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Non-linear interactions among the inertial modes of a rotating fluid can bedescribed by a network of coupled oscillators. We use such a description for anincompressible fluid to study the development of the r-mode instability ofrotating neutron stars. A previous hydrodynamical simulation of the r-modereported the catastrophic decay of large amplitude r-modes. We explain thedynamics and timescale of this decay analytically by means of a single threemode coupling. We argue that at realistic driving and damping rates such largeamplitudes will never actually be reached. By numerically integrating a networkof nearly 5000 coupled modes, we find that the linear growth of the r-modeceases before it reaches an amplitude of around 10^(-4). The lowest parametricinstability thresholds for the r-mode are calculated and it is found that ther-mode becomes unstable to modes with 13<n<15 if modes up to n=30 are included.Using the network of coupled oscillators, integration times of 10^6 rotationalperiods are attainable for realistic values of driving and damping rates.Complicated dynamics of the modal amplitudes are observed. The initialdevelopment is governed by the three mode coupling with the lowest parametricinstability. Subsequently a large number of modes are excited, which greatlydecreases the linear growth rate of the r-mode.Comment: 3 figures 4 pages Submitted to PR

Nonlinear couplings of R-modes: Energy transfer and saturation amplitudes at realistic timescales