Numerical simulations of global three-dimensional (3D), self-gravitatingdiscs with a gap opened by an embedded planet are presented. The simulationsare customised to examine planetary gap stability. Previous results, obtainedby Lin & Papaloizou from two-dimensional (2D) disc models, are reproduced in3D. These include (i) the development of vortices associated with localvortensity minima at gap edges and their merging on dynamical timescales inweakly self-gravitating discs, (ii) the increased number of vortices as thestrength of self-gravity is increased and their resisted merging, and (iii)suppression of the vortex instability and development of global spiral armsassociated with local vortensity maxima in massive discs. The verticalstructure of these disturbances are examined. In terms of the relative densityperturbation, the vortex disturbance has weak vertical dependence whenself-gravity is neglected. Vortices become more vertically stratified withincreasing self-gravity. This effect is seen even when the unperturbed regionaround the planet's orbital radius has a Toomre stability parameter ~10. Thespiral modes display significant vertical structure at the gap edge, with themidplane density enhancement being several times larger than that near theupper disc boundary. However, for both instabilities the vertical Mach numberis typically a few per cent,and on average vertical motions near the gap edgedo not dominate horizontal motions.
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