1 Introduction

Periodicity behaviour and modulation of galactic cosmic ray (GCR) particles have been studied and analysed for several decades, using both ground-based neutron-monitor (NM) data (Gil and Mursula, 2017; Usoskin et al., 2001; Kotzé, 2020) as well as space-based data, e.g. the Advanced Composition Explorer (ACE: Leske et al., 2010; Kotzé, 2021). The transient behaviour of GCR particles is particularly sensitive to solar activity and the resulting influence of the heliospheric-magnetic-field fluctuations and orientations in the local interstellar medium (Potgieter, 2013), leading to the modulation of GCR particles on periodic time scales ranging from hours to several years (Chowdhury, Kudela, and Moon, 2016; Usoskin, 2017). The anomalously low levels of solar activity during both the minima of Solar Cycle 23/24 (SC23/24) and Solar Cycle 24/25 (SC24/25) have resulted in much lower interplanetary magnetic-field strengths, which was accompanied by unusually small tilt-angle values of the heliospheric current sheet. During the minimum of SC24/25, the tilt angle was more than 50% lower than during the minimum of SC23/24. In fact, weaker than usual heliospheric magnetic fields and lower levels of turbulence led to an increase of \(\approx 10\%\) in the cosmic-ray mean free path during SC24/25 in comparison to 2009 (Rankin et al., 2022). This led to the highest recorded levels of GCR intensities (Rankin et al., 2022). The movement of cosmic rays in the heliosphere is determined by both turbulence effects in the magnetic field and gradient and curvature drifts of the large-scale heliospheric magnetic fields Jokipii, Levy, and Hubbard (1977). In addition, cosmic rays are also modulated by the polarity of the solar dipole, giving rise to the well-known 11-year and 22-year cycles. During negative polarity cycles (A < 0, minimum SC23/24), positively charged ions drift inward along the heliospheric Equator and outward through the north and south polar regions. On the contrary, during positive-polarity cycles (A > 0, minimum SC24/25), particles drift outward along the Equator and inward towards the solar polar regions (see Figure 4, Rankin et al., 2022). An investigation by Cane et al. (1999) concluded that the amplitude of recurrent GCR variations is larger during minimum intervals when A > 0 in comparison to minima characterised by A < 0. Several studies have been made on particularly the polarity-dependent behaviour of the 27-day periodicity in GCRs. A large proportion of these investigations focused on the use of NM data (Alania et al., 2001; Gil and Alania, 2001; Alania, Gil, and Modzelewska, 2008), while spacecraft data have been incorporated in studies by McDonald, Webber, and Reames (2010) and Modzelewska and Gil (2021). A study by Gil and Mursula (2017) concluded that the polarity-dependent behaviour of the 27-day periodicity of GCRs as observed by NMs is controlled by a combination of drift and solar-wind convection in the heliosphere. A recent study was carried out by Modzelewska and Gil (2021) on the recurrence rate of the 27-day periodicity in GCR observations of both NMs and spacecraft (ACE, STEREO, and SOHO). They reported greater amplitudes during the A > 0 minimum of SC24/25 (2017 – 2019) in comparison to the A < 0 minimum of SC23/24 (2007 – 2009) as observed by ground-based NMs. However, Modzelewska and Gil (2021) analyzed three-year averaged heavy-ion C, N, O, Ne, Si, and Fe observations by ACE/CRIS during 2007 – 2009 and 2017 – 2019, respectively, and found no clearly defined 27-day polarity-dependent behaviour. The authors further concluded that the GCR recurrence as a result of solar rotation for low-energy (< 1 GeV) particles is more sensitive to diffusion effects, leading to similar amplitudes of the 27-day periodicity for both solar polarity minima. On the other hand, high-energy GCRs (> 1 GeV), as observed by NMs, are modulated by drift effects in the heliosphere, resulting in lower 27-day periodicity amplitudes during negative polarity cycle minima in comparison to positive polarity minima.

Most studies in the past have concentrated on the behaviour of the 27-day synodic rotation period in energetic cosmic particles, while the second harmonic 13.5-day period received far less attention. Previous research results reported in the literature were by Gil and Mursula (2017) investigating the 13.5-day periodicity, by Alania and Shatashvili (1974) on the 9-day periodicity, and by Kotzé (2020) using NM data to investigate the fourth harmonic (6.7 days) of the 27-day periodicity. This article reports for the first time the temporal and energy-dependent behaviour of both the 27-day periodicity and its second (13.5-day) harmonic on an annual interval in proton, C, and O cosmic particles as observed by the ACE and SOHO spacecraft for both SC23 and SC24 during the time interval 2001 to 2019. In addition, NM cosmic-ray observations corresponding to different cut-off rigidities have been analysed annually during the same time interval to investigate the behaviour of both the 27-day and the 13.5-day periodicities. Of particular interest is the behaviour of these periodicities during the unique and anomalous minima of SC23/24 (2008 – 2009) and SC24/25 (2018 – 2019), corresponding to different polarity orientations of the solar dynamo.

2 Data and Analysis

In this investigation, hourly mean proton particle fluxes measured by the SOHO/EPHIN instrument (Müller-Mellin et al., 1995) have been downloaded from the OMNIWeb database (omniweb.gsfc.nasa.gov/ftpbrowser/soho_ephin_flux_hr.html) and converted to daily mean values for each energy interval. The centre of each energy interval has been taken as the representative energy level used in this investigation. In the case of C and O, daily mean cosmic particle fluxes measured by the ACE/CRIS spacecraft instrument (www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_CRIS.html) at seven different energy levels have been used and arranged in yearly intervals from 2001 to 2019. Heliospheric solar-wind data at 1 AU are provided by OMNI (omniweb.gsfc.nasa.gov). Hourly mean neutron-monitor data for the three stations at Oulu, Hermanus, and Tsumeb were obtained from the World Data Center for Cosmic Rays (cidas.isee.nagoya-u.ac.jp/WDCCR/) and subsequently converted to daily mean values. The Lomb–Scargle (Lomb, 1976; Scargle, 1982) periodogram spectrum-analysis technique has been used to determine the behaviour of the 27-day as well as the 13.5-day periodicity for each mean energy level for every annual interval in the mentioned time interval between 2001 and 2019 for the Oulu, Hermanus, and Tsumeb NM observations as well as the GCR modulation parameter \(\zeta \) = BIMF × VSW. In this investigation, the absolute magnitude of \(B\)IMF was determined as \(\sqrt{B_{X}^{2} + B_{Y}^{2} + B_{Z}^{2}}\), where \(B_{X}\), \(B_{Y}\), and \(B_{Z}\) are respectively the \(X-, Y\)-, and \(Z\)-components of the interplanetary magnetic field in GSE coordinates. Similarly, a three-dimensional pattern of the Lomb–Scargle power as a function of energy and time for both the 13.5 and 27-day periodicities in proton, C, and O GCR particles have been determined. All energies reported and utilized in this study were the mean of each of the seven energy intervals for each respective element as measured by the ACE/CRIS as well as the SOHO/EPHIN spacecraft-based instruments. Only results complying with a 95% confidence level are reported and utilized in this study.

3 Results and Discussion

A recent study by Modzelewska and Gil (2021) investigating the recurrence rate of the 27-day periodicity in GCR observations of both NMs and spacecraft (ACE, STEREO, and SOHO), reported greater amplitudes during the A > 0 minimum of SC24/25 (2017 – 2019) in comparison to the A < 0 minimum of SC23/24 (2007 – 2009) NM data, while heavy-ion energetic GCR particle observations (C, N, O, Ne, Si, and Fe) by ACE did not exhibit the same polarity-dependent behaviour, using three-year averaged data. In this article, the main aim is to determine whether the 27-day as well as the 13.5-day periodicities in GCR particles show any energy- and time-dependent behaviour during SC23 and SC24, particularly during the respective minima of these solar cycles. A Lomb–Scargle spectral analysis was performed at each of the various energy intervals for C and O as measured by the ACE/CRIS instrument (Stone et al., 1998) for every year, consisting of daily mean values between 2001 and 2019. A similar analysis was also carried out for all energy intervals of proton energetic cosmic particles between 2001 and 2019 as observed by the SOHO/EPHIN instrument. The main aim was to investigate the temporal behaviour of each periodicity during different phases of both Solar Cycles 23 and 24. Figures 1 and 2 display the Lomb–Scargle powers of the 27-day and 13.5-day periodicities, respectively, between 2001 and 2019 as a function of particle energy for proton GCR particles.

Figure 1
figure 1

Behaviour of the 27-day Lomb–Scargle periodicity as revealed in proton GCR particles observed by SOHO/EPHIN between 2001 and 2019 as a function of energy.

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Figure 2
figure 2

Behaviour of the 27-day Lomb–Scargle periodicity as revealed in proton GCR particles as observed by SOHO/EPHIN between 2001 and 2019 as a function of energy.

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From Figure 1 it is evident that the 27-day periodicity in proton GCR particles shows a clear energy- and time-dependent behaviour. Of particular interest is the difference between SC23 when A < 0 and SC24 when A > 0. In the case of SC24, we notice a substantial (\(\approx 30\%\)) increase in power for lower energies (20 MeV n−1 < E < 50 MeV n−1) in comparison to the minimum of SC23.

Figure 2 reveals that for energies > 70 MeV n−1, the powers of the 13.5-day period during the minimum of SC23/24 exceed that of SC24/25. However, for energies between 20 and 30 MeV n−1, the powers of the 13.5-day period during the positive solar polarity prevailing during Cycle 24, we observe the opposite behaviour. It is also clear from Figure 2 that during Cycle 23, when the solar polarity was negative, that the 13.5-day period was much stronger by \(\approx 35\%\) at higher energies (70 – 90 MeV n−1) than at lower energies (20 – 30 MeV n−1).

A Lomb–Scargle periodogram analysis of C GCR particles as observed by the ACE/CRIS spacecraft instrument between 2001 and 2019 revealed a strong energy as well as a solar-polarity dependence. This is particularly evident for the 27-day periodicity, as shown in Figure 3.

Figure 3
figure 3

Behaviour of the 27-day Lomb–Scargle periodicity as revealed in carbon GCR particles as observed by ACE/CRIS between 2001 and 2019 as a function of energy.

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From Figure 3, we notice that during the minimum of SC34/25 during 2018 – 2019 for energies between \(\approx 100\) MeV n−1 and \(\approx 150\) MeV n-1, the 27-day period exhibit powers that are \(\approx 50\%\) higher in comparison to the minimum of Cycle 23 (2008 – 2009). This is a time interval characterised by positive (A > 0) solar polarity.

The 13.5-day periodicity in C GCR energetic particles on the other hand can be seen in Figure 4. From Figure 4, a strong energy as well as temporal dependence of the 13.5-day periodicity can be observed. Of particular interest is that the power of the 13.5-day period during Cycle 23, characterised by a negative solar polarity, is comparable to the power during Cycle 24, when a positive solar polarity prevailed, but only for energies between \(\approx 110\) MeV n−1 and \(\approx 150\) MeV n−1. However, at lower energies, between \(\approx 70\) MeV n−1 and \(\approx 80\) MeV n−1, the power of the 13.5-day period is noticably higher during the minimum of SC24/25 (2018 – 2019) in comparison to the minimum of SC23/24 (2007 – 2009). From Figures 3 and 4 it is noticeable that the maximum power of the 13.5-day periodicity is approximately 50% weaker than the maximum power of the 27-day period.

Figure 4
figure 4

Behaviour of the 27-day Lomb–Scargle periodicity as revealed in carbon GCR particles as observed by ACE/CRIS between 2001 and 2019 as a function of energy.

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An investigation of the 27-day periodicity behaviour in energetic O GCR particles as measured by the ACE/CRIS spacecraft instrument during 2001 – 2019 also revealed an interesting behaviour. Figure 5 shows the Lomb–Scargle behaviour of the 27-day period as a function of energy between 2001 and 2019. The 27-day period displays a noticeable energy as well as a temporal behaviour with enhanced power during particularly the minimum of SC24/25, a time interval characterised by positive solar polarity (A > 0). These power enhancements are predominantly concentrated around \(\approx 105\) MeV n−1 and \(\approx 215\) MeV n−1.

Figure 5
figure 5

Behaviour of the 27-day Lomb–Scargle periodicity, as revealed in oxygen GCR particles observed by ACE/CRIS between 2001 and 2019, as a function of energy.

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The behaviour of the 13.5-day periodicity in O GCR particles is displayed in Figure 6, showing a strong and clear energy as well as a temporal variation during 2001 – 2019. During Cycle 23, the strongest 13.5-day activity occurs during 2007 – 2010, the minimum of SC23/24, and concentrated at energies between \(\approx 105\) MeV n−1 and \(\approx185\) MeV n−1. On the other hand, during Cycle 24 (A > 0), the strongest power of the 13.5-day period occurs between \(\approx 175\) and \(\approx 195\) MeV n−1 during the time interval 2015 – 2017. During the minimum of SC24/25, we also notice above-average 13.5-day activity at lower energies around \(\approx 85\) MeV n−1; a phenomenon that was not observed during Cycle 23.

Figure 6
figure 6

Behaviour of the 13.5-day Lomb–Scargle periodicity, as revealed in oxygen GCR particles observed by ACE/CRIS between 2001 and 2019, as a function of energy.

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A comparison between the maximum powers of the 27-day (Figure 4) and 13.5-day (Figure 5) periodicities revealed that the 27-day period is approximately double the strength of its second harmonic.

A study of the solar-polarity dependence of ground-based GCR particles, using NM data obtained from stations during the time interval 2001 – 2019 with widely different rigidity cut-off parameters yielded results for both the 27-day as well as the 13.5-day periodicity in line with spacecraft observations. In this study, we used daily mean NM data from Oulu (RC = 0.78 GV), Hermanus (RC = 4.9 GV), and Tsumeb (RC = 9.29 GV) at annual time intervals. The Lomb–Scargle periodogram technique was subsequently applied to obtain periodicity behaviour information in compliance with a 95% statistical confidence level. Results obtained, showing the variability of the 27-day periodicity during Cycles 23 and 24, are shown in Figure 7. A procedure proposed by Vaughan (2005) to test for red noise in the data revealed that white noise is the dominant background.

Figure 7
figure 7

Variability of the 27-day periodicity in GCR observations at Oulu (a), Hermanus (b), and Tsumeb (c) NM stations. Calculated error bars are shown for each result.

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The 27-day periodicity of GCR as revealed by the Oulu, Hermanus, and Tsumeb NM observations show a defined temporal variation during Cycles 23 and 24. In particular, both Hermanus and Tsumeb, which are more sensitive to high-energy GCR particles (> 1 GeV), display clearly higher 27-day powers during Cycle 24, when a positive solar polarity prevailed, in comparison to Cycle 23 with a negative solar polarity. At Oulu, however, with a RC = 0.87 GV, the NM is more sensitive to low-energy GCR particles (< 1 GeV). We observe a much smaller difference between a negative- and a positive-polarity solar cycle, in line with the results reported by Modzelewska and Gil (2021). The 13.5-day periodic behaviour for Oulu, Hermanus, and Tsumeb during Cycles 23 and 24 is shown in Figure 8.

Figure 8
figure 8

Variability of the 13.5-day periodicity in GCR observations at Oulu (a), Hermanus (b), and Tsumeb (c) NM stations. Calculated error bars are shown for each result.

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The 13.5-day periodicity also exhibits a solar-cycle-dependent behaviour with the highest power appearing during Cycle 24, which is characterised by a positive solar-polarity orientation. This demonstrates that both the 27-day as well as the 13.5-day periodicity are subject to the same heliospheric modulation effects prevailing during different solar-polarity intervals. Interplanetary and heliospheric parameters that have been identified in recent studies (Fadaaq and Badruddin, 2021) that play an important role in the modulation of GCR particles are the solar-wind speed [VSW] and BIMF. A Lomb–Scargle periodogram analysis of the modulation parameter \(\zeta \) = BIMF × VSW that reveal the variability of both the 27-day as well as the 13.5-day periodicity can be seen in Figure 9.

Figure 9
figure 9

The Lomb–Scargle power of the 27-day (a) as well as the 13.5-day (b) periodicity in \(\zeta \) = BIMF × VSW. Calculated error bars are shown for each result.

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An analysis of the behaviour of the 27-day periodicity in Figure 9 reveals a distinct difference between the minimum of SC23/24 and the minimum of SC24/25. The average power of the 27-day periodicity during 2017 – 2019 (when A < 0) is \(\approx 50\%\) less than during 2017 – 2019 (when A > 0). The 13.5-day periodicity behaviour during 2001 – 2019 shows no clear difference during Cycle 23 and Cycle 24, corresponding to different solar-polarity orientations.

4 Conclusions

A Lomb–Scargle periodogram spectral analysis has been made to investigate the energy- and time-dependent behaviour of the 27-day as well as the 13.5-day periodicities in proton, C, and O GCR particles as measured by the SOHO/EPHIN and ACE/CRIS spacecraft during SC23 and SC24. In addition, the behaviour of these periodicities has also been studied in GCR observations at ground level corresponding to different particle energies using NM data collected at the Oulu, Hermanus, and Tsumeb stations. A Lomb–Scargle periodogram analysis of the cosmic-ray modulation parameter \(\zeta \) = BIMF × VSW was done to investigate the behaviour of both the 27-day and 13.5-day periodicity using daily means for each annual interval during SC23 and SC24. Particular attention has been paid to the respective cycle minima; SC23/24 (2008 – 2009) and SC24/25 (2018 – 2019). Previous investigations of particularly the 27-day periodicity behaviour of GCR particles as detected by NMs (Gil and Mursula, 2017; Modzelewska and Gil, 2021) concluded that the amplitude of this recurrent oscillation is more enhanced during solar-cycle minima characterised by A > 0 (northward solar dipole) in comparison to minima with A < 0 (southward solar dipole). During cycles with a negative solar polarity, energetic GCR particles are more sensitive to diffusion effects as they migrate inwards towards the Sun along the heliospheric current sheet (Moraal and Stoker, 2010; Rankin et al., 2022), where transient events such as co-rotating interaction regions determine the modulation characteristics of the GCR particles. A study by Modzelewska and Gil (2021) on the behaviour of the amplitudes of the 27-day periodicity in heavy-ion particle GCR measurements (C, N, O, etc.) by ACE/CRIS, concluded that the 27-day period is solar-polarity independent during the minima of SC23/24 and SC24/25. In addition, these authors also concluded that high-energy GCR observations by NMs are more sensitive to large-scale drift effects, resulting in larger amplitudes of the 27-day variation when A > 0 in comparison to A < 0 intervals.

Results obtained in this study clearly show both an energy as well as a solar-polarity dependence of the 27-day as well as the 13.5-day periodicity for protons, C, and O GCR particles observed by ACE/CRIS and SOHO/EPHIN spacecraft during SC23 and SC24. Of particular interest in the case of the 27-day periodicity in protons is the noticeable difference between the SC23 when A < 0 and SC24 when A > 0. During SC24, we notice a substantial (\(\approx30\%\)) increase in power for lower energies (20 MeV n−1 < E < 50 MeV n−1) in comparison to the minimum of SC23. For proton energies > 70 MeV n−1, the power of the 13.5-day periodicity during the minimum of SC23/24 exceeds that of SC24/25 by approximately 30%. The power of the 27-day periodicity behaviour in C and O GCR particles during the minimum of SC24/25 exceeds their powers as observed during the minimum of SC23/24. It is therefore clear that the 27-day periodicity in high-energy, heavy-ion C and O cosmic particles is more sensitive to large-scale drift effects during periods of positive solar-polarity intervals. A similar tendency could also be observed for the 13.5-day periodicity in C, but in O the 13.5-day period exhibited a more energy-dependent behaviour, with enhanced amplitudes (powers) for lower energies around \(\approx 85\) MeV n−1 during SC24, when A > 0 prevailed.

Our results confirm the polarity rule in the behaviour of the power of both the 27-day and 13.5-day periodicities of GCR observations by NM stations in the solar minimum of Cycle 23 and Cycle 24, with larger powers observed during positive polarity intervals. Both Hermanus and Tsumeb stations, which are more sensitive to high-energy GCR particles (> 1 GeV), display clearly higher 27-day powers during Cycle 24 when a positive solar polarity prevailed in comparison to Cycle 23 with a negative solar polarity. At Oulu, however, with a RC = 0.87 GV, the NM is more sensitive to low-energy GCR particles (< 1 GeV), the difference of the powers between the two polarity intervals is much smaller. This indicates that low-energy cosmic rays are more sensitive to enhanced diffusion, resulting in smaller differences between negative and positive solar-polarity intervals, as shown by the results obtained in this investigation.

An investigation by Modzelewska and Alania (2013) showed that the product of the solar-wind speed VSW and the heliospheric magnetic field BIMF has an important influence on the transient modulation of GCR particles, particularly the behaviour of the 27-day as well as the 13.5-day periodicity. Results obtained in this investigation, shown in Figure 9, clearly indicate an enhancement in the power of the 27-day periodicity during the minimum of SC24/25 in comparison to the minimum of SC23/24. We attribute this to the much weaker BIMF during Cycle 24 compared to Cycle 23. However, the behaviour of the 13.5-day periodicity can be regarded as inconclusive, although a substantial increase of the 13.5-day power could be detected during 2016 of Cycle 24 (A > 0).

Results reported in this article reveal strong evidence that the 27-day periodicity in GCR particles, as well as NM observations and interplanetary solar-wind parameters, experience an increase in power during time intervals characterised by a positive solar polarity in contrast to intervals with negative solar polarity. The 13.5-day periodicity also showed similar characteristics, although in the case of protons (SOHO/EPHIN), C (ACE/CRIS), and BIMF × VSW the results showed no clear difference between SC23 and SC24. A strong energy-dependence for both the 27-day and 13.5-day period could also be clearly observed in the GCR analysis results. A recent study (Modzelewska et al., 2020) of the 27-day variations in GCR fluxes during the prolonged 2007 – 2008 solar minimum between SC23 and SC24 using space-based PAMELA (Adriani et al., 2014) and ARINA (Bakaldin et al., 2007) proton data also revealed a strong rigidity (particle energy) dependence. The increase in the power of periodicities during time intervals of positive solar polarity indicates that large-scale drift effects play an important role in the behaviour and modulation of cosmic particles.

To the best of our knowledge, these results, based on annual daily mean observations, have not been reported before in the literature.