  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 & 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 nonlinear 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
nonlinear outcome of the hydrodynamic instability of warped discs, and its
interaction with others such as the magnetorotational instability.
