In this paper we use wavelets and Lomb–Scargle spectral analysis techniques to investigate the changing pattern of the different harmonics of the 27-day solar rotation period of the AE (auroral electrojet) index during various phases of different solar cycles between 1960 and 2014. Previous investigations have revealed that the solar minimum of cycles 23–24 exhibited strong 13.5- and 9.0-day recurrence in geomagnetic data in comparison to the usual dominant 27.0-day synodic solar rotation period. Daily mean AE indices are utilized to show how several harmonics of the 27-day recurrent period change during every solar cycle subject to a 95 % confidence rule by performing a wavelet analysis of each individual year's AE indices. Results show that particularly during the solar minimum of 23–24 during 2008 the 27-day period is no longer detectable above the 95 % confidence level. During this interval geomagnetic activity is now dominated by the second (13.5-day) and third (9.0-day) harmonics. A Pearson correlation analysis between AE and various spherical harmonic coefficients describing the solar magnetic field during each Carrington rotation period confirms that the solar dynamo has been dominated by an unusual combination of sectorial harmonic structure during 23–24, which can be responsible for the observed anomalously low solar activity. These findings clearly show that, during the unusual low-activity interval of 2008, auroral geomagnetic activity was predominantly driven by high-speed solar wind streams originating from multiple low-latitude coronal holes distributed at regular solar longitude intervals.

Geomagnetic activity is driven by solar activity which varies on different
timescales, of which the

Files containing the 1 h AE indices for the period 1960–2014 were
downloaded from the World Data Centre for Geomagnetism in Kyoto, Japan
(

The AE time series from 1960 to 2013.

A distinctive feature in Fig. 1 is the large and seemingly bursty excursions indicating auroral magnetic disturbances as well as very low levels of activity around 2010. The main aim of this study is to investigate the evolution of the 27-day solar rotation period and its different harmonics in the AE time series on an annual basis.

All graphs and plots in this paper were produced using the SigmaPlot
(

Spectral analysis techniques such as wavelets and Fourier applications are well-established methods for revealing periodicities on various timescales in time series. Determining periodicities in incomplete or unevenly sampled time series for a given variable is not directly possible with methods such as fast Fourier transform (FFT) and may require some degree of interpolation to fill in gaps. An alternative is the Lomb–Scargle method (Lomb, 1976; Scargle, 1982), which estimates a frequency spectrum based on a least squares fit of unevenly sampled data to represent a signal as the sum of sine and cosine functions of infinite duration. The Fortran computer algorithm of Press et al. (1992, chapter 13.8) was applied to all daily mean observations for each individual year between 1960 and 2014.

On the other hand wavelet analysis is useful to decompose a time series into
time–frequency space and to establish the modes of variability, particularly
when the time series under investigation contains non-stationary power at
different periods. As the wavelet transform is a localized transform in both
space (time) and frequency, this property can be applied to extract
information from the signal, which is not possible with Fourier techniques
(Daubechies, 1990). For the purposes of this investigation the Morlet
wavelet (Morlet et al., 1982; Torrence and Compo, 1998),

Both above-mentioned methods were used to obtain the spectral contents of the AE index time series at each annual interval. The Lomb–Scargle method in particular served to verify the Morlet wavelet analysis results. Wavelet transforms were also applied by Balasis et al. (2006) to perform a fractal spectral analysis of the 2001 Disturbance storm-time (Dst) index time series at solar maximum, while Zaourar et al. (2013) performed a wavelet-based multiscale analysis of geomagnetic disturbances recorded at various magnetic observatories. Kunagu et al. (2013) also analysed 10 years of CHAMP (CHAllenging Minisatellite Payload) magnetic field data using wavelets and found external field periodicities of interest to global geomagnetic field models.

The correlation between the AE index and various spherical harmonic
coefficients of the Wilcox magnetic field model, at 2.5 solar radii, were
calculated using Pearson's technique. According to this method a linear
correlation between two variables (

Morlet wavelet power spectra were obtained by analysing each individual year's daily mean AE indices from 1961 to 2013 in order to reveal the short-period geomagnetic activity behaviour during various phases of each solar cycle in this interval. This enabled the identification of second (13.5-day), third (9.0-day), and fourth (6.7-day) harmonics of the fundamental 27-day solar rotation period and their individual behaviour during, particularly, solar maximum and solar minimum intervals at the 95 % confidence level. A typical power spectrum for a solar maximum in 1991 is shown in Fig. 2. The dominance of the 27-day recurrence period can clearly be seen.

A Morlet AE power spectrum during solar maximum in 1991 showing the clear domination of the 27-day recurrence period. The top graph shows the daily mean AE indices used in the wavelet analysis for 1991.

A Morlet wavelet power spectrum of the AE index during 2008 showing the dominant behaviour of the 13.5- and 9.0-day periodicities. The top plot shows the daily mean AE indices during 2008 used in the current wavelet analysis.

In contrast to a solar maximum interval, a solar minimum shows that the auroral geomagnetic activities are no longer dominated by the 27-day solar rotation period but that in fact the 13.5- and 9.0-day periodicities are playing a significant role. A typical example of a solar minimum AE wavelet power spectrum for 2008 is shown in Fig. 3.

Lomb–Scargle, being a variant of the Fourier spectral analysis methods, does not have the ability to analyse a time series into time–frequency space and to determine the dominant modes of variability, particularly when these time series contain non-stationary power located at different frequencies. This method was therefore employed to verify the wavelet results and to obtain the frequency (periodicity) content of the AE time series without localization in time. A typical example of Lomb–Scargle spectral analysis for a solar maximum year (2001) and a solar minimum year (2008) can be seen in Fig. 4.

Lomb–Scargle spectral analysis of AE time series for 2001

What is also evident from this investigation is that the 27-day solar rotation period is constantly present for all solar cycles between 1960 and 2014, but during the peculiar solar minimum period of cycles 23–24 the power spectrum is dominated by the second and third harmonics. The 9.0- and 13.5-day periodicities seem to play an important and distinguishable role particularly during the downward phase and minimum periods of several solar cycles since 1960. This was quite evident during 1965, 1985, 1994, 1996, and 2008. A Lomb–Scargle analysis confirmed the wavelet analysis results. In particular for 2008, as shown in Fig. 4, only the 9.0- and 13.5-day periods can be distinguished above the 95 % level. Although a 26- and 28-day periodicity can be identified in the spectrum of the solar magnetic field during 2008, the solar activity is dominated by the 13.5-day period. These figures and similar results obtained between 1960 and 2014 show that the different periodicities tend to appear without a fixed pattern during each individual year. In addition the average power also scales according to the particular harmonic. This phenomenon can most probably be ascribed to the dominance of CMEs during solar maximum geomagnetic storms, while fast streams originating in coronal holes are dominant during solar minimum periods. These results confirm previous findings regarding the behaviour of K indices at both high latitudes and midlatitudes (Kotzé, 2015). Several publications have also appeared in the scientific literature on results obtained by studying the behaviour and influence of solar cycle and solar wind parameters on geomagnetic activity levels (e.g. Crooker et al., 1977; Kojima and Kakinuma, 1990; Mursula, 1999), as well as periodicities in the interplanetary magnetic field (Gonzalez and Gonzalez, 1987) and solar excursion phases during several solar cycles (Mursula and Zieger, 1998). Results obtained by these and other authors showed that the origin of periodicities in geomagnetic activities is closely linked to the domination of solar activity during solar cycle minimum by coronal holes (e.g. Sheeley Jr. et al., 1976), responsible for multi-component high-speed streams, while during solar maximum intervals CMEs are responsible for geomagnetic activity (Gosling et al., 1991).

Pearson linear correlation coefficients were determined between AE indices and all low-degree and low-order spherical harmonic coefficients as a function of Carrington rotation for every year since 1975 when these coefficients were starting to be determined by the Wilcox Solar Observatory. Table 1 shows results obtained for 2008 during the very low minimum of cycles 23–24.

Pearson correlation coefficients between AE and dipole and quadrupole spherical harmonic coefficients of the solar magnetic field for 2008.

From this table we can clearly see a strong correlation between AE and
particularly the sectorial harmonic coefficients

Spectral analysis of the auroral geomagnetic activity as represented by the AE index has been performed for several solar cycles between 1960 and 2014 using techniques such as Morlet wavelet analysis and Lomb–Scargle. In particular it has been revealed that the 27-day solar rotation period dominates above the 95 % confidence level during solar maximum years due primarily to the occurrence of CME-driven magnetic storms. In contrast, solar minimum periods are characterized by the domination of 13.5- and 9.0-day periodicities above the 95 % confidence level, which is a result of geomagnetic storms to a large extent being driven by coronal-hole activities in these intervals. This is particularly evident during the extensive solar minimum period observed during cycles 23–24.

Lei et al. (2008) investigated CHAMP neutral densities for 2005 and revealed
the presence of a 9-day periodicity in the data. A spectral analysis of the
data showed strong evidence that the upper atmosphere is modulated by
rotating coronal holes that have a dominant 27-day periodicity as well (see,
e.g., Fig. 2, Lei et al., 2008) above the 95 % significance level. Similar behaviour was also observed by Thayer et al. (2008) when investigating
thermospheric density oscillations resulting from high-speed solar wind
streams during 2006, reporting periodic oscillations at 4–5-, 6–7-, and
9–11-day periods, with the 7-day period the dominating oscillation. It is
also quite clear (see, e.g., Fig. 2, Thayer et al., 2008) that the 27-day period is still present well
above the 95 % significance level. A separate investigation by Love et
al. (2012) using autocorrelation techniques, clearly identified 6.7- and
9.0-day geomagnetic recurrence intervals in the

Results obtained in this investigation using both wavelet and Lomb–Scargle spectral analysis methods to investigate the behaviour of auroral geomagnetic activity during cycles 23–24 support the findings of all three above-mentioned investigations during 2005 and 2006. In both 2005 and 2006 the AE index, which is a proxy for ionospheric convection behaviour, is characterized by the appearance of 6.7-, 9.0-, and 13.5-day periodicities, while the 27-day synodic oscillation is still the dominating periodicity above the 95 % significance level. However, during 2008 which is characterized by the least number of sunspots since 1913, the spectrum of the auroral geomagnetic activity shows dominant 9.0- and 13.5-day periodicities well above the 95 % confidence level, while the power of the 27-day periodicity is significantly reduced in comparison to these prominent higher harmonics. It is worth mentioning that the power of 27-day period is now below the 95 % significance level during 2008 as can be seen in Fig. 4. This new finding shows that during the unusual low-activity interval of 2008 that auroral geomagnetic activity was predominantly driven by high-speed solar wind streams originating from multiple low-latitude coronal holes distributed at regular solar longitude intervals.

A study by Richardson et al. (2001) concluded that storms associated with
high-speed solar wind streams originating in coronal holes are most
prevalent during the decay and minimum phase of the solar cycle which are
generally small or medium in size in comparison to solar maximum conditions.
The declining phase of cycle 23 and the following unusually low and extended
minimum of cycles 23–24 were different to previous solar cycles (e.g.
Russell et al., 2010). A previous investigation of K indices (Kotzé, 2015)
also revealed that the solar magnetic field was dominated by the 13.5-day
period during the solar minimum of cycles 23–24. This and the findings in
this paper show that the peculiar behaviour of geomagnetic activity in the
auroral region can most probably be attributed to the non-axisymmetric
heliosphere that drives the observed recurrences. Since the Sun controls the heliosphere, we can safely conclude that the solar dynamo
was in a state of unusual asymmetry during that time. The strong Pearson
correlations we obtained between auroral activity as measured by the AE
index and particularly the low-degree sectorial spherical harmonics is
therefore an indication of the non-axisymmetric structure in the solar
poloidal magnetic field having a profound influence on the heliosphere. The
indications are strongly in favour of the sectorial structure in the solar
magnetic field being responsible for the dominant 13.5- and 9.0-day auroral
geomagnetic activity seen during minimum 23–24, supporting the investigation
by Love et al. (2012) using

Files containing the 1 h AE indices for the period 1960–2014 were
downloaded from the World Data Centre for Geomagnetism in Kyoto, Japan
(

The efforts in running the auroral magnetic observatories to measure and archive data for the derivation of AE indices are greatly appreciated. The topical editor, G. Balasis, thanks the two anonymous referees for help in evaluating this paper.