# Modulation of Solar Wind Impact on the Earth's Magnetosphere during the Solar Cycle
## 2 Methodology
The appropriate choice of \(\Psi(t)\), characterized by its shape, is important for specific applications. For turbulence analysis, several well-localized mother wavelets, such as Morlet [24; 25; 26], or Haar [27], are typically used, while, for phase analysis, Hermitian [28] wavelets can be selected. In this investigation, the Paul-wavelet transform was selected as a mother wavelet (\(\Psi_{n}(t)\in\mathbb{C}\), and \(n=4\) is the order), since they are better resolved in the time domain. This family behaves like a Ricker wavelet at low order with respect to its real part, while it approaches a Morlet wavelet for high orders [21]. Moreover, the peak frequency in the spectral domain just equals the wavelet order \(n\)[29]. Turbulence causes kinetic and magnetic energy, mostly contained in the large scales, to cascade via non-linear interactions to smaller scales, where they are eventually dissipated [19; 30]. Such processes generate strongly irregular field fluctuations that may mask the fine structure of the temporal evolution of \(E(t)\). In light of this, to enhance the contribution of the small- or meso-scale structures that superpose to the SW turbulent background, the total energy spectrum \(E_{t}(u,t)\equiv E_{v}(u,t)+E_{b}(u,t)\) was compensated with the background spectrum, defined through the global wavelet spectrum [21] as \(E_{0}(u)=\int E(u,t)dt\), at each scale \(u\). An example of the non-dimensional energies \(E_{v}(t)\) and \(E_{b}(t)\) is reported in the last three panels of Figure 1 for the CME recorded during the period 15 July 2012 06:00:00\(\rightarrow\)17 July 2012 05:00:00 (left panels), and 6 August 2011 22:00:00\(\rightarrow\)7 August 2011 22:00:00 (right panels). The shape and dynamical properties of the magnetosphere are strongly regulated by the SW through a direct exchange of energy, mass and momentum [31]. Geomagnetic storms are caused by magnetic reconnection between southward directed IMF and the oppositely directed geomagnetic field at the low-latitude magnetopause region of the dayside magnetosphere. The solar wind magnetic flux is then transferred into the magnetosphere, inflating the magnetotail and storing the energy there, until the pressure build-up cannot be maintained and reconnection on the night-side releases the stored energy, transferring and depleting the magnetic flux back to the day-side. Additionally, a finite time is required for the magnetospheric perturbation to propagate to the magnetosphere-ionosphere current system. Therefore, the two datasets should be synchronized, considering the different specific regions of space where the measurements were collected. Moreover, only a finite subset of wavelet scales \(u\in[u_{\text{min}},u_{\text{max}}]\) will be taken into account, for both maximum and minimum solar activity periods. This range was accurately selected for reconstructing the total dimensionless energy spectrum:  
\[E(t)=\int_{u_{\text{min}}}^{u_{\text{max}}}E(u,t)[E_{0}(u)]^{-1}du. \tag{5}\]  
Finally, it should be pointed out that the downsampling of the data was performed for the energy spectrum \(E(t)\). The lower limit of the integral Equation (5) is simply dictated by the resolution of the \(Dst\) index and was set to \(t_{\text{min}}=1\) h. On the other hand, the upper limit is extrapolated from the cross-correlation (CC) of the two variables (\(\mathcal{C}_{E,Dst}\)) as a function of the temporal lag \(\ell\), defined:  
\[\mathcal{C}(\ell)=\frac{\sum_{i=1}^{N-\ell}(E_{i}-\langle E\rangle))(Dst_{i+ \ell}-\langle Dst\rangle)}{\sqrt{\sigma_{E}}\sqrt{\sigma_{Dst}}}, \tag{6}\]  
where \(\langle E\rangle\), and \(\langle Dst\rangle\) are the means of the two populations, \(\sigma_{E}\) and \(\sigma_{Dst}\) are their associated variances, \(\ell\) is the temporal lag and \(N\) the total number of observations. In Figure 2 the CC is shown for various lag times ranging from \(\ell=1\) h to \(\ell=1000\) days, for both the solar minimum (blue symbols) and the solar maximum (red symbols) periods.