Temporal evolution and correlations of optical activity indicators measured in Sun-as-a-star observations

(Abridged) We perform a detailed study of the main optical activity indicators (Ca II H & K, Balmer lines, Na I D$_{\rm 1}$ D$_{\rm 2}$, and He I D$_{\rm 3}$) measured for the Sun using the data provided by the HARPS-N solar-telescope feed at the Telescopio Nazionale Galileo. The value of the solar rotation period is found in all the activity indicators, with the only exception being H$\delta$. The derived values vary from 26.29 days (H$\gamma$ line) to 31.23 days (He I). From an analysis of sliding periodograms we find that in most of the activity indicators the spectral power is split into several "bands" of periods around 26 and 30 days, that might be explained by the migration of active regions between the equator and a latitude of $\sim$ 30$^{\circ}$, spot evolution or a combination of both effects. In agreement with previous works a typical lifetime of active regions of $\sim$ ten rotation periods is inferred from the pooled variance diagrams. We find that H$\alpha$, H$\beta$, H$\gamma$, H$\epsilon$, and He I show a significant correlation with the S index. Significant correlations between the contrast, bisector span, and the heliocentric radial velocity with the activity indexes are also found. We show that the full width at half maximum, the bisector, and the disc-integrated magnetic field correlate with the radial velocity variations. The correlation of the S index and H$\alpha$ changes with time, increasing with larger sun spot numbers and solar irradiance. A similar tendency with the S index - radial velocity correlation is also present in the data. Our results are consistent with a scenario in which higher activity favours the correlation between the S index and the H$\alpha$ activity indicators and between the S index and radial velocity variations.