Local and global dynamics of warped astrophysical discs

Astrophysical discs are warped whenever a misalignment is present in the system, or when a flat disc is made unstable by external forces. The evolution of the shape and mass distribution of a warped disc is driven not only by external influences but also by an internal torque, which transports angular momentum through the disc. This torque depends on internal flows driven by the oscillating pressure gradient associated with the warp, and on physical processes operating on smaller scales, which may include instability and turbulence. We introduce a local model for the detailed study of warped discs. Starting from the shearing sheet of Goldreich and Lynden-Bell, we impose the oscillating geometry of the orbital plane by means of a coordinate transformation. This warped shearing sheet (or box) is suitable for analytical and computational treatments of fluid dynamics, magnetohydrodynamics, etc., and it can be used to compute the internal torque that drives the large-scale evolution of the disc. The simplest hydrodynamic states in the local model are horizontally uniform laminar flows that oscillate at the orbital frequency. These correspond to the non-linear solutions for warped discs found in previous work by Ogilvie, and we present an alternative derivation and generalization of that theory. In a companion paper, we show that these laminar flows are often linearly unstable, especially if the disc is nearly Keplerian and of low viscosity. The local model can be used in future work to determine the non-linear outcome of the hydrodynamic instability of warped discs, and its interaction with others such as the magnetorotational instability.