On the equations of warped disc dynamics

The ID evolution equations for warped discs come in two flavours: For very viscous discs, the internal torque vector G is uniquely determined by the local conditions in the disc, and warps tend to damp out rapidly if they are not continuously driven. For very inviscid discs, on the other hand, G becomes a dynamic quantity, and a warp will propagate through the disc as a wave. The equations governing both regimes arc usually treated separately. A unified set of equations was postulated recently by Martin et al., but not yet derived from the underlying physics. The standard method for deriving these equations is based on a perturbation series expansion, which is a powerful, but somewhat abstract technique. A more straightforward method is to employ the warped shearing box framework of Ogilvie & Latter, which so far has not yet been used to derive the equations for the wave-like regime. The goal of this paper is to analyse the warped disc equations in both regimes using the warped shearing box framework, to derive a unified set of equations, valid for small warps, and to discuss how our results can be interpreted in terms of the affine tilted-slab approach of Ogilvie.