Saturation of the r-mode instability

Rossby waves (r-modes) in rapidly rotating neutron stars are unstable because of the emission of gravitational radiation. As a result, the stellar rotational energy is converted into both gravitational waves and r-mode energy. The saturation level for the r-mode energy is a fundamental parameter needed to determine how fast the neutron star spins down, as well as whether gravitational waves will be detectable. In this paper we study saturation by nonlinear transfer of energy to the sea of stellar "inertial'' oscillation modes that arise in rotating stars with negligible buoyancy and elastic restoring forces. We present detailed calculations of stellar inertial modes in the WKB limit, their linear damping by bulk and shear viscosity, and the nonlinear coupling forces among these modes. The saturation amplitude is derived in the extreme limits of strong or weak driving by radiation reaction, as compared to the damping rate of low-order inertial modes. In the weak driving case, energy can be stably transferred to a small number of modes, which damp the energy as heat or neutrinos. In the strong driving case, we show that a turbulent cascade develops, with a constant flux of energy to large wavenumbers and small frequencies where it is damped by shear viscosity. We find that the saturation energy is extremely small, at least 4 orders of magnitude smaller than that found by previous investigators. We show that the large saturation energy found in the simulations of Lindblom and coworkers is an artifact of their unphysically large radiation reaction force. In most physical situations of interest, for either nascent, rapidly rotating neutron stars or neutron stars being spun up by accretion in low-mass X-ray binaries (LMXBs), the strong driving limit is appropriate and the saturation energy is roughly Er-mode/(0.5 Mr*(2)Omega(2)) similar or equal to 0.1 gamma(gr)/Omega similar or equal to 10(-6) (nu(spin)/10(3) Hz)(5), where M and r* are the stellar mass and radius, respectively, gamma(gr) is the driving rate by gravitational radiation, Omega is the angular velocity of the star, and nu(spin) is the spin frequency. At such a low saturation amplitude, the characteristic time for the star to exit the region of r-mode instability is greater than or similar to 10(3) - 10(4) yr, depending sensitively on the instability curve. Although our saturation amplitude is smaller than that found by previous investigators, it is still sufficiently large to explain the observed period clustering in LMXBs. We find that the r-mode signal from both newly born neutron stars and LMXBs in the spin-down phase of Levin's limit cycle will be detectable by enhanced LIGO detectors out to similar to 100 - 200 kpc.