mean areas of sunspot groups (i.e. RWA, RNA, and RSA here) at the
epochs of maxima of Solar Cycles 21 and 22 (see also jj19). As we can see in
Table 1, except in the case of SNA, this property exists in the case of all
remaining parameters. The value of SNA of SSGA Cycle 21 is larger than that of
SSGA Cycle 22.

According to the Gnevyshev–Ohl rule (G–O: go48), an odd-numbered sunspot cycle
is larger than its preceding even-numbered sunspot cycle. However, sunspot-
cycle pair (22, 23) violated this rule, i.e. $R_{\rm M}$ of Sunspot Cycle 22
is larger than that of Sunspot Cycle 23. As we can see in Table 1, WSA values
of WSGA cycle pair (22, 23) and SNA values of SSGA cycle pair (22, 23)
satisfied the G–O rule and the corresponding pairs of each of the remaining
all parameters violated the G–O rule (here the G–O rule in WSA and SNA
tentatively means that the cycle-to-cycle modulations in WSA and SNA imitate
the G–O rule of solar cycles).

Figure 2 shows the linear relationships between WSA pairs of even- and odd-
numbered WSGA cycles and between SNA pairs of even- and odd-numbered SSGA
cycles. Besides the obtained linear equation, the values of $\chi^{2}$ and the
corresponding probabilities (PROB), and rms (root-mean-square deviation) are
also shown in Figure 2. The probabilities of the corresponding values of
$\chi^{2}$ of the best-fit linear relationships are somewhat small. Note that
the $\chi^{2}$-probability is a scalar (between zero and unity) giving the
probability that a correct model would give a value equal or larger than the
observed $\chi^{2}$. A small value of PROB indicates a poor fit, perhaps
because the errors are underestimated.

By using the linear relationships shown in Figure 2 we get the values $1348\pm
161$ msh and $542\pm 177$ msh for WSA of WSGA Cycle-25 and SNA of SSGA Cycle
25, respectively. These values are larger than the corresponding observed
values of WSGA Cycle 24 and SSGA Cycle 24 implying satisfaction of the G–O
rule. By using the predicted value of WSA and the existence of reasonably good
correlations (good linear relationships) between WSA and SAM, SAM and SWA, SWA
and WAM, WAM and RWA, and RWA and $R_{\rm M}$ we obtained the values for WAM
of WSGA Cycle 25 and $R_{\rm M}$ of Sunspot Cycle 25, which were found to be
much larger than the corresponding values of the respective 24th cycles. We
have not given them here because the aforementioned predicted values of WSA of
WSGA Cycle 25 and SNA of SSGA Cycle 25 are to some extent unreliable. It
should be noted that without prior knowledge about non-violation of the G–O
rule by an upcoming even- and odd-numbered solar cycle pair, by using the G–O
rule it is not possible to predict the amplitude of the odd-numbered solar
cycle.

We calculated the cosine fits to the values of the parameters of WSGA, NSGA,
and SSGA Cycles 12 – 24 given in Table 1. In Figures 3 – 6, we show the best-
fit cosine curves. In Table 3, we give the results obtained by extrapolating
the best-fit cosine curves of different parameters, i.e. the value of period,
the predicted corresponding 25th cycle value of a parameter, and the values of
rms and $\chi^{2}$. The cosine fit of each parameter seems to be reasonably
good because only one (mostly the data point of Cycle 19) or at most two data
points are outliers (away from the one-rms level). In principle, we can
extrapolate the best-fit cosine curves for several cycles. However, the
$\chi^{2}$ values are large, i.e. the cosine best fits are not very accurate.
Therefore the corresponding results are only suggestive rather than
compelling. Hence we restricted our conclusions to only the predictions for
the corresponding 25th cycle (although the values obtained for the
corresponding 26th cycle are also shown in Figures 3 – 6).

We can see in Table 3 that the periodicity in a parameter of the northern
hemisphere is $\approx$132 years, whereas it is 54 – 66 years in a parameter
of the southern hemisphere, i.e. approximately half of that of the northern
hemisphere. This is consistent with the similar results noticed in the earlier
analysis (jj19). Morlet-wavelet analysis were also suggested that a
$\approx$51-year periodicity in the 13-month smoothed area of sunspot groups
of the northern hemisphere is much weaker than that of the southern hemisphere
(jj20).

The cosine fits to the data of RWA, WAM, NWA, and SWA are found to be much
more uncertain, and therefore we do not show them. We get the corresponding
25th cycle values of these parameters from the predicted values of the
parameters given in Table 3.
${\rm RWA}={\rm RNA}+{\rm RSA}=1317\pm 123$ msh,
${\rm WAM}={\rm WNA}+{\rm WSA}=1402\pm 124$ msh,
${\rm NWA}={\rm NAM}+{\rm NSA}=1082\pm 100$ msh,
${\rm SWA}={\rm SNA}+{\rm SAM}=1354\pm 116$ msh.

The rms appears large for the corresponding predicted value of a parameter of
Solar Cycle 25 given Table 3. However, the range of values of the parameter is
large and the predicted value is at the minimum level of the corresponding
long-term cycle (Gleissberg cycle). The uncertainties in the values of RWA,
WAM, NWA, and SWA obtained above are reasonably small. They are determined as
$\sqrt{({\rm rms}_{\rm{}_{N}}^{2}+{\rm rms}_{\rm{}_{S}}^{2})/13}$, where ${\rm
rms}_{{}_{\rm N}}$ and ${\rm rms}_{{}_{\rm S}}$ are the rms values of northern
and southern hemispheres’ parameters, respectively.

We also determined the best-fit cosine function to the values of $R_{\rm M}$
of SN Cycles 12 – 24. It is shown in Figure 7, and the details are also given
in the last row of Table 3. The obtained value for $R_{\rm M}$ of SN Cycle 25
is slightly larger than that of SN Cycle 24, but the former has a large
uncertainty (rms value). Moreover, the cosine fit of $R_{\rm M}$ is not good,
since the corresponding $\chi^{2}$ is very large.

Figure 3.: Continuous curve represents the best-fit cosine function to the
values (filled circles) (a) of RNA and (b) of RSA at the epochs of maxima of
Sunspot Cycles 12 – 24, i.e the values of NSGA and SSGA at $T_{\rm M}$. The
dotted curve (red) represents the one-rms level. The extrapolated portion is
shown as a dashed curve and the filled squares (red) represent the predicted
values of RNA and RSA at the maximum epochs of Sunspot Cycles 25 and 26. The
period (in number of solar cycles) of the cosine function is also shown. See
also the details given in Table 3.f3

Figure 4.: Continuous curve represents the best-fit cosine function to the
values (filled circles) (a) of WNA and (b) of WSA of WSGA Cycles 12 – 24, i.e.
the values of NSGA and SSGA at $T_{\rm W}$. The dotted curve (red) represents
the one-rms level. The extrapolated portion is shown as a dashed curve and the
filled squares (red) represent the predicted values of WNA and WSA at the
maximum epochs of WSGA Cycles 25 and 26. The period (in number of solar
cycles) of the cosine function is also shown. See also the details given in
Table 3.f4

Figure 5.: Continuous curve represents the best-fit cosine function to the
values (filled circles) (a) NAM, i.e. the maximum values of NSGA Cycles 12 –
24 and (b) NSA, i.e. the values of SSGA at $T_{\rm N}$. The dotted curve (red)
represents the one-rms level. The extrapolated portion is shown as a dashed
curve and the filled squares (red) represent the predicted values of NAM and
NSA at the maximum epochs of NSGA Cycles 25 and 26. The period (in number of
solar cycles) of the cosine function is also shown. See also the details given
in Table 3.f5

Figure 6.: Continuous curve represents the best-fit cosine function to the
values (filled circles) (a) SNA, i.e. the values of NSGA at $T_{\rm S}$, and
(b) SAM, i.e. the maximum values of SSGA Cycles 12 – 24. The dotted curve
(red) represents the one-rms level. The extrapolated portion is shown as a
dashed curve and the filled squares (red) represent the predicted values of
SNA and SAM at the maximum epochs of SSGA Cycles 25 and 26. The period (in
number of solar cycles) of the cosine function is also shown. See also the
details given in Table 3.f6 Figure 7.: Continuous curve represents the best-
fit cosine function to the values (filled circles) of $R_{\rm M}$, i.e. the
values of the amplitudes of Sunspot Cycles 12 – 24. The dotted curve (red)
represents the one-rms level. The extrapolated portion is shown as a dashed
curve and the filled squares (red) represent the predicted values of SN at the
epochs of maxima of Sunspot Cycles 25 and 26. The period (in number of solar
cycles) of the cosine function is also shown. See also the details given in
Table 3.f7 Table 3.: The periods of best-fit cosine functions of the different
parameters given in Table 1 and the parameter values ( msh, in the case of a
parameter of area) in Solar Cycle 25 predicted by extrapolating the best-fit
cosine curves. The corresponding values of rms and $\chi^{2}$ are also given.
Note that the average period of solar cycles is $11.03\pm 1.18$-year (Pesnell,
2018.)

Parameter | Period | Period | Pred. Cycle 25 | rms | $\chi^{2}$
---|---|---|---|---|---
| [cycles] | [years] |  |  |
RNA | $\approx 12$ | $\approx 132$ | 491 | 255 | 66
RSA | $\approx 6$ | $\approx 66$ | 826 | 358 | 118
WNA | $\approx 12$ | $\approx 132$ | 522 | 283 | 69
WSA | $\approx 5.8$ | $\approx 64$ | 880 | 345 | 69
NAM | $\approx 12$ | $\approx 132$ | 627 | 294 | 46
NSA | $\approx 4.9$ | $\approx 54$ | 455 | 210 | 64
SNA | $\approx 12$ | $\approx 132$ | 461 | 242 | 47
SAM | $\approx 5.5$ | $\approx 61$ | 893 | 342 | 67
$R_{\rm M}$ | $\approx 12.3$ | $\approx 136$ | 123 | 33 | 141

table3

Figure 8.: Scatter plot of RWA during Sunspot Cycles 12 – 24 versus WAM during
WSGA Cycles 12 – 24. The continuous line represents the linear least-square
best-fit to the data. The dotted line (red) represents the one-rms level. The
obtained linear equation and values of the corresponding $r$, rms, $\chi^{2}$,
and PROB are given. The filled square (red) represents the predicted value of
RWA, i.e. the value of WSGA at the epoch of $R_{\rm M}$ of Solar Cycle 25.f8
Figure 9.: predicted values of the mean sunspot-group area at $T_{\rm M}$,
$T_{\rm W}$, $T_{\rm N}$, and $T_{\rm S}$, i.e. at the epochs of maxima of the
25th SN, WSGA, NSGA, and SSGA cycles, respectively (filled symbol) and the
corresponding observed values of the 24th cycles (open symbol) versus the
maximum epochs (the differences between the epochs of $X$-axis are not equal).
The symbols square, star, and circle represent the corresponding WSGA, NSGA
and SSGA values, respectively. First column: RWA, RNA, and RSA at the epochs
of maximum of SN Cycles 24 and 25, Second column: WAM, WNA, and WSA at the
maximum of WSGA Cycles 24 and 25, Third column: NWA, NAM, and NSA at the
maximum of NSGA Cycles 24 and 25, and Fourth column: SWA, SNA, and SAM at the
maximum of SSGA Cycles 24 and 25. The filled triangle represents the value of
RWA obtained from the WAM–RWA relationship shown in Figure 8.f9

Figure 8 shows the correlation between WAM and RWA. Obviously, there exists a
high correlation between WAM and RWA. The corresponding linear least-squares
best-fit is highly statistically significant ($\chi^{2}=2$, PROB = 0.998).
Using the above predicted value of WAM in WAM–RWA relationship shown in Figure
8, we get the value $1348\pm 96$ msh for RWA of SN Cycle 25, which seems to be
more reliable than the value of RWA obtained above by summing the values
predicted for RNA and RSA (note that the value of $\chi^{2}$ corresponding to
the best-fit cosine function of RSA is very high). However, the two values are
almost equal.

In Figure 9 we compare the predicted values of the parameters at the epochs of
the maxima of the 25th SN, WSGA, NSGA, and SSGA cycles with the observed
values of the corresponding 24th cycles. As we can see in this figure there is
an indication that except at the peaks of both the NSGA Cycles 24 and 25 where
the activity in the northern hemisphere looks to be negligibly larger than
that in southern hemisphere, in all the remaining other occasions, including
at the epoch of $R_{\rm M}$, the southern hemisphere is dominant. The
difference between the predicted values of all northern and southern
hemispheres’ parameters of the 25th cycles at a given epoch is not significant
with respect to the large uncertainties in these values, suggesting that there
exists no significant north–south asymmetry in Solar Cycle 25, whereas most of
the corresponding differences of the 24th cycle seem to be statistically
significant. The overall pattern of Solar Cycle 25 would be closely similar to
that of Solar Cycle 24. The value predicted here for RWA is much larger than
the value $\approx$701 msh of RWA (i.e. $A_{\rm W}$) predicted by Javaraiah
(2021) by a different method, which is much smaller than the RWA of Solar
Cycle 24. The above predicted values of RWA and WSA are reasonably smaller
than the corresponding values of Solar Cycle 24, and there are no significant
differences in the corresponding values of WNA. The predicted value of WAM of
Solar Cycle 25 is also slightly smaller than that of Solar Cycle 24,
suggesting that on average Solar Cycle 25 in sunspot-group area would be
smaller than the corresponding Solar Cycle 24. However, the uncertainties in
the predicted values of Solar Cycle 25 are large. Therefore all the above
results are only suggestive rather than compelling.

Figure 10.: Scatter plot of 13-month smoothed monthly mean northern
hemisphere’s sunspot number (${\rm SN}_{\rm N}$) versus NSGA during the period
1992 – 2017 (300 data points). The continuous line (red) represents the linear
least-squares best-fit to the data. The dotted line (red) represents the one-
rms level. The obtained linear equation and the values of the corresponding
$r$, rms, $\chi^{2}$, and PROB are given. The predicted values of ${\rm
SN}_{\rm N}$ at the epochs of $R_{\rm M}$, WAM, NAM, and NSA of the 25th SN,
WSGA, NSGA, and SSGA cycles are also shown.f10 Figure 11.: Scatter plot of
13-month smoothed monthly mean southern hemisphere’s sunspot number (${\rm
SN}_{\rm S}$) versus SSGA during the period 1992 – 2017 (300 data points). The
continuous line (red) represents the linear least-squares best fit to the
data. The dotted line (red) represents the one-rms level. The obtained linear
equation and the values of the corresponding $r$, rms, $\chi^{2}$, and PROB
are given. The predicted values of ${\rm SN}_{\rm S}$ at the epochs of $R_{\rm
m}$, WAM, SNA, and SAM of 25th SN, WSGA, NSGA, and SSGA cycles are also
shown.f11 Figure 12.: Scatter plot of 13-month smoothed monthly mean whole-
sphere (total) sunspot number (${\rm SN}_{\rm T}$) versus WSGA during the
period 1874 – 2017 (1713 data points). The continuous line (red) represents
the linear least-squares best-fit to the data. The dotted line (red)
represents the one-rms level. The obtained linear equation and the values of
the corresponding correlation coefficient $r$, rms, $\chi^{2}$, and PROB are
given. The predicted values of ${\rm SN}_{\rm T}$ at the epochs of $R_{\rm
M}$, WAM, NAM, and SAM of the 25th SN, WSGA, NSGA, and SSGA cycles are also
given. The value of ${\rm SN}_{\rm T}$ obtained by using RWA predicted from
the WAM–RWA relationship shown in Figure 8 is also given.f12 Table 4.:
Predictions for the values of SN in northern hemisphere (${\rm SN}_{\rm N}$),
southern hemisphere (${\rm SN}_{\rm S}$), and whole sphere (${\rm SN}_{\rm
T}$) at the epochs $T_{\rm M}$, $T_{\rm W}$, $T_{\rm N}$, and $T_{\rm S}$ of
the maxima of 25th SN, WSGA, NSGA, and SSGA cycles, respectively, by using in
the linear relationships shown in Figures 10 – 12 the values of the different
parameters predicted from their respective best-fit cosine functions. The
symbol a indicates that the value of RWA is obtained from WAM–RWA relationship
shown in Figure 8 (note that ${\rm SN}_{\rm T}$ at $T_{\rm M}$ is the same as
$R_{\rm M}$). In the last column of the bottom panel, we give the values of
${\rm SN}_{\rm T}$ obtained by summing the corresponding predicted values of
${\rm SN}_{\rm N}$ (top panel) and ${\rm SN}_{\rm S}$ (middle panel).

Predictions for the values of ${\rm SN}_{\rm N}$ |
---|---
At the epoch | Predicted value | Used value of |
$T_{\rm M}$ | $45\pm 4.4$ | RNA |
$T_{\rm W}$ | $48\pm 4.4$ | WNA |
$T_{\rm N}$ | $57\pm 4.4$ | NAM |
$T_{\rm S}$ | $43\pm 4.4$ | SNA |
Predictions for the values of ${\rm SN}_{\rm S}$ |
At the epoch | Predicted value | Used value of |
$T_{\rm M}$ | $73\pm 6.8$ | RSA |
$T_{\rm W}$ | $78\pm 6.8$ | WSA |
$T_{\rm N}$ | $41\pm 6.8$ | NSA |
$T_{\rm S}$ | $79\pm 6.8$ | SAM |
Predictions of for the values of ${\rm SN}_{\rm T}$ |
At the epoch | Predicted value | Used value of | ${\rm SN}_{\rm N}+{\rm SN}_{\rm S}$
$T_{\rm M}$ | $127\pm 12$ | RWA | $\approx 118$
$T_{\rm W}$ | $135\pm 12$ | WAM | $\approx 126$
$T_{\rm N}$ | $106\pm 12$ | NWA | $\approx 98$
$T_{\rm S}$ | $131\pm 12$ | SWA | $\approx 122$
$T_{\rm M}$ | $130\pm 12$ |  RWAa |

table4

Figure 13.: Predicted values of SN given in Table 4 at $T_{\rm M}$, $T_{\rm
W}$, $T_{\rm N}$, and $T_{\rm S}$, i.e. at the epochs of maxima of 25th SN,
WSGA, NSGA, and SSGA cycle, respectively (filled symbols) and the observed
values of SN of the corresponding 24th cycles (open symbols) versus the
maximum epochs (the differences between the epochs of $X$-axis are not equal).
The symbols square, star, and circle represent ${\rm SN}_{\rm T}$, ${\rm
SN}_{\rm N}$, and ${\rm SN}_{\rm S}$ values, respectively (note that ${\rm
SN}_{\rm T}$ at $T_{\rm M}$ is the same as $R_{\rm M}$). The first, second,
third, and fourth columns contain the values of ${\rm SN}_{\rm T}$, ${\rm
SN}_{\rm N}$, and ${\rm SN}_{\rm S}$ at the maximum epochs of ${\rm SN}_{\rm
T}$, WSGA, NSGA, and SSGA Cycles 24 and 25, respectively. The cross represents
the value of ${\rm SN}_{\rm T}$ obtained by summing the predicted values of
${\rm SN}_{\rm N}$ and ${\rm SN}_{\rm S}$. The filled triangle represents the
value of $R_{\rm M}$ obtained by using the value of RWA obtained from WAM–RWA
relationship shown in Figure 8f13

As shown in Figures 5 – 7 of Javaraiah (2021), here we also attempted to
determine the linear relations between RWA and $R_{\rm M}$, RWA and RNA, and
RWA and RSA. Here in the calculations of linear least-squares fit the errors
in the values of both abscissa and ordinate are taken into account, whereas in
that earlier paper, only errors in the ordinate values were considered. None
of the corresponding linear least-squares best fits was found to be reasonably
statistically significant, that is, the corresponding values of $\chi^{2}$ are
found to be very large and, obviously the corresponding values of PROB are
found to be very small ($\leq$0.07). If we use in Equation 5 of Javaraiah
(2021) the values $\approx$1317 msh and $\approx$1348 msh obtained/predicted
above for RWA (i.e. $A_{\rm W}$ in that paper), then we get $129\pm 19$ and
$131\pm 19$, respectively, for $R_{\rm M}$ of Sunspot Cycle 25. Each of these
values of $R_{\rm M}$ of Sunspot Cycle 25 is slightly larger than the observed
value of $R_{\rm M}$ of Sunspot Cycle 24. As already mentioned above the
predicted values of RWA of Solar Cycle 25 are considerably smaller than the
observed value of RWA of Solar Cycle 24. However, this comparison seems to be
not genuine because the latter is an outlier (far from one-rms level) in the
$A_{\rm W}$–$R_{\rm M}$ linear relationship shown in Figure 6 of Javaraiah
(2021).

We determined the best-fit linear relation between the 13-month smoothed
monthly mean values of sunspot-group area and SN. Figures 10, 11, and 12 show
the relations between NSGA and ${\rm SN}_{\rm N}$, SSGA and ${\rm SN}_{\rm
S}$, and WSGA and ${\rm SN}_{\rm T}$, where the subscripts N, S, and T
indicate north, south, and total. All these linear relationships are
statistically highly significant (the values of PROB are high). By using in
these relations the values predicted above for RWA, RNA, RSA, WNA, WSA, NAM,
and SAM we get/predict the values of ${\rm SN}_{\rm N}$, ${\rm SN}_{\rm S}$,
and ${\rm SN}_{\rm T}$ (it is nothing but $R_{\rm M}$ at the epoch of $R_{\rm
M}$/RWA) of Sunspot Cycle 25 and the corresponding values at the maxima of
25th NSGA cycle, SSGA cycle, and WSGA cycle. The predicted values are also
shown in Figures 10 – 12 and listed in Table 4, and in Figure 13 we compare
these predicted values of Sunspot Cycle 25 with the corresponding observed
values of Sunspot Cycle 24. The values shown in Figure 13 for $R_{\rm M}$ of
Solar Cycle 25 are considerably larger than the value ($86\pm 18$) predicted
earlier by Javaraiah (2021) by using the value predicted for RWA from a
different method. As we can see in Figure 13 the values predicted here for
$R_{\rm M}$ of Solar Cycle 25 also appear to be slightly larger than that of
Solar Cycle 24 (the corresponding difference is negligible if we take the
uncertainty into account). There is also a suggestion that the activity in the
southern hemisphere is dominant in Solar Cycle 25, similarly to the case of
sunspot-group area shown in Figure 9, except at the peak of NSGA, in all the
remaining cases including the case of $R_{\rm M}$. The southern hemisphere’s
peak may coincide with whole sphere’s peak. In fact, the overall pattern of
the predicted values of Solar Cycle 25 closely resembles to that of Solar
Cycle 24. Opposite to that seen in Figure 9 for the case of sunspot-group
area, in Figure 13 there is a suggestion that on average Sunspot Cycle 25
would be slightly larger than Sunspot Cycle 24. The reason for this difference
is not clear, but one possibility is that the ratio of small to large sunspots
during the maximum of Solar Cycle 25 would be larger than that during the
maximum of Solar Cycle 24. However, the differences between Solar Cycles 24
and 25 in sunspot-group area and also between Sunspot Cycles 24 and 25 are
statistically insignificant. On the other hand properties of sunspot number
and sunspots area cycles are not exactly the same. For example, the well-known
Waldmeier effect of sunspot cycles is not present in the cycles of sunspot-
group area (dgt08; jj19). In several solar cycles, there exist some
differences in the maximum epochs of sunspot-number and sunspot-area cycles
and there are also differences in the relative heights of the peaks of some
sunspot-number and sunspot-area cycles.

Figure 14.: The continuous curve represents the best-fit cosine function to
the values of the relative north–south asymmetry (a) in RWA, i.e.
(RNA$-$RSA)/(RNA$+$RSA), (b) in WAM i.e. (WNA$-$WSA)/(WNA$+$WSA), (c) in NWA,
i.e. (NAM$-$NSA)/(NAM$+$NSA), and (d) in SWA, i.e. (SNA$-$SAM)/(SNA$+$SAM).
The dotted curve (red) represents the one-rms level. The period (in cycles) of
the best-fit cosine function is also shown. The dashed portion of the curve is
obtained by extrapolation and the filled squares (red) represent the predicted
value of the north–south asymmetry in the corresponding parameters of Solar
Cycles 25 and 26.f14 Figure 15.: Open square (filled square) and open circle
(filled circle) represent the values of north–south asymmetry in the observed
(predicted) sunspot-group area and SN shown in Figures 9 and 13, respectively,
at $T_{\rm M}$, $T_{\rm W}$, $T_{\rm N}$, and $T_{\rm S}$, i.e. at the epochs
of maxima of the 24th (25th) ${\rm SN}_{\rm T}$, WSGA, NSGA, and SSGA cycles,
respectively (no real-time differences are indicated along the $X$-axis). The
filled triangle represents the values of the north–south asymmetry in the
sunspot-group area at the maximum epochs of the 25th ${\rm SN}_{\rm T}$, WSGA,
NSGA, and SSGA cycles obtained by extrapolating the best-fit cosine curves
shown in Figure 14.f15

As already mentioned above, because the cosine best-fits of most of the
parameters are not very accurate, although the area–SN linear relationships
have small rms values it may be still difficult to draw a definite conclusion
from these predicted values about the dominant hemisphere. On the other hand,
uncertainty in the relative north–south asymmetry of any two quantities is
much smaller than those of the corresponding absolute values (jg97). Therefore
we calculated the relative north–south asymmetry in RWA, WAM, NWA, and SWA and
determined the cosine fit to the relative north–south asymmetry in each of
these quantities. It would help, besides to find the long-term periodicity in
north–south asymmetry of each quantity, to predict the north–south asymmetry
in the corresponding quantities of future cycles. Figures 14a, 14b, 14c, and
14d show the best-fit cosine curves of the relative north–south asymmetry
(RNA$-$RSA)/(RNA$+$RSA) in RWA, (WNA$-$WSA)/(WNA$+$WSA) in WAM,
(NAM$-$NSA)/(NAM$+$NSA) in NWA, and (SNA$-$SAM)/(SNA$+$SAM) in SWA,
respectively. The best-fit cosine curves suggest the existence of
$\approx$6-cycle (66-year), $\approx$6-cycle (66-year), $\approx$4.5-cycle
(50-year), and $\approx$5.5-cycle (61-year) periodicities for the north–south
asymmetry in RWA, WAM, NWA, and SWA, i.e. at the epochs of $R_{\rm M}$, WAM,
NAM, and SAM, respectively. We have extrapolated the best-fit cosine curves
and predicted the relative north–south asymmetry in these quantities at the
maximum epochs of the 25th SN, WSGA, NSGA, and SSGA cycles. We find the values
$-0.08\pm 0.19$, $-0.14\pm 0.17$, $0.19\pm 0.16$, and $-0.07\pm 1.0$ for the
north–south asymmetry in RWA, WAM, NWA, and SWA of these cycles, respectively.
Most of these values do not significantly differ from zero. Figure 15 shows,
besides these values, the values of north–south asymmetry in sunspot group
area and sunspot number shown in Figures 9 and 13, respectively, that is, in
this figure essentially we compare the predicted values of the aforementioned
25th cycle with the observed values of the corresponding 24th cycle. As we can
see in this figure, the pattern of the predicted value of 25th cycle is almost
the same as that of the observed values of the 24th cycle, that is there is a
suggestion that except at the epoch of NAM, in all the remaining cases,
including at the epoch of $R_{\rm M}$, the southern hemisphere is dominant
during both Solar Cycles 24 and 25. There is a suggestion that overall the
asymmetry in Solar Cycle 25 would be smaller than that of Solar Cycle 24.

##  4 Discussion and Conclusions

In most of the solar cycles, activity in the northern and southern hemispheres
peaks at different times, that is at least one of the hemispheric maxima
frequently does not coincide with the whole sphere (total) maximum. Typically,
one hemisphere peaks well before the other, and the maximum of the whole-Sun
area or sunspot number can be the peak in one hemisphere, or the peak in the
other hemisphere, or something in between. Predictions for each hemisphere’s
maximum and the corresponding north–south asymmetry of a solar cycle may help
to understand the mechanisms of the solar cycle, the solar-terrestrial
relationship, and solar-activity influence on space weather. Here we analysed
the GPR and DPD sunspot-group data during 1874 – 2017 and studied the cycle-
to-cycle variations in the values of WSGA, NSGA, and SSGA at the epochs of the
maxima of Sunspot (SN) Cycles 12 – 24 and at the epochs of the maxima of the
12th to 24th WSGA, NSGA, and SSGA cycles. We find that, except in the cases of
SSGA at the peak of WSGA and NSGA at the peak of SSGA, in all the remaining
cases the G–O rule is violated by the solar cycle pair (22, 23). Using the G–O
rule, we obtained a value of SSGA at the maximum of WSGA Cycle 25 that is much
larger than the value of SSGA at the maximum of WSGA Cycle 24. Similarly, we
obtained a value of NSGA at the maximum of SSGA Cycle 25 that is also found to
be larger than the value of NSGA at the maximum of SSGA Cycle 24. However,
these predictions are somewhat unreliable because the G–O rule cannot be used
to predict the amplitude of an odd-numbered solar cycle without prior
knowledge about non-violation of this rule by the corresponding pair of even-
and odd-numbered solar cycles. We determined the best-fit cosine functions to
the values of each of the parameters given in Table 1. We find that there
exists a $\approx$132-year periodicity in all the northern hemisphere’s
parameters and the 54 – 66-year periodicity in all the southern hemisphere’s
parameters. By extrapolating the best-fit cosine curves we make cautious
predictions for the values of the parameters of the corresponding 25th cycles.
The sums of the predicted values of the parameters of northern and southern
hemispheres represent the predictions for the corresponding value of the whole
sphere. It is found that the amplitudes of the 25th WSGA and SSGA cycles would
be to some extent smaller than those of the corresponding 24th cycle. The
amplitude of 25th NSGA cycle would be almost the same as that of the
corresponding 24th cycle. The value of WSGA at the maximum epoch of Sunspot
Cycle 25 would also be smaller than that at the maximum epoch of Sunspot Cycle
24, i.e. the overall Solar Cycle 25 in sunspot-group area would be smaller
than Solar Cycle 24 in sunspot-group area. We obtained a good linear
relationship between the 13-month smoothed monthly mean SN and sunspot-group
area in the whole sphere and also in the northern and the southern
hemispheres. By inputting the predicted values of RWA, RNA, and RSA in these
relationships we obtained the values of SN at the epochs of maxima of the 25th
SN, WSGA, NSGA, and SSGA cycles. The obtained value ($130\pm 12$) of $R_{\rm
M}$ of Sunspot Cycle 25 is found to be slightly larger than that of Sunspot
Cycle 24. Except at the maximum of NSGA Cycle 25 where the north is found to
be dominant in strength of activity, the values of SN at the maxima of WSGA
Cycle 25, SSGA Cycle 25, and Sunspot Cycle 25 the south is found to be
dominant. The cosine best-fits to the values of the corresponding north–south
asymmetry in WSGA, NSGA, and SSGA at the maxima of the 12th to 24th SN, WSGA,
NSGA, and SSGA cycles suggest the existence of 50 – 66-year periodicities in
the north–south asymmetry and the same aforementioned pattern of hemispheric
dominance. Overall, the present analysis suggests that the amplitude of Solar
Cycle 25 would be slightly larger than that of Solar Cycle 24 and the activity
in the southern hemisphere would be dominant.

All the parameters of the solar cycles of a hemisphere have the same
periodicity, implying that different parameters of the hemisphere are at
different phases of the same long-period cycle; that is, the epochs of
different parameters of the northern hemisphere represent different phases of
the same cosine wave of $\approx$132-year period and the different parameters
of the southern hemisphere represent different phases of the same cosine wave
of 54 – 66-year period. The large difference in the periods of the cosine
waves of northern and southern hemispheres’ activity may be responsible for
varying phase differences between 11-year cycles of northern and southern
hemispheres’ activity. It may be worth finding why there exists a large
difference in the long-term periodicities in northern and southern
hemispheres’ activity (which is beyond the scope of the present analysis).

The best-fit cosine functions of sunspot-number data have already indicated
that Solar Cycle 25 is a reasonably small cycle and constitutes the minimum of
the upcoming new Gleissberg cycle (jbu05; jj17). A number of studies on long-
term variations in solar activity by using different techniques have also
predicted Dalton-minimum-like characteristics around Solar Cycle 25 (jj15;
goa16; cob21). In the present analysis, we also find the same. However,
whether Solar Cycle 25 will be smaller or larger than the reasonably small
Solar Cycle 24 is yet to be established. Many authors predicted the amplitude
of Solar Cycle 25 by using different techniques (for recent reviews see pet20;
nandy21). Some authors predicted a small amplitude, some others predicted a
large amplitude for Solar Cycle 25, and a larger number of authors predicted
that it would be approximately the same size as Solar Cycle 24.

It is widely believed that the strength of polar fields at the minimum of an
upcoming solar cycle is a good physically oriented precursor to predict the
strength of the solar cycle. From this method and in flux-transport dynamo
models it is predicted that Solar Cycle 25 would have the same size or
slightly larger than that of Solar Cycle 24 (nandy21; kumar21) and our
prediction here closely agrees with it. As already mentioned earlier we found
a specific method (jj07), and using it we predicted a small value for the
amplitude of Solar Cycle 25, strongly implying that Solar Cycle 25 will be
considerably smaller than Solar Cycle 24 (jj15; 2017; 2021). However, the
predictions made in the present analysis by using a basic method suggest that
Solar Cycle 25 in sunspot-group area would be slightly smaller than Solar
Cycle 24 in sunspot-group area, but Sunspot Cycle 25 would be slightly larger
than Sunspot Cycle 24. A reason for this discrepancy in our present and
earlier predictions for Sunspot Cycle 25 is not known to us. There are some
inconsistencies in both earlier and present analyses. Regarding the earlier
method, in the case of Solar Cycle 24 its second SN peak (coinciding with the
strong southern hemisphere’s peak, see Figure 1) is taken into account because
it is substantially larger than its first peak (coinciding with the weak
northern hemisphere peak), whereas in the case of most of the previous cycles
the respective first peak is taken into account because it is larger than the
corresponding second peak. In addition, since the uncertainties in the
measured areas of sunspot groups are not known, in the earlier analyses the
linear least-squares fit calculations were not included the uncertainties in
the area-sums (sums of the areas of sunspot groups; see Section 1). Here we
repeated the earlier calculations by taking 10 % of the area-sums for their
uncertainties (note that $R_{\rm M}$ values have 5 % – 10 % uncertainties).
None of the best-fit linear relations, namely between the area-sum and $R_{\rm
M}$ and area-sum and RWA, etc. are found statistically significant (PROB
values are found considerably small). However, the 10 % uncertainties in the
area-sums seem to be underestimated. To get a reasonably large PROB it seems
necessary to use 20 – 25 % of the area-sums for their uncertainties, which may
be too unrealistic. In the present analysis the results of best-fit cosine
functions are suggestive rather than compelling (best-fit cosine functions
have to some extent large $\chi^{2}$-values). The predictions made from our
earlier method are based on the existence of a strong connection (high
correlation) between consecutive cycles. The values predicted here from the
long-term cyclic trends in the data might be to some extent overestimated.
Further investigations are needed to conclude whether our present or earlier
predictions will be right or wrong. On the other hand the value predicted in
the present analysis for the amplitude of Solar Cycle 25 is only slightly
larger than that of the reasonable small Solar Cycle 24 and considerably
smaller than the average amplitude 178.7 of solar cycles (pesnell18). Hence in
the present analysis we confirmed that the beginning of the upcoming
Gleissberg cycle would take place around Solar Cycle 25 (see the cosine curve
in Figure 7).

Our present predictions (see Table 4) for Solar Cycle 25 are somewhat
comparable to the following predictions made by different authors using
different techniques. Rigozo et al. (2011) by extrapolating the sunspot-number
time-series spectral components estimated a maximum sunspot number of 132.1
for Solar Cycle 25. Cameron, Jiang, and Schüssler (2016) by using a flux-
transport dynamo model simulated a value of dipole moment around 2020, i.e.
around the beginning of Solar Cycle 25, and suggested that Solar Cycle 25 will
be of moderate amplitude, not much higher than that of Solar Cycle 24. Pesnell
and Schatten (2018) applied a precursor method by using the SODA Index, which
is determined by polar magnetic fields and the spectral index, predicted a
large amplitude ($135\pm 25$) for Solar Cycle 25. Jiang et al. (2018) used a
solar surface flux-transport model and predicted that the peak sunspot number
of Solar Cycle 25 is 125$\pm$ 32\. Petrovay et al. (2018) using coronal green-
line data as a precursor predicted 130 for the amplitude of Solar Cycle 25.
Bhowmik and Nandy (2018) using magnetic-field evolution models for the Sun’s
surface and interior predicted that Solar Cycle 25 will be similar or slightly
stronger than Solar Cycle 24. Du (2020) using the rate of decrease in the
smoothed monthly mean sunspot number over the final three years of Solar Cycle
24 predicted $130.0\pm 31.9$ for the amplitude of Solar Cycle 25. Kumar et al.
(2021) used the polar-precursor method and predicted $126\pm 3$ for the
amplitude of Solar Cycle 25. Ahluwalia (2022) using data for sunspot numbers
and geomagnetic indices (aa/Ap) and their heuristic methodology inferred that
Solar Cycle 25 may be as active as Solar Cycle 24 ruling out a Dalton-like
minimum in the 21st century. Upton and Hathaway (2018) using an advective flux
transport code found evidence that during Solar Cycle 25 the southern
hemisphere will be more active than the northern hemisphere. Gopalswamy et al.
(2018) used a precursor method (microwave-imaging observations) and predicted
that the sunspot-number peaks of southern and northern hemispheres during
Sunspot Cycle 25 will be 89 and 59, respectively. These indicate a
considerable north–south asymmetry corresponding to the amplitude of Solar
Cycle 25 and that activity would be dominant in the southern hemisphere.
Werner and Guineva (2020) using auto-regressive models of different order
predicted 117 for the amplitude of Solar Cycle 25 and that solar activity
would be dominant in the southern hemisphere. Pishkalo (2021) using the
absolute values of the mean polar magnetic fields during the two-year interval
just before the cycle minimum in northern and southern hemispheres as
precursors predicted $66\pm 17$ and $83\pm 21$ for northern and southern
hemispheres’ amplitudes of Solar Cycle 25. Labonville, Charbonneau, and
Lemerle (2019) from a data-driven solar cycle model predicted a low amplitude
for Solar Cycle 25 and the northern hemisphere about 20 % more than the south
southern hemisphere. Our earlier analysis (jj21) also suggested that at the
epoch of maximum of the Solar Cycle 25 the activity will be about 20% more in
northern hemisphere than that in southern hemisphere, whereas our present
analysis suggests that the southern hemisphere would be dominant. According to
the international panel to forecast Solar Cycle 25, the amplitude of this
cycle would be 115, which is almost the same as the amplitude of Solar Cycle
24 (see www.swpc.noaa.gov/news).

### Acknowledgments

The author thanks the anonymous reviewer for useful comments and suggestions.
The author acknowledges the work of all the people who contribute to and
maintain the GPR and DPD Sunspot databases. The sunspot-number data are
provided by WDC-SILSO, Royal Observatory of Belgium, Brussels.

## Declarations

## Conflict of interest

The author declares that he has no conflicts of interest.

## Data Availability

All data generated or analysed during this study are included in this article.

## References

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