## 6 Sea Storm Effect Analysis
### Reflective Effect of Castel Dell'Ovo on Wave Motion  
The presence of Castel dell'Ovo represents an enhancing factor on overall wave motion in front of Via Partenope. The interaction between waves and Castel dell'Ovo basin is comparable to a surging process (Figure 19), occurring on very steep seabed and leading to reflective processes. To estimate the influence of reflected energy, we consider the dynamics occurring in the local coastal setting. For the evaluation of the energy carried by a surface wave, we apply the simplified study for a sine wave (Figure 20) and Equation (16). \[\text{z}\left(\text{x}\right)=\text{a}\sin\left(\frac{2\text{T}\text{I}}{ \lambda}.\text{x}\right). \tag{16}\]  
The potential energy V of the infinitesimal mass of fluid dm = \(\rho\) dxdz increases moving from \(-\text{z}\) to \(+\text{z}\):  
\[\text{dV}=\text{dm}\cdot\text{g}\cdot\left(\text{z}-\left(-\text{z}\right) \right)=2\text{g}\text{g}\text{z}\text{ d}\text{x}\text{d}\text{z}. \tag{17}\]  
The total variation in potential energy of the fluid is equal to:  
\[\text{V}=\int_{\text{x}=0}^{\text{x}=\text{\lambda}/2}\int_{\text{z}=0}^{ \text{z}=\text{a}\,\sin 2\text{\pi}\text{x}/\text{\lambda}}\text{z}\text{ d}\text{z}\text{ d}\text{x}=\frac{1}{4}\rho\text{ g}\,\text{\lambda}\,\text{a}^{2}. \tag{18}\]  
Figure 19: Hydrodynamic motion occurring on the western side of the Castel dell’Ovo. Water masses climb up the western bastion of Castel dell’Ovo and then are reflected back towards the seaside. This means that almost no energy dissipation occurs. Assuming the principle of energy equipartition (the change in average kinetic energy is equal to the change in average potential energy), we may conclude that the total energy of a surface wave per unit of wavefront width and per unit of wavelength \(\lambda\) is equal to:  
\[\lambda=\frac{2\mathrm{V}}{\lambda}=\frac{1}{2}\,\mathrm{\rho g}\mathrm{g}^{2}, \tag{19}\]  
where J\(\lambda\) is an energy flux and is measured in Jm-2. The reflection processes on the western bastion of the Castel dell'Ovo involve an energy transfer from Supplementary S2 (L2 = 180 m wide) to Supplementary S1 (L1 = 300 m wide) (Figure 18b). The reflection processes are not perfect:  
\[\mathrm{E}_{\mathrm{tot}}=\mathrm{E}_{1}+\mathrm{k}\mathrm{E}_{2}, \tag{20}\]  
where 0 \(<\) k \(<\) 1. We calculate the energy flux relative to Supplementary S1 ed Supplementary S2, and then, under the hypothesis that the amount of reflected energy is equal to 50% of total energy impacting on Castel dell'Ovo, we evaluate the influence in terms of surplus of energy and greater wave height. The calculation indicates a surplus of about 25% in energy flux. Waves also show an increase in height of about 10-15%. It is possible to evaluate the maximum absolute wave height, add to the lifted sea level, and estimate the relation between coastal infrastructures and wave motion:  
\[\mathrm{H}_{2}=\mathrm{H}_{1}/2+0.125\;\mathrm{H}_{1}/2=2.80+0.125\cdot 2.80=3.15 \;\mathrm{m}, \tag{21}\]  
\[\mathrm{R}_{\mathrm{max,abs}}=3.15+1.08=4.23\;\mathrm{m}. \tag{22}\]  
Based on the analysis of the storm wave interaction with Via Partenope wall vestment, this wave crest height (\(\mathrm{R}_{\mathrm{max,abs}}\) = +4.23 m a.s.l.) will be considered to characterize the exceptional nature of the sea storm event. The maximum run up (\(\mathrm{R}_{\mathrm{max}}\)) triggered by extreme swells (T = 100 years) in front of Via Caracciolo and Via Partenope was evaluated as \(\mathrm{R}_{100}\) = 3.95 m a.s.l. by the Autorita di Bacino nord-occidentale della Campania [43]. The comparison between \(\mathrm{R}_{\mathrm{max}}\) and \(\mathrm{R}_{100}\) indicates that the event occurred on 28 December 2020 is larger than the one statistically generated for a return period of 100 years. A fundamental role was played by Castel dell'Ovo in determining a reflection of the wave energy that has resulted in a surplus wave height of 0.35 m which, in turn, caused the recent event to exceed the one evaluated for T = 100 years.