  We study the damping of the gravitational radiation-driven f-mode instability
in rotating neutron stars by nonlinear bulk viscosity in the so-called
supra-thermal regime. In this regime the dissipative action of bulk viscosity
is known to be enhanced as a result of nonlinear contributions with respect to
the oscillation amplitude. Our analysis of the f-mode instability is based on a
time-domain code that evolves linear perturbations of rapidly rotating
polytropic neutron star models. The extracted mode frequency and eigenfunctions
are subsequently used in standard energy integrals for the gravitational wave
growth and viscous damping. We find that nonlinear bulk viscosity has a
moderate impact on the size of the f-mode instability window, becoming an
important factor and saturating the mode's growth at a relatively large
oscillation amplitude. We show similarly that nonlinear bulk viscosity leads to
a rather high saturation amplitude even for the r-mode instability. In
addition, we show that the action of bulk viscosity can be significantly
mitigated by the presence of superfluidity in neutron star matter. Apart from
revising the f-mode's instability window we provide results on the mode's
gravitational wave detectability. Considering an f-mode-unstable neutron star
located in the Virgo cluster and assuming a mode amplitude at the level allowed
by bulk viscosity, we find that the emitted gravitational wave signal could be
detectable by advanced ground-based detectors such as Advanced LIGO/Virgo and
the Einstein Telescope.
