Geomagnetic activity occurs when there are disturbances in the Earth's magnetosphere caused by an enhanced solar-wind–magnetosphere energy transfer. The most known types of geomagnetic activities are the geomagnetic storms (Gonzalez et al., 1994), magnetic substorms (Akasofu, 1964) and HILDCAA (high-intensity long-duration continuous AE activity) (Tsurutani and Gonzalez, 1987). HILDCAA event is a kind of auroral activity and its cause is associated with the southward interplanetary magnetic field (IMF) Bz component of Alfvén waves (Tsurutani and Gonzalez, 1987; Tsurutani et al., 2004). These events are defined by four criteria: (i) the AE index has to attain at least one peak value ≥1000 nT, (ii) the event must last for a minimum of 2 d, (iii) the AE index cannot drop below 200 nT for more than 2 h at a time, and (iv) the event cannot occur during the main phase of a geomagnetic storm (Dst < -50 nT).

In the first studies that were done about HILDCAAs (Tsurutani et al., 1990, 2004), the question of whether these events could be a kind of continuous substorm arose. Tsurutani et al. (2004) proved the opposite, since those authors found that some substorms occurred with little or no changes in the AE index, which is the major characteristic observed during HILDCAA events. Thus, HILDCAAs are not necessarily geomagnetic substorms, although they may contain substorms in some cases. Another characteristic of HILDCAAs that distinguishes them from substorms is the aurora formed by these events. During HILDCAA events the auroras are weak or moderate, distributed in the whole auroral zone and can last several days, while during substorms, auroras are confined in small regions and last only 15 min (Guarnieri, 2006).

HILDCAAs are events of low Dst intensity, when compared with geomagnetic storms. However, the integrated energy input from particle precipitation into the ionosphere during the whole duration of a HILDCAA event can be bigger than the energy of moderate geomagnetic storms recovery phase (Guarnieri, 2005). Recent studies (Hajra et al., 2014; Mendes et al., 2017) of the energy transfer from solar wind to the magnetosphere and ionosphere during HILDCAAs have shown that two mechanisms are responsible for the energy and matter transfer, namely the magnetic reconnection (Dungey, 1961) and viscous interaction (Axford and Hines, 1961). Hajra et al. (2014) have shown the main process of solar wind energy dissipation in the magnetosphere during HILDCAA is by Joule heating in the auroral region, which is responsible for 67 % of the input energy. It was seen that the coupling between solar wind and the magnetosphere during these events occurs mainly in periods ≤8 h (Souza et al., 2018). Similar periods were observed in most energetic periods in the AE index and Bz IMF component. Those are comparable to the periods observed in the interplanetary Alfvén waves present in high-speed streams (HSSs) from coronal holes (Smith et al., 1995; Souza et al., 2016).

The magnetotail has an important role in the energy transfer from solar wind to the inner magnetosphere, since the energy of this process is stored in the tail (Lopez, 1990). During geomagnetic substorms, the solar wind energy is stored in the magnetotail during the initial phase and then in the expansion phase it is suddenly released and deposited in the ionosphere (Akasofu, 1981, 2013). This storage of energy can be interpreted as asymmetric ring currents and magnetotail currents, which respond to the periods of substorms that were observed as 2–4 h (Sitnov et al., 2001). The aim of this work is to identify the main periodicities present in the geomagnetic field Bx component in the magnetotail during HILDCAA events, and also to study at which periods the energy is preferentially transferred from the solar wind to that region during these geomagnetic disturbances using wavelet analyses.

In order to develop this study, geomagnetic field Bx component data obtained by the fluxgate magnetometer (FGM) (Balogh et al., 2001) from the satellite SC4 of the Cluster constellation (https://csa.esac.esa.int/csa-web/#search, last access: 7 October 2019) with 4 s resolution were used. The Bx component corresponds to the Sun–Earth line direction, and it is an indicator of when the Cluster spacecraft crosses the magnetic equator in the tail and is also an indicator for the stretching or depolarization of the tail field (e.g. see Korth et al, 2006; Echer et al., 2017). Nine events were selected (one in August 2003, two in September 2003, one in October 2003, one in September 2004, one in August 2005, two in October 2006 and one in September 2007), which correspond to the events when Cluster crossed the plasma sheet during the occurrence of HILDCAAs. In this analysis AE and IMF Bz component data were also used. The AE indices at 1 min time resolution were obtained from the World Data Center for Geomagnetism, Kyoto, Japan (http://wdc.kugi.kyoto-u.ac.jp/, last access: 30 September 2019), and the IMF Bz data (1 min) were obtained from the OMNI web (http://omniweb.gsfc.nasa.gov/, last access: 30 September 2019). In this paper Cluster and IMF data are in the Geocentric Solar Ecliptic (GSE) coordinate system.

Table 1 shows some relevant information about the HILDCAA events that were analyzed, where the start and end times of each event are shown, as well as the data intervals analyzed for the Cluster magnetotail crossings.

Figure 1 shows the time series during the HILDCAA event that occurred between 03:38 UT on 15 October to 18:35 UT on 22 October 2003 (event 4 in Table 1). Panel (a) shows the IMF Bz component, (b) the Cluster tail Bx component, and (c) the AE index. From this figure it is possible to observe HIDLCAA features in the IMF Bz and AE index data, where Alfvénic fluctuations can be observed in the IMF Bz data. In the AE time series, the values of the index can reach peaks higher than 1000 nT. The blue line in the AE index panel corresponds to the 200 nT value, where we can note that the AE index is lower than this value only for short intervals of time. The interval on which the Cluster crossed the magnetotail during the event is marked with the red rectangle on the panels.

The intervals of the Cluster crossing the magnetotail during HILDCAAs selected for our study were identified by analyzing the orbit of the spacecraft during the HILDCAA events, where these nine events were obtained. The Cluster orbit in the XZ plane for the HILDCAA plotted in Fig. 1 is shown in Fig. 2, where the Cluster magnetotail crossing is marked in pink on the orbit panel.

Non-stationary time series can be analyzed by wavelet transform (WT), a powerful mathematical tool that is able to show the temporal variability of the power spectral density (Morettin, 1992). Wavelet functions ψt are generated by a simple function called wavelet mother, shown in Equation 1, which suffers expansion ψt→ψ2t and translations ψt→ψt+1 in time, giving rise to those wavelet functions well known as wavelet daughters (Torrence and Compo, 1998). 1ψa,bt=1aψt-ba, where a represents the scale associated to the dilation and contraction of the wavelet, and b is the temporal location, which relates to the translation in time.

The WT applied on f(t) time series is defined as follows: 2TWa,b=∫fxψa,b*tdt, where f(x) is the time series, ψa,bt is the wavelet function and ψa,b∗t represents the complex conjugate thereof.

The wavelet function can be characterized by using two types of functions, the continuous and discrete ones (Daubechies, 1992). The discrete wavelets are used for decomposition of time series in frequency, which is useful for the filtering process. Among the most known functions are the Meyer, Daubechies and Haar functions (Grinsted et al., 2004). The continuous wavelet functions allow the separation of phase and amplitude components associated with the signal; therefore they are generally used for analysis of time series. The most common continuous wavelet functions are the Mexican hat and the Morlet functions (Torrence and Compo, 1998; Addison, 2018). In this paper, both a discrete and a complex wavelet function were used, the Haar function in order to remove long-term trends in the data and the Morlet function for the periodicity identification.

As Cluster spacecraft travels through large spatial and latitudinal ranges, where the Earth's magnetic field has strong dependence, the magnetic field sampled by Cluster data has large spatial variations. These are noted in time as a long-term trend, as can be observed in Fig. 1c, in the geomagnetic Bx component. These long-term effects observed in the geomagnetic field Bx component data can affect the results of the periodicities identified by wavelet analysis. In order to remove it, a technique of long-term trend extraction using the Haar wavelet function was applied in the data. The Haar wavelet function is the simplest type of wavelet and has been used since 1910 (Haar, 1910; Porwik and Lisowska, 2004). The Haar function is defined as a complete orthogonal system of functions which have a dyadic dilatation a=2j and with translations in discrete steps (b=2-jk) (Porwik and Lisowska, 2004; Bolzan et al., 2009; work in progress). The Haar wavelet function is shown in Eq. (3): 3Ψj,kt=2j2,2-jk≤t<2-jk+12,-2j2,2-jk+12≤t<2-jk+1,0,otherwise, where j and k are integers.

The detrending method consists of applying a Haar wavelet transform function in the data, which decomposes them in dyadic scales, after identifying the scales where the long-term periodicities are present; these scales are removed from the data, in such a way that the main characteristics of the time series are maintained (Bolzan et al., 2019). Figure 3 shows an example where this technique was applied in the geomagnetic field Bx component in the magnetotail during a HILDCAA event that occurred from 22:19 UT on 5 August to 22:54 UT on 7 August 2005 (event 6, Table 1). Panel (a) shows the original time series where the long trend can be observed, and (b) shows the time series after filtering the scales and removing the long-term trend.

After detrending the long-term trend in the data, the Morlet wavelet was applied with the aim of identifying the main periodicities present. The Morlet wavelet is a plane wave modulated by a Gaussian envelope and can be described by Eq. (4) (Torrence and Compo, 1998): 4ψt=eiξ0te-t22, where ξ0 represents the dimensionless frequency. In this work this value of parameter was set up as 6.0, since its shape gives good localization in time (Torrence and Compo, 1998).

The global wavelet spectrum (GWS) shows the integrated energy in time for each frequency. Consequently, the GWS is useful for the identification of the most energetic frequencies and periods in a time series. The GWS is given as follows: 5GWS=∫TWa,b2db. This study also aims to identify the periods in which the energy transfer from the solar wind to the magnetotail and auroral zone is more efficient during HILDCAA events. In order to obtain that, the cross-wavelet transform (XWT) was used. The XWT (Eq. 6) is constructed from two continuous WT of two time series, which allows analysis of the correlation between the time series as a function of the signal period and its temporal evolution identifying their power in common (Grinsted et al., 2004; Bolzan et al., 2012). 6Wyxa,b=Wxa,bWya,b*, where Wx and Wy represent the WT applied to the time series x(t) and y(t), and (*) represents the complex conjugate of the WT.

In order to study the main periods with higher correlation between two time series the global correlation spectrum (GCS) is computed. The GCS can be obtained by rewriting Eq. (5) as follows: 7GCS=σxσyσx+σy∫Wxya,b2db, where σx and σy represent the variances of the time series x(t) and y(t), respectively.

After removing the long-term trend of the Cluster Bx data, the WT was applied on the detrended Bx series in order to identify the main periodicities in the geomagnetic Bx component in the magnetotail. The XWT was also computed for the identification of the periods with higher correlation between the Cluster Bx component and IMF Bz component and also between the Cluster Bx component and AE index.

The main goal of this paper was to identify the main periods of HILDCAA events in the magnetotail, as well as to find the major periods of energy transfer from solar wind to the magnetotail and also from the magnetotail to the auroral region. The main results obtained here were as follows

The most energetic periods were observed for T≤4 h in the Bx component in the magnetotail during HILDCAAS. This result is coincident with cyclic substorm periods observed by Korth et al. (2006).

The characteristic periods of energy transfer from solar wind to the magnetotail observed in the XWT were also observed mainly for ≤4 h, in 62.9 % of the identified periods. Echer et al. (2017) also found similar periods in the study of periods of HSS-storm- and substorm-driven events in the magnetotail.

The energy transfer process between magnetotail and auroral region during HILDCAA events has also been shown to be more efficient in periods between 2 and 4 h, with 45 % of the periods identified using XWT analysis. The correlation between the Bx geomagnetic field component and the AE index is higher for the local type of distribution, which means that the energy transfer should preferentially occur in localized regions in the magnetotail or due to a short, burst-like event in the tail.

Therefore, this work presented some characteristics of HILDCAA events in the magnetotail, which were shown to be consistent with results found by other authors studying substorm- and HSS-storm-driven events, which are related to HILDCAA occurrence. Thus the magnetotail responds stronger to IMF fluctuations during HILDCAAS at 2–4 h scales, which are typical substorm periods. These results indicate that energy transfer from solar wind to magnetosphere during HILDCAA events occurs preferentially through intermittent magnetic reconnection between the fluctuating IMF Bz field and Earth's magnetopause diurnal fields, at timescales of 2–4 h. This scale is most likely the high-speed stream Alfvén wave main oscillation periods that affect Earth's magnetosphere.