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Two mechanisms for nonlinear mode saturation of the r-mode in neutron starshave been suggested: the parametric instability mechanism involving a smallnumber of modes and the formation of a nearly continuous Kolmogorov-typecascade. Using a network of oscillators constructed from the eigenmodes of aperfect fluid incompressible star, we investigate the transition between thetwo 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 withdetuning< 0.002 (out of a total of approximately 10^9 possible couplings).  The lowest parametric instability thresholds for a range of temperatures arecalculated and it is found that the r-mode becomes unstable to modes with13<n<15. In the undriven, undamped, Hamiltonian version of the network the rateto achieve equipartition is found to be amplitude dependent, reminiscent of theFermi-Pasta-Ulam problem. More realistic models driven unstable bygravitational radiation and damped by shear viscosity are explored next. Arange of damping rates, corresponding to temperatures 10^6K to 10^9K, isconsidered. Exponential growth of the r-mode is found to cease at smallamplitudes, approximately 10^-4. For strongly damped, low temperature models, afew modes dominate the dynamics. The behavior of the r-mode is complicated, butits amplitude is still no larger than about 10^-4 on average. For hightemperature, weakly damped models the r-mode feeds energy into a sea ofoscillators that achieve approximate equipartition. In this case the r-modeamplitude settles to a value for which the rate to achieve equipartition isapproximately the linear instability growth rate.Comment: 18 Pages 14 Figure

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Nonlinear coupling network to simulate the development of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>r</mml:mi></mml:math>mode instability in neutron stars. II. Dynamics

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