Responses and Periodic Variations of Cosmic Ray Intensity and Solar Wind Speed to Sunspot Numbers

To investigate the periodic behaviour and relationship of sunspot numbers with cosmic ray intensity and solar wind speed, we present analysis from daily data generated from 1995 January to 2018 December. Cross-correlation and wavelet transform tools were employed to carry out the investigation. .e analyses confirmed that the cosmic ray intensity correlates negatively with the sunspot numbers, exhibiting an asynchronous phase relationship with a strong negative correlation. .e trend in cosmic ray intensity indicates that it undergoes the 11-year modulation that mainly depends on the solar activity in the heliosphere. On the other hand, the solar wind speed neither shows a clear phase relationship nor correlates with the sunspot numbers but shows a wide range of periodicities that could possibly be connected to the pattern of coronal hole configuration. A number of short and midterm variations were also observed from the wavelet analysis, i.e., 64–128 and 128–256 days for the cosmic ray intensity, 4–8, 32–64, 128–256, and 256–512 days for the solar wind speed, and 16–32, 32–64, 128–256, and 256–512 days for the sunspot numbers.


Introduction
It is well known that various of solar activities are closely associated with solar magnetic field processes and the study of the long-term evolution of solar activities are useful for the understanding of the solar atmosphere and the dynamo theories (e.g., [1][2][3][4]). e sunspot numbers (SSN), being the most important index of the solar activities, have been widely studied together with other indices [5][6][7][8][9][10][11][12][13].
Cosmic rays are highly energetic particles striking the earth from space. ey can be originated from two primary sources: heliospheric and galactic cosmic rays (GCR). e solar wind is electrically charged, and the energised particles can move at the speed of about 400 km·s − 1 freely in the heliospheric space. Both the cosmic ray intensity and the solar wind speed (SWS) are closely related to solar activity variations. ey exhibit variations over different periods of time [11,[14][15][16][17][18][19][20][21][22].
Cosmic ray intensity (CRI) varies with numerous causes including the solar wind parameters and sunspots numbers. Mishra et al. [23] showed that solar wind velocity has a strong positive correlation with CRI during solar cycle 21. According to Jokipii and omas [24], variation in the observational angular parameter of heliographic equator could cause considerable changes in the galactic cosmic ray intensity. Long-term cosmic ray evolution can indicate solar cycle effect. Some attributes of cosmic rays and their behaviour in 11-year cycle with the interplanetary magnetic field (IMF) showed decrease in GCR flux that corresponds to an increase in the IMF intensity [25]. Ahluwalia [26] attributed the decrease in observed galactic cosmic ray flux to the increase in residual modulation within the heliosphere using GCR data from 1937 to 1998. It was also noted that there was no increase in SSN observed during the same time period of observation. Fujimoto et al. [6] observed changes of CRI with solar activity; Hempelmann and Weber [27] used Fourier analysis of the time series and suspected that the cosmic rays have a link with solar activity following their observations that showed significant peaks of 10.7, 22.4, and 14.9 years.
Yan et al. [28] investigated the phase relationship between SSN, flare index, and solar radio flux using crosscorrelation analysis. Pérez-Peraza et al. [29] presented evidence of the existence of cosmic ray fluctuations with a periodicity of around 30 years. Li et al. [22] and Li et al. [30] investigated the periodicities in daily SWS and observed different periods in low and high wind speeds.
ere have been many diverse studies carried out to understand several mechanisms in our solar system, which would make useful tools in the knowledge and understanding of our space weather and protection of artificial outer space objects. Oloketuyi et al. [31] investigated the influence of sunspot group numbers on the flare classes, finding that different class of flares respond differently to sunspot group emergencies.
is study, however, investigated the behaviour and responses of CRI observed at the earth and SWS to the solar magnetic activity using SSN. e present study is an attempt to discover new periodicities and relationship between SSN and CRI with SWS that will broaden our understanding of cosmic ray modulations within the heliosphere in respect to the solar activities.

Data.
e data used in this study were obtained from different sources between 1995 January and 2018 December. e period of investigation covers solar cycles 23 to the present cycle 24 using daily data. e obtained data were used to investigate the periodic variation of SSN with CRI and SWS.
(1) Daily cosmic ray: the CRI data used in this study are pressure-corrected and obtained from the Cosmic Ray Station of the University of Oulu/Sodankyla Geophysical Observatory. e data are set to 1440 minutes auto resolution and can be downloaded from http://cosmicrays.oulu.fi/. Figure 1(   sunspot numbers is available at http://www.sidc.be/ silso/datafiles. e daily SSN distribution is presented in Figure 1(c).
(3) Daily solar wind: the daily SWS data used was generated by several spacecraft orbiting the Earth and can be obtained from https://spdf.gsfc.nasa.gov/pub/data/ omni/low_res_omni/omni_m_daily.dat. e daily distributions of SWS is presented in Figure 1(e).

Methods.
To achieve the objective of detecting the responses and significant periodicities of CRI and the SWS to SSN in the investigated solar cycles, the cross-correlation analysis, continuous wavelet transforms, and the wavelet coherence methods were employed.

Cross-Correlation Analysis.
e cross-correlation analysis (CCA) is a well-known method used to find where two signals match. Several authors have used this method. e coefficient of cross-correlation between the two data series is defined as [32] in which p represents the mean value of SSN, q represents the mean value for CRI or SWS, δ p and δ q stand for their respective standard deviations. Positive coefficient Δ means that the time series of SSN leads the other parameters while negative coefficient means lag.

Wavelet Transform Methods.
Wavelet analysis is a computational tool that helps in decompositions of varying signals into time and frequency dissections. is method helps in noise reduction from a signal by using the approximation process. e Continuous Wavelet Transform (CWT) has been noted to be a powerful tool for detecting the localised and quasiperiodic oscillations [33,34]. e wavelet coherence (WTC), an extended form of Continuous Wavelet Transform, is used in the analysis of time-frequency relationship between twotime varying signals [33,[35][36][37][38][39]. ey reveal similarities in the states of two systems and allow the study of the synchronisation or phase difference in two-time series data [40]. e cross-wavelet transform (XWT) for the two-time varying signals SSN x { } and CRI or SWS y can be defined as e X x and X y designate the continuous wavelet transform of the time-varying signals SSN x { } and CRI or SWS y . e * designates complex conjugation. e complex argument arg X xy can be considered as a localised relative phase between the SSN x { } and CRI or SWS y in the time-frequency domain which is the phase angle difference between them [33]. e continuous wavelet transforms of a continuous function x(t) relative to a real-valued wavelet ψ(t) is described by [41] as where in which s and τ are called scale and translation parameters, respectively. W ψ (s, τ), represents the wavelet transform coefficients and ψ is the fundamental mother wavelet. Wavelet Coherence (WTC) functions as a correlation coefficient, whereby the areas of high common power between two-time varying data are revealed in their time and frequency domains. is technique is unique and useful in the computation of time and frequency of signals such that it distinguishes significant coherence even in their common low power [33]. e importance of WTC is due to the circumstance that wavelet cross-spectrum looks not good for testing of the interrelation between two progressions [42,43]. e wavelet coherence of two time-varying signals A and B can be defined as in which S represents smoothing operator in both time and frequency components, X A represents the wavelets for the time-varying data of the SSN and X B for CRI or SWS. X AB is the cross wavelet. e cone of influence (COI) defines the wavelet power for a discontinuity at the edge decreases by a factor e − 2 [33]. e wavelet analysis was employed on all the data sets separately for the selected solar cycles and the combined solar cycles for all solar activity indices used. e Morlet Wavelet function is represented by A plane sine wave with amplitude derived in time from the Gaussian function where ω 0 is a nondimensional frequency. ω 0 � 6 was adopted to give a good balance of spectral and temporal resolution [33,36]. For this study, 95 percent confidence level has been applied in the analysis [33]. Table 1 shows the correlation coefficients obtained between the daily SSN with CRI and SWS. e analysis shows that CRI is negatively correlated with SSN for the two solar cycles.

Results and Discussions
e cross-correlation analysis shows strong anticorrelations with coefficients of − 0.72 and − 0.73 for cycles 23 and 24, respectively. However, the analysis for SWS with SSN shows inconsequential positive correlations. Figure 1 shows the distributions and trends for SSN, CRI, and SWS. e trends were calculated using a 13-day moving average. From this figure, a significant increase in CRI was observed during the solar minima while decreasing at the Advances in Astronomy 3 peaks of the solar cycles.
e trends perfectly reveal the variations they exhibit in long-term responses to SSN. e CRI shows periodicity similar to the 11-year solar cycle in the opposite while the SWS shows irregular modulations in response to the solar activities. is observation in CRI could be attributed to the varying amplitude of the heliospheric magnetic field at the solar maximum reducing the influx of galactic cosmic rays entering the solar system and vice versa at solar minima due to the low solar magnetic activity. is shows that CRI is free and constantly abundant outside the heliosphere, but its influence and modulations in the heliosphere are solely dependent on solar activities. Figure 2 shows the cross-correlation analyses between daily SSN and daily CRI and SWS. e abscissa in each frame shows the lag or time shift with respect to SSN. e negative values designate backward shifts or lag, and if positive, it designates otherwise. e vertical dash line in each frame indicates no shift or phase boundary. From the cross-correlation analysis in Figure 2(a), CRI shows a strong negative correlation with SSN for the solar cycle 23. It lags behind SSN with coefficients of − 0.72. Moreover, CRI also shows anticorrelation with SSN with a coefficient of − 0.73 in cycle 24 as shown in Figure 2 Figure 3 shows the correlation analysis for the overall period. e analysis shows the overall behaviour of CRI and SWS with respect to SSN. e time series of CRI lags behind SSN at a correlation coefficient of -0.73 shown in Figure 3(a), while SWS is uncorrelated with SSN with a weak coefficient of 0.06 as shown in Figure 3  is the cone of influence (COI) which represents the 95 percent confidence level. is was introduced to minimise errors in the areas liable to have edge effects due to discontinuity, and it is padded with zeros. e GWS results, representing the power variation with period, are presented at panels (c). Figure 4 shows the wavelet analysis for the daily CRI. e analysis shows periodicities of 64-128 and 128-256 days. e most prominent periodicity is 128-256 days. is periodicity appeared in both solar cycles with a peak value of 2.47 × 10 6 on the global wavelet spectrum. e wavelet spectrum also shows that CRI is more dominant in cycle 23 with more variations and larger amplitudes. e wavelet analysis for the daily SWS is presented in Figure 5. A wide range of periodicities were observed. e most significant periodicity is 256-512 days which appeared in both cycles and peaked at 7.53 × 10 5 on the global wavelet spectrum. Periodicity of 4-8 days is also significant with 6.3 × 10 4 on the global wavelet spectrum analysis. Other noticeable periodicities include 32-64 days in 1996, 1999-2001, 2003-2005, and 2017, while periodicity of 128-256 days was observed in both cycles. For the SWS wavelet spectrum, both cycles seem to be moderately substantial in occurrence. e scale-average time series shows SWS has its higher amplitudes at the descending phase of cycle 23. Figure 6 shows the periodicities for the daily SSN. e wavelet spectrum for SSN shows that solar cycle 24 is weaker compared to cycle 23. e amplitude of SSN is much higher during the solar cycle 23.

Wavelet Coherence (WTC).
is study also employed the wavelet coherence method, an extended tool of wavelet transforms. e wavelet coherence analysis was used to examine the phase relationships CRI and SWS independently have with SSN. e results from the analyses are shown in Figure 7. e confidence level of wavelet coherence analysis presented is above 95 percent. Figure 7(a) shows the WTC analysis for SSN and CRI while Figure 7(b) for SSN and SWS. e directions of the arrows indicate relative phase relationships existing between them. If the arrows point towards right-hand (left-hand), they indicate that SSN is in phase (antiphase) with CRI or SWS. When the arrows are pointing upward (downward), they indicate that the phase relationship is leading (lagging). e white dash line in each panel is the cone of influence (COI) introduced.
In Figure 7(a), SSN shows an asynchronous phase relationship with CRI. e yellow regions indicate a strong anticorrelation or antiphase relation at higher periodicities between 64 and 256 days in both cycles. e blue regions However, there is no other significant phase synchronisation at any point in the analysis. is could be attributed to their different and independent sources in the interplanetary space.   observed were mostly around descending and ascending phases of the solar cycles. Largely, the uneven phase interactions at different periodicities could be ascribed to the pattern of configuration in the coronal hole and the solar wind outflow from different sources due to the solar magnetic structures [44,45]. e cross-correlation analysis agrees with previous studies. Forbush [46] and [47] confirmed the anticorrelation of CRI with SSN using data from different stations between 1937 and 1952. Mishra [48] used monthly data of Grouped    Advances in Astronomy Solar Flare Index (GSF), instead of SSN and CRI for solar cycles 20 to 23. It was observed that CRI negatively correlated with solar activities in cycles 20 to 23. Kane [15] and [11] also found that CRI is anticorrelated with SSN. However, Tiwari et al. [49] suggested that the strength of IMF characterised quantitive effect on the cosmic ray.   Advances in Astronomy number of variations were found: ∼9, ∼14, ∼75, ∼99, ∼200 days, ∼1.4 years, ∼1.7 years. ese periodicities were observed in annual, semiannual, and triannual variations.
Sunspots are formed as a result of the sun's strong magnetic activities. ese activities of solar magnetic field can lead to the heating of solar corona resulting in the escape of solar wind from the coronal hole. It was noted that the differences in SWS variations could be attributed to their different sources and configuration of coronal hole. Dunzlaff et al. [52] and Lario and Roelof [53] suggested that different coronal hole structures could lead to different co-rotating interaction region structures which could possibly also be applicable.
Joshi et al. [54] showed periodicities of (∼175, 133, 113, 104, 84, 63 days) in sunspot activity of solar cycle 23. Mendoza and Velasco-Herrera [55] showed the wavelet analysis of all the sunspot groups. e significant periodicities in the global spectrum are about 2, 1.  [31] found that the B flares responded differently from the other flares to sunspot group numbers by having cycles around 5 years of variation.

Conclusions
We have investigated the temporal and periodic variations of CRI, and SWS with SSN in the solar cycles 23 and 24 using cross-correlation and wavelet transforms. e results obtained are summarized as follows.
We found that CRI undergoes 11-year solar cycle within the heliosphere, which is greatly influenced mainly by solar activities. e cycle formed has its peak at the solar minimum and vice-versa. e present study also confirmed that the daily sunspot numbers and CRI are negatively correlated. e anticorrelations observed from the cycles are highly significant. SWS was found to be uncorrelated with SSN.
e wavelet analyses show a wide range of periodicities. e observed periodicities for the daily CRI include 64-128 and 128-256 days. e most prominent periodicity is 128-256 days which appeared in both cycles analysed. e observed daily SWS exhibits a wide range of periodicities. e obtained periodicities include 4-8, 32-64, 128-256, and 256-512 days which appeared in both cycles and the most significant periodicity with a peak value of 7.53 × 10 5 on the global wavelet spectrum. e wavelet analysis also shows that most of the periodicities for the daily sunspot numbers were obtained in solar cycle 23. e obtained periodicities mostly appeared briefly include 16-32, 32-64, 128-256, and 256-512 days. e most significant periodicity is 256 days which appeared in both solar cycles. e wavelet spectrum shows solar activity cycle 24 is weaker compared to cycle 23.
Recently, Singh [59] studied the short-term variations of SWS, CRI, interplanetary magnetic field, SWS, solar radio flux, and geomagnetic Ap Index. e study investigated solar cycles 20 to 24 covering the polarity reversal period and found periodicities of 2. 5, 4.5, 9, 14.5 [60] investigated five solar cycles from 1965 to 2018 for new periodicities for the CRI, sunspots, Bz-component of the interplanetary magnetic field, and geomagnetic Ap Index. Periodicities observed are 5.5-, 6-, and 9-year and 13.9-day periods, including the 27-day, 11-year periods, and the 1.7-and 2.9year periodicities which are found to be integral multiples of the Rieger and QBO periods, respectively. e analysis from the wavelet coherence also confirms that CRI and SSN correlate negatively. e analysis also shows that CRI lagged behind sunspot numbers. Largely, they exhibit an asynchronous phase relationship, a clear indication that CRI in the interplanetary space responds negatively to solar activity. On the other hand, SWS does not show a clear phase relationship with SSN but largely irregular phase interaction, a phenomenon that could be best described as noise, which could be connected to irregularities in coronal holes configurations where the high-speed solar wind originates from. However, the availability of a physically meaningful phase definition depends crucially on the appropriate choice of the reference frequency [61]. If we want to investigate the phase relationship between different solar activity indicators, we should be careful in choosing the reference periodic scales [62]. e 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 [63]. e anticorrelation relationship of SSN with the CRI could be attributed largely to the influx of galactic cosmic ray into the heliospheric space, and the response of CRI shows that it could be useful in investigating the solar activities and other parameters like solar flares and the CMEs in the solar system. However, there is a need to make further investigation on solar magnetism and its mechanisms which are the primary source and causes of solar-related phenomena.

Conflicts of Interest
e authors declare that they have no conflicts of interest.