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

Abstract

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 $n\ensuremath{\le}30$ 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 ${10}^{6}\text{ }\text{ }\mathrm{K}$ to ${10}^{9}\text{ }\text{ }\mathrm{K}$, is considered. Exponential growth of the $r$ mode is found to cease at small amplitudes $\ensuremath{\approx}{10}^{\ensuremath{-}4}$. 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 ${10}^{\ensuremath{-}4}$ 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.