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The hydrodynamical response of a tilted circumbinary disc: linear theory and non-linear numerical simulations

In this paper we present an analytical and numerical study of the response of a circumbinary disc subject to the tidal-forcing of a binary with a fixed circular orbit. We consider fluid discs with a range of thicknesses and binaries with a range of mass ratios, orbital separations and inclination angles. Our numerical simulations are implemented using a SPH code. For our unperturbed disc models, we write a scaling relation for the shear viscosity and deduce that the disc thickness cannot be varied without affecting the viscosity in these kinds of SPH disc models. It is found that maintainance of an inner cavity owing to the tidal truncation of the disc is effective for non-zero orbital inclinations. Also we show that our model discs may precess approximately like rigid bodies, provided that the disc is able to communicate on a length scale comparable to the inner boundary radius by either sonic or viscous effects, in a sufficiently small fraction of the local precession period. Furthermore, the disc precession period may tend to infinity if the disc outer edge is allowed to become arbitrarily large, the disc suffering only a modest quasi-steady warp near the inner boundary. When the disc response is linear, or weakly non-linear, the precession periods and the forms of warping that we measure yield reasonable quantitative agreement with the analytical expressions that we derive from a linear response calculation. For a stronger disc response the results can agree poorly with our linear analysis, although some qualitative features of the response remain intact. We show that the response of a disc of non-interacting particles is qualitatively different from this. The work presented in this paper is of relevance to a number of astrophysical phenomena of current interest in star and planet formation.