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 modes 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-modes instability window we provide results on the modes 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.
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