The oscillation and stability of differentially rotating spherical shells: The initial value problem

Abstract

An understanding of the oscillations of differentially rotating systems is key to many areas of astrophysics. It is of particular relevance to the emission of gravitational waves from oscillating neutron stars, which are expected to possess significant differential rotation immediately after birth or binary merger. In a previous paper we analysed the normal modes of a simple system exhibiting differential rotation. In this complementary paper we address the initial value problem for the same simple model using both analytical methods and numerical time evolutions. We derive a necessary and sufficient condition for dynamical shear instability. We discuss the dynamical behaviour of the continuous spectrum in response to an initial perturbation, and show that certain singular solutions within the continuous spectrum appear physically indistinguishable from the discrete modes outside the continuous spectrum.

14 pages, 4 figures, minor additions, accepted for publication in MNRAS

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• Non-axisymmetric instability and fragmentation of general relativistic quasi-toroidal stars
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• Regularizing the r-mode problem for nonbarotropic relativistic stars
• Latitudinal Shear Instabilities during Type I X-ray Bursts
• Inertial waves near corotation in 3D hydrodynamical disks
• Gravitational Wave Astronomy
• The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regions I. Free waves
• The Kinematic Richness of Star Clusters - II. Stability of Spherical Anisotropic Models with Rotation