Nonlinear coupling network to simulate the development of the r mode instability in neutron stars. II. Dynamics

Two mechanisms for nonlinear mode saturation of the r mode in neutron stars have been suggested: the parametric instability mechanism involving a small number of modes and the formation of a nearly continuous Kolmogorov-type cascade. Using a network of oscillators constructed from the eigenmodes of a perfect fluid incompressible star, we investigate the transition between the two regimes numerically. Our network includes the 4995 inertial modes up to n ≤ 30 with 146 998 direct couplings to the r mode and 1 306 999 couplings with detuning <0.002 (out of a total of approximately 10^9 possible couplings). The lowest parametric instability thresholds for a range of temperatures are calculated and it is found that the r mode becomes unstable to modes with 13 < n <15. In the undriven, undamped, Hamiltonian version of the network the rate to achieve equipartition is found to be amplitude dependent, reminiscent of the Fermi-Pasta-Ulam problem. More realistic models driven unstable by gravitational radiation and damped by shear viscosity are explored next. A range of damping rates, corresponding to temperatures 10^6  K to 10^9  K, is considered. Exponential growth of the r mode is found to cease at small amplitudes ≈10^(−4). For strongly damped, low temperature models, a few modes dominate the dynamics. The behavior of the r mode is complicated, but its amplitude is still no larger than about 10^(−4) on average. For high temperature, weakly damped models the r mode feeds energy into a sea of oscillators that achieve approximate equipartition. In this case the r-mode amplitude settles to a value for which the rate to achieve equipartition is approximately the linear instability growth rate.