Abstract
ABSTRACT Understanding oscillations and waves in astrophysical fluid bodies helps to elucidate their observed variability and the underlying physical mechanisms. Indeed, global oscillations and bending modes of accretion discs or tori may be relevant to quasi-periodicity and warped structures around compact objects. While most studies rely on linear theory, observationally significant, non-linear dynamics is still poorly understood, especially in Keplerian discs for which resonances typically demand a separate treatment. In this work, we introduce a novel analytical model which exactly solves the ideal, compressible fluid equations for a non-self-gravitating elliptical cylinder within a local shearing sheet. The aspect ratio of the ring is an adjustable parameter, allowing a continuum of models ranging from a torus of circular cross-section to a thin ring. We restrict attention to flow fields which are a linear function of the coordinates, capturing the lowest order global motions and reducing the dynamics to a set of coupled ordinary differential equations (ODEs). This system acts as a framework for exploring a rich range of hydrodynamic phenomena in both the large amplitude and Keplerian regimes. We demonstrate the connection between tilting tori and warped discs within this model, showing that the linear modes of the ring correspond to oppositely precessing global bending modes. These are further confirmed within a numerical grid based simulation. Crucially, the ODE system developed here allows for a more tractable investigation of non-linear dynamics. This will be demonstrated in a subsequent paper which evidences mode coupling between warping and vertical motions in thin tilted rings.
Highlights
1.1 Astrophysical motivationDiscs appear ubiquitously in the zoo of astrophysical phenomena and occur in a range of systems
In this paper we have presented a novel analytical framework for exploring a range of hydrodynamic phenomenon, with particular application to torus oscillations and warped disc theory
The tilting ring within the local model can be interpreted as a precessing inclined disc as seen from a global view, with both prograde and retrograde motion depending on whether the tilt and shear are in phase or anti-phased
Summary
Discs appear ubiquitously in the zoo of astrophysical phenomena and occur in a range of systems. The Hubble Space Telescope images of TW Hya have captured a shadow moving around the outer disc regions at a rate of 22.7◦yr−1 – faster than any feature possibly advected with the flow This suggests an inner tilted precessing disc is blocking light from the central T Tauri star (Debes et al 2017). ALMA detection of CO molecular line emission allows for a kinematic study of the gas flow which is effectively modelled with a warped inner structure (Rosenfeld et al 2012) These warps inferred from shadows are even more exaggerated in transition discs where large radial gaps divide the inner and outer regions and they present extreme inclination differences. The resonances between the vertical and radial motions make this regime trickier to understand but all the more important for the dynamical evolution
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