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Understanding oscillations and waves in astrophysical fluid bodies helps toelucidate their observed variability and the underlying physical mechanisms.Indeed, global oscillations and bending modes of accretion discs or tori may berelevant to quasi-periodicity and warped structures around compact objects.While most studies rely on linear theory, observationally significant,nonlinear dynamics is still poorly understood, especially in Keplerian discsfor which resonances typically demand a separate treatment. In this work weintroduce a novel analytical model which exactly solves the ideal, compressiblefluid equations for a non-self-gravitating elliptical cylinder within a localshearing sheet. The aspect ratio of the ring is an adjustable parameter,allowing a continuum of models ranging from a torus of circular cross-sectionto a thin ring. We restrict attention to flow fields which are a linearfunction of the coordinates, capturing the lowest order global motions andreducing the dynamics to a set of coupled ordinary differential equations(ODEs). This system acts as a framework for exploring a rich range ofhydrodynamic phenomena in both the large amplitude and Keplerian regimes. Wedemonstrate the connection between tilting tori and warped discs within thismodel, showing that the linear modes of the ring correspond to oppositelyprecessing global bending modes. These are further confirmed within a numericalgrid based simulation. Crucially, the ODE system developed here allows for amore tractable investigation of nonlinear dynamics. This will be demonstratedin a subsequent paper which evidences mode coupling between warping andvertical motions in thin tilted rings.

Nonlinear dynamics of hydrodynamic tori as a model of oscillations and bending waves in astrophysical discs