Nonlinearr-modes in neutron stars: Instability of an unstable mode

Abstract

We study the dynamical evolution of a large amplitude r-mode by numerical simulations. R-modes in neutron stars are unstable growing modes, driven by gravitational radiation reaction. In these simulations, r-modes of amplitude unity or above are destroyed by a catastrophic decay: A large amplitude r-mode gradually leaks energy into other fluid modes, which in turn act nonlinearly with the r-mode, leading to the onset of the rapid decay. As a result the r-mode suddenly breaks down into a differentially rotating configuration. The catastrophic decay does not appear to be related to shock waves at the star's surface. The limit it imposes on the r-mode amplitude is significantly smaller than that suggested by previous fully nonlinear numerical simulations.

r-mode instability: Analytical solution with gravitational radiation reaction

“…[19,20], where an approximate analytical evaluation of nonlinear effects was performed. Differential rotation drifts induced by r-modes were also found in numerical simulations of nonlinear r-modes carried out both in general relativistic hydrodynamics [21], and in Newtonian hydrodynamics [22][23][24]. Differential rotation was also reported in a model of a thin spherical shell of a rotating incompressible fluid [25].…”

“…To compare our expressions with those of Refs. [22][23][24] we establish the connection between the two approaches in this Appendix.…”

Differential rotation and r-modes in magnetized neutron stars

“…As a result the theory in third order can be much more complicated than the one in second order described here. Furthermore, for very large amplitudes α 3, which can be studied only numerically, r-modes decay nonlinearly and strong differential rotation develops (Gressman et al 2002;Kastaun 2011). Such a strong differential rotation (angular velocities in the range 0.5 .…”

Nonlinear coupling network to simulate the development of the r-mode instability in neutron stars. I. Construction

“…There have been three attempts at direct hydrodynamical calculations [6] [7] [8]. These simulations (1) assume that the background star has uniform initial rotation, (2) use approximate linear eigenfunctions as initial perturbations, and (3) assume either a large initial amplitude, or else artificially increase the radiation reaction force enough to ensure that a large amplitude is reached relatively quickly.…”

“…Gressman et al [8] performed a Newtonian hydrodynamical simulation for non-viscous fluid flow very similar to [7] using a high resolution shock capturing scheme to accurately integrate their equations. They began by amplifying an initially weak (l = m = 2) r-mode at 9 × 10 7 times the radiation reaction rate, to amplitudes of order unity, at which point they ceased driving the mode and observed its decay.…”