  Stellar magnetic activity induces both distortions and Doppler-shifts in the
absorption line profiles of Sun-like stars. Those effects produce apparent
radial velocity (RV) signals which greatly hamper the search for potentially
habitable, Earth-like planets. In this work, we investigate these distortions
in the Sun using cross-correlation functions (CCFs), derived from intensive
monitoring with the high-precision spectrograph HARPS-N. We show that the RV
signal arising from line-shape variations on time-scales associated with the
solar rotation and activity cycle can be robustly extracted from the data,
reducing the RV dispersion by half. Once these have been corrected,
activity-induced Doppler-shifts remain, that are modulated at the solar
rotation period, and that are most effectively modelled in the time domain,
using Gaussian Processes (GPs). Planet signatures are still best retrieved with
multi-dimensonal GPs, when activity is jointly modelled from the raw RVs and
indicators of the line width or of the Ca II H and K emission. After GP
modelling, the residual RVs exhibit a dispersion of 0.6-0.8 m/s, likely to be
dominated by signals induced by super-granulation. Finally, we find that the
statistical properties of the RVs evolve significantly over time, and that this
evolution is primarily driven by sunspots, which control the smoothness of the
signal. Such evolution, which reduces the sensitivity to long-period planet
signatures, is no longer seen in the activity-induced Doppler-shifts, which is
promising for long term RV monitoring surveys such as the Terra Hunting
Experiment or the PLATO follow-up campaign.
