The time-dependence of non-planar accretion discs

Equations are derived which describe the time-dependence of a viscous, tilted accretion disc subject to external torques. We use a full hydrodynamical treatment and linearize the equations in terms of a small tilt. We show that previous attempts at deriving these equations are inadequate due to the inconsistency of some of the assumptions made. The behaviour of a tilted viscous disc is not simple to describe. Viscous forces do tend to flatten the disc, but they also produce torques within the disc which induce local precession of disc elements. The main complication is that the horizontal motions induced in the disc by the tilt are resonantly driven so that their magnitude and phase are governed by the viscosity. This leads to the disc being smoothed on a time-scale faster than the usual viscous one. In order to obtain a simple diffusion equation governing the tilt angle β, it is necessary in a Keplerian disc that the viscosity parameter α be greater than the opening angle of the disc (H/R). If this condition is not satisfied one must give consideration to the proper dynamical modes of the disc, and the simple approach to the problem adopted here and in previous work is inadequate.