Nonlinear Couplings of R-modes: Energy Transfer and Saturation Amplitudes at Realistic Timescales

TL;DR

This paper models the nonlinear interactions of inertial modes in rotating neutron stars to understand r-mode instability saturation, showing that large amplitudes are unlikely and that mode coupling leads to complex amplitude dynamics.

Contribution

It introduces a network of nearly 5000 coupled oscillators to analyze r-mode instability development and saturation in neutron stars, providing new insights into amplitude limits and mode interactions.

Findings

R-mode amplitude saturates around 10^(-4) due to mode coupling.

Large amplitude r-modes are unlikely at realistic damping rates.

Complex modal dynamics emerge from the coupled oscillator network.

Abstract

Non-linear interactions among the inertial modes of a rotating fluid can be described by a network of coupled oscillators. We use such a description for an incompressible fluid to study the development of the r-mode instability of rotating neutron stars. A previous hydrodynamical simulation of the r-mode reported the catastrophic decay of large amplitude r-modes. We explain the dynamics and timescale of this decay analytically by means of a single three mode coupling. We argue that at realistic driving and damping rates such large amplitudes will never actually be reached. By numerically integrating a network of nearly 5000 coupled modes, we find that the linear growth of the r-mode ceases before it reaches an amplitude of around 10^(-4). The lowest parametric instability thresholds for the r-mode are calculated and it is found that the r-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 rotational periods are attainable for realistic values of driving and damping rates. Complicated dynamics of the modal amplitudes are observed. The initial development is governed by the three mode coupling with the lowest parametric instability. Subsequently a large number of modes are excited, which greatly decreases the linear growth rate of the r-mode.