Generalised warped disk equations

The manner in which warps in accretion disks evolve depends on the magnitude of the viscosity. For small viscosity $(\alpha <H\mathrm{/}R)$ the warp evolves in a wave-like manner; for large viscosity $H\mathrm{/}R<\alpha \ll 1$ it evolves diffusively. Here $\alpha$ is the viscosity parameter and $H\mathrm{/}R$ the disk aspect ratio. Currently there is no simple set of equations which describes the evolution in both regimes. In this paper we describe a possible solution to this problem and introduce a set of one-dimensional equations that describe the evolution of a warped disk that are applicable in both high and low viscosity regimes for arbitrary tilts, but small warps.