# Modulation of Solar Wind Impact on the Earth's Magnetosphere during the Solar Cycle
## 2 Methodology  
To study the impact of solar activity on \(Dst\), three magnetohydrodynamic (MHD) quantities were investigated, i.e., the kinetic energy \(E_{v}=\int\mathbf{V}^{2}\ d^{3}\mathbf{r}\), the magnetic energy \(E_{b}=\int\mathbf{B}^{2}\ d^{3}\mathbf{r}\), and the total energy \(E_{t}\equiv E_{v}+E_{b}\), where \(\mathbf{B}=\mathbf{b}/\sqrt{4\pi\rho}\) is the magnetic field in Alfven (velocity) units, with \(\rho\) the solar wind mass density, and \(\mathbf{V}\) the solar wind velocity [13; 19]. Such quantities were obtained from in situ measurements with resolution \(\Delta t=92\) s [17; 18]. The measurements were downsampled to \(\Delta t=3600\) s in the last step of the analysis so that comparison with the \(Dst\) index could be performed using data with a similar sampling time. Because of the strongly statistical nature of the analysis and the robustness of the wind database, no pre-processing of the data was necessary. In the first three rows of Figure 1 are reported the temporal evolution for the total magnetic field \(|B|\), the solar wind bulk velocity \(V\), and the solar wind density \(\rho\) for two CME events observed during the period 15 July 2012 06:00:00\(\rightarrow\)17 July 2012 05:00:00 (\(Dst=-139\), left panels), and 6 August 2011 22:00:00\(\rightarrow\)7 August 2011 22:00:00 (\(Dst=-115\), right panels) [20]. The vertical dashed lines represent the CME duration. MHD quantities can be investigated in terms of local contributions in the frequency and time domains, by expanding \(\mathbf{V}\) and \(\mathbf{B}\) through a wavelet transform (WT). The main advantage of using WT compared to Fourier methods is the ability to extract frequency information from a complex time series using temporal windows of variable size \(u\)[21]. WT transforms a signal into a series of coefficients associated to different scales and with discrete energies, offering a powerful method to quantify how energy is spatially and  
Figure 1: From top to bottom: temporal evolution of the total magnetic field \(|B|\), solar wind bulk velocity \(V\), and the solar wind density \(\rho\) for two samples measured by Wind spacecraft: left column 15 July 2012 06:00:00\(\rightarrow\)17 July 2012 05:00:00 \(Dst=-139\), right column 6 August 2011 22:00:00\(\rightarrow\)7 August 2011 22:00:00 \(Dst=-115\). The vertical dashed lines represent the CME duration. In the last three panels, the temporal evolution of the integrated quantities \(E_{v}\), \(E_{b}\), and \(E_{t}=E_{v}+E_{b}\) is reported. temporally distributed on multiple scales [15; 16; 21; 22]. In light of this, the integral relations for the kinetic and magnetic energy can be redefined as:  
\[E_{v}(u,t) = \frac{1}{2}\sum_{i=1}^{3}\bar{V_{i}}(u,t) \tag{1}\] \[E_{b}(u,t) = \frac{1}{2}\sum_{i=1}^{3}\bar{B_{i}}(u,t), \tag{2}\]  
where \(\bar{V_{i}}(u,t)\) and \(\bar{B_{i}}(u,t)\) are the WT of the velocity and magnetic field vector components \(V_{i}\to i\doteq\{v_{x}(t),\ v_{y}(t),\ v_{z}(t)\}\) and \(B_{i}\to i\doteq\{b_{x}(t),\ b_{y}(t),\ b_{z}(t)\}\), \(u\) and \(t\) are the wavelet scale (local time scale) and time defining the domain of the WT, and the sum is extended over the entire field-containing region. The continuous WT of the MHD variables is defined by the convolution of \(V_{i}(t)\) and \(B_{i}(t)\) with an analyzing wavelet function \(\Psi_{u,s}(t)\), based on a scheme where \(\Psi_{u,s}(t)\) is scaled and shifted along the analyzed signal \(F(t)\) (being \(F(t)\equiv E_{v}(t)\) or \(F(t)\equiv E_{b}(t)\)) according to [23]:  
\[W(u,s) = \langle F(t),\Psi_{u,s}(t)\rangle=\int_{-\infty}^{\infty}F(t) \cdot\Psi_{u,s}^{\star}(t)dt \tag{3}\] \[\Psi_{u,s}(t) = \frac{1}{\sqrt{s}}\Psi\left(\frac{t-u}{s}\right), \tag{4}\]  
where \((\star)\) indicates a complex conjugate, and \(\Psi u,s(t)\) is the kernel function stemming from a specific wavelet family \(\Psi(t)\), \(u\) is the scale parameter (stretching), and \(s\) determines the location of the wavelet (shift on the time axis) [21].