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

Figure 1

Temperature dependence of the lowest three parametric instability thresholds of the $r$ mode to the inertial modes.

Figure 2

Numerical solutions to the three-mode problem with the lowest (${\gamma }_{\beta },{\gamma }_{\gamma }\to 0$) parametric instability threshold for a variety of temperatures: (a) $T=5\times {10}^6\mathrm{K}$ (Chaotic motion), (b) $T=9\times {10}^6\mathrm{K}$ (two point periodic cycle), (c) $T=1.5\times {10}^7\mathrm{K}$ (approaches one point periodic cycle), and (d) $T=2\times {10}^7\mathrm{K}$ (diverging solution). The actual (${\gamma }_{\beta },{\gamma }_{\gamma }\ne 0$) parametric instability thresholds are indicated with dashed lines.

Figure 3

Evolution of the Hamiltonian energy for temperatures in the range $T=2.1\times {10}^7\mathrm{K}$ to $T=3.7\times {10}^7\mathrm{K}$ in increments of $2\times {10}^6\mathrm{K}$. Plot (a) shows the diverging three-mode system with the lowest parametric instability. Plot (b) shows the system used in (a) with the coupling that yields the second lowest parametric instability threshold added. Plot (c) shows the case with the three lowest parametric instability thresholds included. Dashed horizontal lines indicate the linear energy corresponding to the parametric instability thresholds.

Figure 4

Evolution of the modal amplitudes for the systems discussed in Fig. 3 for the case with $T=3.7\times {10}^7\mathrm{K}$. Dashed lines indicate the parametric instability amplitudes. The $n=3$, $m=2$ $r$ mode is indicated in black, other colors are chosen to match the color coding for the instability thresholds used in Fig. 1.

Figure 5

Evolution of the Hamiltonian system including the 1 306 999 couplings with $\delta w<0.002$. An initial excitation of $1.3\times {10}^{-4}$ is placed in the principal $r$ mode. $n=m+1$ $r$ modes are indicated in black; all other modes are indicated in light gray/cyan on the lower panel. The parametric instability thresholds are indicated with dashed lines. The relative energy change during the simulation is indicated on top.

Figure 6

Approach to equipartition of the Hamiltonian system. The time evolution of the spectral entropy for various initial $r$-mode amplitudes: The dashed line is $c_{\alpha }(0)=5\times {10}^{-4}$ (blue), the solid line is $c_{\alpha }(0)=1.3\times {10}^{-4}$ (cyan), and the dot-dashed line is $c_{\alpha }(0)=0.8\times {10}^{-4}$ (red).

Figure 7

Modal dynamics for a Hamiltonian system with an initial amplitude $c_{\alpha }(0)=0.8\times {10}^{-4}$ (bottom). The $n=m+1$ $r$ modes are indicated in black. The distribution of the maximum modal amplitude obtained with respect to mode number is shown on top.

Figure 8

Top panels: Time evolution of large strongly damped systems for temperatures (a) $T=5\times {10}^6\mathrm{K}$, (b) $T=9\times {10}^6\mathrm{K}$, (c) $T=1.5\times {10}^7\mathrm{K}$, and (d) $T=2\times {10}^7\mathrm{K}$. Bottom panels: Corresponding distribution of averaged modal amplitudes that rise above the initial noise level. In the top panels, the parametric instability thresholds are indicated with dashed lines.

Figure 9

Modal spectra for strongly damped system, $T=9\times {10}^6\mathrm{K}$ (black) and an undamped system, “equipartition solution” (light/cyan). Initial noise level indicated with a dashed line.

Figure 10

Evolution of the ($n=3$, $m=2$) $r$ mode (bottom) and total energy (top) for a $T=5\times {10}^7\mathrm{K}$ system with different detuning criteria. The parametric instability thresholds are indicated by horizontal dashed lines.

Figure 11

Comparison of the integration times achieved, average time-step sizes and number of couplings included for various detuning criteria. All evolutions were run on 16 processors of the NCSA Platinum cluster for a period of 24 hours. The temperature was chosen as $T=5\times {10}^7\mathrm{K}$.

Figure 12

Evolution of the $T=5\times {10}^7\mathrm{K}$ system with $\delta w<0.002$. The parametric instability thresholds are indicated by horizontal dashed lines.

Figure 13

Evolution of the $T=5\times {10}^8\mathrm{K}$ system with $\delta w<0.002$. The parametric instability thresholds are indicated by horizontal dashed lines. Modal amplitudes are indicated below and the total Hamiltonian energy on top.

Figure 14

Evolution of the $T=5\times {10}^7\mathrm{K}$, $e=0.82$ system with $\delta w<0.002$. The parametric instability thresholds are indicated by horizontal dashed lines. Modal amplitudes are indicated below and the total Hamiltonian energy in the top panel.