The stability of accretion tori – II. Non-linear evolution to discrete planets

Abstract

Hawley has shown through two-dimensional computer simulations that a slender torus in which a linear Papaloizou-Pringle (PP) instability with azimuthal wavenumber m, is excited evolves non-linearly to a configuration with m nearly disconnected 'planets'. We present an analytical fluid equilibrium that we believe represents his numerical planets. The fluid has an ellipsoidal figure and is held together by the Corio lis force associated with the retrograde fluid motion. There is a bifurcation between the torus and planet configurations at precisely the vorticity below which the PP instability switches on. Although the solution is three-dimensional, there is perfect hydrostatic equilibrium and the motion is entirely two-dimensional. We analyse the linear modes of the analytical planet and find that there are numerous instabilities, though they are not as violent as the PP instability in the torus. We also discuss the energy and vorticity of neutral modes, and we argue that when the torus breaks up into planets, neutral modes with negative energy and non-zero vorticity are excited in order to conserve total energy and specific vorticity. We speculate that the fluid in Hawley's simulations may be approaching two-dimensional turbulence.