  We study the dynamical evolution of the gravitational-wave driven instability
of the f-mode in rapidly rotating relativistic stars. With an approach based on
linear perturbation theory we describe the evolution of the mode amplitude and
follow the trajectory of a newborn neutron star through its instability window.
The influence on the f-mode instability of the magnetic field and the presence
of an unstable r-mode is also considered. Two different configurations are
studied in more detail; an N = 1 polytrope with a typical mass and radius and a
more massive polytropic N = 0.62 model with gravitational mass M = 1.98 Msun.
We study several evolutions with different initial rotation rates and
temperature and determine the gravitational waves radiated during the
instability. In more massive models, an unstable f-mode with a saturation
energy of about 1e-6 Msun c^2 may generate a gravitational-wave signal which
can be detected by the Advanced LIGO/Virgo detector from the Virgo cluster. The
magnetic field affects the evolution and then the detectability of the
gravitational radiation when its strength is higher than 1e12 G, while the
effects of an unstable r-mode become dominant when this mode reaches the
maximum saturation value allowed by non-linear mode couplings. However, the
relative saturation amplitude of the f- and r-modes must be known more
accurately in order to provide a definitive answer to this issue. From the
thermal evolution we find also that the heat generated by shear viscosity
during the saturation phase completely balances the neutrinos' cooling and
prevents the star from entering the regime of mutual friction. The evolution
time of the instability is therefore longer and the star loses significantly
larger amounts of angular momentum via gravitational waves.
