Semiannual variation of Pc5 ULF waves and relativistic electrons over two solar cycles of observations: comparison with predictions of the classical hypotheses

. Pc5 ULF (ultra-low frequency) waves can energize electrons to relativistic energies of >2MeV in geostationary orbits. Enhanced ﬂuxes of such electrons can induce operational anomalies in geostationary satellites. The variations of the two quantities in time scales ranging from days to solar cycles are thus of interest in gauging their space weather effects over different time frames. In this study, we present a statistical analysis of two 11-year solar cycles (Cycle 22 and 23) of data comprising the daily relativistic electron ﬂuence observed by GOES geostationary satellites (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Geostationary (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Environment Satellites (GOES) and daily Pc5 ULF indicates 27-day periodicity in both parameters for all phases, a periodicity is most in the declining and late-declining phase. a 9-day and 13-day periodicity some a superposed epoch analysis performed to scrutinize Semiannual Variation (SAV), shows ﬂuence near the equinoxes is one order of magnitude higher than near solstices and Pc5 wave power is 0.5 orders of magnitude higher near the equinoxes than near the solstices. We then evaluate three possible SAV mechanisms (which are based on the Axial, Equinoctial, and Russel & McPherron effect) to determine which one can best explain the observations. Correlation of the proﬁles of the observational curves with those of the angles that control each of the SAV mechanisms suggests that the Equinoctial mechanism may be responsible for the SAV of electron ﬂuence while both the Equinoctial and the Russell & McPherron mechanisms are important for the SAV of Pc5 ULF wave power. Comparable are obtained when functional dependencies of the main angles instead of the angles superposed curves ﬂuence and Pc5 ULF wave power to calculate least-square ﬁts ﬁxed semiannual of Canada’s Anik-E1 E-2

However, since the orbit of the satellite is the same day by day, flux component that results from the orbit configuration will also be the same day by day. As a consequence, when the superposition is applied, this flux component will not affect the semiannual pattern that will be very similar in the three cases.
To prove this point, I have replicated Figure 4 in (McPherron et al., 2009) with Fluence data in our work, and the result is in Figure 1 at the end of this document. To improve the comparison, Figure 4 in (McPherron et al., 2009) has also been included at the end of this document.
Dashed line in Figure 1 shows the median of the Fluence ratio as a function of the DOY. The Fluence ratio is defined as the 27-day running average divided by 365-day running averages of the Fluence values in Solar Cycle 22.
In spite of the use of a different data set, the curve in Figure 1 is very similar to the one in the paper and shows clearly the Semiannual Variation.
It should be mentioned at this point that if we would like to study diurnal variations, UT variations of φ and θ introduced in the manuscript should also be considered. These are represented in Figure 2 at the end of this document. The proper quantity to use when working with daily values is the mean daily value of each angle as we have done in our work ( Figure 8 of the manuscript).
The publications referenced above are the following: - Su, Y.-J., J. M. Quinn, W. R. Johnston, J. P. McCollough, & Starks M. J. (2014). Specification of > 2 MeV electron flux as a function of local time and geomagnetic activity at geosynchronous orbit, Space Weather, 12, 470-486, doi:10.1002/2014SW001069. -McPherron R.L., Baker D.N., & Crooker N.U. (2009. Role of the Russell-McPherron effect in the acceleration of relativistic electrons. Journal of Atmospheric and Solar-Terrestrial Physics, Volume 71, Issue 10-11, p. 1032-1044, doi:10.1016/j.jastp.2008 Page 4 In lines 7 to 11, the choice to include data from the magnetosphere nightside is briefly explained. It would be noteworthy to add that a premidnight peak has been observed in GOES magnetic field data by Huang et al. (2010) and is likely the consequence of storm as well as substorm activity driven by tail processes, including substorm injections and dampened oscillatory flow in the plasma sheet. Lyons et al. (2002) has argued that ULF waves that strongly perturb the plasma sheet are a key component of tail dynamics during periods of enhanced convection. These ULF waves occasionally have amplitudes as large as plasma flow changes that occur in association with auroral zone disturbances, such as substorms. The publications referenced above are the following: Response: As a starting point, the objective was to evaluate periods and specifically study the Semiannual Variation considering powers at all local times together. Repeating the superposition and autocorrelation analyses to the powers of specific local times could give information about where the periods are produced. However, this does not mean that the main conclusions of the manuscript are invalid because the periods and the Semiannual intensity modulation are still clearly present in the daily values as they were used.
Moreover, excluding nigh-time powers should not change the results much because as it is pointed out in (Kozyreva et al., 2007), the correlation coefficient between ULF indices calculated for 00-24 and for 03-18 MLTs is very high at ~0.95, meaning that the substorm contribution to the daily Pc5 power would have been minor.
Response: We will change the axes terminology. Thank you.

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In lines 15 and 16, the authors note that, during 1996, relativistic electron fluence shows a different trend in Figures 2 and 4. However, how this is different from relativistic electron fluence observed during the remaining time series analyzed has not been described.

Response:
We thank the reviewer for pointing this out. We indeed need to clarify why 1996 looks different. The different behavior of fluence values in 1996 is related to the distinct semiannual variation pattern of that year, as alluded to earlier in Figure 2. We will add this information after the sentence in lines 15 and 16 of Page 11.

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In line 4, it is not clear to me and perhaps the reader why the choice of displaying relativistic electron fluence and Pc5 ULF wave power has been selected to be displayed at intervals of three days. Would the choice of a longer or shorter intervals make a difference in the variation observed through the year?
Response: No, the variation through the year is the same. Displaying the curves with a 3day interval helps to improve the visualization since there are five time series plotted together in this figure. This information will be added to the Figure description.
In line 9, could the cut-off value in the condition |tn -tn+1| < "small value" checked before every iteration be provided?
Response: Yes, the value was 1E-14 that is reached after five iterations approximately. This information will be added to the text.

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In lines 16 to 19, the authors suggest that increases in Pc5 ULF wave power has been linked to relativistic electron fluence enhancements during individual events. However, I could not understand from the context whether geomagnetic storms are meant by individual events. In addition, references to such studies have not been provided.
Response: By individual events we meant relativistic electron enhancements analyzed individually. Many enhancements take place during geomagnetic storms but they are not exclusively restricted to storm periods as it is pointed out in (Reeves et al., 2003). The paragraph will be rephrased to clarify this point.
The publication referenced above is the following: - Reeves, G. D., McAdams, K. L., Friedel, R. H. W., and O'Brien, T. P. ( 2003). Acceleration and loss of relativistic electrons during geomagnetic storms, Geophys. Res. Lett., 30, 1529, doi:10.1029 The relationship with solar wind speed could also be discussed at this point along with seasonal variations in relativistic electron fluence and Pc5 ULF wave power. In the past, Lukianova et al. (2016) had looked into variations of solar wind speed over several solar cycles over the last 100 years. Several studies have suggested that the solar wind speed is a dominant driver of relativistic electron fluxes in the outer radiation belt (e.g. Kellerman & Shprits, 2012,Paulikas & Blake, 1979. Furthemore, enhanced Pc5 ULF wave activity has associated with higher solar wind flow speed in the recovery phase of storms leading to enhanced electron fluxes (e.g. Georgiou et al., 2018, Mann et al., 2004. The publication referenced above are the following: Response: Thanks for this comment. A brief discussion of the relationship between electron fluxes, Pc5 ULF wave power and solar wind speed will be added. An interesting point is that solar wind speed does not present a recognizable semiannual pattern as electron fluxes and Pc5 ULF wave powers do. In fact, this is a strong argument to discard the Axial hypothesis. The reviewer may check Figure 4 in (McPherron et al., 2009) (that is at the end of this document) that shows a superposed epoch analysis for solar wind speed in which no seasonal pattern can be identified.
In (Lukianova et al., 2017) they calculate monthly linear regressions between DH (disturbed values of H geomagnetic component) and V (solar wind velocity) as: V = a DH + b. Then, they plot coefficients a and b (Figure 2) showing that a clear Semiannual Variation can be observed. This supports the idea that solar wind speed does not have any seasonal pattern in the following manner.
Since it is known that H component has a Semiannual Variation (see for example (Azpilicueta et al., 2012) ), if V would have this variation the coefficients of the fits should not show any seasonal pattern because the slope and intercept value would not vary much from month to month. So the fact that the slope and intercept value seasonally change is a consequence of a seasonality in DH and a lack of a seasonal behavior in V.
The publication referenced above is the following: - Azpilicueta, F., and Brunini, C. (2012), A different interpretation of the annual and semiannual anomalies on the magnetic activity over the Earth,J. Geophys. Res.,117,A08202,doi:10.1029/2012JA017893.   (McPherron et al., 2009) We would like to thank the reviewer for valuable comments. They have been perused carefully and responses to all of them are shown below. Our feedback for each comment are in the corresponding "Response" in red italics.
In this study the authors aim at presenting a detailed study of the correlation between PC5 ULF waves and enhancements of MeV electrons at GEO orbit. The follow the first study from Lam et al. (2017), and provide evidences of annual and semi-annual variability over two consecutive solar cycles. Moreover, they present insights to identify the major origins of these variabilities. The study is well detailed and numerous aspects are discussed. However, even if the authors rely on the previous study from Lam et al. (2017), t he new findings are not enough highlighted, and conclusions do not provide fully new assets. I would recommend this work for publication after a few major revisions. I detail in the following these points. These four points are put in context in Sections 3.1.2 and 5 where they are compared and discussed with results obtained in previous works.

Response
We will modify the text so that the main results are clear for the reader.
Major remarks: 1-In Lam et al. (2017) the correlation is computed between electron fluxes and PC5 pulsations. Even if it is not the point in this study, I am thinking if the authors could discuss more these correlations, in particular in section 4.1. Figure 9 could benefit from more detailed cross-correlation between fluence and PC5 waves. As mentioned in the title of the manuscript, the reader is waiting for more details on such correlation in my mind. However, a brief discussion of the results in (Lam, 2017) will be added since it could be valuable to highlight the main results of the manuscript.
2-Moreover, it is compared here with only > 2Mev electrons fluences. Do the authors tried to use the lower energy channel (>650keV electrons)? This may also add some discussion on the energization induced by these waves as well as radial diffusion, as a function of energy, as it has been discussed in some previous studies (see for example Lejosne et al., 2013).
3-One last major remark is (maybe naïve), why do the authors only discuss the power of the PC5 waves? Wouldn't it be interesting to discuss the correlation with fluence and solar cycle according to their modes (toroidal or poloidal as they tend to induce different effects on electrons trapped at GEO orbit, and as their sources may differ)?

Response:
The suggestion of studying low energy electrons as well as poloidal and toroidal modes would certainly improve the knowledge of the influence of Pc5 ULF waves on magnetospheric electrons. We are considering to pursue such topics in our future studies.
However, we think the study of periods and Semiannual Variation in both sets of observations used in this manuscript is long enough and self sufficient to present it in a paper as it is.

List of relevant changes
• The acronyms in the abstract were expanded and the typographical errors were corrected.
• Information about the data has been added.
• "Pc5 power" was changed by the correct terminology "Pc5 ULF wave power" throughout the paper. However, the shorter version "Pc5 power" was also used.
• Section 5 has been modified clarifying paragraph where "individual events" is mentioned. Also, a discussion of solar wind speed influence on relativistic electrons in the context of the Semiannual Variation has been added.
• Section 6 has been re-written in order to clarify general conclusions of the paper.
For a list of more specific results the reader can look at Sections 3.1.2, 4 and 5.
Reviewers may find below a marked-up version of the manuscript.
Semiannual variation of Pc5 ULF waves and relativistic electrons over two solar cycles of observations: comparison with predictions of the classical hypotheses Satellites :::::::: (GOES) and daily Pc5 :::: ULF ::::: wave power derived from auroral zone magnetic observatories in Canada. Firstly, an autocorrelation analysis is carried out, which indicates 27-day periodicity in both parameters for all solar phases, and such a periodicity is most pronounced in the declining and late-declining phase. Also, a 9-day and 13-day periodicity , though not present in all the years, are seen in some years. Then, a superposed epoch analysis is performed to scrutinize Semiannual Variation (SAV), which shows fluence near the equinoxes is one order of magnitude higher than near solstices and Pc5 :::: ULF 10 :::: wave : power is 0.5 orders of magnitude higher near the equinoxes than near the solstices. We then evaluate three possible SAV mechanisms (which are based on the Axial, Equinoctial, and Russel & McPherron effect) to determine which one can best explain the observations. Correlation of the profiles of the observational curves with those of the angles that control each of the SAV mechanisms suggests that the Equinoctial mechanism may be responsible for the SAV of electron fluence while both the Equinoctial and the Russell & McPherron mechanisms are important for the SAV of Pc5 :::: ULF :::: wave : power. Comparable

1
Relativistic electrons with energies >2 MeV can penetrate the surface of a satellite and cause internal charging that can induce satellite operational anomalies, as conclusively demonstrated by Wrenn (1995). Internal charging by relativistic electrons not only causes satellite operational anomalies that are a nuisance to satellite operators, but can also render the complete failure of a satellite, as exemplified by the consecutive outages of Telesat Canada's Anik-E1 and E-2 geostationary satellites on 20 25 January 1994 that wreaked havoc in communication across Canada for hours (Baker et al., 1994a, b;Lam et al., 2012). There are other serious satellite incidents due to internal charging by relativistic electrons such as the Anik-E1 failure on 26 March 1996 (Baker et al., 1996). The intensification of relativistic electrons that can cause satellite problems have been shown to be associated with Pc5 ULF (ultra-low frequency) waves (Rostoker et al., 1998;Mathie and Mann, 2001;Mann et al., 2004;Simms et al., 2014;Lam, 2017). The acceleration mechanisms of relativistic electrons attributable to Pc5 ULF waves can be 30 due to magnetic pumping (Borovsky, 1986;Liu et al., 1999), drift-resonant acceleration (Elkington et al., 1999), transit-time acceleration (Summers and Ma, 2000), and the popular radial diffusion (e.g., Falthammar, 1968;Schulz and Lanzerotti, 1974;Perry, 2005;Ozeke et al., 2014). No matter what the actual acceleration mechanism or process is, Lam (2017) has shown that Pc5 ::: ULF ::::: wave : power has the potential of predicting relativistic electrons that can harm satellites. It is, therefore, pertinent to peruse the time variations of Pc5 ULF waves and relativistic electrons together in detail in order to appraise their space weather 35 effects over different time assemblies.
In this work we analyze both ground-based Pc5 magnetic pulsations ::: ULF ::::: wave ::::::: powers, which are a manifestation of Pc5 ULF waves, and relativistic electrons at geostationary orbit, focusing on their time variations from a few days to a Solar Cycle (SC). An extended analysis is carried out for a particular kind of variation known as the Semiannual Variation (SAV). SAV is an annual phenomenon, characterized by maximum levels of activity near equinoxes and minima near solstices and it can 40 be detected in a diverse set of solar-terrestrial measurements Brunini, 2011, 2012;Vichare et al., 2017;Bai et al., 2018), including relativistic electrons of the outer Van Allen belt Li et al., 2001;Kanekal et al., 2001) and ULF waves (Sanny et al., 2007;Rao and Gupta, 1978). In the first case Baker et al. (1999) used measurements of both the low-altitude SAMPEX and high-altitude POLAR spacecraft to calculate quarterly averages centered at the equinoxes and solstices. They found that the fluxes were nearly three times higher at the equinoxes than at solstices which means a 45 semiannual modulation in these measurements (McPherron et al., 2009). Moreover, SAMPEX observations were also used by Kanekal et al. (2010) to study the dependence of the SAV in relativistic electrons with a wide range of L-shells covering the descending and ascending parts of a SC. Their results showed that the flux peaks were delayed about 30 days from the times of the nominal equinoxes during the descending phase. But in the ascending phase, the lag times were asymmetrical for both equinoxes.

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In the case of ULF waves, Sanny et al. (2007) examined the seasonal and diurnal pattern of ULF wave powers, using magnetic measurements from Geostationary Environment Satellites (GOES) sensors. They studied Pc3, Pc4 and Pc5 pulsations, which all clearly exhibit the June/July minimum. They also identified a strong local minimum in Pc4 band power around noon, whereas the minima of the Pc5 and Pc3 bands appeared to be distributed on the dayside. All the frequency bands had elevated (1978). They found the SAV to be particularly evident in the morning hours, close to 8 ± 1 h LT.
There are three mechanisms that are commonly referred to in the literature to explain the SAV and each one seems to be controlled by an angle. The first mechanism is known as the Axial hypothesis and the angle is the Earth's heliographic latitude.
This angle reaches maximum absolute values about 14 days before the equinoxes (see Table 5) when the Earth approaches high-speed solar wind regime such as sunspot region (Cortie, 1912) or coronal holes. The high solar wind speed originating 60 from these regions might be the driver of the enhancements in the activity. On the contrary, the Earth crosses regions of slowspeed solar wind approaching the solstices, at the proximity of the Sun's equator and then there is minimum activity (Phillips et al., 1995).
The second mechanism is known as the Russell & McPherron (RM) hypothesis (Russell and McPherron, 1973), which establishes that there is a varying probability of a southward directed component of the Interplanetary Magnetic Field throughout the 65 year. This leads to different probability of magnetic reconnection between the Interplanetary Magnetic Field and the terrestrial magnetic field lines at the nose of the magnetopause. Near the equinoxes(solstices) the probability is maximum(minimum).
The last mechanism is known as the Equinoctial hypothesis (Bartels, 1932). Boller and Stolov (1970) showed that in 70 theory, the Kelvin-Helmholtz instability originated by the viscous-like interactions between the solar wind and the magnetosphere along the flanks of the magnetosphere, predicts a semiannual pattern with instability maxima(minima) near the equinoxes(solstices). This is thought to be the physical process behind the Equinoctial theory. The controller angle is the one delimited by the SW direction and the Earth's dipole.
A main objective of this work is to test which one of these mechanisms better predicts the SAV that we find in Pc5 pulsations 75 :::: ULF :::: wave ::::::: powers and in relativistic electrons. The procedure involves the comparison between observational curves and the shape of the relevant angles of each mechanism. This method has been applied before to look for the dominant mechanism in the geomagnetic activity (Roosen, 1966;Cliver et al., 2002) finding that the Equinoctial and RM effects are the dominant ones and the Axial effect is the least important. This paper not only extends their work in magnetic activity in terms of Pc5 magnetic pulsations :::: wave :::::: powers : but also includes relativistic electrons in geostationary orbits. The consolidation of the two 80 quantities in a single study on their semiannual variations ::::: SAVs and other periodicities over two solar cycles elucidates their space weather effects under different temporal contexts.  onboard NOAA's ::::::: National ::::::: Oceanic ::: and ::::::::::: Atmospheric :::::::::::::: Administration's :::::::: (NOAA) GOES. GOES are in geostationary orbit about 35790 km above Earth's surface in the equatorial plane at 6.6 R E .

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The CANMOS observatories selected to calculate Pc5 power (see Table 2) are located in the Canadian auroral zone close to the footprints of magnetic field lines threading GOES in order to relate ground magnetic variations with relativistic electrons near geostationary orbit. As can be seen from the last column of Table 2 the data come primarily from Fort Churchill station (FCC) that is located at a geographic longitude of 94.1 • W which is approximately midway between GOES-East and GOES-West. Where there were gaps or spikes in FCC data, YKC ::::::::: Yellowknife :::::: (YKC) : data were used. When both FCC and YKC data 115 were absent or not usable, PBQ ::::: Poste ::::::::::: de-la-Baleine :::::: (PBQ) data were used. When data from FCC, YKC, and PBQ were not available, data from BLC and CBB ::::: Baker ::::: Lake ::::: (BLC) :::: and ::::::::: Cambridge :::: Bay :::::: (CBB) : at the fringe of the auroral zone near the cusp were used to fill in the data gap. It can be seen from Table 2 that FCC and YKC together cover ∼94 % of the total days processed with FCC contributing most of the data.
Many studies have been carried out using Pc5 power derived from a single magnetic station in the auroral oval as in this study 120 (Glaβmeier, 1988;Trivedi et al., 1997;Mathie and Mann, 2001;Mann et al., 2004). As pointed out by Lam (2017) trends delineated in Figure 1 of (Lam, 2017), whose daily values exclude midnight sector contributions, as mentioned earlier.
The thick black lines are the 365-day moving average of the Pc5 power and fluence. The smoothed sequence of daily Sunspot Numbers has also been added (orange curve) to represent the SC which is useful when making reference to variations of the 130 parameters to a specific SC phase.
The smoothed curves of Pc5 power and fluence can be used to highlight the underlying trends. For example, they indicate high levels during the descending phases of both cycles. Differences in trends at different phases of a SC can also be seen.
Although there appears to be minor variations in the trends between Pc5 power and fluence (e.g. Pc5 increasing while fluence  Lam (2017), who concluded that Pc5 power can potentially be used to predict electron fluence 2 to 3 days in advance before the enhancements in electron fluence at geostationary orbit and also that the lag is smaller for extremely high fluence values.
Besides showing the relationship between Pc5 power and electron fluence, Figure 2 also indicates that in 1996, fluence values clearly demonstrate SAV pattern, which is not readily discernible at first glance when looking at other years. Furthermore, both years show regular variations in the two parameters. The SAV and the regular variations as exemplified here will be further 145 investigated statistically in the sections below.

Autocorrelation functions
In order to investigate the dominant periodicities in Pc5 power and electron fluence, we calculated the autocorrelation function (ACF) of the logarithm of both parameters for specific years corresponding to different phases of a solar cycle. To establish 150 whether a value of correlation at a certain lag was significant or not, a criterion based on a Student's test (or "t-test") on the correlation coefficient r was adopted. Following (Rodgers and Nicewander, 1988), the hypothesis of null correlation (r = 0) is 7 rejected when r satisfies: where N is the length of sequence in days, and t is the quantile of a Student's distribution (t-distribution) with N −2 degrees of 155 freedom and a significance level of 1%. If the hypothesis can not be rejected, r is statistically equivalent to zero and considered not significant. On the contrary, in the late descending phase the transitions between the peaks at multiples of 27 differs in both parameters.
In fluence, we can deduce that the 27-day variation acts more like a spike owing to the flat correlation values in between solar rotations whereas in Pc5 power the lower harmonic with a period of ∼ 9 days in 2008 is evident.
In the minimum phase, ACFs in fluence exhibit continuously moderate correlation values above CVC between lags 0 and  phase, 1996 shows the persistent 27-day peak, which is present during other phases as mentioned above, and that peak is also present in 2009, though at a lower r.

A synopsis of the periodicities of Pc5 power and electron fluence over two solar cycles
The analysis developed in Section 3.1.1 provides a partial view of the periodicities as it only relates to specific years in  During the declining phase of the SC, CIRs are particularly prominent as a result of the expansion of CHs to lower latitudes, generating a well developed sector structure in the heliospheric magnetic field. In this SC phase the ACFs of Pc5 power and 210 fluence show the strongest values of correlation at 27-day lag as well as the clearest 27-day periodicity that repeats for several solar rotations. However, the fact that the ACFs peaks above CVC occur not only during the descending phase but also during other phases suggests that the 27-day variation in Pc5 power and fluence could also be due to smaller irregularities, other than CHs, capable of persisting for more than a solar rotation in the corona.
The peaks with a 9-day period seen in 2008 for Pc5 power (shown clearly in Figure 3 There are some previous reports of the 9-day recurrence in solar variables. For example, Ram et al. (2010) developed a comprehensive analysis of the solar rotation period and its subharmonics in the fractional area that CHs occupy at a fixed region of the Sun and also in the solar wind velocity. They found that both parameters exhibit subharmonics with a period of 9 220 days during the declining and minimum phase of SC 23. Also, Temmer et al. (2007) and Lei et al. (2008) studied the prominent 9-day periodicity in the solar wind velocity on 2005 probing that it was caused by a triad of CHs separated by ∼ 120 • in heliographic longitude that were active for several rotations. So the 9-day periodicity that we find in Pc5 power and fluence seems to be supported by prior investigations.
Finally, note that 1996 in fluence shows a different behavior than all the other years. This is evident when looking at this 225 particular year in Figures 2 and 4. ::: The ::::::: different ::::::: behavior ::: of :::::: fluence ::::: values :: in ::::: 1996 :: is :::::: related :: to ::: the :::::: distinct :::::::::: semiannual :::::: pattern :: of ::: that ::::: year, :: as :::::: alluded :: to :::::: earlier :: in :::::: Figure  years) for each DOY to use in the calculation of the median. The results can be seen in Figure 6 which shows the superposed curve as black lines on the left and right panel corresponding to Pc5 power and electron fluence respectively. We chose the median over the mean for the superposition, since it is not skewed so much by extremely large or small values, and so it may 235 give a better approximation to the "typical" value for each DOY. The upper and lower limits of the gray band mark the quartiles.
The 30-day running average of the curves with the median is also added in the figure (green line for Pc5 power and red line for electron fluence) and will be referred to as Pc5 SAV and Fl SAV . The 30-day moving average serves to diminish the strong 27-day variation since this is the most prevalent periodicity in both Pc5 power and fluence values, as shown in Section 3.1. Pc5 SAV and Fl SAV demonstrate a clear SAV with maxima around the equinoxes and minima near solstices. Although not 240 as clear, the SAV pattern can also be seen in the curves associated with the median and quartiles. The peak-to-peak variation of Fl SAV is of one order of magnitude approximately, and of ∼ 0.5 orders of magnitude for Pc5 SAV . There are differences as well as similarities in the SAV of Pc5 power and fluence, and they will be discussed in Sections 4.1 and 4.3 below. Those sections will explore in more detail the phases and profiles of the SAV in both parameters, but more importantly they will be compared with the phases and profiles predicted by the three classical hypotheses (introduced in Section 1) so that the dominant 245 mechanism can be ascertained.

Annual profiles
In this Section we compared the profiles of the angles that govern each SAV mechanism (introduced in Section 1) with the profiles of Pc5 SAV and Fl SAV . For the axial hypothesis we considered the daily values of the Earth's heliographic latitude (ψ).
For the equinoctial hypothesis we used daily mean values of the angle delimited by z GSM and the Earth's dipolar axis denoted 250 by φ that is equivalent to the magnetic solar declination (with the same annual variation). Finally, for the RM effect we took the daily mean values of the angle between the z GSM and z GSEq axes that is measured in the y-z plane of both coordinates systems (GSM and GSEq), referred to as θ. Figure 7 shows schematically these three angles in the Sun-Earth environment where the gray plane is the solar equatorial plane.  in some works (Lockwood et al., 2016). This causes |θ| to reach slightly different values at the maxima. Considering GSEq over GSE also delays the location of the maxima for several days and is more consistent with the original definition of the RM effect reported in Russell and McPherron (1973) (see for example Figure 4 on that paper). respectively. FLSAV and Pc5SAV have also been added, plotted at 3-day intervals. ::: The ::::: 3-day :::::: interval :::: helps :: to :::::: improve ::: the :::::::::: visualization :: of :: the ::: five :::: time ::::: series.
To compare the shape of the angles with FL SAV and Pc5 SAV we applied a 30-day running average to the curves in Figure 8.
In fact, between DOYs 200 and 250, Pc5 SAV almost overlaps the smoothed inverted |φ| curve. Between DOYs 1 and ∼60, FL SAV and Pc5 SAV reach higher values than the curves of the angles. In addition, as can also be seen in Figure 6, FL SAV shows sharper maxima than Pc5 SAV .

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Some authors have used these three angles (or similar ones) in the past to test SAVs detected on magnetic indices. For example, Roosen (1966) used ap index from 1932 to 1966 and determined that the annual pattern of the smoothed index presents greater similarity with the smoothed Equinoctial angle than with the smoothed Axial angle. Cliver et al. (2002) extended that comparison utilizing the 30-day smoothed patterns of the three angles and the aa magnetic index from 1868 to 1998 obtaining high values of correlation with the smoothed |θ| but specially with the smoothed inverted |φ|. 275 We calculated the correlation values between our observational curves (FL SAV and Pc5 SAV ) and the smoothed angles and the results are summarized in Table 3. The equinoctial hypothesis seems to dominate the SAV in fluence since the correlation value between the smoothed |φ| and FL SAV profiles reaches the minimum value of −0.87, meaning that they anti-correlate very well. There is a lower fidelity of FL SAV with the RM profile (r = 0.82). As regards as Pc5 SAV , the profiles of both In principle, it should be possible to use the profiles of the three angles to determine which is the dominant mechanism, but 285 a better approximation may be achieved by considering functional dependencies of each angle. In this Section we evaluate functions of φ or θ proposed by different authors (Svalgaard, 1977;Perreault and Akasofu, 1978) in the past on studying the SAV in geomagnetic activity. Svalgaard (1977) pointed out that the am magnetic index can be fitted empirically using an expression for the magnetic field near a dipole, parameterized in terms of the controller angle of the equinoctial theory. The angular part of Svalgaard's function 290 in terms of φ as defined in this work is S(φ) = 1 + 3 cos 2 (90 • − φ) −2/3 .
The angle θ of the RM hypothesis is considered in the "Akasofu" parameter (Perreault and Akasofu, 1978) that is usually utilized to characterize the energy brought by the SW to the magnetosphere. In addition to the SW and Interplanetary Magnetic field quantities involved in this proxy, the angular dependence is of the form Ak(θ) = sin 4 (θ/2). Finch and Lockwood (2007) determined that functions with this angular dependence are very successful on quantifying terrestrial disturbance levels on 295 timescales of 1 day.
We correlated S(φ) and Ak(θ) with FL SAV and Pc5 SAV and the results are shown in Table 4. The correlation values of S(φ) are slightly better than to just using |φ| and the opposite occurs for Ak(θ) (see Table 3). However, all the correlation values are very similar to the ones obtained in Section 4.1 so no additional conclusions can be drawn.

Dates of maxima and minima 300
To continue the comparison with the three classical hypotheses, we determined the dates of maxima and minima of the SAV in fluence and Pc5 power and compared them with the corresponding dates of maxima and minima predicted by the three hypotheses.  Table 4. Correlation coefficients between functional dependencies of the angles (S(φ) and Ak(θ), read text for details) and observational curves (Pc5SAV and FLSAV ). First, we applied a non-linear least square fit with five parameters to the superposed median curves (black curves) of Figure   6. The following function was used: with fixed annual and semiannual periodicities and the fitted parameters A 0 , A a , α a , A sa and α sa . f (t) is plotted in Figure 10 as a green(red) curve in the left(right) panel that corresponds to the Pc5 power(fluence) fit. The other curves of Figure 10 are the median and quartiles as were presented in Figure 6.
Both fits follow the semiannual trend of the superposed median curves very well. In fact, the coefficient that modulates the 310 amplitude of the annual variation is very low for both cases being A a = −0.06 for the fluence fit and A a = 0.04 for the Pc5 power fit. But in the semiannual term they are higher: A sa = −0.23 and A sa = −0.12 for fluence and Pc5 power respectively.
An interesting characteristic that f (t) reveals is that the minima on June/July and on December/January are not symmetric in both fits. The minimum of f (t) on June/July is lower than the minimum on December/January for Pc5 power and the opposite occurs with the fluence fit. On the contrary, f (t) does not present this asymmetry for the maxima in both cases.

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Once f (t) was defined, we looked for the t values that obey df (t)/dt = f (t) = 0 i.e. the times of maxima or minima of f (t) referred to as t max,min , which will be used as the times of the maxima-minima of the SAV in fluence and Pc5 power. We applied the so called "Newton-Raphson" method (Ypma, 1995) which is a classic method implemented to find zeros of a function.
The advantage of calculating t max,min with this procedure is that Equation 3 also serves to estimate the errors in the determi-325 nation of t max,min because once t max,min is determined, we can interpret Equation 3 as having t expressed as a function of the parameters for values near t max,min , i.e t = F (A 0 , A a , α a , A sa , α sa ). Then, error propagation can be used in the determination of t with F . The maxima and minima dates (t max,min ) with their uncertainty interval 2σ t are shown in table 5. The table also shows the dates of maxima and minima predicted by the three mechanisms and which one of them falls into the uncertainty interval.
The best prediction of the SAV minima in fluence is given by the Equinoctial hypothesis. This mechanism is also the best 330 one in estimating the September maximum with just one day of difference between the observed and predicted date. However, the three mechanisms fail to predict the March maximum in fluence that falls between the Equinoctial and RM predictions.
Note that if the peaks and valleys times expected for the equinoctial mechanism are shifted forward 4 days as in (Kanekal et al., 2010), the fluence times of maxima/minima fall into the equinoctial uncertainty interval. This time shift was attributed by Li et al. (2001) and Kanekal et al. (2010) to finite solar wind speed (∼ 440 km s −1 ).

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For the SAV in Pc5 power it is not possible to find a dominant effect since the RM and the equinoctial theory give the best predictions for one maximum and one minimum but not both.
The results of this Section agree with the results found in the profiles analysis of Section 4.1. The equinoctial effect seems to be dominant in the generation of the SAV in fluence and both equinoctial and RM effects might be equally important for the SAV of Pc5 power.

Discussion
The previous sections have demonstrated a clear SAV in both parameters analyzed in this work. As a result of the length of the observations (two complete SCs of daily values) we were able to recover the background semiannual intensity variation in electron fluence and in Pc5 power. In the first case, this variation can be seen clearly in the red curve of the right-hand  Table 5. Dates of maxima and minima for |φ|, |θ| and |ψ| and for the fits (f (t)) of the superposed median curve of Pc5 power and fluence.
side panel of Figure 6 (Fl SAV ). Fl SAV reaches ∼ 7.5 near equinoxes and ∼ 6.5 near solstices that is equivalent to a difference 345 of one order of magnitude approximately. This means that there is a higher probability of internal charging on satellites near equinoxes then being more plausible for them to suffer operational anomalies. It also illustrates the way that the SAV influences space-based technologies.
In the study of the dominant effects, we found that the Equinoctial mechanism is dominant in the SAV of fluence and both the Equinoctial mechanism and the RM effect play equally relevant roles in the SAV of Pc5 pulsations :::::: powers. These conclusions These results differ from previous ones reported in (Kanekal et al., 2010). Analyzing SAMPEX electron flux data, they found a more prominent role for the RM effect. However, they considered fluxes in the heart of the outer radiation belt (L 355 4) to evaluate the leading mechanism and not at GEO. So it is possible that different mechanisms may control the SAV in relativistic electrons in different regions of the magnetosphere. Another reason why we obtain different results could be that in (Kanekal et al., 2010) they used 10 years of daily values (from 1993 to 2002) which are less than half of the measurements processed in this work. Longer time spans of the data make the statistics more representative. In the case of the SAV of Pc5 pulsations :::::: powers, we were not able to find prior reports studying the controller mechanism to compare with our findings.