Features of the Galactic Cosmic Ray Anisotropy in the Solar Cycle 24 and Solar Minima 23/24 and 24/25

We study the role of drift effect in the temporal changes of the anisotropy of galactic cosmic rays (GCRs) and the influence of the sector structure of the heliospheric magnetic field on it. We analyze the GCRs anisotropy in the Solar Cycle 24 and solar minimum 23_24 with negative polarity for the period of 2007-2009 and near minimum 24_25 with positive polarity in 2017-2018 using data of global network of Neutron Monitors. We use the harmonic analyses method to calculate the radial and tangential components of the anisotropy of GCRs for different sectors (plus corresponds to the positive and minus to the negative directions) of the heliospheric magnetic field. We compare the analysis of GCRs anisotropy using different evaluations of the mean GCRs rigidity related to Neutron Monitor observations. Then the radial and tangential components are used for characterizing the GCRs modulation in the heliosphere. We show that in the solar minimum 23_24 in 2007-2009 when negative, the drift effect is not visibly evident in the changes of the radial component, i.e. the drift effect is found to produce 4 % change in the radial component of the GCRs anisotropy for 2007-2009. Hence the diffusion dominated model of GCRs transport is more acceptable in 2007-2009. In turn, near the solar minimum 24_25 in 2017-2018 when positive, the drift effect is evidently visible and produce 40% change in the radial component of the GCRs anisotropy for 2017-2018. So in the period of 2017-2018 diffusion model with noticeably manifested drift is acceptable. The results of this work are in good agreement with the drift theory of GCRs modulation, according to which during negative (positive) polarity cycles, a drift stream of GCRs is directed toward (away from) the Sun, thus giving rise to a 22-year cycle variation of the radial GCRs anisotropy.


Introduction
Galactic cosmic rays (GCRs) measured at Earth are modulated in the heliosphere by expanding solar wind (SW). Modulation of GCRs is governed by the convection of SW, inward diffusion of GCRs on the irregularities of heliospheric magnetic field (HMF), drift on the regular HMF and adiabatic cooling of GCRs particles due to extension of SW. GCRs intensity detected near Earth contains the various long and short term quasi-periodic variations, see e.g. On the background of the long term solar modulation: 22 years and 11 years, connected with the global HMF and the solar activity (SA) cycle, respectively, the short term modulation effects sporadic (e.g. Forbush decreases) and recurrent (≈27 days) also take place.
This paper focuses on an anisotropic part of GCRs flux, reflected in the solar diurnal variation (≈24 hours wave), being a consequence of the equilibrium state between inward diffusion of GCRs in the HMF in the heliosphere and outward convection by the SW. The mechanism of the solar diurnal anisotropy was developed by (Ahluwalia and Dessler, 1962;Krymsky, 1964;Parker, 1964) based on the anisotropic diffusion-convection modulation theory of GCRs transport in the heliosphere, further developed by (Gleeson andAxford, 1967, 1968).
The anisotropy vector A is expressed in terms of the stream of GCRs and spatial density gradient, as (Gleeson, 1969): where N is the GCRs density, = ∇ is the density gradient, S is the stream of GCRs, K is the diffusion tensor representing the diffusion and drift effects of GCRs, V and U are speeds of the GCRs particles and of the SW, respectively, C is the Compton-Getting factor.
The components of the free-space anisotropy vector A=[Ar, Aθ, A] in spherical coordinates centered on the Sun are in the following form (Riker and Ahluwalia, 1987 where, K, KII, and KT are perpendicular, parallel and drift diffusion coefficients of GCRs, respectively; Gr  , G  and G   are the radial, heliolatitudinal and heliolongitudinal gradients of GCRs; sign (-) and (+) correspond to the HMF lines toward (negative) and away (positive) from the northern hemisphere of the Sun; V and U are speeds of the GCRs particles and of the SW, respectively;  is the angle between the HMF lines and the Earth-Sun line; C=1.5 for GCRs sensitive to neutron monitors (NMs) energy. It can be seen from the system of Equations 2 -4, that the GCRs anisotropy (recorded by NMs on the Earth) must be due to diffusion, convection and drift of GCRs particles in the regular HMF of the interplanetary space. It is, on average, 0.3-0.4%. At the same time, the fraction of the particle drift effect can reach 0.05-0.1% of the average anisotropy (Alania et al., 1983(Alania et al., , 1987(Alania et al., , 2001Alania, Bochorishvili, and Iskra, 2003). It is necessary to distinguish two types of drift effect in the GCRs anisotropy. The first type, due to the gradient and curvature of the HMF, can be identified in changes in the average anisotropy of GCRs in different periods of solar magnetic cycle qA>0 and qA<0 (for qA>0, the HMF lines come out from the northern hemisphere of the Sun, and for qA<0, they enter the northern hemisphere of the Sun). The second type of drift, due to the existence of local spatial gradients of the GCRs, is manifested in different sectors of the HMF. This component of drift is most notably linked to the heliospheric current sheet (HCS). The HCS is the surface where the polarity of the Sun's magnetic field changes from south to north and vice versa. The tilt angle (TA) is referred to the latitudinal extension (or waviness) of the HCS. We suppose that this drift effect can be manifested due to the sector structure of HMF for the minima epochs of SA, when the TA of the HCS is <10-15 degrees, i.e. the Earth is located near the weakly wavy HCS. Revealing this type of drift may be difficult due to changes in SW speed and corotating interaction region (CIR) areas when the fast SW catches up slow SW (Richardson, 2018) disrupting the stability of the HMF sector structure. The absence of a significant azimuthal gradient in GCRs density when changing the HMF sign indicates a limited HCS thickness. This thickness should be many times smaller than the Larmor radius of GCRs particles with energy range of 1-10 GeV. If the width of the HCS were of the order of the Larmor radius or greater, then the Earth, moving in orbit around the Sun, would pass areas with isotropic and anisotropic diffusion and we would observe a significant azimuthal GCRs density gradient (Alania and Dzhapiashvili, 1979). However, in order to detect the first type of the drift effect in GCRs anisotropy, it is possible to average data over a long period of time (for example, over a period of ≈ 11 years). Whilst a clear identification of second type of drift effect is difficult on the background of changes in SW parameters for a short period of time comparable with the durations of the positive and negative sectors of the HMF. Investigation of the role of drift effect in the temporal changes of the anisotropy of GCRs is very important from the point of view of the GCRs particle transport theory in the heliosphere (e.g. Siluszyk, Wawrzynczak, and Alania, 2011; Siluszyk, Iskra, and Alania, 2015). On the one hand, it allows to understand the behavior of GCRs particles transport in the heliosphere Jokipii, 1983, 1998;Potgieter and Moraal, 1985;Burger and Potgieter, 1989;Potgieter, 1995 Bieber (2010) found that the higher energy NMs contribute more to the 11 year phase variation of GCRs anisotropy due to the diffusion process. Sabbah (2013) reported that the diurnal phase of higher energy NMs shifts towards earliest hours; this is connected with outward convection by SW that increases the radial component of daily variation more than the azimuthal component.
The aim of this work is to continue research on the role of drift effect on temporary changes in GCRs anisotropy. We analyze in detail the anisotropy behavior in the Solar Cycle 24, especially during the solar minimum 23/24 in 2007-2009, when qA<0 in comparison to the period 2017-2018 being near the solar minimum 24/25 when qA>0. The most important issue is: based on anisotropy analysis to determine the contribution of drift effect in GCRs modulation especially near solar minimum periods for Solar Cycle 24. This paper is organized as follows: in Section 1 we have presented a short introduction and description of anisotropy of GCRs. Section 2 introduced data and methods used to calculate the GCRs anisotropy. In Section 3 we have presented experimental results and discussion. Section 4 concludes the paper.

Data and Methods
We use the hourly data of GCRs intensity for NMs with geomagnetic cut-off rigidities Rc<5 GV. For calculation of the GCRs anisotropy we follow the procedure from ( First for calculations of the daily radial r A and tangential  A components of the diurnal variation of the GCRs intensity for each NM, we use the harmonic analyses method (e.g. Gubbins, 2004;Wolberg, 2006): where: where i y designates the hourly GCRs intensity data, k is the consecutive harmonic of the Fourier extension. The daily radial r A and tangential  A components of the diurnal variation of the GCRs intensity were calculated by means of normalized and detrended (excluding 25 hours trend) hourly data of the pressure corrected GCRs intensity as the first (k=1, 2p=24 hours) harmonic of the Fourier extension.
We exclude from consideration the diurnal amplitudes >0.7% as an anomalous events related to the transient disturbances in the interplanetary space, generally connected with Forbush decreases and we do not take into account HMF sectors with duration less than 4 days. A number of the excluded days is less than 2-3% from the total number of days used for analyses.
Next, in order to find local components of diurnal variation we rotate the vector by geographic longitude geo  of each NM station. Furthermore, we convert the radial Ar and tangential A components of the GCRs diurnal variation into the radial Ar and azimuthal A components of the GCRs anisotropy into the heliosphere (free space), (for details see Modzelewska and Alania, 2018), dividing the diurnal components by coupling coefficients (CC). The CCs represent the ratio of the diurnal variation to the corresponding amplitude of the anisotropy of cosmic rays in the heliosphere (Dorman, 1963;Yasue, Sakakibara, and Nagashima, 1982).
Calculated components of the diurnal variation of the GCRs intensity are corrected due to influence of the Earth magnetic field (Dorman et al., 1972;Dorman, 2009), taking into account the asymptotic cone of acceptance characteristic for each NM station (Shea, Smart, and Mc Cracken, 1965;Shea et al., 1967) for the rigidity to which NMs respond. The asymptotic cones of acceptance for each NM station were calculated using the FORTRAN TJ2000 program developed by Shea et al.
Finally we correct the GCRs anisotropy for Compton-Getting (CG) effect, due to Earth orbital motion (Ahluwalia and Ericksen, 1970;Hall, Duldig, and Humble, 1996). CG correction is a vector with amplitude 0.045 cos asym  and phase 6h, where asym  is the asymptotic latitude of each NM.
The GCRs intensity measured by ground-based NMs is based on the concept of an integrated GCRs flux above the magnetic rigidity threshold that depends on the geomagnetic cut-off typical for each NM. In this case there arises a question what is the real energy of the measurement. It is natural to introduce the "effective energy/rigidity" typical for GCRs detector, here NM. In the literature this term has different meanings: fixed energy/rigidity ≈ 10 GV for a NM, (Belov, 2000)

Experimental Results and Discussion
Solar Cycle 24 was much less active than, for example, Solar Cycle 23, being rather similar to cycles at the beginning of the previous century. During this relatively weak cycle not many impulsive events happened on the Sun, but we could observe the prolonged solar minimum 23/24 with good established regular structure of the HMF. The current Solar Cycle 24 has now more or less finished its descending phase. At the beginning of 2018 (≈ April), the Sun showed indications of the reverse magnetic polarity sunspot announcing the Solar Cycle 25 (Phillips, 2018). The poleward reversed polarity sunspots suggest that a transition to Solar Cycle 25 is in progress and indicating that we may be in-or near the minimum between Solar Cycles 24 and 25. So it is of special importance to study the features of the GCRs anisotropy characteristics through the almost whole Solar Cycle 24 with special importance of the comparison of the drift effect in GCRs anisotropy during solar minimum 23/24 and near the solar minimum 24/25 with different polarity periods of solar magnetic cycle.
Here we present the time variation of the GCRs anisotropy for Oulu NM (as an example) through the almost whole Solar Cycle 24, 2007Cycle 24, -2018. We compare the values of the amplitude, phase, radial and azimuthal components of the GCRs anisotropy using two approaches: median Rm (Ahluwalia et al., 2015) and effective Ref (Gil et al., 2017) rigidity of response. The errors for annual values are estimated as the standard deviation from monthly data. The calculated results of the GCRs anisotropy for effective and median rigidity are in good agreement, the difference is in scope of the error bars. Figure 1   In Figures 2abc are presented average GCRs anisotropy vectors from following NMs: Apatity, Oulu, Newark, Thule, Fort Smith, Inuvik, Kerguelen, Terre Adelie and Nain (see Table 1 Table 2 presents the radial and azimuthal components, magnitude of the GCRs anisotropy and the phase relative to 18h. Uncertainty of this data is estimated as the Poisson statistics using standard deviations for separate NMs devided by √ , where n is the number of NMs used in averaging for each period. As far the drift stream is evidently pronounced in the radial component we present in Table 2 Figure 4b and Table 3 show averaged over HMF sectors the anisotropy vectors for different directions of the global HMF i.e.   One can see that the drift anisotropy vector Adr (red colour in Figure 5) for the qA>0 polarity is preferentially directed to the Sun (to the direction of 12h in the harmonic diagram), and for the qA<0 polarity the drift anisotropy vector Adr (blue colour in Figure 5) is directed out of the Sun. These empirical outcome are in good correspondence with the drift theory of modulation of GCRs (Jokipii, Levy, and Hubbard, 1977). Based on that theory in the qA>0 cycle 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 qA<0 cycle exists the opposite direction of the drift stream of GCRs. Table 4

in total dr
A is more than ≈ 86% for qA>0 and almost ≈ 99% in qA<0. The difference between anisotropy A for qA>0 and qA<0 polarities is produced by the drift of GCRs due to the curvature and gradient of the regular HMF and is the source of the long term variation of the anisotropy A of GCRs (Forbush, 1969;Ahluwalia, 1988). The results presented in this paper will be used in future work concerning the modeling of the transport of GCRs particles as the crucial parameters characterizing the solar modulation of GCRs by the SW in the heliosphere.

Conclusions
Using data from several neutron monitor stations of different magnetic rigidity cut-offs, we have studied the role of drift in the temporal variations of the GCRs anisotropy and its influence on the sector structure of the heliospheric magnetic field. Using the harmonic analysis method, we have analyzed the GCRs anisotropy in the qA<0 solar minimum period of 2007 -2009 (Cycle 23/24) and near the qA>0 minimum period of 2017-2018 (Cycle 24/25). Our main results can be summarized as follows:

i)
We have compared the analysis of the GCRs anisotropy using the median Rm and effective Ref rigidity of neutron monitor response. We have found that the differences of the GCRs anisotropy parameters calculated for these two approaches: median and effective rigidity are rather negligible. ii) The global drift due to gradient and curvature of the heliospheric magnetic field is manifested in the radial component Ar of the anisotropy A of GCRs. For the qA<0 cycle the Ar is directed away from the Sun, but for the qA>0 magnetic cycle the Ar is directed to the Sun. This type of drift effect is the source of the 22-year variation of the anisotropy of GCRs. iii) The drift effect due to the sector structure of the heliospheric magnetic field in the anisotropy A of GCRs (during the minima epochs of solar activity) is caused by the heliospheric current sheet and is visible in the azimuthal component of GCRs anisotropy.