1 Introduction

The modulation of galactic cosmic rays (GCRs) in the heliosphere is a very complex process, closely related to the activity of the Sun. So far it has not been possible to create a uniform theory of long-term GCR modulation in the heliosphere. Potgieter and Ferreira (2001) proposed the time-dependent model that included the propagating diffusion barrier as the main mechanism of the 11-year and 22-year variations of the GCR. Bieber and Matthaeus proposed two theories: the BAM theory in 1997 (Bieber and Matthaeus, 1997) and later the nonlinear guiding center theory (NLGCT) (Matthaeus et al., 2003), and Shalchi suggested a weakly nonlinear theory (WNLT) (Shalchi and Schlickeiser, 2004). Recently, Engelbrecht et al. (2022) studied the behavior of the diffusion parameters for low-energy electrons. Burger, Nel, and Engelbrecht (2022) analyzed the turbulence properties based on the spectral features of magnetic field observations for 1973 – 2020.

Based on the quasi-linear theory (QLT) developed by Jokipii (1971), Toptygin (1985) and Shalchi (2009) showed that there should be a relationship between the parallel diffusion coefficient \(K_{\mathit{II}}\) and the rigidity \(R\) of GCR particles, as \(K_{\mathit{II}} \propto R^{\beta}\) for the rigidity range of neutron monitors (NMs). Based on the QLT, the parameter \(\beta \) is determined as \(\beta = 2 - \nu \), where \(\nu \) is the exponent of the power spectrum density (PSD) of the heliospheric magnetic field (HMF) turbulence. Lockwood and Webber suggested (Lockwood and Webber, 1984) that the major part of the 11-year variation is the result of the accumulative effects of Forbush decreases. In turn, Webber and Lockwood (1988) showed that drift effects play a significant role in the GCR modulation process; however, other effects could be equally important. Le Roux and Potgieter (1995) made a simulation of complete 11- and 22-year modulation cycles for GCR in the heliosphere using a model with a combination of drift and global merged interaction regions.

As was shown in the many works (Alania et al., 2001; Alania, Bochorishvili, and Iskra, 2003; Siluszyk et al., 2005; Alania, Iskra, and Siluszyk, 2008, 2010; Siluszyk, Iskra, and Alania, 2014; Iskra, Siluszyk, and Alania, 2015; Siluszyk et al., 2018; Siluszyk and Iskra, 2020), the general mechanisms of long-term variation of GCRs are caused by a redistribution of large-scale structures of the magnetic turbulence in the solar wind (SW) in various epochs of solar activity (SA). These large-scale structure of the magnetic turbulence and its redistribution are determined by two independent proxies: exponent \(\gamma \) of the power-law rigidity spectrum of the GCR intensity variation \(\delta D(R)/D(R)\) \((\delta D(R)/D(R) \propto R^{- \gamma})\) and the exponents \(\nu _{z}\) and \(\nu _{y}\) of the PSD of the \(Bz\) and \(By\) components of the HMF turbulence (PSD \(\approx f^{- \nu}\)), where \(f \) is the frequency.

In Iskra, Siluszyk, and Alania (2015), Iskra et al. (2019), it was found that the delay time exists between the changes of the SA and other parameters characterizing the conditions in the interplanetary space and the GCR intensity, and that the amplitudes of the GCR modulation significantly vary for different 11-year cycles (Nagashima and Morishita, 1980). Based on works on the mechanism of long-term variations in GCR intensity, it can be concluded that each cycle is different and modulation processes may take place in different ways. There are periods where the main modulating factor is diffusion, and other times a drift. There are also periods where we have a combination of diffusion and drift. An important tool in estimating the role of drift is the analysis of the GCRA in various HMF sectors and different polarities of the Sun’s magnetic field, which also allows one to determine the parameters needed for modeling of GCR propagation in various cycles of SA.

The solar diurnal variation of the GCRA is the result of the fact that the diffusion flux due to the existence of the GCR gradient is balanced by the convective flux of the SW. The mechanism of the solar diurnal GCRA is based on the anisotropic GCR modulation theory in the heliosphere (Ahluwalia and Dessler, 1962; Krymsky, 1964; Parker, 1964; Gleeson and Axford, 1967, 1968).

This picture was later developed to include the effects of particle drifts in the large-scale magnetic field (Jokipii, Levy, and Hubbard, 1977; Kota and Jokipii, 1983).

Recently, the GCRA has also been used to classify different classes of Forbush decreases (e.g. Lingri et al., 2022), study the uniqueness of Dome C NM, being the only existing NM accepting GCRs originating from the off-equatorial region (Gil et al., 2022), or investigate the scaling features of GCR transport throughout the solar cycle using the Hurst exponent (Modzelewska et al., 2021).

The aim of this work is to study the nature of the large-scale structure of the HMF turbulence in the solar wind in various cycles of solar activity and within each cycle on the basis of the analysis of GCRA in 1996 – 2020. We determined the contributions of drift and diffusion processes to galactic cosmic-ray transport in the heliosphere from year to year starting from 1996 until 2020 (Solar Cycles 22 – 25). Furthermore, we updated the calculation of parameters that characterize the GCR modulation from 2014 to 2020.

The article consists of four main sections. In Section 1 we present a brief introduction to the GCR modulation mechanism and the role of particle drift in particle propagation, and estimate its role on the basis of the analysis of the anisotropy in different HMF sectors and in different polarities of the Sun’s magnetic field. In Section 2, we briefly recall the methods that were previously used in the works of Modzelewska et al. (2019) to calculate the GCRA and Ahluwalia et al. (2015) to estimate the modulation parameters. In Section 3, we present experimental results and a discussion. Section 4 is a summary of the work.

2 Methods

The purpose of this work is to analyze the GCR anisotropy from year to year, starting from 1996 (the last year of the Solar Cycle 22) with positive polarity (\(A>0\)), throughout the Solar Cycles 23 (1997 – 2008) and 24 (2009 – 2019) with positive (\(A>0\)) and negative polarity (\(A<0\)), and the beginning of the Solar Cycle 25 (i.e. 2020) with positive polarity (\(A>0\)). We show in which periods of SA diffusion or drift dominate in the process of GCR particle transport, or both are important for the modulation of GCR particles in the heliosphere.

The second important goal of this work is to identify the second kind of drift in the GCRA due to the motion of GCR particles along the heliospheric current sheet alternating through the negative and positive sectors of HMF.

Using the algebraic system of equations for the GCRA components, we calculate different parameters important for the modeling of GCR propagation in the heliosphere, such as: the ratios perpendicular \(K_{\bot}\) to parallel \(K_{\mathit{II}}\) and drift \(K_{T}\) to \(K_{\mathit{II}}\) diffusion coefficients.

The parameters characterizing the NMs that were used to calculate the components of the GCRA are shown in Table 1. The expression \(\int _{R_{c}}^{R_{\max}} R^{-\gamma} W(R,h)\,\mathrm{d}R\) is called the integral coupling coefficient (\(\mathit{CC}\)) connecting the intensity of the secondary GCR with the primary one, \(W \left ( R,h \right )\) is the coupling function for rigidity \(R\) at height \(h\), \(R_{c}\) and \(R_{\max}\) are the cut-off and maximum rigidities, respectively. The integral coupling coefficients were determined by Yasue et al. (1982) for various maximum rigidities \(R_{\max}\) of the modulation and various power-law exponents \(\gamma \) on the rigidity \(R\) of the spectrum of GCR intensity variation. In our calculations, the CCs for the GCRA calculations were taken for \(\gamma = 0\) and \(R_{\max} = 100\) GV at the minimum, maximum, and intermediate level of SA, respectively.

Table 1 Parameters characterizing the NMs.
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We converted the radial \(a_{r}\) and azimuthal \(a_{\varphi}\) components of the diurnal GCR variation into the radial \(Ar\) and azimuthal \(A_{\varphi}\) components of the GCR anisotropy in the heliosphere (Dorman, 1963; Yasue et al., 1982) dividing the radial and azimuthal amplitudes of the components of the anisotropy recorded on Earth by the appropriate coupling coefficients:

$$\begin{aligned} & A_{r_{j}} = \frac{a^{E}_{r_{j}}}{\int_{R_{c}}^{R_{\max}} R^{-\gamma} W (R, h_{j})\,\mathrm{d}R} \\ & A_{\varphi_{j}} = \frac{a^{E}_{\varphi_{j}}}{\int_{R_{c}}^{R_{\max}} R^{-\gamma} W (R, h_{j})\,\mathrm{d}R} \end{aligned}$$

where \(a_{rj}^{E}\), \(a_{\varphi j}^{E} \) are the amplitudes of the radial and azimuthal components of the anisotropy for the \(j\)th NM on Earth and \(A_{rj}\), \(A_{\varphi j} \) in the heliosphere. For more details of the GCRA calculations see, e.g., Modzelewska et al. (2019) and Modzelewska and Alania (2018).

An energy-integrating GCR detector is typically described by its ‘effective rigidity’, which may have different meanings in the literature, such as: fixed rigidity of 10 GV for a NM (Belov, 2000) or the median rigidity of the NM response (Ahluwalia et al., 2015; Ahluwalia and Fikani, 2007). However, these concepts may vary with a solar cycle. In Gil et al. (2017) and Asvestari et al. (2017), a concept of the effective rigidity was proposed, defined as the variability of the GCR flux at this energy being directly proportional to the detector count rate. For a comparison of anisotropy analysis for median and effective rigidity approaches for NMs, see Modzelewska et al. (2019). We follow the procedure for GCRA calculation based on the works by Alania, Bochorishvili, and Iskra (2003), Modzelewska and Alania (2018), and Modzelewska et al. (2019) using the approach of the effective rigidity Ref, unlike the median energy approach.

Based on the hourly data from six NMs with cut-off rigidities \(R_{c}<5\) GV, using harmonic analyses the components radial \(Ar\) and tangential \(A\varphi \) were calculated for each year in the last year of the even Solar Cycle 22, the odd Cycle 23, the even Cycle 24, and beginning of the odd Cycle 25. We used pressure- and efficiency-corrected data from the Neutron Monitor Data Base (NMDB, http://www01.nmdb.eu/) and for Moscow NM in 2019 and 2020 from IZMIRAN (http://cr0.izmiran.ru/mosc/). In those years, when the data of a given monitor were not available (e.g. Kiel in 2018 – 2020), this NM was not taken into account for the calculation of the average anisotropy for a given year. Therefore, in 2018 – 2020, the average anisotropy was determined from five NMs.

We used a system of equations for components of the GCRA (Alania et al., 1987; Riker and Ahluwalia, 1987; Alania et al., 2001):

$$\begin{aligned} A_{r}^{\pm} =&3 \left [ \mathit{CU} - K_{rr} G_{r}^{\pm} \pm K_{T} G_{\theta}^{\pm} \sin \varPsi + \left ( K_{\coprod} - K_{\bot} \right ) G_{\varphi}^{\pm} \sin \varPsi \cos \varPsi \right ] / V \end{aligned}$$
(1)
$$\begin{aligned} A_{\theta}^{\pm} =&-3 \left [ \pm K_{T} G_{r}^{\pm} \sin \varPsi + K_{\bot} G_{\theta}^{\pm} \pm K_{T} G_{\varphi}^{\pm} \cos \varPsi \right ] / V \end{aligned}$$
(2)
$$\begin{aligned} A_{\varphi}^{\pm} =&3 \left [ \left ( K_{\coprod} - K_{\bot} \right ) G_{r}^{\pm} \sin \varPsi \cos \varPsi \pm K_{T} G_{\theta}^{\pm} \cos \varPsi - K_{\varphi \varphi} G_{\varphi}^{\pm} \right ] / V \end{aligned}$$
(3)

where, \(K_{T}\), \(K_{\bot}\), and \(K_{\mathit{II}}\) are drift, perpendicular, and parallel diffusion coefficients of GCRs, respectively; \(K_{rr} = K_{\mathit{II}} \cos^{2} \varPsi + K_{\bot} \sin^{2} \varPsi \), \(K_{\varphi \varphi} = K_{\mathit{II}} \sin^{2} \varPsi + K_{\bot} \cos^{2} \varPsi \), \(G_{r\, \pm}\), \(G_{\theta \, \pm}\) and \(G_{\varphi \, \pm}\) are the radial, heliolatitudinal, and heliolongitudinal gradients of GCRs; \(C=1.5\) for GCRs sensitive to NM rigidity; \(\varPsi \) is the angle between the HMF lines and the Earth-Sun line; \(U\) and \(V\) are the velocity of the SW and of the GCR particles, the signs (−) and (+) correspond to the different solar polarity periods: negative and positive intervals of the HMF, respectively.

In the work of Alania et al. (1983a, 1983b), the system of Equations 1 – 3 was considered, assuming the equality of the GCR gradients in opposite sectors of the HMF. Previously, Ahluwalia and Dorman (1997) and Alania et al. (2001), Alania, Bochorishvili, and Iskra (2003), and recently Ahluwalia et al. (2015) and Modzelewska et al. (2019) performed similar calculations.

In this work, we discuss the nature of the change in the irregularity of the HMF and determine the role of the drift of GCR particles in the heliosphere from year to year, starting from 1996 until 2020 (i.e. Solar Cycles 22 – 25) and, continuing the research by Ahluwalia et al. (2015), we update the parameters characterizing the GCR modulation up to 2020 using an effective energy approach.

We performed calculations for the positive and negative sectors with durations no shorter than four days (HMF sector boundaries are taken from http://wso.stanford.edu/SB/SB.Svalgaard.html), and we also excluded changes in diurnal variation amplitude that exceeded 0.7%. This strict data selection criterion significantly reduces statistics. However, the effect of a given HMF sector on the particle motion is detected more reliably, i.e., the GCR particle drift in the regular HMF is determined relatively unambiguously.

We averaged the data annually, i.e., over a period much longer than the solar rotation period (≈27 days), and therefore we can assume that the heliolongitudinal gradient is equal to zero (\(G_{\varphi \, \pm} =0\)). After a simple transformation the system of Equations 1 – 3 with \(G_{\varphi \, \pm} =0\) takes the form:

$$\begin{aligned} A_{r}^{\pm} =&3 K_{\mathit{II}} \left [ \frac{\mathit{CU}}{K_{\mathit{II}}} - \left ( \cos^{2} \varPsi + \alpha \sin^{2} \varPsi \right ) G_{r}^{\pm} \pm \alpha _{1} G^{\pm} \sin \varPsi \right ] /V \end{aligned}$$
(4)
$$\begin{aligned} A_{\theta}^{\pm} =&-3 K_{\mathit{II}} [\pm \alpha _{1} G_{r}^{\pm} \sin \varPsi + \alpha G^{\pm} ]/ V \end{aligned}$$
(5)
$$\begin{aligned} A_{\varphi}^{\pm} =&3 K_{\mathit{II}} \left [ \left ( 1- \alpha \right ) G_{r}^{\pm} \sin \varPsi \cos \varPsi \pm \alpha _{1} G^{\pm} \cos \varPsi \right ] / V \end{aligned}$$
(6)

where \(\alpha \) and \(\alpha _{1}\) are the ratios of the perpendicular \(K_{\bot}\) and drift \(K_{T}\) diffusion coefficients to the parallel diffusion coefficient (\(K_{\mathit{II}}\)), respectively. The system of Equations 4 – 6 can be rewritten as:

$$\begin{aligned} A_{r}^{+} + A_{r}^{-} =& \left [ 6\mathit{CU}-3 K_{\mathit{II}} \left ( \cos^{2} \varPsi + \alpha \sin^{2} \varPsi \right ) \left ( G_{r}^{+} + G_{r}^{-} \right ) + 3 \alpha _{1} K_{\mathit{II}} \left ( G^{+}_{\theta} - G^{-}_{\theta} \right ) \sin \varPsi \right ] /V \\ \end{aligned}$$
(7)
$$\begin{aligned} A_{r}^{+} - A_{r}^{-} =&[-3 K_{\mathit{II}} \left ( \cos^{2} \varPsi + \alpha \sin^{2} \varPsi \right ) \left ( G_{r}^{+} - G_{r}^{-} \right ) + 3 \alpha _{1} K_{\mathit{II}} \left ( G^{+}_{\theta} + G^{-}_{\theta} \right ) \sin \varPsi ]/V \end{aligned}$$
(8)
$$\begin{aligned} A^{+}_{\theta} + A^{-}_{\theta} =&-3 K_{\mathit{II}} [ \alpha _{1} \left ( G_{r}^{+} - G_{r}^{-} \right ) \sin \varPsi + \alpha \left ( G^{+}_{\theta} + G^{-}_{\theta} \right ) ]/V \end{aligned}$$
(9)
$$\begin{aligned} A^{+}_{\theta} - A^{-}_{\theta} =&-3 K_{\mathit{II}} [ \alpha _{1} \left ( G_{r}^{+} + G_{r}^{-} \right ) \sin \varPsi + \alpha \left ( G^{+}_{\theta} - G^{-}_{\theta} \right ) ]/V \end{aligned}$$
(10)
$$\begin{aligned} A^{+}_{\varphi} + A^{-}_{\varphi} =&3 K_{\mathit{II}} [(1- \alpha ) \left ( G_{r}^{+} + G_{r}^{-} \right ) \sin \varPsi \cos \varPsi + \alpha _{1} \left ( G^{+}_{\theta} - G^{-}_{\theta} \right ) \cos \varPsi ]/V \end{aligned}$$
(11)
$$\begin{aligned} A^{+}_{\varphi} - A^{-}_{\varphi} =&3 K_{\mathit{II}} [(1- \alpha ) \left ( G_{r}^{+} - G_{r}^{-} \right ) \sin \varPsi \cos \varPsi + \alpha _{1} \left ( G^{+}_{\theta} + G^{-}_{\theta} \right ) \cos \varPsi ]/V \end{aligned}$$
(12)

When using the obtained values of the components \(A_{r}^{+}\), \(A_{r}^{-}\), \(A^{+}_{\varphi}\), \(A^{-}_{\varphi}\), \(A^{+}_{\theta}\), \(A^{-}_{\theta}\) of the GCRA to solve the system of Equations 7 – 12, the following circumstance should be pointed out: while the components \(A_{r}^{+}\), \(A_{r}^{-}\) and \(A^{+}_{\varphi}\), \(A^{-}_{\varphi}\) are determined quite reliably, there is uncertainty in the calculation of the components \(A^{+}_{\theta}\) and \(A^{-}_{\theta} \). This difficulty is associated with the possible existence of north-south asymmetries in the intensity and heliolatitudinal GCR gradient.

Thus, only \(A_{r}^{+}\), \(A_{r}^{-}\), \(A^{+}_{\varphi}\), and \(A^{-}_{\varphi}\) the GCRA components can be used to solve the system of Equations 7 – 12. However, in this case, the number of unknowns is greater than the number of equations, and some simplifications should be made. In this case, we used the procedure previously used by Alania et al. (1983a, 1983b), Ahluwalia and Dorman (1997), and Alania, Bochorishvili, and Iskra (2003) assuming equality of gradients in different sectors of the HMF (\(G_{r}^{+} = G_{r}^{-}\) and \(G^{+}_{\theta} = G^{-}_{\theta}\)).

Then the system of Equations 7 – 12 will have the following appearance:

$$\begin{aligned} A_{r}^{+} - A_{r}^{-} =& 6 \alpha _{1} K_{\mathit{II}} G_{\theta} \sin \varPsi /V \end{aligned}$$
(13)
$$\begin{aligned} A_{r}^{+} + A_{r}^{-} =& \left [ 6\mathit{CU}-6 K_{\mathit{II}} \left ( \cos^{2} \varPsi + \alpha \sin^{2} \varPsi \right ) G_{r} \right ] /V \end{aligned}$$
(14)
$$\begin{aligned} A^{+}_{\varphi} + A^{-}_{\varphi} =&6 K_{\mathit{II}} (1- \alpha ) G_{r} \sin \varPsi \cos \varPsi /V \end{aligned}$$
(15)
$$\begin{aligned} A^{+}_{\varphi} - A^{-}_{\varphi} =&6 K_{\mathit{II}} \alpha _{1} G_{\theta} \cos \varPsi ]/V \end{aligned}$$
(16)

The \(\varPsi \) parameter is the angle between the Sun–Earth lines of the HMF and the line determined based on direct solar wind velocity \(U\) measurements in interplanetary space from the equation:

$$ \mathrm{tg} \varPsi = \frac{\varOmega R_{0}}{U} $$
(17)

where \(\varOmega \) is the angular velocity of the Sun, and \(R_{0}\) is the distance from the Sun to Earth.

From Equations 13 – 16 we can find \(\alpha \) and \(\alpha _{1} \) and the products \(K_{\mathit{II}} G_{r}\), \(K_{\mathit{II}} G_{\theta}\).

Using the relationship \(K_{\mathit{II}} = \frac{\lambda _{\mathit{II}} V}{3}\) between the mean free path \(\lambda _{\mathit{II}} \) and the parallel diffusion coefficient \(K_{\mathit{II}}\), we can determine the products \(\lambda _{\mathit{II}} G_{r} \) and \(\lambda _{\mathit{II}} G_{\theta}\).

Knowing \(\alpha \) allows us to determine \(\lambda _{\mathit{II}} \) or \(K_{\mathit{II}} \) and parameter \(\omega \tau \) from the following dependencies (Chapman and Cowling, 1970; Forman and Gleeson, 1975), where \(\omega \) and \(\tau \) are the cyclotron frequency and the mean time between diffusive collisions of GCR particles, respectively:

$$\begin{aligned} \alpha =& \frac{K_{\bot}}{K_{\mathit{II}}} = \frac{1}{1+( \omega \tau )^{2}}; \quad \omega \tau = \frac{\lambda _{\mathit{II}}}{r_{L}} \end{aligned}$$
(18)
$$\begin{aligned} \alpha =& \frac{1}{1+( \frac{\lambda _{\mathit{II}}}{r_{L}} )^{2}} \end{aligned}$$
(19)

where \(K_{\bot}\) and \(r_{L}\) are the GCR perpendicular diffusion coefficient and the Larmor radius, respectively.

The Larmor radius was estimated on the basis of the effective rigidity of registration by NMs with a cutoff rigidity ≤5 GV and the average value of the magnetic field induction \(B\) in Earth’s orbit at different periods of SA: \(r_{L} = \frac{R}{Bc}\), where \(c\), the speed of a relativistic particle, approximately equals the speed of light. In turn having \(\lambda _{\mathit{II}}\) or \(K_{\mathit{II}}\) we can calculate \(G_{r}\ G_{\theta}\) from previously calculated products \(\lambda _{\mathit{II}} G_{r}\ \lambda _{\mathit{II}} G_{\theta}\) or \(K_{\mathit{II}} G_{r}\), \(K_{\mathit{II}} G_{\theta} \).

Thus, summing up this part of the work, on the basis of the solution of the above equations and using data of the \(A_{r}^{\pm} \) and \(A^{\pm}_{\varphi} \) anisotropy components in different sectors of the HMF, various parameters important in the modulation process were obtained, such as the parameters: \(\alpha \), \(\alpha _{1}\), \(K_{\mathit{II}}\), \(\varPsi \), \(G_{r}\), \(G_{\theta}\) in different periods of solar activity.

3 Experimental Results and Discussion

3.1 Temporal Changes of GCR Modulation Parameters

We have updated the modulation parameters for Solar Cycle 24 for the period 2008 – 2020 as a continuation of the results obtained by Ahluwalia et al. (2015). Figure 1 shows the changes in these parameters at different periods of solar activity in 2008 – 2020. Figure 1a-b presents the timelines of parameters \(\alpha \) and \(\alpha _{1}\), respectively, \(\alpha \) varies from 0.15 to 0.45 and \(\alpha _{1}\) from 0.35 – 0.50 for NM data, confirming the previous results (Ahluwalia et al., 2015). There is no clear dependence of parameters \(\alpha \) and \(\alpha _{1}\) on solar activity or solar magnetic polarity. Only in 2019 – 2020 one can see a rapid increase reaching values of ≈0.45 and ≈0.5, respectively. Figure 1c presents the time variation of the parallel diffusion coefficient \(K_{\mathit{II}}\). The \(K_{\mathit{II}}\) seems to be anti-correlated with the solar cycle, being in agreement with results of Zhao et al. (2018). Its values tend to be higher near the solar minima 2008 – 2010 and 2018, but in 2019 – 2020 one can see a rapid decrease connected with the change in parameter \(\alpha \). Figure 1d-e presents the estimation of radial \(G_{r}\) and latitudinal \(G_{\theta} \) gradients in 2008 – 2020. Radial gradient \(G_{r}\) changes according to the 11-year solar activity cycle with larger values for solar maximum. Latitudinal gradient \(G_{\theta} \) varies according to the 22-year solar magnetic cycle, with near 0 and negative values for negative polarity, next changing sign to be positive in positive polarity. Figure 1f presents the estimated \(\varPsi \) angle, its values range from ≈ 41 – ≈480 with high values in 2009 and 2020 and lower values near the period 2015 – 2017.

Figure 1
figure 1

The annual values of the parameters: (a) \(\alpha \), (b) \(\alpha _{1}\), (c) \(K_{\mathit{II}}\), (d) \(G_{r}\), (e) \(G_{\theta}\), and (f) \(\varPsi \) in 2008 – 2020.

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3.2 Temporal Variations of the Vector of GCRA in 1996 – 2020

Table 2 shows the magnitudes of the azimuthal and radial components of the GCRA averaged in the HMF sectors, the ratio of the module of the \(A_{r}\) component to the average absolute value \(A_{v}\) (\(A_{v}\) is the absolute value of the GCRA in the period 2009 – 2019), and the phase pointed to 18h (as a reference hour) for 1996 – 2008 (last year of the Solar Cycle 22 and Solar Cycle 23) with various signs of the HMF, that is, for 1996 – 2000 when \(A>0\), and 2003 – 2008 when \(A<0\). In the last line of Table 2 we have the average values for the whole Solar Cycle 23.

Table 2 Magnitudes of the azimuthal and radial components and value of the total GCRA, the ratio of the module of the radial component to the total value of the GCRA, the phase pointed to 18h (as a reference hour) from year to year for 1996 – 2008 (last year of the Solar Cycle 22 and Solar Cycle 23) to the different signs of the HMF, i.e., for 1996 – 2000 when \(A>0\) and 2003 – 2008 when \(A<0\), and the average values of the above parameters for the entire Solar Cycle 23. Errors for \(A\varphi \) and \(Ar\) components of the GCRA are in the range of 0.01-0.02%. Colors determine magnetic polarity: positive periods (\(A>0\)) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.
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The same is shown in Table 3 only for the period 2009 – 2012 when \(A<0\) (Solar Cycle 24), and for the period 2015 – 2020 when \(A>0\) (Solar Cycle 24 and the first year of the Solar Cycle 25). In the last line of Table 3 we have the average values for the whole Solar Cycle 24.

Table 3 The same as in Table 2 for the period 2009 – 2020 (Solar Cycle 24 and the first year of the Solar Cycle 25) with different signs of the HMF, i.e., for 2009 – 2012 (\(A<0\)) and 2015 – 2020 (\(A>0\)) and the average values of the above parameters for the whole Solar Cycle 24. The errors for the radial \(\mathrm{A} \varphi \) and \(Ar\) components are in the range of 0.01-0.02%. Colors determine magnetic polarity: positive periods (\(A>0\)) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.
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From Table 2 we can see that at the end of the Solar Cycle 22 and in the Solar Cycle 23 for the period with positive polarity of the global HMF, i.e., for 1996 – 2000 in the variation of the radial component the drift effect is clearly visible (from \(A_{r} = -0.18\%\) to \(A_{r} = -0.02\%\)). The phases are moved to the earlier hours compared to 18h and vary from 2 h 24 min to 11 min.

During this period the diffusion effect with drift is clearly visible.

In turn, for the period 2003 – 2008 with the negative polarity of the HMF the \(A_{r}\) component responsible for the drift effect is small and changes from 0.06% to 0.003%, which is the lowest value at the end of the Solar Cycle 23. The phases are shifted towards the later hours compared to 18 h and vary from 31 min to 6 min. Only in 2008 the phase is shifted slightly towards earlier hours and is equal to 2 min. During this period the diffusion dominated model of GCR transport is more acceptable. The drift effect, pronounced in the radial component of the GCRA, did not play a significant role in the modulation process in this period. We observe a similar phenomenon in Solar Cycle 24 and at the beginning of the Solar Cycle 25 (see Table 3). In the Solar Cycle 24 for the period with negative polarity, i.e., for 2009 – 2012, the \(A_{r}\) anisotropy component responsible for the drift effect is small and changes from 0.04% to 0.02%. The phases are also small and vary from shifting towards earlier hours in 2009 and 2010 (13 min and 8 min, respectively) to later hours in 2011 and 2012 (24 min and 8 min, respectively). During this period the diffusion dominated model of GCR transport is more acceptable too. In the period with positive polarity, i.e., for 2015 – 2020 (Solar Cycle 24 and the beginning of Solar Cycle 25) the \(A_{r}\) anisotropy component changes from 0.04% to 0.18% and phases are moved to the earlier hours compared to 18h and vary from 21 min to 2 h 45 min. During this period, the diffusion model with clearly manifested drift in the radial component of the GCRA is acceptable.

Table 4 shows the (\(A_{\varphi \mathit{AV}} \)) and (\(A_{r \mathit{AV}} \)) components, the total \(( A_{\mathit{AV}} ) \), the ratio of the module \(A_{ r\, \mathit{AV}}\) to the \(A_{\mathit{AV}}\), and the phases of the GCRA for 1996 – 2008 (last year of the Solar Cycle 22 and Solar Cycle 23), that is for 1996 – 2000 when \(A>0\), and 2003 – 2008 when \(A<0\), and for 2009 – 2020 (Solar Cycle 24 and the first year of the Solar Cycle 25), that is for 2009 – 2012 when \(A<0\), and 2015 – 2020 when \(A>0\). The last rows show the averaged values of the above-mentioned parameters for the whole Solar Cycles 23 and 24.

Table 4 The average values of the azimuthal \(( A_{\varphi AV} )\) and radial (\(A_{\ r\ \mathit{AV}} \)) components, the total \(( A_{\ \mathit{AV}} ) \), the ratio of the module \(A_{\ r\ \mathit{AV}}\) to the \(A_{\ \mathit{AV}}\), and the phases of the GCRA for 1996 – 2008 (the last year of the Solar Cycle 22 and Solar Cycle 23), that is for 1996 – 2000 when \(A>0\), and 2003 – 2008 when \(A<0\), and for 2009 – 2020 (Solar Cycle 24 and the first year of the Solar Cycle 25), that is for 2009 – 2012 when \(A<0\), and 2015 – 2020 when \(A>0\). Colors determine magnetic polarity: positive periods (\(A>0\)) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.
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From Table 4 we can see that the anisotropy of the average radial component of the GCRs in the ascending period of SA (1996 – 2000) for \(A>0\) is \(-0.107\%\), then decreases to a value of 0.029% in the descending part of SA (2003 – 2008), and continues to decrease to a value of 0.006% in the ascending part of SA (2009 – 2012) for \(A<0\). Then it rises again to a value of 0.109% in the descending part of SA (2015 – 2020) for \(A>0\). In the Solar Cycle 23 and the last year of the Solar Cycle 22 for \(A>0\), the average anisotropy vector was shifted towards the earlier hours compared to 18 h (the shift is 1 h and 6 min). For \(A<0\), the average anisotropy vector is shifted toward the later hours compared to 18 h (the shift is 12 min). In Solar Cycle 24 and the first year of the Solar Cycle 25, it is similar that for \(A<0\) the average anisotropy vector is shifted towards the later hours compared to 18 h (the shift is 6 min) and for \(A>0\) the average anisotropy vector is shifted toward the earlier hours compared to 18 h (the shift is 1 h and 18 min). The results presented in the Table 4 for averaged values, confirm the thesis that in periods with positive polarity of the global magnetic field diffusion and partly drift are responsible for GCR modulation in the heliosphere (the ratios of the absolute radial components to the average magnitudes of the GCRA are equal 27.78% and 32.89%, respectively). While during periods with negative polarity, the drift effect is very small (the ratios of the absolute radial components to the average magnitudes of the GCRA are equal 6.17% and 1.37%, respectively) and mainly diffusion is responsible for GCR modulation.

Figure 2a shows the changes in the directions of the anisotropy vector starting from the shift towards the earlier hours and ending with shifts towards the later hours compared to 18h in the last year of the Solar Cycle 22 and Solar Cycle 23. In turn, Figure 2b shows changes in the directions of the anisotropy vector starting from shifts towards later hours and ending with shifts towards earlier hours in the Solar Cycle 24 and the beginning of the Solar Cycle 25.

Figure 2
figure 2

(a) Harmonic diagrams of the GCRA from year to year for the periods 1996 – 2008 (the last year of Solar Cycle 22 and Solar Cycle 23), (b) the same for 2009 – 2020 (Solar Cycle 24 and the beginning of Solar Cycle 25). Colors determine magnetic polarity: positive periods (\(A>0\)) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.

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Figure 3 presents the average anisotropy vectors of GCRs for the period (a) 1996 – 2008 (last year of Solar Cycle 22 and Solar Cycle 23) with various signs of the HMF, i.e., for 1996 – 2000 when \(A>0\), and 2003 – 2008 when \(A<0\), and (b) the same for 2009 – 2020 (Solar Cycle 24 and the beginning of the Solar Cycle 25) for 2009 – 2012 when \(A<0\), and 2015 – 2020 when \(A>0\). In turn Figures 4 and 5 show temporal changes of the \(Ar\) (Figure 4a) and \(A\varphi \) (Figure 4b) components, the total magnitude (Figure 5a), and phase (Figure 5b) of the GCRA in 1996 – 2020.

Figure 3
figure 3

(a) Harmonic diagrams of the average GCRA vectors for the period 1996 – 2008 (last year of the Solar Cycle 22 and Solar Cycle 23) with various signs of the HMF, i.e., for 1996 – 2000 when \(A>0\) and 2003 – 2008 when \(A<0\). (b) The same for 2009 – 2020 (Solar Cycle 24 and the beginning of the Solar Cycle 25) for 2009 – 2012 when \(A<0\) and 2015 – 2020 when \(A>0\). Colors determine magnetic polarity: positive periods \(A>0\) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.

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Figure 4
figure 4

Temporal changes of the \(Ar\) (a) and \(A\varphi \) (b) components of the GCRA in 1996 – 2020.

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Figure 5
figure 5

Temporal changes of the total magnitude (a) and phase (b) of the GCRA in 1996 – 2020.

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From the harmonic diagrams (Figure 2a – b and Figure 3a – b, and Figure 4a – b and Figure 5a – b), we can see that temporary changes in the \(Ar\) component and phase of the GCRA are subject to a 22-year periodicity. The phases are shifted towards earlier hours for positive polarity of the global HMF while for negative polarity towards later hours compared to 18 h.

Our results reflect well the theory of GCR modulation including diffusion and drift in the heliosphere proposed by Jokipii, Levy, and Hubbard (1977).

The module of the \(A\varphi \) of the GCRA is always greater than the module of the \(Ar\). Therefore, both \(A_{\varphi}\) and the value of the total \(A\) of the GCRA behave similarly and are subject to an 11-year periodicity, which was shown for previous solar cycles (e.g. Bieber and Chen, 1991; Alania et al., 2005).

3.3 Temporal Changes of the Drift Vector from the GCRA in 1996 – 2020

The drift vectors \(A_{dr}\) from the GCRA were calculated as in Modzelewska et al. (2019), i.e., Solar Cycles 23 and 24, using the equation \((A(A<0)-A_{\mathrm{av}}23)\) and \((A(A>0)-A_{\mathrm{av}}24)\) for 1996 – 2008 and the same for 2009 – 2020, respectively.

In Tables 5 and 6 and Figures 6a – b and 8a – b we present \(A_{dr}^{r}\) and \(A_{dr}^{\varphi} \) components, and \(A_{dr}\) of the GCRA from year to year for 1996 – 2008 (last year of Solar Cycle 22 and Solar Cycle 23) and for 2009 – 2020 (Solar Cycle 24 and the beginning of Solar Cycle 25) with different signs of the HMF, i.e., for 1996 – 2000 when \(A>0\), 2003 – 2008 when \(A<0\), 2009 – 2012 when \(A<0\), and 2015 – 2020 when \(A>0\). In turn, in Table 7 and Figure 7, we present the same, but for the averaged values: \(A_{dr\, \mathrm{av}}^{r}\) and \(A_{dr\, \mathrm{av}}^{\varphi} \) components, and \(A_{dr\, \mathrm{av}}\), as well the ratios of the module \(A_{dr\, \mathrm{av}}^{r} \) to the \(A_{dr\, \mathrm{av}}\) and \(A_{dr\, \mathrm{av}}^{\varphi} \) to the \(A_{dr\, \mathrm{av}}\), respectively, and phases for different directions of the HMF.

Figure 6
figure 6

Harmonic diagrams of the GCR drift vectors from year to year for the periods: (a) 1996 – 2008 (last year of Solar Cycle 22 and Solar Cycle 23) with various signs of the HMF, i.e., for 1996 – 2000 when \(A>0\), 2003 – 2008 when \(A<0\), and (b) for 2009 – 2020 (Solar Cycle 24 and the beginning of the Solar Cycle 25) for 2009 – 2012 when \(A<0\) and 2015 – 2020 when \(A>0\). Colors determine magnetic polarity: positive periods (\(A>0\)) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.

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Figure 7
figure 7

The same as in Figure 6a – b for the averaged values of the drift vectors.

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Table 5 The values of the \(A_{dr}^{r}\) and \(A_{dr}^{\varphi} \) components, and the value of the total drift vector \(A_{dr}\) from the GCRA from year to year for the period 1996 – 2008 (last year of the Solar Cycle 22 and Solar Cycle 23) with different signs of the HMF, that is, for 1996 – 2000 when \(A>0\), and 2003 – 2008 when \(A<0\). Colors determine magnetic polarity: positive periods (\(A>0\)) shown in red, negative (\(A<0\)) periods in blue, reversal time in yellow.
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Table 6 The same as in Table 5 for the period 2009 – 2020 (Solar Cycle 24 and the beginning of Solar Cycle 25).
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Table 7 The same as in Tables 5 and 6 only for averaged values of \(A_{dr}^{r}\) and \(A_{dr}^{\varphi} \) for 1996 – 2020.
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The temporal changes of the radial (Figure 8a) and azimuthal (Figure 8b) components of the drift vector are similar to the temporal changes of the \(A_{r}\) and \(A_{\varphi}\) components of the GCRA, respectively.

Figure 8
figure 8

(a) Temporal changes of \(A_{dr}^{r}\) and (b) \(A_{dr}^{\varphi} \) components of the drift vector from the GCRA in 1996 – 2020.

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The radial component \(A_{dr}^{r}\) of the drift vector undergoes a 22-year periodicity, while the \(A_{dr \,\mathrm{av}}\) component the 11-year one. From Table 7 and Figure 7 we can see that the averaged drift vectors are shifted towards the later hours compared to 18 h for 2003 – 2008 and 2009 – 2012 (shifts are 3 h 51 min and 3 h 58 min, respectively) when \(A<0\), i.e., the anisotropy vectors are directed to 21 h 51 min and 21 h 58 min. In turn, for periods 1996 – 2000 and 2015 – 2020, the drift vectors are shifted towards earlier hours compared to 12h (shifts are 3 h 28 min and 3 h 8 min, respectively) when \(A>0\), i.e., the anisotropy vectors are directed to 8 h 32 min and 8 h 52 min, respectively.

The contributions of the module of the \(A_{dr\, \mathrm{av}}^{r} \) components to the \(A_{dr\, \mathrm{av}}\) for the periods 2003 – 2008 and 2009 – 2012 are 84.6% and 86.3%, respectively, when \(A<0\) and are greater than the contributions of the module of the \(A_{dr\, \mathrm{av}}^{r}\) components to the value of \(A_{dr\, \mathrm{av}} \) for 1996 – 2000 and 2015 – 2020 and are 78.8%, 73.2%, respectively, when \(A>0\). While for the azimuthal components we observe the opposite situation. The contributions of the \(A_{dr\, \mathrm{av}}^{\varphi} \) components to the value of \(A_{dr\, \mathrm{av}}\) for 1996 – 2000 and 2015 – 2020 are 61.5% and 68.1%, respectively, when \(A>0\), and for 2003 – 2008 and 2009 – 2012 are 53.3% and 50.6%, respectively, when \(A<0\).

The results presented in Table 4 for averaged values confirm the thesis that in periods with positive polarity of the global magnetic field, diffusion and partly drift are responsible for GCR modulation in the heliosphere (the ratios of the module of the \(A_{r}\) components to the \(A_{\mathit{AV}}\) of the GCRA are equal 27.78% and 32.89%, respectively), while during the periods with negative polarity the drift effect is very small (the ratios of the module of the \(A_{r}\) components to the \(A_{\mathit{AV}}\) of the GCRA are equal 6.17% and 1.37%, respectively) and mainly diffusion is responsible for GCR modulation.

Detailed studies of the GCRA from year to year in different periods of SA and different polarities of the global HMF also explain some problems of GCR propagation and modulation, e.g., delay time between sunspot number SSN and intensity of GCR.

It was shown in Iskra et al. (2019) that the delay time, e.g., for the period 2000 – 2012 (Solar Cycles 23 and 24) when \(A<0\) is 13 – 14 months and is greater than the delay time (4 months) for the period 1990 – 1999 (Solar Cycles 22 and 23) when \(A>0\). On the basis of the above results, regarding temporal changes in anisotropy in different periods of SA and different polarities of the global HMF, we can conclude that the main reasons for different delay times values for different polarities of global HMF are essential temporal rearrangements of the structure of the HMF turbulence. The drift of GCRs caused by the gradient and curvature of the HMF is an additional factor that strengthens this phenomenon.

4 Conclusions

Summarizing the above results, it can be concluded that:

  1. i)

    The temporary changes of the \(A_{r}\) component responsible for the drift effect and phase of GCRA are subject to a 22-year periodicity. The phases are shifted towards earlier hours for positive polarity (\(A>0\)) of the global HMF while for negative polarity (\(A<0\)) in the direction to later hours, compared to 18 hour (as a reference hour). These results reflect well the theory of GCR modulation including diffusion and drift.

  2. ii)

    The module of the \(A_{\varphi}\) component of the GCRA is always greater than the module of \(A_{r}\). Therefore, both \(A_{\varphi}\) and the value of the total \(A\) of the GCRA behave similarly and are subject to an 11-year periodicity.

  3. iii)

    The radial component \(A_{dr}^{r}\) of the drift vector undergoes a 22-year periodicity, while the \(A_{dr}^{\varphi}\) component an 11-year one.

  4. iv)

    The ratios of the module of the \(A_{r}\) components to the \(A_{\mathit{AV}}\) of the GCRA in 1996 – 2000 when \(A>0\) is 27.8%, while in 2003 – 2008 when \(A<0\) is 6.2% in the last year of the Solar Cycle 22 and in the Solar Cycle 23. In turn, in 2009 – 2012 when \(A<0\) in the Solar Cycle 24 this ratio is 1.4%, while in the 2015 – 2020 for \(A>0\) in Solar Cycle 24 and the beginning of the Solar Cycle 25 this ratio is 32.9%.

  5. v)

    The contributions of the radial drift components to the total drift for the periods 2004 – 2008 and 2009 – 2012 are 84.6% and 86.3%, respectively, when \(A<0\), and are greater than the contributions of the radial components to the value of the total drift value for periods 1996 – 2000 and 2015 – 2020 when they are 78.8% and 73.2%, respectively, when \(A>0\). While for the azimuthal components we observe the opposite situation. The contributions of the azimuthal components to the total drift for the periods 1996 – 2000 and 2015 – 2020 are 61.5% and 68.1%, respectively, when \(A>0\), and for the periods 2003 – 2008 and 2009 – 2012 they are 53.3% and 50.6%, respectively, when \(A<0\).

  6. vi)

    The average radial component of the drift vector for \(A>0\) is directed towards 12h, i.e., a drift stream of GCRs caused by the gradient and curvature of the HMF is preferentially coming from the polar regions to the helioequatorial region and is directed away from the Sun. For the \(A<0\) polarity, the GCR opposite direction of the drift stream occurs, i.e., the radial component of the drift vector is directed to 24 h. These results are consistent with the drift theory of modulation of GCRs.

  7. vii)

    The values of the \(A_{r}\) and \(A_{\varphi}\) components of the GCRA from year to year for various signs of the HMF were used to calculate the parameters that characterize the GCR modulation in interplanetary space. We have updated the modulation parameters up to 2020 as a continuation of results obtained in Ahluwalia et al. (2015). We found no clear dependence of the parameters \(\alpha \) and \(\alpha _{1}\) on solar activity or on solar magnetic polarity. Only in 2019 – 2020 one can observe the rapid increase to values of ≈0.45 and ≈0.5, respectively. We plan to apply the calculated modulation parameters in the modeling of GCR transport in the heliosphere in different periods of SA and for both polarities of the global heliospheric magnetic field.