Nonlinear couplings between r-modes of rotating neutron stars

The r-modes of neutron stars can be driven to instability by gravitational radiation. While linear perturbation theory predicts the existence of this instability, linear theory cannot provide any information about the nonlinear development of the instability. The subject of this paper is the weakly nonlinear regime of fluid dynamics. In the weakly nonlinear regime, the nonlinear fluid equations are approximated by an infinite set of oscillators that are coupled together so that terms quadratic in the mode amplitudes are kept in the equations of motion. In this paper, the coupling coefficients between the r-modes are computed. The stellar model assumed is a polytropic model in which a source of buoyancy is included so that the Schwarzschild discriminant is nonzero. The properties of these coupling coefficients and the types of resonances possible are discussed in this paper. It is shown that no exact resonance involving the unstable l = m = 2 r-mode occurs and that only a small number of modes have a dimensionless coupling constant larger than unity. However, an infinite number of resonant mode triplets exist that couple indirectly to the unstable r-mode. All couplings in this paper involve the l > \m\ r-modes that only exist if the star is slowly rotating. This work is complementary to that of Schenk and coworkers in 2002, who consider rapidly rotating stars that are neutral to convection.