In order to determine the main periodicities occurring in HSS and in the magnetotail and auroral regions, the wavelet transform (WT) was employed to the IMF, magnetotail and auroral data.

The WT allows us to investigate how the energy is distributed in time and period or frequency ranges in non-stationary time series (Torrence and Compo, 1998). The wavelet functions (known as “daughter wavelets”) are generated from a simple generating function (called wavelet-mother): ψa,b=1aψt-ba for a,b∈Z and a≠0, which undergoes expansions/contractions and translations in time. The WT transforms the signal f(t) into a function with two variables (scale and time), represented by WT(a,b) from Eq. (1): 1WT(a,b)=∫f(t)ψa,b(t)*dt, where a is the dilation coefficient, b is the coefficient of translation in time, and ψa,b(t) represents the complex conjugate of the mother-wavelet function.

The Cross-Wavelet Transform (XWT) is constructed from two continuous WT of the variable time series (Grinsted et al., 2004; Bolzan, 2005) and can be obtained by Eq. (2). 2XWTx,y(a,b)=Wy(a,b)Wx(a,b)* In Eq. (2), Wx and Wy represent the WT applied to the time series x(t) and y(t), and (^*) represents the complex conjugate of the WT.

The periods with high energy or with higher correlation can be detected using the Global Wavelet Spectrum (GWS), which can be computed by GWS=∫|WT(a,b)|2db for the main periodicities. For the identification of the periods with higher correlation, the GWS is obtained by GWS=σxσyσx+σy∫|XWTxy(a,b)|2db, where σx and σy represent the variances of the time series, x(t) and y(t), respectively.

The wavelet power persistence can be classified into four types: Local, Intermittent, Quasi-Continuous, and Continuous, based on their temporal duration and physical significance. Local distribution indicates power restricted to short intervals, representing transient phenomena like bursty bulk flows. The Intermittent characteristic describes features appearing and disappearing at intervals, reflecting recurrent or stochastic processes. The Quasi-continuous distribution denotes signals present for most of the duration, suggesting a sustained but modulated response to solar wind drivers. Continuous classification refers to power persisting throughout the entire interval, signifying a steady-state, stable coupling between the solar wind and the magnetosphere (Souza et al., 2016).

In order to study the turbulence of space plasmas, one powerful technique is to apply Fourier transform to the data and obtain a power law spectrum for different frequency ranges. The Fourier spectrum of a time series is fitted by a power-law function as P(f)=cfα, where P(f) is the Fourier spectral power, c is a constant, f is the frequency, and the α exponent is the power law index (Bolzan and Echer, 2014). While the spectral power gives an indication of the amplitude of turbulence fluctuations in different regions and frequency ranges, the power law index indicates how steep is the spectrum, i.e., how fast it changes with frequency, and how the rate of energy is transferred from large to small scales (Bruno et al., 2005). In this study, the Fourier analysis was used in order to obtain the spectral index from the solar wind, magnetotail and auroral index data.

Figure 3 shows the wavelet spectrogram (map) applied to the IMF Bz component (top panel) and magnetotail Bx component (middle panel) and AE index (bottom panel) during a moderate geomagnetic storm main phase interval between 00:00 UT and 19:09 UT, 24 October 2002 (Event 06 from Table 1). In the upper panel, the left figure shows the wavelet spectrum of the IMF Bz and the right figure presents the global wavelet spectrum (GWS). In the GWS, three periods with higher energy can be identified. The first, around 0.5 h, can be characterized as having local distribution, since higher energy (integrated wavelet power) is observed in the wavelet spectrum only in a small interval between ∼ 10:00 and ∼ 15:00 UT. The second period is of 1.5 h and also presents local distribution, located in a similar region of the first period (from ∼ 10:00 to ∼ 16:00 UT). The last observed period is around 3.3 h, and in the wavelet spectrum it can be observed continuously for a long time interval, from around ∼ 05:00 UT to more than 16:00 UT, characterizing a quasi-continuous behavior. The quasi-continuous nature of the 3.3 h period can indicate large-scale fluctuations in the solar wind's driving force. This is related to the characteristics of the high-speed stream (HSS) following the CIR, which can contain continuous fluctuations that drive the magnetosphere over a long duration (Borovsky and Denton, 2006).

In the middle panel of Fig. 3, the wavelet transform periodogram for the magnetotail Bx is shown. As for the IMF Bz wavelet spectrum, upper panel, the left figure shows the wavelet spectrum and the right panel shows the GWS. Four significant periods were identified in the GWS: ∼ 0.4, ∼ 1.3, 2.4 and 4.2 h. From the wavelet spectrum one can see that local distribution is observed in the two first periods, which are found from about 03:00–05:20 UT and from about 01:00–08:30 UT, respectively. The other two periods show quasi-continuous distribution, where the period of 2.4 h is the most energetic. The strong energy at these periods suggests that the magnetotail's internal dynamics, possibly related to current sheet flapping or other global oscillations, are being excited and maintained by the external solar wind driver (Runov et al., 2005). The wavelet analysis of the AE index (bottom panel) shows a single, significant quasi-continuous period around 3 h. This timescale is consistent with the substorm cycle (Borovsky et al., 1993; Korth et al., 2011).

The coincidence of this dominant period with the quasi-continuous periodicities found in both the solar wind (IMF Bz) and magnetotail (Bx) provides evidence of a direct and efficient coupling mechanism between the external driver and the magnetospheric response. It indicates that fluctuations in the solar wind (IMF Bz) are directly linked to magnetotail dynamics (Bx) and subsequently drive auroral substorm activity (AE).

We also applied the wavelet transform to the solar wind and magnetospheric data during the main phase intervals of the events from Table 1 when the Cluster spacecraft remained at least 10 h in the magnetotail. A total of 6 events were analyzed (events: 6, 7, 9, 14, 26 and 30). The number of events analyzed for each variable varies due to gaps in the available data of IMF Bz and magnetotail Bx components. For the IMF Bz and magnetotail Bx, the WT was applied to data in 5 events: 6, 7, 9, 14 and 30 for IMF Bz and 6, 7, 9, 14 and 26 for magnetotail Bx. The statistical result of this analysis is shown in the Section below along with the results of the recovery phase.

As mentioned before, the WT was also applied to the CIR-driven geomagnetic storm recovery phases during intervals when Cluster was in the magnetotail. Figure 4 shows one example, where the WT was performed for event 18 of Table 1, which occurred from 19:00 UT on 31 August to 08:40 UT on 1 September 2005 (intense storm).

Similarly as it was shown in Fig. 3, the upper panel shows WT for the IMF Bz component, the middle panel for the magnetotail Bx and the bottom panel shows the WT applied to the AE index. Again, the left panels correspond to the wavelet spectrum and the right panels to the global wavelet spectrum.

The GWS, shown in the top panel in Fig. 4, for IMF Bz, shows that two significant spectral peaks are observed, both with local distributions (left panel). The first period, near 1.0 h, is localized between 19:00 and 23:00 UT. The second one is observed at 2.0 h, with higher energy (integrated wavelet power) concentrated between 23:00 UT on 31 August and ∼04:20 UT on 1 September 2005.

For the magnetotail Bx, shown in the right middle panel, the GWS also shows only one significant period, very close to the periodicity observed for IMF Bz, 2.1 h. However, for the magnetotail Bx component, the energy distribution has a quasi-continuous variation (left middle panel), and is observed from the beginning of Cluster interval until about 05:00 UT of 1 September. In the AE index, (bottom panels), the wavelet analysis shows a period around 3.6 h in the GWS, which also has a quasi-continuous distribution, from the beginning of the interval to the period at about 06:00 UT on 1 September 2002. Other short periods can be seen in the GWS, but in the wavelet distribution (left bottom panel), the higher energy is observed outside the cone of influence (COI), which delimits the region that is not affected by edge effects. Therefore they were discarded.

While the IMF Bz shows localized, high-energy periods at 1.0 and 2.0 h, the magnetotail Bx and AE index display quasi-continuous dominant periods at 2.1 and 3.6 h, respectively. This continuous nature suggests that even in the recovery phase, the magnetosphere is being driven by sustained solar wind fluctuations, a typical characteristic of HSS (Tsurutani and Gonzalez, 1987; Souza et al., 2018).

We followed the same criteria used to compute WT for both the main and recovery phases. For the Cluster magnetotail's periods this interval should be ≥ 10 h. For the recovery phase, a much higher number of events followed this criterion (29 events). It is worth mentioning that for some cases only one or two variables meet the criterion due to the absence of available data. Figure 5 shows the histograms for period ranges, with bins of 2 h, for the three variables for the main phase (left panels) and recovery phase (right panels). This figure shows that, except for the magnetotail Bx, longer periods begin to be observed in the recovery phases, right top and bottom panels. For the IMF Bz component histograms, main phase (left panels) and recovery phase (right panels), we can see the same dominant ranges of periods for both geomagnetic storm phases. Periods ≤2h represent 50.0 % of the main periods during the main phase, and 54.2 % of the periods during the recovery phase. The range between 2 and 4 h corresponds to more than 35 % of the periods for the main phase, and 31.3 % for the recovery phase. These two ranges are also predominant for the magnetotail Bx periodicities. During HILDCAA events (Franco et al., 2019), the stronger magnetotail response to the IMF fluctuations are typically found in the substorm periodicity range (2–4 h) (Borovsky et al., 1993; Franco et al., 2019).

For the magnetotail Bx, although the range between 2 and 4 h corresponds to 25 % of the periods for the main phase, and 29.2 % for the recovery phase, periods≤2h represent 68.7 % of the periods during the main phase and 61.1 % for the recovery phase. From our present results, it seems that during CIR-driven geomagnetic storms the magnetotail has a stronger response to lower periods than during HILDCAA events. This could occur because the prolonged duration and wave-driven nature of HILDCAAs favor longer characteristic periods, while the shorter, more intense driving of the CIR main phase emphasizes the shorter, internal magnetotail response periods. Further, these results show us that more than 80 % of the most energetic periods in the IMF Bz and magnetotail Bx during geomagnetic storms driven by CIRs are found for periods shorter than 4 h, independently of the storm phase. Similar periods (1.8–3.1 h) were found by Echer et al. (2017) in a study of the solar wind – magnetotail coupling during HSS and CIR intervals.

The AE index analysis revealed that periods between 2 and 4 h are the most prominent during both the main (42.9 %) and recovery (53.7 %) phases of geomagnetic storms. However, there are notable differences between these phases. During the main phase, periods between 4 and 6 h are the second range most often observed, followed by periods shorter than 2 h. In contrast, during the recovery phase, no significant periods shorter than 2 h were observed. Instead, the distribution of higher-energy periods seems to extend to longer durations, with the 4–6 and 6–8 h ranges sharing the same percentage of periods. This difference may be attributed to the occurrence of HILDCAA events during the recovery phase (Kamide et al., 1998). These events are often associated with Alfvénic trains in the IMF, and typically have periods between 4 and 12 h (Souza et al., 2016). While the increased number of analyzed recovery phase events could potentially influence the period distribution, we believe this is not the primary cause of the observed distribution. This is supported by the fact that the spread in periods was not observed for the magnetotail Bx component but it was noted only in the IMF Bz component distribution, which oscillations are usually due to Alfvén waves during CIR intervals. Furthermore, the absence of shorter periods during the recovery phase cannot be solely explained by the higher number of analyzed cases.

Figure 6 represents the energy distribution across the main identified periods for each variable and period range during the main and recovery phases. Considering all ranges, the continuous behavior dominates the IMF Bz energy distribution during the main phase with 35.7 % of the cases, which is also observed for individual ranges, except for the range≤2h, at which local distribution dominates. During the recovery phase, the local and intermittent distributions are the most common observed energy distribution types, while the continuous distributions represent only 3.6 % of the most energetic periods identified. Further, note that during the main phase, the local characteristic dominates the range with the larger occurrence of energetic periods (≤2h), while during the recovery phase, the intermittent behavior is dominant for this period range. This distinction in energy distribution characteristics is significant, as the main phase of geomagnetic storms is frequently characterized by enhanced substorm activity and the most intense occurrence of magnetotail dipolarizations (Lee and Min, 2002).

Near Earth, these events can be associated with localized fast plasma jets during intense substorms, which may justify the local distribution observed in the most energetic periods during the main phase (Nakamura et al., 2002, 2017). The quasi-continuous behavior observed in the recovery phase was also seen in the IMF Bz component during HILDCAA events. For shorter periods (≤2h), an intermittent distribution was observed, which might be related to the events that occur during geomagnetic storm recovery phases (Souza, 2015; Souza et al., 2016).

For the magnetotail Bx component, the local type distribution of the periodicities was the most often observed for both main and recovery phases considering all ranges. The same characteristic is observed for the range with the higher occurrence of the main periods (≤2h) with 72.7 % for the main phase, which could also be related to localized fast plasma jets in the magnetotail (Nakamura et al., 2017). The same predominant distribution was also observed for the recovery phases (66.7 %). This result differs from those found in a previous magnetotail study during HILDCAAs (Franco et al., 2019), that showed that more than 50 % of the magnetotail Bx periods presented quasi-continuous and intermittent distributions, which is a characteristic of substorms. In the AE index, quasi-continuous and intermittent distributions dominate for all ranges in the main phase. Intermittent behavior was also observed for periods ≤ 2 h. In the recovery phases, the quasi-continuous characteristic is more frequent for all ranges, which is also seen for the main range, between 2 and 4 h. Again, these results differ from that observed from HILDCAAs ( Souza et al., 2016), which presented intermittent energy distribution as the most observed type of periodicity in the wavelet spectrum. As it was perceived during supersubstorms studied by (Hajra et al., 2023), during geomagnetic storms, in short periods, the local and intermittent behavior is dominating while for longer periods, quasi-continuous and continuous distributions are more often observed.

The cross-wavelet results applied for the interval of the magnetotail Cluster crossing during the main phase of the event 6 (24 October 2002) of CIR-driven geomagnetic storm presented in Table 1 are shown in Fig. 7. For this same event the wavelet transform was applied (presented in Fig. 3). The cross wavelet spectrum is presented in the left panels: IMF Bz component and magnetotail Bx (upper panel), IMF Bz component and AE index (middle panel) and magnetotail Bx and AE index (bottom panel). The GWS is shown in the right panel of the respective figure. Then the XWT between IMF Bz and magnetotail Bx (top panel) and between magnetotail Bx and AE index (bottom panel) shows three similar main periods of correlation around: 1.30, 2.60 and 4.20 h. The local distribution dominates these periods, except for the period near 2.60 h of the bottom panel (magnetotail Bx×AE index), where quasi-continuous behavior is observed. In the middle panel (IMF Bz×AE index) only one significant period is seen, which shows local distribution and a period of 3.21 h. Note that in the cross-wavelet spectrum, the region with higher correlation occurs between 08:00 and 15:00 UT for the top and bottom panels, which is also observed in the middle panel, but with a shift in time (10:00 and 16:30 UT). These similar dominant periods (1.30, 2.60 and 4.20 h) of correlation between solar wind-magnetosphere-auroral region oscillations can indicate resonant modes of the coupling of the solar-wind–magnetosphere–ionosphere system, and represent the time scales at which the energy transfer is most efficient. This supports the idea of a linked chain of events: solar wind fluctuations drive magnetotail oscillations, which in turn drive substorms and auroral activity at the same time scales.

Following the WT analysis approach, we initially planned to apply the cross-wavelet transform to the 6 events with data intervals of at least 10 h. However, as not all analyzed variables met the duration criterion within these events, the number of events suitable for cross-wavelet analysis varied depending on the availability of IMF Bz and magnetotail Bx data. The XWT was applied between IMF Bz and magnetotail Bx in only 4 events (6, 7, 9 and 14 from Table 1). For IMF Bz and the AE index, it was applied in 5 events (6, 7, 9, 14 and 30 from Table 1), and for magnetotail Bx and the AE index, it was applied in 5 events (6, 7, 9, 14 and 26 from Table 1). The results of this analysis are presented in the following section, combined with the recovery phase analysis results.

Figure 8 shows the cross-wavelet transform applied to the event 18 (31 August–1 September 2005) from Table 1, the same event shown in Fig. 4. The left panels show the cross wavelet spectrum between the IMF Bz component and magnetotail Bx (upper panel), IMF Bz component and AE index (middle panel) and between magnetotail Bx and AE index (bottom panel). The right panel shows the GWS, where we can note that the period near 2.28 h is present in the three cases. These intervals show quasi-continuous behavior for the top and bottom panels, and local correlation distributions for the middle panel (IMF Bz×AE index). Further, periods with higher correlation are observed near 0.80 h and ∼3.5h in the GWS of the middle (IMF Bz×AE index) and bottom (magnetotailBx×AE index) panels. For both cases, the periodicity around 0.80 h shows local distribution, while the 3.5 h shows a quasi-continuous distribution. In the middle panel, (IMF Bz×AE index), an intermittent higher correlation distribution is found for the period of 1.31 h.

In Fig. 9, histograms with the range of periods with higher correlation obtained from the XWT are presented for the main and recovery phases. As it was seen in the wavelet analysis, in the main phase the periodicities are restricted for periods shorter than 8 h, while during the recovery phase, except for the magnetotail Bx×AE index analysis, longer periods are observed. Despite that, even for the recovery phases, more than 95 % of the periods are lower than 8 h. Similar periods that were found for the IMF Bz WT analysis were obtained in the correlation between the IMF Bz and magnetotail Bx (upper panel). Most of the higher correlation intervals, ∼54% for the main phase and ∼44% (recovery phase), occurred for shorter periods (≤ 2 h). Further, the second range (2–4 h) also represents about 30 % of the periods with higher correlation for main (∼31%) and recovery (∼33%) phases. These results are in agreement with those results that were found in the XWT analysis performed by (Franco et al., 2019) for HILDCAAs events.

A significant difference can be observed between the main and the recovery phases in the periods of correlation between the IMF Bz and the AE index. Although the dominant range is the same (2–4 h) in both phases, during the recovery phase, periods≤2h represent more than 40 % of periods while in the main phase they represent only 21.4 %. Furthermore, considering ranges with longer periods, the main phase has more than 21 % of the periods of correlation between 4 and 8 h, while less than 15 % of the periods of correlation during the recovery phase are found for periods>4h. This change reflects a transition from external, sustained driving to the magnetosphere's internal process of unloading stored energy through a series of more frequent, cyclic substorms. Further, the dominance of periods≤4h in the recovery phase coincide with cyclic substorm periods (Korth et al., 2011). The periodicity with higher correlation observed in the range between 10 and 12 h may be related to Alfvén waves present in the IMF Bz during periods of HILDCAAs in the magnetotail (Smith et al., 1995; Souza et al., 2016).

In the analysis of the periods at which the energy stored in the magnetotail is transferred to the auroral region (XWT between magnetotail Bx×AE index), the histogram with the periods of correlation distribution is presented in the lower panels of Fig. 9. In the main phase (left panel), the percentage of periods of correlation per range decreased as the periods increased. About 47 % of the periods with higher correlation are ≤2h. The range between 2–4 h corresponds to almost 30 % of the periods, followed by 17.65 % (4–6 h) and 5.88 % in the range between 6 and 8 h. In the recovery phase (right panel) the range 0–2 h and the range 2–4 h showed the same percentage of periods (41.93 %). The others ∼16% intervals are found in the range between 4–6 h, indicating that all the periods with higher correlation are shorter than 6 h. As the periods with higher energy from the previous analysis, the periodicities with higher correlation values occur for substorms.

Figure 10 shows the results with the classification of the type of correlation distribution. Considering all periods, the local distribution is dominant in three investigated cases: IMF Bz× magnetotail Bx, IMF Bz×AE index and magnetotail Bx×AE index (also intermittent) during the main phase. For the recovery phase, only the analysis between IMF Bz and magnetotail Bx data did not show the predominance of local distribution, with the intermittent behavior as the most commonly observed, probably related to substorm events.

Results from the main phase are summarized in Table 2, and the spectral indices for event 7 are shown in Fig. 11a). According to Table 2, considering all analyzed events, it is possible to observe that mean values of the spectral indices were (-1.62±0.04) for the IMF Bz, (-1.70±0.22) for the magnetotail Bx and (-1.92±0.38) for the AE index. Recently, Rakhmanova et al. (2020) studied the dependence of the spectral indices over plasma turbulence and showed that both the solar wind parameters and bow shock geometry have a great influence on the variation on the spectral index values. This fact shows that these values are sensible to this kind of parameters and the geometry from the space environment. These dependencies from the space geometry for the spectral indices are due to the formation of several kinds of phenomena, such as Alfvén waves, and also the coherent structures inside the plasma turbulence. This dependence of the spectral indices over the space environment was also demonstrated in Neptune day side magnetosheath fluctuations during Voyager-2 flyby (Echer et al., 2023) and during solar-wind high-speed streams (HSSs) at high heliolatitudes during the last two solar-cycle minima (SCM) (Echer et al., 2022).

Bolzan (2005) showed that the coherent structures present in solar wind time series increase (in absolute value) the spectral indices. The idea is that, into the turbulence flow, these organized structures are responsible to lead the energy from large to small scales in the fast and energetic way like a cascade as proposed by (Kolmogorov, 1941). According to Müller and Biskamp (2000), MHD turbulence is driven by the Alfvén wave effect, which couples small-scale velocity and magnetic fluctuations. This results in an energy spectrum consistent with a k-5/3 law (the K41 law), which also aligns with observations of solar wind turbulence. Thus, our results from spectral indices mean values found for the AE index are consistent with previous works, where the difference found between -1.67=-5/3 and value=-1.92 are due to the intermittency phenomena present in the MHD turbulence.

An interesting aspect observed in this study is the difference found between spectral index mean values obtained for the AE index time series and the solar wind-magnetotail time series (IMF Bz and magnetotail Bx). Three important points are noted: first, the mean values for the IMF Bz and magnetotail Bx series matches the K41 law which corresponds to k-5/3; second, despite the standard deviation, there is a notable difference between these spectral indices for these both kinds of time series; third, the mean values for the AE index are in agreement with results from (Tsurutani et al., 1990). Tsurutani et al. (1990) suggested that high amplitude fluctuations in the southward IMF component will generate flux variations in the magnetotail and may be related to the increase of the ionospheric conductivity, which can be associated with the higher spectral index values found for AE index.

In a previous work, similar results were obtained for the study of the Earth's magnetotail variability during supersubstorms (Hajra et al., 2023). In that study it was found that the IMF Bz exhibits the lowest spectral index (compared to the magnetotail Bx and SML auroral indices), thus, corroborating our results. It is important to mention that the solar wind turbulence is significantly amplified after crossing the bow shock, and plasma flows are generated in the magnetosheath region. In the nightside, eddy vortices are generated due to those large-scale magnetospheric flows, which can drive the magnetotail turbulence (Antonova and Stepanova, 2021).

Similarly to the analyses conducted for the main phase intervals, we performed the Fourier Transform in all data in order to obtain the spectral indices during the recovery phase. An example is shown in Fig. 11b, which displays the spectral indices obtained for the parameters analyzed during Event 1. Table 3 shows the results where the mean values found for IMF Bz, magnetotail Bx and AE index were -1.60±0.15, -1.95±0.51 and -2.23±0.20, respectively. For the IMF Bz index, the spectral index is lower in the recovery phase than during the main phase,while for the magnetotail and auroral data, higher mean absolute values are found, where they were -1.95±0.51 and -2.23±0.20, respectively. A sharp break is observed in the magnetotail Bx and AE index PSDs during the main phase, which is not seen during the recovery phase (Fig. 11). We may conjecture that during the recovery phase the geomagnetic field promotes a fast transfer of the energy from the large to small scales in order to leave the magnetosphere to return to almost steady state.

We also have separated the spectral indices from AE indices time series and magnetic ones (IMF Bz and Bx) where the results were, respectively, -2.23±0.2 and -1.72±0.38. Note, again, the magnetic time series, IMF Bz and Bx, following the K41 law, corroborating results from (Müller and Biskamp, 2000). Furthermore, these results, similar to those observed in the main phase, suggest a characteristic of AE index time series: a distinct behavior compared to magnetic time series, as previously discussed in (Hajra et al., 2023). Also, despite the large standard deviation, it is possible to recognize that the mean values of the spectral indices for the recovery phases are greater if compared with those noted during the main phase. We can associate this result with the presence of the strong turbulence present during the recovery phase. It is just in this phase that the turbulent phenomena need to act in order to promote the transferring of energy from large to small scales like a Kolmogorov cascade.