1 Introduction

Galactic cosmic rays (GCRs) are energetic charged particles mostly having an out of solar system origin, far away from the heliosphere. The high-energy GCRs (\(10^{8}\mbox{--}10^{20}~\mbox{eV/nucleon}\)) may reach the Earth, interact with atomic nuclei in the atmosphere, initiate nuclear-electromagnetic cascades there, and produce secondary elementary particles such as neutrons that can be measured by ground-based neutron monitors (NMs) or other detectors. The heliosphere is a cavity in the interstellar plasma controlled by the continuously emitted solar wind and by a heliospheric magnetic field (HMF) (Balogh and Erdős 2013). The size of the heliosphere is about 120 AU as indicated by Voyager 1 spacecraft (Stone et al. 2013) and may vary in the course of the ∼ 11 year sunspot cycle (e.g., Pogorelov et al. 2013, and references therein). Although GCRs form a nearly stable and isotropic background of high-energy radiation in the local interstellar space, their intensity is modulated with the solar activity level of the ∼ 11 year sunspot cycle because GCR particles travel through the magnetized interplanetary medium, which is composed of an interplanetary magnetic field emanating from the Sun and changing with the solar cycle.

The spatial and temporal distribution of IMF throughout the heliosphere controls the transport of energetic particles within. It is described in terms of outward solar-wind convection, pitch angle diffusion, guiding-center drifts, and energy change by expansion or compression of the interplanetary plasma structures (e.g. Jokipii 1971; Potgieter 2013; Bazilevskaya et al. 2014b). GCRs also exhibit a distinct ∼ 11-year cyclic variation, anti-correlated with sunspot activity with some time lag. The investigation of GCR modulation and time lag with solar, interplanetary, and geomagnetic indices within the heliosphere is a subject of intensive research with the aim to assess the continuously changing behavior of the solar magnetic field and its influence on the propagation of charged particles during odd/even solar cycles. Previous studies regarding the phenomenon of time lag for different solar cycles using different solar and geomagnetic data revealed several interesting results (e.g., Usoskin et al. 1998, 2001; Cliver and Ling 2001; Minarovjech and Kudela 2004; Badruddin, Singh and Singh 2007; Sabbah and Rybanský 2006; Sabbah and Kudela 2011; Mavromichalaki, Paouris, and Karalidi 2007; Ahluwalia et al. 2010; Chowdhury, Dwivedi, and Ray 2011; Kane 2011; Paouris et al. 2012; Owens, Usoskin, and Lockwood 2012; Potgieter et al. 2014; Aslam and Badruddin 2015 and references therein). Like the solar parameters, GCRs also exhibit different quasi-periodic oscillations like ∼ 9 day, ∼ 13.5 days, ∼ 27 day, Rieger and Rieger-type periods (130 – 180 days) and quasi-biennial oscillations (QBOs) (Sabbah and Kudela 2011; Gil and Alania 2012; Modzelewska and Alania 2013; Kudela, Ananth, and Venkatesan 1991, 2010; Mavromichalaki et al. 2003; Laurenza et al. 2012; Vecchio et al. 2010; Chowdhury et al. 2010a; Kato et al. 2003 etc.). Moreover, GCRs are the main source of ionization in the atmosphere at altitudes below 55 – 60 km. Since the GCR fluxes are strongly modulated by solar activity, they could be a translator of solar influence on the Earth’s environment and have a strong impact on space-borne instruments. As the GCR controlling solar parameters change from cycle to cycle and even within the same cycle, it is very important to study the modulation and periodicities of GCRs to assess the behavior of the continually changing Sun. The previous solar cycle, Cycle 23, showed a number of peculiar features in the behavior like a double peak with a Gnevyshev gap, in GCRs a pronounced bimodal modulation, analyzed in solar Cycles 20 to 22 e.g. by Storini, Laurenza, and Fujii (2003) and Krainev et al. (1999), slow reversal of polar field (Dikpati et al. 2004); prolonged deep minimum which was stretched to ∼ 3 years, of which 70 – 71 % days were without sunspots, record-low values of HMF (∼ 28 % weaker) both at low and high latitudes (Smith and Balogh 2008) as well as Ulysses spacecraft measurement of slow solar wind flux, significantly less dense, cooler, with less mass and momentum compared with the 22/23 solar minimum (McComas et al. 2008). However, the heliospheric current sheet (HCS) during this minimum was less flat than during the previous two minima. During this epoch, the GCRs’ intensity reached the record-highest level in the whole era of neutron-monitor observations. A number of researchers studied different properties of the solar magnetic field and GCRs during the prolonged minimum via observations and numerical models (Paouris et al. 2012; Chowdhury, Kudela, and Dwivedi 2013; McDonald, Webber, and Reames 2010; Leske et al. 2013; Cliver, Richardson, and Ling 2013; Alania, Modzelewska, and Wawrzynczak 2014; Zhao et al. 2014; Aslam and Badruddin 2012; Potgieter et al. 2014; Heber et al. 2009; Bazilevskaya et al. 2013 and references therein) but their opinion in some cases are quite different. Chowdhury, Kudela, and Dwivedi (2013) indicated an anomalous modulation of GCRs during the descending phase (\(qA < 0\)), the GCRs’ recovery was much faster than those of the solar parameters with negative time lag. According to Mewaldt et al. (2010) a weak IMF strength and interplanetary turbulence level produced a sustained increase in the estimated the GCRs’ parallel mean free path and also increased the GCR drift velocities. Moraal and Stoker (2010) concluded that a weak IMF during the minima of Cycle 23/24 caused higher cosmic-ray diffusion coefficient producing a record high GCR intensity. On the other hand Cliver, Richardson, and Ling (2013) argued that a weak IMF strength is related to the huge intensity of GCRs and not related to the reduction of tilt angle of the HCS. Zhao et al. (2014) argued that due to a low magnetic turbulence, parallel diffusion increases and the perpendicular diffusion in the polar direction decreases, which might be an additional possible mechanism for the record high GCR intensity in the solar minimum phase of Cycle 23/24. Solar Cycle 24 has started much later than expected, and is going sluggishly toward its maxima. Richardson (2013) has reported a weak geomagnetic activity in the rising branch of this cycle; Pacini and Usoskin (2015) have indicated some unusual pattern of the GCR modulation in this cycle. At the time of writing, when solar Cycle 24 is on the verge of the maximum phase, the time is ripe to study the modulation of GCRs in the rising phase of this cycle. The present communication reports on a detailed study of solar and heliospheric modulation of GCRs during about six years of the start of Solar Cycle 24 and investigates the intermediate-term periodicities present in the GCR time series during this epoch.

2 Data and Methods

To study the GCRs’ modulations and quasi-periodicities during the ascending phase of Cycle 24 (January 2009 – August 2014), pressure-corrected daily, monthly, and 27-day-averaged GCR intensity values obtained at the Oulu neutron-monitor station (cut-off rigidity 0.81 GV; 65.05° N; 25.47° E) were used ( http://cosmicrays.oulu.fi ). The solar, heliospheric, and geomagnetic parameters used in this study are: i) sunspot numbers (SSN), ii) sunspot areas (SSA) of the whole solar disk, iii) the 10.7-cm solar radio flux, iv) modified coronal index (MCI), v) the tilt angle of heliospheric current sheet (HCS), vi) coronal mass ejections (CMEs), vii) the geomagnetic index Ap, and viii) the interplanetary magnetic field (IMF).

The monthly data of sunspot numbers of the whole solar disk were taken from Royal Observatory of Belgium ( http://sidc.oma.be/silso/ ). The sunspot area time series were taken from the NASA/Marshall Space Flight Center ( http://solarscience.msfc.nasa.gov/greenwch.shtml ). The SSN and SSA data represent solar magnetic fields in the photosphere. The F10.7-cm (2800 MHz) solar radio flux originating from the Sun’s chromosphere and the lower corona data were taken from the Penticton solar radio flux database ( http://radbelts.gsfc.nasa.gov/RB_model_int/Psi_database.html ). The modified coronal index (MCI) time series, which characterize the activity of the “Sun as a star” and represent the total irradiance of the green corona emitted by the Sun’s visible hemisphere, has been taken from the web site of the Slovak Central Observatory in Hurbanovo ( http://www.suh.sk/obs/vysl/MCI.htm – described in the paper by Rybanský 1975).

The heliospheric current sheet, which takes an active role in the GCR modulation (Potgieter 2013) is the layer where the polarity of the solar magnetic field changes from north to south. The HCS extends throughout the Sun’s equatorial plane in the heliosphere (Bazilevskaya et al. 2014a, 2014b). Its waviness is defined by the tilt angle (\(\Lambda\)), which varies cyclically over the solar cycle, so that the HCS is nearly flat around the solar minimum period and reaches its maximum values of above 70° around the solar maximum phase. We have considered the 27-day rotation-wise \(\Lambda\) of HCS (Classical model) from January 2009 – August 2014 (Carrington rotations 2079 to 2154) from the Wilcox Solar Observatory ( http://wso.stanford.edu/Tilts.html ). The monthly CME data have been taken from the SOHO LASCO CME catalog ( http://cdaw.gsfc.nasa.gov/CME_list/ ). Here, we have excluded narrow CMEs (width < 30°) in our calculations.

The geomagnetic index Ap and IMF (Carrington rotations from 2394 to 2471) were taken from the OMNIWEB database ( http://omniweb.gsfc.nasa.gov/ ) and the MAG instrument onboard the ACE spacecraft ( http://www.srl.caltech.edu/ACE/ASC/ ), respectively.

To study the time lag, we have calculated the cross-correlation coefficients (CC) between solar, interplanetary, and geomagnetic parameters and GCR data with various time lags (\(0, {\pm}1, {+}2,\dots, {\pm}30\) months). Each of these data sets was shifted forward or backward by one month or one Bartels solar rotation (27 days for Ap, \(\Lambda\) and IMF data) with respect to the neutron-monitor time-series data set recorded at Oulu. Then the correlation coefficient was calculated after shifting by the time lag (\(L\)) corresponding to the best correlation (cf. Figure 3) of the full time span.

To identify the periodicities present in the GCR data set and their time variations, we use the Morlet wavelet technique. The Morlet wavelet function is a plane wave modulated by a Gaussian and the non-dimensional frequency \(\omega_{0}\), which defines the value of the periodicity within the wavelet itself. We have

$$ \Psi(\eta)=\pi^{-1/4}\mathrm{e}^{-i\omega_{0}\eta}\mathrm{e}^{-\eta^{2}/2}. $$
(1)

Here, the non-dimensional frequency has been chosen as 6 (Torrence and Compo 1998) which gives reasonable spectral and temporal resolution as well, \(\eta\) (Fourier time period) is \(1.033s\) where \(s\) is the Morlet time scale.

The significance levels for the wavelet power spectra (WPS) were similarly calculated by assuming a mean background spectrum modeled with a univariate lag 1 autoregressive process considering a red noise background. In a red noise spectrum, the discrete Fourier power spectrum, after normalization, is given by

$$ P_{k}= \frac{1-\alpha^{2}}{1+\alpha^{2}-2\alpha\cos(2\pi k/N)}. $$
(2)

Here \(k = 0, \dots, N/2\) is the frequency index, \(N\) is the number of data, and \(\alpha\) is the lag 1 serial correlation coefficient. When \(\alpha = 0\), we obtain the white-noise spectrum with an expectation value of 1 at all frequencies.

The wavelet transform suffers from edge effects at both ends of the time series and gives rise to a cone of influence (COI). These edge effects, caused by padding zeros at the beginning and at the end of the data series, lead to a power reduction within the COI. The thin black contours within the COI show the periods above 95 % confidence level (Torrence and Compo 1998). Both the local and the global wavelet spectra are displayed for the data set we study.

The local wavelet spectrum contains information on the time interval of the detected frequencies/periods. The global wavelet spectra, being integrated over time, are analogous to the Fourier spectra and are helpful to detect the dominant frequencies/periods present in the time series. However, the wavelet technique is quite laborious and not always completely unambiguous. The tabulated time intervals should therefore be seen as informative estimates and not as exact lifetimes (De Moortel, Munday, and Hood 2004).

The original GCR data series have a long-term tendency with a very strong periodicity of around the end of the time interval. This end scale periodicity reduces the confidence levels of the other short- and intermediate-term periodicities. Therefore to dig out significant short-term periodicities in any time profile \(f (t_{i})\), we have used the following method.

First, the time profile of the low-frequency (long-period) component \(f^{\mathrm{sm}}(t_{i})\) is found by smoothing the original signal time profiles \(f (t_{i})\) using a moving-average method with a suitable interval \(\tau\). After smoothing, the high-frequency (short-period) component is picked out by subtracting the low-frequency component from the original signal, obtaining

$$f^{\mathrm{hf}}_{\tau} (t_{i}) = f (t_{i})- f^{\mathrm{sm}}_{\tau} (t_{i}). $$

Mid-term periodicities in the resulting time series \(f^{\mathrm{hf}}_{\tau}(t_{i})\) are investigated by using wavelet techniques.

To avoid the appearance of false periodicities due to filtering, we have repeated the procedure for a broad range of smoothing intervals \(\tau\) from 1 to 17 months (30 – 510 days). Any spectral component that really belongs to the signal is clearly seen in a wavelet spectrum and keeps its period for all the intervals (“\(\tau\)”s).

The monthly or 27-day means of all our data sets are illustrated in Figures 1(a) – (i). We have also plotted the monthly mean of GCRs against all the parameters we study to investigate the primary nature of the hysteresis curve for the rising branch of Cycle 24 and also calculated the correlation coefficient between GCRs and the solar, interplanetary, and heliospheric parameters.

Figure 1
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(a) Monthly average GCR counts from Oulu NM station for Cycle 24 (Jan. 2009 – Aug. 2014); (b) monthly average sunspot number of the whole solar disk for Cycle 24 (Jan. 2009 – Aug. 2014); (c) same as (b) but for sunspot area on the full disk; (d) same as (b) but for modified coronal index data; (e) same as (b) but for 10.7 cm. solar radio flux; (f) same as (b) but for CMEs; (g) 27-day average value of IMF; (h) 27-day average value of HCS and (I) 27-day average data of Ap for Cycle 24.

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3 Results

The time series of GCRs and solar/geomagnetic parameters for the ascending phase of solar Cycle 24 are shown in Figures 1(a) – (i), which provide the following information:

  1. 1)

    The monthly mean value of the GCR intensity increases sharply from the beginning of 2009 and continues up to the end of the year 2009. After February/March 2010, the intensity of the GCRs gradually decreased, but the trend is different for 2010 to 2012.5 from 2012.5 to 2014.

  2. 2)

    SSN, SSA, 10.7 cm flux and MCI shows two prominent peaks; one ∼ July – August 2011 and the other ∼ February – March 2014. The height of the second peak is larger than the first one. On the other hand, Ap and IMF show peaks about mid-2012 and in early 2014. For both of these parameters, the peak in 2012 is higher than the peak in 2014. Thus it seems that Cycle 24 has entered into its maximum phase from 2014 and like the previous Cycle 23, the ongoing cycle is also a double-peaked cycle. But it is interesting that for solar photospheric activities like SSN/SSA, the strength of the second peak is higher than that of the first one and it is an unusual one, which was not observed in the previous double-peaked Solar Cycles 22 and 23.

  3. 3)

    The number of CMEs gradually increased after 2010 and reached a peak about the early phase of 2014. On the other hand, the tilt of HCS was ∼ 22° at the beginning of the cycle and reduced to ∼ 10° in the middle of 2009. Afterward, the tilt gradually increased and during the period of 2011 – 2014, its value varied from 70° to 75°. This high value of \(\Lambda\) of HCS indicates that Cycle 24 is about to end its maximum and the rise of \(\Lambda\) is similar to the previous two solar cycle. More on the evolution of the heliospheric magnetic field (HMF) as well as on the relations between HMF and CMEs can be found in a paper by Owens and Forsyth (2013). The complex pattern in the relation between the solar cycle in the heliosphere and cosmic rays for several solar cycles is discussed e.g. by Bazilevskaya et al. (2014b) and the references therein.

Figure 2 shows the results of the hysteresis loop and Figure 3 the cross-correlation and time lag between GCRs and different solar/heliospheric/geomagnetic activity indicators between 2009 and 2014, respectively. The average correlation coefficient (without time lag) between SSN and GCRs is \({\sim}\,{-0.867}\), between SSA and GCRs \({\sim}\,{-0.770}\), \({\sim}\,{- 0.859}\) for 10.7 cm solar flux and GCRs; between IMF and GCRs \({\sim}\,{-0.895}\); \(\Lambda\) of HCS and GCRs is \({\sim}\,{-0.847}\); between MCI and GCRs \({\sim}\,{-0.899}\); and between GCRs and CMEs \({\sim}\,{-0.873}\) and \({\sim}\,{-0.572}\) with Ap. Thus except for the geomagnetic index Ap, all other solar and heliospheric parameters are highly correlated with the GCR intensity in the current cycle.

  1. (a)

    Figure 3 shows the cross-correlations and time lags between GCRs and different solar, geomagnetic, and interplanetary activity indicators between 2009 and 2014 with their statistical errors. Table 1 presents the values of cross-correlation coefficient and time lag between GCRs and different parameters during the period of study. When the peak of the curve is nearly flat, then we have considered the first and the second highest cross-correlation values and the corresponding time lags.

    Table 1 Cross-correlation coefficients between GCR and various indices with corresponding time lags (\(L\)) for the 24th Solar Cycle.
    Full size table
  2. (b)

    Quasi–periodicity in the GCRs’ time series:

    Figure 4(a) – (c) shows the power spectrum of the daily GCR time series recorded by the Oulu station for the ascending phase of Cycle 24 (2009 – 2014) obtained with the Morlet wavelet method. The thin black contours in the local wavelet spectrum represent the periods with > 95 % confidence level. The periods in the range of 16 – 32 days including the solar rotation period are prominent from the beginning of 2011 to 2014 [Figure 4(a)(i)] and prior to this it was less significant. This indicates that the solar rotational period was absent in the time phase when the GCR intensity was at its highest value. Periods in the range of 32 and 75 days were present approximately from January to June 2012. Periods in the range of 50 – 75 days also appeared in the middle of 2013 and at the end of 2013. A contour of length between 60 and 170 days was localized from early 2011 to the end of 2012. This contour shows the existence of the best-known Rieger period of length 150 – 160 days and this period was prominent mainly in the phase from the middle of 2011 to the middle of 2012 [Figures 4(b)(i) and 4(c)(i)]. The global wavelet power spectrum [Figures 4(a)(ii) and 4(b)(ii)] also exhibits pronounced peaks in the range of 19 – 37 days and 100 – 175 days; this is consistent with the result of the local wavelet spectra. The quasi-periodicity in the range of 240 – 490 days was present around early 2011 to the middle of 2014 and also dominant in the global wavelet spectra. Quasi-biennial oscillations (QBOs) having characteristic periods between ∼ 0.6 – 4 yr are well-known variations in solar activity, interplanetary parameters, and geomagnetic indices as well as GCR data (Bazilevskaya et al. 2014a). The period in the range of ∼ 1.3 yr was significant in the solar magnetic-field data during the maxima and declining phase of the last eight solar cycles and this is assumed to be generated at the base of the convection zone, being an inherent feature of the global solar magnetic field (Obridko and Shelting 2007; Howe et al. 2000). Our wavelet power spectra confirm the presence of this well-known QBO of ∼ 1.3 yr localized from the end of 2010 to the middle of 2012; after that its power diminished due to COI [Figure 4(c)(i)] and this period is also confirmed in the global wavelet spectrum [Figure 4(c)(ii)].

Figure 2
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(a) – (h) Hysteresis plots between GCRs and different solar, heliospheric, and geomagnetic parameters for Solar Cycle 24.

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Figure 3
figure 3figure 3figure 3

(a) Cross-correlation and time lag between monthly SSN and GCRs for Cycle 24; (b) Same as (a) with SSA and GCRs; (c) Same as (a) with 10.7 cm solar flux and GCRs; (d) Same as (a) with MCI and GCRs; (e) Same as (a) with CMEs and GCRs; (f) Same as (a) with IMF and GCRs; (g) Same as (a) with HCS and GCRs; (h) Same as (a) with Ap and GCRs.

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Figure 4
figure 4figure 4

Pairs of panels showing local and global wavelet spectra of GCRs for Jan. 2009 – Aug. 2014 in successive periodicity ranges: (a) in the range of 8 – 32 days for studying short-term periodicities; (b) in the range of 32 – 256 days for studying mainly Rieger-type quasi periodicities; and (c) in the range of 128 – 512 days for studying QBOs; (In all the panels, the 95 % confidence levels in the local wavelet spectra are shown by thin black contours and those in the global wavelet spectra are shown by blue dash-dot lines. The thin black contours within the COI show the periodicities above 95 % confidence level.

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4 Discussion and Conclusions

In this work we have made a detailed study of the time lag between GCRs and different solar, geomagnetic, and heliospheric parameters as well as the mid-term periodicities of the GCR intensity for about six years after the start of the current solar cycle, Cycle 24. We have also calculated the correlation coefficient between GCRs and different solar, geomagnetic, and heliospheric parameters and found that the correlation coefficient with all parameters is very high. The correlation coefficient between GCRs and \(\Lambda\) of HCS is ∼ 0.84, which shows that good agreement exists between the long-term cosmic-ray intensity and the tilt of HCS in all periods during the whole history of cosmic-ray observations with NMs since 1953 (Belov 2000). The time lag between the GCR intensity and the solar variables, such as SSN, SSA, CMEs, and MCI, is remarkably large, reaching a value of 10 – 17 months. On the other hand, the heliospheric variable IMF shows a pronounced peak correlation at zero time lag and then decreases steadily. However, the correlation with the HCS tilt also peaks at zero and then has broken descending slope (a sort of “knee”) at nine months. The 10.7 cm solar flux and the geomagnetic index, Ap, show the maximum correlation at zero months. These results are consistent with the results for the earlier cycles (Mavromichalaki, Paouris, and Karalidi 2007; Paouris et al. 2012; Chowdhury, Kudela, and Dwivedi 2013 and references therein). It is interesting to note that although the 10.7 cm solar flux is a proxy of the solar cycle variation, there is a broad difference between the time lags of this activity indicator (originating mainly from solar chromosphere) and the activities of the photosphere (SSN/SSA). A similar result was found in the ascending phase of the previous odd numbered Cycle 23 (Chowdhury, Kudela, and Dwivedi 2013). In summary, the time lags of the GCR intensity with respect to the solar parameters are remarkably large ranging from 0 to 17 months for the ascending phase of the current solar cycle, whereas for the geomagnetic and heliospheric parameters no significant time lag was found. However, it is noteworthy that the nature of the modulation of the GCRs is quite different in Cycle 24 comparing to the nature of previous two even solar Cycles 20 and 22 as in those two past cycles the time lag between GCRs and SSN was much lower, varying from two to four months (Mavromichalaki, Belehaki, and Rafios 1998). It was found that the time lag between GCRs and SSN is longer in odd solar cycles (19, 21, and 23) than the even cycles (20 and 22) (Dorman and Dorman 1967; Nagashima and Morishita 1980; Usoskin et al. 1998; Singh, Badruddin, and Ananth 2005). Kane (2014), after analyzing the GCR data from Newark NM station for the first three years of Cycle 24, detected a broad hysteresis loop and indicated a long time lag of at least 10 months for the maximum phase of Cycle 24, which is unusual for an even number solar cycle and the present study is consistent with that prediction. Our plot of the hysteresis curves also indicate a broad area, which supports the time-lag findings. However, this matter will be clearer a few years later, after analyzing the data of the whole solar Cycle 24.

The solar magnetic field reverses at each solar activity maximum resulting in the 22-year Hale cycle (Jokipii and Thomas 1981; Potgieter 2013). The orientation of the field (polarity) is defined to be positive when the field lines are oriented outward from the Sun in the northern hemisphere and negative when the field lines are oriented outward in the southern hemisphere. A positive polarity field is denoted as an \(A > 0\) epoch and a negative field as an \(A < 0\) epoch.

We have investigated data from a mixed phase of \(A < 0\) (2009 to middle of 2013) and \(A > 0\) (middle of 2013 to the end of this study) regarding the polarity states of the Sun. According to the drift scenario of charged particle propagation in the heliosphere, when the solar polarity is positive (\(A>0\)), then positively charged particles enter the heliosphere through the polar regions (Potgieter, Burger, and Ferreira 2001), and when the solar polarity is negative (\(A <0\)), then GCRs enter the inner heliosphere through the equatorial region along the HCS. Under such conditions these particles will be more readily affected by the heliospheric current sheet and the propagating diffusion barriers associated with solar activity, mainly confined to near equatorial regions.

Here we analyze only Cycle 24. There was a reversal in the solar magnetic-field polarity. From the data used here (2009 – 2014.8) only a small part belongs to the reversed polarity (\(A>0\)) in comparison with a large portion from the beginning of that interval (\(A<0\)). The north pole completed the polarity reversal in June 2012 and the south pole polarity reversal took place in July 2013. Checking the change in the correlation coefficient studied for opposite polarity intervals requires a longer time series after 2014. For “global” parameters (SSN, SSA, HCS tilt angle, MCI) there is a tendency toward lower correlation coefficients and longer time lags, while for “local” ones (measured at 1 AU, near Earth as Ap and IMF) it is the opposite. However, to distinguish unambigously between the influence of “global” and “local” parameters on the CR modulation is not straightforward from an analysis of one solar cycle. In addition, the unresolved question of the high correlation coefficient and no time lag with radio emission from the Sun (10.7 cm wavelength), contrary to SSN, remains.

GCR transport in the heliosphere is described by the Parker transport equation (Parker 1965; Gleeson and Axford 1968) and these charged particles suffer from four distinct transport effects:

  1. i)

    An outward convection is caused by radially directed solar-wind velocity.

  2. ii)

    Curvature and gradient drift effects occur as the global heliospheric magnetic field varies systematically over large scales.

  3. iii)

    An adiabatic energy change takes place depending on the sign of the divergence of the solar-wind velocity.

  4. iv)

    The spatial diffusion is caused by the scattering by random magnetic irregularities.

Therefore, in each of the four terms of the GCR transport equation, directly or indirectly, either the solar-wind velocity [\(V\)] or the heliospheric magnetic field [\(B\)] is involved (Heber 2013; Kóta 2013 and references therein). We propose, as Solar Cycle 24 progresses, and the magnetic activity of the Sun rises, it simultaneously increase the strength of IMF, CMEs, and other outer propagating diffusive barriers (like Global Merged Interaction Regions, GMIRs, introduced by Burlaga, McDonald, and Ness 1993) during the ascending/maximum phase of this cycle and hence we detected a large time lag. It is also interesting that the lag between GCRs and SSN/SSA/MCI is large, but no lag was found with IMF and Ap. GCRs at the Earth’s orbit are modulated by the global distribution of the IMF over the whole heliosphere, so GMIRs can explain the time lag with (local) solar activity. IMF (B) is characteristic just of the inner heliosphere. The same for Ap (local, Earth magnetosphere). Our analysis shows that the time profile of GCRs measured at the Earth’s orbit is related to “some combination” of local transient phenomena (IMF, maybe the magnetosphere) and global heliospheric phenomena (dependence on solar activity reflecting (with some “precursor” as SSN is) the distribution of IMF throughout heliosphere). It is possible that some indices such as SSN, SSA, and MCI are global solar indices, whereas others such as Ap and IMF are limited to the position of Earth and ∼ 1 AU, respectively. Usoskin et al. (1998) suggested that the GCR modulation is defined mainly by the global indices because of their complicated transport in the heliosphere, consistent with our results in the present analysis.

Our power spectral decomposition method has detected significant periods in the range of 8 – 32 days including a solar rotational period of ∼ 27 days. The existence of 24 – 31 days quasi-periodicities due to solar rotation, in the GCR time series, is a good indicator of the global sectorial pattern of heliospheric magnetic fields (Antalová, Kudela, and Rybák 2001; Gil and Alania 2010). Recently the presence of second (∼ 13.5-day) and third (∼ 9-day) harmonics of the ∼ 27-day periodicity in the GCR data at different cut-off rigidities and their coupling with the interplanetary parameters have been analysed by Sabbah and Kudela (2011). Periods in the range of 38 – 40 days, ∼ 55 days and ∼ 78 days were detected in cosmic-ray time series of Haleakala and Climax stations in different times during the declining phase of Cycle 22 as well as in the ascending phase of Cycle 23 (Caballero and Valdés-Galicia 2001) and our analysis also detected these types of quasi-periodicities. The present study shows the existence of the best-known Rieger and Rieger-type periods of in the length of 130 – 170 days prominent mainly from the middle of 2011 to the middle of 2012. To the best of our knowledge, this is the first report on the presence of a Rieger-type periodicity in GCR data during Cycle 24. Oscillations in this group were pronounced in different solar and GCR data during Cycle 23 (Chowdhury, Jain, and Awasthi 2013; Chowdhury et al. 2010a, 2010b; Kudela et al. 2010; Singh, Gautam, and Badruddin 2012, and references therein). Recently, Chowdhury et al. (2015) has made an extensive time series analysis of different solar and interplanetary data for the first five years of Cycle 24 and detected a prominent presence of solar rotational as well as Rieger-type periods during this epoch. Our present analysis also exhibits the presence of some other QBOs within the range of 240 – 490 days including the well-known ∼ 1.3 yr period. In terms of quasi-biennal oscillations we assume a rather wide interval of the time scales of 0.6 – 4 years, consistently with the article by Bazilevskaya et al. (2014a). However, here we discuss only selected quasi-periodicities in that range.

The QBOs are considered as one of the basic variations of solar activity indices on the scale shorter than 11 years and probably they are intrinsic properties of the Sun related to the solar dynamo mechanism. The solar QBOs are translated into the heliosphere through the solar open magnetic flux (Lockwood 2001) and permanently exist in the heliospheric parameters and GCR data. Vecchio et al. (2012) studied QBOs in the GCR intensity and showed that these groups of oscillations are actually responsible for the Gnevyshev Gap phenomenon as well as for the step-like decreases typical for the GCRs’ modulation. However, the origin and nature of the QBOs in the Sun and their influence on the heliospheric phenomena including GCR modulation remain not understood fully and need further investigation.

Mavromichalaki et al. (2003) studied the periodic fluctuation of GCR data obtained from the Climax Neutron Monitor station for the time interval from 1953 to 1996 and noted different periodicity peaks at 11, 7.5, 5.5, 2, 1.7, and 1 years, as well as at 8, 6, 4, and 3 months; in the integral power spectra over all these years those of 1.7 year and 5.1 months are reported. Kane (2005) found CR periodicities in the QBO range matching those of solar indices. Thus, in the present investigation we have detected some periodicities which were obtained by several studies from various cosmic-ray stations in different previous solar cycles. However, the mechanisms behind the periodicities are still unknown. The intensity of GCRs inside the heliosphere is highly modulated by global solar magnetic activities and eruptive events. Solar flares and CMEs change the conditions in the interplanetary medium, introducing magnetic irregularities, which in turn change the GCR intensity. GCR intensity variations are well correlated with the variation of the interplanetary magnetic field (Kudela et al. 1993; Burlaga, McDonald, and Ness 1993; Cane et al. 1999). One nanotesla (nT) increase in IMF magnitude leads to a ∼ 0.2 % decrease in GCR intensity. So, IMF changes play an important role in the increase or decrease of GCR intensity. Gil and Alania (2013) concluded that the ∼ 27 day period and its harmonics are created by joint processes of the turbulent solar magnetic dynamo and differential rotation of the Sun and this is also prominent in GCR fluctuations. Kudela, Ananth, and Venkatesan (1991) suggested that there are two distinct groups of cosmic-ray periods caused by different physical mechanisms. The large-scale variations are caused by the solar activity dynamics, whereas the short-scale ones are caused by transient effects in interplanetary space. This depends on the fact that the short- and intermediate-term periods have different probabilities of occurrence in different epochs. Hence, further studies of GCR data from different neutron-monitor stations in different phases of the solar activity cycle are required to collect more information and advance our understanding of the reasons behind the GCR periodicities.

The cosmic-ray modulation is a very complex phenomenon which occurs all over the heliosphere and depends on several factors. No single solar or heliospheric parameter, however sophisticated, can account for the cosmic-ray variations.

The understanding of the solar modulation of GCRs is based on the standard model of diffusion, convection, and the adiabatic deceleration effect, where the IMF lines and \(\Lambda\) including drift processes determine the path of individual particles through the heliosphere. This leads to characteristic differences between adjacent solar cycles due to the different solar polarity states. The main determining parameter of particle transport is the gyroradius, which is inversely proportional to the IMF strength. An increase of the IMF strength leads to a decrease of the transport path and the diffusion coefficient and, consequently, an increase in the GCR modulation. However, this solar cycle is somehow different from the previous even-numbered cycles for which we have found a long time lag in the GCR modulation, QBOs, and Gnevyshev gap. A more detailed study considering the data from other NM stations is under way.