The Periodic and Temporal Behaviors of Solar X-Ray Flares in Solar Cycles 23 and 24

Solar activity demonstrates a wide range of periodicities on different timescales. These periodic variations are related to the solar magnetic field generated as a result of solar dynamo (Pesnell 2015). It has been established that solar cycles undergo an 11-yr cycle variation and a 22-yr cycle of magnetic field polarity activity. These periodic behaviors influence variations of all solar parameters on different timescales ranging from days to years (Schwabe 1844; Nayar 2006; Kilcik et al. 2010; Mendoza & Velasco-Herrera 2011; Du 2012; Zou & Li 2014; Li et al. 2016, 2017).

There have been several studies done to ascertain periodicities of different solar parameters, e.g., Rieger et al. (1984) discovered 153-day periodicity in solar soft X-ray (SXR) flare occurrence. Yan et al. (2011) investigated the long-term solar flare activity and found that the smoothed monthly peak fluxes of C, M, and X solar flares have time lags of 1 month, 5 months, and 21 months, respectively, with respect to sunspot numbers in the solar cycle 23. McCloskey et al. (2016) studied the flaring rates using the sunspot group McIntosh classifications and found that flaring rate increases with greater degrees of upward evolution.

Bai (2003) analyzed solar flare occurrence for the mid-range periodicities between solar cycles 19 and 23. There are several periodicities found across the cycles. It is important to detect mid-range flare periodicities and to accurately calculate their statistical significances, because flare periodicities can provide information on properties of the Sun (Bai 2003). Dimitropoulou et al. (2008) argued that periodicities of solar parameters from 150 to 160 days, also known as Rieger Type Periodicities (RTP) are present in lower energetic solar flares, and they also linked the X-ray flare to the mechanism behind RTPs to be the Rossby Type Waves using Lou's theoretical model (Lou 2000).

Understanding solar flare periodicities would be a useful tool for prediction of space weather. A number of authors have carried out investigations on solar flares at different periods of time using different methods. Therefore, it is important to study the periodic variations and distributions of solar flares using daily generated data for the most recent two cycles. In this study, we investigate the individual flare class relationship with the daily count of sunspot group numbers.

The data archived on the National Oceanic and Atmospheric Administration (NOAA) website are available from 1996 July 31 to 2016 December 30 covering solar cycle 23 as well as the ascending, maximum, and most of the descending phase of cycle 24. The data was generated by the XRS instrument onboard the GOES satellites. The solar flare data are classified into A, B, C, M, and X, each is divided into scales between 1 and 9 according to peak flux except X that has no upper limit. This study considers B, C, M, and X flares. Their daily count numbers are used to carry out the study.

2.1. Data

2.2. Methods

To study the temporal behavior and periodicities of the SXR flare based on the available archived data, the cross-correlation analysis (CCA) and the continuous wavelet transform (CWT) tool with its extensions were employed.

2.2.1. Cross-correlation Analysis

The CCA is a popularly used method to detect the degree of how two signals are related (Hagino et al. 2004; Deng et al. 2012). The cross-correlation coefficient between the distributions of the two-data series is calculated as

$$\begin{eqnarray}&&\mathrm{CC}\left(\bigtriangleup \right)=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{n}\left[a\left(i\right)-\left\{a\right\}\right][b\left(i+\bigtriangleup \right)-\left\{b\right\}]}{(n-1){\delta }_{a}{\delta }_{b}},\end{eqnarray} \tag{ 1 }$$

where {a} and {b} in the equation stand for the mean value of each of the solar flare classes (B, C, M, X) and the sunspot group numbers respectively. The ${\delta }_{a}$ and ${\delta }_{b}$ stand for their respective standard deviations. Δ is time delay. Positive Δ means the solar flare class leads the sunspot group numbers and vice versa.

2.2.1.1. Probable Error of Correlation Coefficient

The Probable Error (PE) is used to determine how accurate and reliable the values of the coefficients obtained are; the correlation is said to be certain when the value of "r" is six times more than that of the PE; this shows that the value of "r" is significant. The PE is defined as (Odell 1926; Soper 1913)

$$\begin{eqnarray}&&{\rm{P}}.{\rm{E}}\ \left(r\right)=0.6745\displaystyle \frac{1-{r}^{2}}{\sqrt{N}}\end{eqnarray} \tag{ 2 }$$

• N = number of observations
• r = coefficient of correlation.

2.2.2. Wavelet Transform Methods

Wavelet analysis is also a well-known tool for analyzing varying signals in their time and frequency domains. The CWT has been noted to be a powerful tool for detecting the localized and quasi-periodic oscillations (Grinsted et al. 2004; Singh & Badruddin 2014). The continuous wavelet extensions, the cross-wavelet transform (XWT) and wavelet coherence (WTC), are useful for investigating the relationship in time-frequency space between two time series (Oliver et al. 1998; Torrence & Compo 1998; Fligge et al. 1999; Grinsted et al. 2004; Polygiannakis et al. 2003; Velasco et al. 2008). They can reveal similarities in the states of the two systems and allow the study of the synchronization or phase difference in two time series (Marwan et al. 2002b). The CWTs of a signal x(t) are defined by (Chui 1992)

$$\begin{eqnarray}&&X\left(b,a\right)=| a{| }^{-1/2}{\int }_{-\infty }^{+\infty }x\left(t\right)\overline{\varphi }\left(\displaystyle \frac{t-b}{a}\right){dt}.\end{eqnarray} \tag{ 3 }$$

where φ is the Morlet wavelet function (Schmitz-Hübsch & Schuh 2003) and is represented as

$$\begin{eqnarray}\begin{array}{rcl}\varphi \left(t\right) & = & \displaystyle \frac{\exp ({ipt})}{\sqrt{2\pi }}\left[\exp \left(-\displaystyle \frac{{t}^{2}}{2{\sigma }^{2}}\right)\right.\\ & & -\,\left.\sqrt{2}\exp \left(-\displaystyle \frac{{t}^{2}}{2{\sigma }^{2}}\right).\exp \left(-\displaystyle \frac{{p}^{2}{\sigma }^{2}}{4}\right)\right]\,,\end{array}\end{eqnarray} \tag{ 4 }$$

where σ is the Morlet decay parameter, p is variable (usually >5) as frequency parameter, a is the dilation parameter ($a\ne 0$), and b is the translation parameter. For p = 2π, the dilated Morlet wavelet oscillation period equals a (Popinski & Kosek 1994).

The cross-wavelet transform (XWT) of the two time signals A and B can be defined as

$$\begin{eqnarray}&&{X}^{\mathrm{AB}}={X}^{{\rm{A}}}{X}^{{\rm{B}}* },\end{eqnarray} \tag{ 5 }$$

where ${X}^{{\rm{A}}}$ and ${X}^{{\rm{B}}}$ represent the CWT of the time signals A and B. The * represents complex conjugation. The complex argument arg ${X}^{\mathrm{AB}}$ can be taken as a local relative phase between A and B in the time-frequency domain; that is, the phase angle difference of A and B (Grinsted et al. 2004).

The continuous Fourier transform (CFT) of the Morlet wavelet is given by the formula (Chui 1992)

$$\begin{eqnarray}\begin{array}{rcl}\overline{\varphi }\left(\omega \right) & = & \sigma \left[\exp \left(-\displaystyle \frac{{\left(\omega -p\right)}^{2}{\sigma }^{2}}{2}\right)\right.\\ & & -\,\exp \left(-\displaystyle \frac{{\left(\omega -p\right)}^{2}{\sigma }^{2}}{4}\right)\exp \left(-\displaystyle \frac{{p}^{2}{\sigma }^{2}}{4}\right).\end{array}\end{eqnarray} \tag{ 6 }$$

It has a quasi-compact provision for both time and frequency.

The formula for coefficients $X\left(b,a\right)$, which involves CFT of the signal $\overline{x}$ (ω) and the wavelet function $\overline{\varphi }$ (ω) reads

$$\begin{eqnarray}&&X\left(b,a\right)=\displaystyle \frac{1}{2\pi }| a{| }^{-1/2}{\int }_{-\infty }^{+\infty }\overline{x}\left(\omega \right)\overline{\varphi }(a\omega )\exp ({ib}\omega )d\omega .\end{eqnarray} \tag{ 7 }$$

The WTC computes how coherent common oscillatory components of two time signals are in the time-frequency domain. The degree of WTC can be defined between two CWTs to distinguish significant coherence even with common low power (Grinsted et al. 2004). The importance of WTC is due to the circumstance that wavelet cross-spectrum seems poor for testing of the interrelation between two progressions (Marwan & Kurths 2002a; Maraun & Kurths 2004). The WTC of two time signals A and B can be defined as

$$\begin{eqnarray}&&{R}_{n}^{2}\left(s\right)=\displaystyle \frac{| S({s}^{-1}{X}_{n}^{\mathrm{AB}}\left(s\right)){| }^{2}}{S({s}^{-1}| {X}_{n}^{{\rm{A}}}\left(s\right){| }^{2}\ast S({s}^{-1}{X}_{n}^{{\rm{B}}}\left(s\right){| }^{2}},\end{eqnarray} \tag{ 8 }$$

where S stands for smoothing operator. Because Wavelet is not localized completely due to the suffering of edge artefacts, we therefore consider the introduction of a cone of influence (COI). The COI defines the wavelet power for a discontinuity at the edges decreases by a factor ${e}^{-2}$ (Grinsted et al. 2004). The wavelet analysis was employed on each of the flare classes. The Morlet Wavelet function is represented by

$$\begin{eqnarray}&&{\psi }_{0}\left(\eta \right)={\pi }^{-1/4}{e}^{i{\omega }_{0}\eta }{e}^{-{\eta }^{2}/2}.\end{eqnarray} \tag{ 9 }$$

It is a plane sine wave with amplitude derived in time from the Gaussian function where ${\omega }_{0}$ is a non-dimensional frequency. We adopted ${\omega }_{0}=6$ because it can maintain well-balanced spectral and temporal resolutions (Torrence & Compo 1998; Grinsted et al. 2004). The level of confidence of the wavelet transform power is derived by the insignificant hypothesis assumption of the existence of the global power spectrum (Torrence & Compo 1998). For this work, we applied the 95% confidence level, and the details can be found in (Grinsted et al. 2004).

Table 1 shows the values obtained from the cross correlations and PEs analyses. The C-class flares show the most significant positive correlations for the monthly data analyzed. This is followed by M flares and X flares, respectively. The B class shows anti-correlation with the sunspot group numbers. The PEs show that all correlation coefficients are significant for all data sets used in this study.

Table 1.  The Minimum, Mean, Maximum and Sum Values Obtained from Each of the Solar Flare Classes (B, C, M, and X), and the Sunspot Group Number for the Daily Data

Flare Class	Minimum Number of Flares(Per Day)	Mean Values of Flares	Maximum Number of Flares(Per Day)	Total Number of Flares	Correlation with SSG (Monthly)	Probable Error (Monthly)	Time delay Obtained (${\boldsymbol{\Delta }}$) (Years)
B	0	1.791	23	13351	−0.277	0.0397	−2
C	0	2.776	26	20699	0.868	0.0106	0
M	0	0.287	10	2141	0.629	0.0260	0
X	0	0.022	3	163	0.284	0.0376	−3
Total	0	4.876	33	36354	0.753	0.0187	⋯
SSG	0	4.406	19	32852	1	0	⋯

Note. The probable errors (PE) show that all correlation coefficients are significant for all data sets used in this study. Positive time delay (Δ) means the solar flare class leads the sunspot group numbers and vice versa.

Download table as:  ASCIITypeset image

Figure 1 shows the 13-month smoothed monthly numbers comparisons between the flares (blue lines) and the sunspot group numbers (black lines). Figure 1(a) shows B-class flare to be in phase from the decline phase of cycle 23 to the ascending phase of cycle 24. It is also observable that its deviation begins when sunspot group numbers rise or fall around 100 for both solar cycles. The flare class shows an antiphase relationship with the sunspot group numbers. This is shown in Table 1. Figure 1(a) shows that B flares exhibit asynchronization with the sunspot group numbers at solar maximum in both solar cycles. The flares also show their deepest minimum simultaneously with the sunspot group numbers at solar minimum as shown in Figure 1 (panel a).

Zoom In Zoom Out Reset image size

The C-class flare in Figures 1(b) and 2 show a high consistency with the sunspot group numbers. This shows that the C-class could be more useful for studying the evolution of sunspots more than other classes. The M and X flares have low occurrence rates as shown in Table 1.

Zoom In Zoom Out Reset image size

Figure 2 shows yearly distributions for both flares and the sunspot group numbers daily counts. The figure confirms the antiphase relationship of B-class flares with the sunspot group numbers. It is observed that B flares show a cycle around 5 yr with ascending and descending phase features similar to the 11-yr solar cycle. It has its peaks around 1998, 2004, and 2005 in cycle 23, and in 2010, 2011, and rising in 2016 in cycle 24. These are the times when the solar cycles are off the peak, either in the ascending or descending phase. Also, this is evident from both Figures 1(a) and 2(a). Table 1 also shows that the B-class flares are negatively correlated with the sunspot group numbers.

Figure 3 shows the cross-correlation analyses. The abscissa designates the time delay with respect to the sunspot group numbers. From the analysis in Figure 3, panel (a), B class has a time delay of 2 yr with respect to the sunspot group numbers at a correlation coefficient of −0.277. This signifies that the sunspot group numbers lead B flares by 2 yr. The levels of phase relationships of C, M, and X solar flares with the sunspot group numbers are shown in Figures 3(b)–(d). The C-class flares show zero time delay, indicating phase synchronization at a peak correlation coefficient of 0.868. This phase synchronization is evident in Figure 1(b). The M-class flares also exhibit phase synchronization with the sunspot group numbers. The correlation coefficient is 0.629. The X-class is characterized by irregular peaks with the two highest peaks at correlation coefficients of 0.284 and 0.283 and with time delays at −3 and 1 yr, respectively, as shown in Figure 3(d). The correlation coefficients presented in Table 1 are significant at 95%.

Zoom In Zoom Out Reset image size

3.1. The Wavelet Analysis

3.1.1. Morlet CWT

The Morlet wavelet transform was employed with the red noise approximation on the entirety of the data used. The results are presented in Figures 4–7. The upper panels of the figures show the scale-average time series distributions while the lower panels show wavelet power spectra.

Zoom In Zoom Out Reset image size

The wavelet power spectra techniques allow computation of time-frequency coherence analyses that reveal the dominant variation in both the spectra. The color distribution provides information on the corresponding variation at the different levels of spectral power at a different time in the wavelet power spectrum. The blue and white color areas correspond to regions with low power while the yellow and black color areas are associated with the regions of larger power. The plots are represented at the 95% confidence level. The COI is also introduced to the area where edge effects cannot be overlooked. The COI is introduced to the wavelet power spectra regions padded by zeros.

From Figure 4, the B flares show a significant periodicity around 256 days in the wavelet power spectrum. This is noticeable from 1996 to 2016, except around 2002 to 2004 where it exhibits a 64-day periodicity. This is around the solar maximum to the descending phase. It also shows periodicity around 32–64 days between 2005 to 2007. There is also a variation of 16 days in the latter part of 2006. Between 2009 and 2011, it shows periodicity around 64 days. Figure 4(a) shows B flares to be more present in solar cycle 23 compared to cycle 24, where it recorded the highest amplitudes at the descending phase of the cycle.

Figure 5 shows C flares having periodicities around 7, 64, and 300 days. Periodicity around 300 days is the most significant, which appears in both solar cycles except during the solar minimum. Periodicity around 64 days is more pronounced in cycle 24 and also between 2001 and 2003 in cycle 23. Figure 5(a) shows C flares to be more present in cycle 23 where it has its highest amplitudes at the ascending phase of the cycle.

Zoom In Zoom Out Reset image size

Figure 6 shows periodicities for the M flares having around 32, 64, 128, and 256 days. The most significant is 64 days, noticeable around 2000 to early 2003 in cycle 23 and 2013 to 2014 in cycle 24. The periodicities around 16–32 days is noticeable from 1999 to 2004 and 2013 to 2014. A periodicity of 128 days is noticeable between 2000 and 2001 in cycle 23. Figure 6(a) shows M flares to be more present in cycle 23 and to have the highest amplitude in 2001 when cycle 23 was at its peak. The X-flare occurrence constitutes 0.45% at 163 occurrences between 1996 July and 2016 December. It recorded a positive correlation with the sunspot group numbers, mostly occurring during the peak time of solar cycles. Figure 7 reveals periodicities around 25, 32, 40, and 64 days in 1998, 2000 and 2003, 2002, and 2005, respectively, with the highest amplitude and most present in cycle 23 as observed from Figure 7(a). The analysis presented is above the 90% confidence level.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

3.1.2. Wavelet Coherence (WTC)

The extended wavelet transforms tool, the WTC, was also employed to analyze the phase relationships between each of the solar flares (B, C, M, and X) and the sunspot group numbers. The results are presented in Figure 8. The arrows designate the relative relationships existing between the sunspot group numbers and the solar flares. When the arrows point toward the right-hand side, this indicates they are in phase, and to the left indicates otherwise. Arrows pointing up show that the first process is lagging behind the second process, and when pointing down, it shows the first process is leading the latter. The analysis is above the 95% confidence level.

Zoom In Zoom Out Reset image size

Figure 8(a) shows the phase relationship between B flares and the sunspot group numbers. In most cases, it can be observed that the arrows pointing left show that they are in an asynchronous relationship at various periodicities. The yellow regions displace strong correlations while the blue regions correspond to weak correlation. The negative correlation is significant for both cycles 23 and 24. This shows B-class flares are produced not relative to evolution of sunspots. However, the sunspot group numbers and the B-class flares exhibit a synchronous relationship. This is noticeable from 2004 to 2010 at various periodicities. The flares exhibit phase synchronization with the sunspot group numbers at periodicity around 4 to 16 days between 2004 and 2006, and at 4 to 64 days between 2006 and early 2010. This was observed in Figure 1(a). The positive correlation shows that B-class flares are produced relative to sunspot evolutions around descending and ascending phases between solar cycles 23 and 24. There is a strong negative correlation around 4 days periodicity from 2011 to 2015. This happens shortly after positive correlation from the descending phase of cycle 23 until the ascending phase of the cycle 24. This could be attributed to an increase in solar magnetic activities that leads to the production of higher-energy flares overshadowing the weaker ones.

From the Figure 8(b), the C-class flares show that they are in phase synchronization with the sunspot group numbers. The correlation between them is a strong positive correlation in most regions. The yellow region at periodicities between 0.25 to 2 days shows the high level of phase synchronization. This could be attributed to the rate of C-class flares occurring faster than others in response to sunspot evolution. This phenomenon could be the cause for the flare accounting for 56.9% of the total flares produced. This was observed in Figure 1(b). However, this method has shown where both signals are asynchronous based on daily data analyzed compared to 13-month smoothed data that could not detect a daily trend. The phase relationship varies with time on periodic scales, with increased uniformity at smaller periodicities compared to larger periods. A similar situation is observed in Figure 8(c) for M flares. The periodicities around 4–8 days show synchronization leading in cycle 24, while it is a mixed situation between synchronous and asynchronous in cycle 23 for the same range of periodicity. However, it is noticeable in Figure 1(c) that they follow the same trend in medium-term periodicity. Figure 8(d) shows an X-class flare in asynchronous phase with sunspot group numbers around 2005 at periodicity 2–8 days, and in synchronous phase around 2011 at 4–8 days periodicity. However, its medium-term trend from Figure 1(d) shows that it declined in cycle 24 around 2013.

Analyses from Figure 8 show phase mixing phenomenon mostly at the larger periodicities. This occurs typically with B-class flares and the sunspot group numbers. The greater the phase mixing, the noisier in the wavelet transform coherence. It is noticeable that strong phase synchronization exists much more often at smaller periodicities across the cycles between the flares and the sunspot group numbers. In order to investigate the phase relationship between different solar activities, caution must be used in choosing the reference periodic scales (Deng et al. 2012). The availability of a physically meaningful phase definition depends crucially on the appropriate choice of the reference frequency (Donner & Thiel 2007; Deng et al. 2012). The low-frequency modes can be considered as a long-term trend and the high-frequency modes as a stochastic component that is not random but amplitude modulated (Carbonell et al. 1993).

In this paper, we have studied the periodic variations and distributions of the solar X-ray flare classes and the sunspot group numbers using cross-correlation and wavelet transform tools. The results observed are summarized as follows:

We found that B flares negatively correlated with the sunspot group numbers while C, M, and X are positively correlated. The C and M solar flares are in phase with the sunspot group numbers while X solar flares are characterized with irregular peaks. The correlations between the solar X-ray flares could be attributed to the vital relationship between the magnetic properties of the solar active regions and the flare-production area in the solar atmosphere. The B-class flare is synchronous with sunspot group numbers when the number of sunspot groups is around 100 and asynchronous when it rises above 100. The phase relationship analyses show C and M to be more synchronous with sunspot group numbers, compared to B-class flares. The last part of the ascending phase of cycle 24 shows X flares to be in antiphase, which could be attributed to the fewer magnetic solar activities at that time.

The B flares show a cyclic variation around 5 yr. The cycle has its lowest points at the peak of the solar cycle and at the solar minimum. In other words, the B flares occurrence rate is low during solar maximum. The flares show periodicities around 32, 64, and 256 days, with 256 days periodicity being the most significant. Periodicities of 7, 64, and 300 days were noticed in C, with 300 days periodicity being the most significant, while the M flares show periodicities around 32, 64, and 256, with 64 days periodicities being the most significant. The X flares exhibit periodicities around 25, 32, and 40 days.

Considering other studies, Gao & Xu (2016a) found C-class flare periods appear dominant in the maximum phase of cycle 23 but more periods of a wide range in the decline phase of cycle 22, while M-class flare appears dominant for both cycles 22 and 23 at cycle maxima. The study further shows that X-class flare exhibits the wide range of periodicities at the maximum phase of solar cycle 22 and the decline phase of cycle 23. Our observations agree with some of these results in cycle 23. The study, however, shows that there would be an increase in X-class flare numbers in cycle 25.

Gao & Zhong (2016b) compared the temporal behavior of B-, C-, M-, and X-class flares using monthly smoothed flare numbers and found the B class to be in complete antiphase with all other flares. The study also shows C-class flare numbers in a small decreasing trend from cycle 22 to 24 at the peak values, while M- and X-class flares frequency reduce by almost 50% and 70%, respectively, during cycle 23 compared to cycle 22; this trend remains until cycle 24. Gao & Xu (2016a) analyzed monthly data for the C, M, and X flares. For the C-class flare, many periodicities were observed in cycles 22 and 23 with dominant periodicities in cycle 23. The M-class flares recorded more periodicities in cycle 22 compared to 23. The periodicity longer than 16 months almost disappeared in cycles 23 and 24 as observed in cycle 22. In cycle 22, nearly all periods were found for the X-class flares while the fewest periodicities were observed in cycle 24.

Studies conducted by Kilcik et al. (2014) and Lefèvre & Clette (2011) reveal the occurrence rate of sunspots in cycles 22 to 23. Large sunspots showed no substantial difference while the smallest sunspots were more than 50% during solar cycle 23 compared to cycle 22.

Yan et al. (2011) observed the smoothed monthly peak fluxes of C-, M-, and X-class solar flares with time lags of 1 month, 5 months, and 21 months, respectively, with respect to sunspot numbers in cycle 23. Eren & Kilcik (2017) found that the C-, M-, and X-class flares have the highest correlation with the large group sunspot counts with correlation coefficients of 0.79, 0.74, and 0.4, respectively.

Bai (2003) found periodicities of different ranges between solar cycle 19 and 23. He discovered 33.5 and 129 days for cycle 23. Other periodicities include: 51 days from cycle 19; 85 and 129 days from cycle 20; and 153 days from cycle 21. Periodicities of 51, 78, 104, and 129 days, in addition to the 154-day period, can often be detected in flare and sunspot records, as well as close to integral multiples (by factors of 2, 3, 4, 5, 1, and 6) of 25.8 days (Bai & Sturrock 1991). Kilcik et al. (2010) observed a periodicity 62 days for cycle 23. Ataç & Özgüç (2006) found 64-, 83-, and 125-day periodicities for the daily flare index data for the northern, southern hemispheres, and the full disk. Kiliç (2009) found no significant periodicity for the flare index data analyses for cycle 23. Özgüç et al. (2004) reported periodicities at 27 days and 33.8 days for cycle 21, 25.6 and 30.2 days for cycle 22, and 37.5 days for cycle 23. Kilcik et al. (2010) found 152 days for cycle 21, 73 days for cycle 22, and 62 days for cycle 23.

The flares can be re-classified into two groups: B and C as background flares as low magnetic activity flares for defining less active solar regions, and M and X as high magnetic activity flares for defining the flare energy.

The following factors could be accountable for the mechanism and fundamental periodicity of solar flares. The flares originate from different sources of solar active regions, which could be one of the causes of their periodic variations. Sunspot groups are the main cause for solar flares. Solar flares have been attributed to complex sunspot groups with magnetic flux twisting as the fundamental mechanism. Mendoza & Velasco-Herrera (2011) suggested that periodicities could be the product of chaotic quasi-periodic processes and not of stochastic processes. Periodic emergence of magnetic flux within already formed sunspot groups enhances the magnetic complexity of the active region, which gives rise to similar periodic variations in solar flares. On the other hand, the periodic emergence of flux away from developed sunspot groups will not be reflected in the periodic behavior of solar flares (Ballester et al. 2002, 2004).

Eren et al. (2017), Kilcik et al. (2011), and Norquist (2011) agreed that the largest and most complex sunspot groups display the highest flare-production potential. Eren et al. (2017) also stated that M- and X-class flares are mostly limited to the large and complex active regions based on long and comprehensive time series. Bai & Sturrock (1993) anticipated that an obliquely rotating structure of wave patterns could be responsible for the periodicity. The findings based on the longitude distribution of major flares for cycles 19–22 exhibit a strongly bimodal distribution in a coordinate system rotating with a period of 25.5 days about an axis tilted by 40° with respect to the solar rotation axis. McCloskey et al. (2016) observed the flaring rate to be highest for upward evolution from the larger, more complex Zurich classes, which are the bipolar and large sunspot groups.

In conclusion, our result is consistent with some other studies in short and intermediate periodicities. The M- and X-class flares are strongly connected to sunspot groups during cycle peak times. It is suggested by Ballester et al. (2002) that the appearance of the periodicity in high-energy flares is triggered by the appearance of the periodicity in the photospheric magnetic flux linked to regions having strong magnetic fields. The difference in periodic variations and distributions of flare classes with sunspot group numbers could be attributed to a magnetic flux system of sunspot groups. Mendoza & Velasco-Herrera (2011) suggested that periodicities could be the result of chaotic quasi-periodic processes. However, in order to shed more light on mechanisms responsible for these short and intermediate periodicities, there is a need for more investigation on solar interior and flare-producing solar active regions.

The authors express their sincere appreciation to the referee for the constructive comments, which have greatly improved the original manuscript. We also appreciate Katie Daugherty for proofreading the manuscript and her contribution. The authors also express their gratitude to the provider of the data used; National Oceanic and Atmospheric Administration (NOAA) and to the wavelet software provided by C. Torrence and G. Compo, http://paos.colorado.edu/research/wavelets/. This study was supported by the CAS-TWAS President's Fellowship and the National Science Foundation of China (NSFC: 11533009 and 11603074). This work is also funded by the grants from the Key Laboratory of Geospace Environment, CAS, University of Science and Technology of China, and the One Belt and One Road scientific project of the West Light Foundation, CAS.