Hydrodynamics of warps in the local model of astrophysical discs

We show how the local approximation of astrophysical discs, which is the basis for the well-known model of the shearing box, can be used to study many aspects of the dynamics of warped discs. In the local model, inclination of the orbit of a test particle, with respect to the reference orbit, corresponds to a vertical oscillation of the particle at the orbital frequency. Warping of a disc corresponds to a locally axisymmetric corrugation of the mid-plane of the disc that oscillates vertically at the orbital frequency, while evolution of the warp corresponds to a modulation of the complex amplitude of the vertical oscillation. We derive a conservation law for this amplitude that is the local equivalent of the conservation of angular momentum and therefore governs the evolution of the warp. For length scales that are long compared to the vertical scale height of the disc, the known non-resonant and resonant regimes of warp dynamics, including the diffusive and wave-like regimes of Keplerian discs, occur in the local model in the same way as in a global view of warped discs.