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 which 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 both large wavenumber, damped by shear viscosity, and small frequency, damped by bulk viscosity. We find the saturation energy is extremely small, at least four orders of magnitude smaller than that found by previous investigators. We show that large saturation energy found in the simulations of Lindblom et al. (2001a,b) 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 (LMXB’s), the strong driving limit is appropriate and the saturation energy is roughly Er−mode/(0.5Mr 2 ∗Ω 2) ≃ 0.1γgr/Ω ≃ 10(νspin/10 Hz)5, where M and r∗ are the stellar mass and radius, γgr is the driving rate by gravitational radiation, Ω is the angular velocity of the star, and νspin 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 >∼10 years, depending sensitively on the instability curve. The spin-down torque exerted on the neutron star by gravitational radiation reaction can still balance the accretion torque, leading to the observed period clustering in LMXB’s. Gravitational waves from r-modes in either young neutron stars or LMXB’s are completely undetectable. Subject headings: stars: neutron — gravitational waves — turbulence — stars: oscillations