Pc5 ULF (ultra-low frequency) waves can energize electrons to relativistic energies of

Relativistic electrons with energies

In this work we analyze ground-based Pc5 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 be detected in a diverse set of solar–terrestrial measurements

In the case of ULF waves,

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 d before the equinoxes (see Table

The second mechanism is known as the Russell and McPherron (RM) hypothesis

The last mechanism is known as the equinoctial hypothesis

A main objective of this work is to test which one of these mechanisms better predicts the SAV that we find in Pc5 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

As internal charging by relativistic electrons on satellites located at geosynchronous orbit is a function of integrated flux over a time period, we use daily fluence values, which are an accumulation of fluxes over 24 h, to represent the electron variations in this work. Specifically, we analyzed fluences of relativistic electrons with energies

The data span SCs 22 and 23 from June 1987 to December 2009. The same data suite has been used previously to study Pc5 ULF wave powers and relativistic electrons by

Table

Satellite data used in this study.

To study ULF waves in the Pc5 frequency band, we generate a time series of daily Pc5 ULF wave powers referred to as Pc5 powers, using Canadian geomagnetic data collected by the Canadian Magnetic Observatory System

The CANMOS observatories selected to calculate Pc5 power (see Table

Many studies have been carried out using Pc5 power derived from a single magnetic station in the auroral oval, as in this study

Coordinates of CANMOS observatories used in this work.

Figure

The thick black lines are the 365 d 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 referencing the variations of the 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 an SC can also be seen. Although there appear to be minor variations in the trends between Pc5 power and fluence (e.g., Pc5 increasing while fluence is decreasing in the early portion of SC 23 and Pc5 leveling while fluence is depressing around the SC 22 maximum), the gross features of their evolution, in both SCs, appear to be similar.

To see the relationship between Pc5 power and electron fluence, Fig.

Besides showing the relationship between Pc5 power and electron fluence, Fig.

Fluence (red curve) and Pc5 power (green curve) values in 1996

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 whether a value of correlation at a certain lag was significant or not, a criterion based on a Student's test (or “

Figures

Since

Autocorrelation functions (ACFs) of Pc5 power for different phases of the SC 22

ACFs of electron fluence for different phases of the SC 22

In the ascending phase, Pc5 power shows two clear peaks that exceed the CVCs around the 13 and 29 d lag in 1987. The

During the maximum phase, the peaks present in the ascending phase cannot be seen in Pc5 power as all the variations in 1989 and 2001 (Fig.

During the descending and late descending phases, the ACFs show not only the strongest values of correlation at the 27 d lag but also high values at

Although the peaks at multiples of 27 are quite sharp in the descending phase, the ACFs have smooth transitions between positive and negative values over the course of 27 d, suggesting that the solar rotation generates a 27 d variation with a sinusoidal-like pattern in these years, as seen in the smooth progression of the anticorrelation values near the

In the minimum phase, ACFs in fluence exhibit continuously moderate correlation values above CVCs between lags 0 and

The analysis developed in Sect.

The 2D plots of ACFs of every year. In the horizontal axis is the year and in the vertical axis is the lag (between 0 and 120 d). Panels

From the almost continuous horizontal line of high correlation values centered at 27 d lag in all the panels, we can infer that the 27 d periodicity is the most prominent regular periodicity detected in Pc5 daily power as well as in fluence. In fact, all years, except 1988 (ascending phase), 1998 (ascending phase) and 2001 (maximum phase), have values above CVCs around the 27 d lag. The years of 1994, 1995, 2006, 2007 and 2008 exhibit the strongest 27 d recurrence pattern with the highest correlation values. All these years belong to the descending or late descending phase. The enhancements at multiples of 27 are also very clear. As Fig. 5c–d show, they are dominant in the descending phase and absent in the ascending and maximum phase for both parameters.

The present understanding of the effects of the 27 d variation that the solar rotation generates in the geospace environment can be used in the interpretation of our results. The regions known as corotating interaction regions

During the declining phase of the SC, CIRs are particularly prominent as a result of the expansion of coronal holes (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 fluence show the strongest values of correlation at a 27 d lag and the clearest 27 d periodicity that repeats for several solar rotations. However, the fact that the ACF peaks above CVCs occur not only during the descending phase but also during other phases suggests that the 27 d 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 d period seen in 2008 for Pc5 power (shown clearly in Fig.

There are some previous reports of the 9 d recurrence in solar variables. For example,

Finally, note that 1996 in fluence shows a different behavior to all the other years. This is evident when looking at this particular year in Figs.

In order to investigate the SAV in Pc5 power and electron fluence, we performed a superposed epoch analysis of the logarithmic daily values of both parameters using the entire suite of two solar cycles of data. The zero epoch was simply the first DOY, and we calculated the median for each DOY from DOY 1 to DOY 365 (the extra day corresponding to leap years was not used due to its negligible effect on the results). Owing to the length of the observations, there are

The 30 d running average of the curves with the median is also added in Fig.

Superposed epoch analysis of the logarithmic values of Pc5 power

Pc5

In this section we compared the profiles of the angles that govern each SAV mechanism (introduced in Sect.

Figure

Absolute value of the angles that might control the SAV.

To compare the shape of the angles with FL

Some authors have used these three angles (or similar ones) in the past to test SAVs detected on magnetic indices. For example,

Smoothed absolute value of the angles that control the SAV.

We calculated the correlation values between our observational curves (FL

In principle, it should be possible to use the profiles of the three angles to determine which is the dominant mechanism, but a better approximation may be achieved by considering the functional dependencies of each angle. In this section, we evaluate the functions of

The angle

We correlated S

Correlation coefficients between the smoothed angles of the main semiannual hypotheses (

Correlation coefficients between functional dependencies of the angles (S

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.

First, we applied a nonlinear least-square fit with five parameters to the superposed median curves (black curves) of Fig.

Superposed epoch analysis of the logarithmic daily values of Pc5 power

Both fits follow the semiannual trend of the superposed median curves very well. In fact, the coefficient that modulates the amplitude of the annual variation is very low for both cases, being

Once

We have

Calling

The advantage of calculating

The best prediction of the SAV minima in fluence is given by the equinoctial hypothesis. This mechanism is also the best one for estimating the September maximum with just 1 d 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 peak and valley times expected for the equinoctial mechanism are shifted forward 4 d, as in

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 Sect.

Dates of maxima and minima for

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 Fig.

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 powers. These conclusions are reached by all the correlation values calculated in Sect.

These results differ from previous ones reported in

The potential of Pc5 power to predict relativistic electron enhancements, combining a set of individual enhancements

High solar wind speed has been found to correlate well with relativistic electron fluxes enhancements

An interesting point is that solar wind speed does not show a semiannual pattern. This statement can be probed by calculating an annual superposed curve of solar wind speed, as it is shown in Fig. 4 of

Finally, other regular variations have also been studied in this work by means of the ACF calculation. The main periodicities displayed by Pc5 power and fluence were tracked year by year along two complete 11-year SCs, demonstrating that the 27 d period can be observed in every phase of the SC. And this period is most prominent during the declining phase when high correlations at multiples of 27 were also observed. On the contrary, the 27 d period is less recognizable in the ascending and maximum phase.

To summarize, this study demonstrates that Pc5 ULF waves and relativistic electrons both vary, with multiple timescales, due to the intrinsic periods of the Sun's dynamics and also those periodicities that result from considering the Sun–Earth system as a whole.

The 11-year solar cycle variation and the 27 d periodicity are associated with the intrinsic periods of the Sun. Enhanced electron levels were found during the declining phase of a solar cycle, as previously reported in other studies. The 27 d periodicity of electrons presented in this study is related to the recurrence of high-speed solar wind streams due to solar rotation.

The semiannual period results from the combination of the periodic dynamic of the Sun and the Earth throughout the year. We have determined the most plausible SAV mechanisms to account for the observations. Similar SAV mechanisms and similar periodicities in both Pc5 power and electrons indicate that Pc5 ULF waves play an important role in energizing electrons, as attested to by other studies.

Pc5 wave power data that were derived from ground magnetic data recorded by NRCan’s CANMOS are available upon request (hlam@nrcan.gc.ca).

FLP conceived the paper, carried out most of the analysis and wrote the paper. FA provided the code for the analysis in Sect. 4.3. HLL provided the data and helped with the writing of the paper. Both coauthors helped with the interpretation of the results, read the paper and commented on it.

The authors declare that they have no conflict of interest.

The authors thank the producers of the GOES energetic electron data, which were downloaded from NOAA and the National Geophysical Data Center (NGDC). The authors also thank the developers of the International Radiation Belt Environment Modeling (IRBEM) library that was used to calculate the theoretical angles used in this work

This paper was edited by Vincent Maget and reviewed by two anonymous referees.