The Papaloizou-Pringle Instability

The stability of tori with constant specific angular momentum l has been of interest since the work of Papaloizou and Pringle (1984), who established that these tori are unstable to non-axisymmetric global modes. It has since been demonstrated that although other angular momentum distributions are also unstable, the constant-l tori are the most susceptible to the PPI. Subsequent work, notably Narayan, Goldreich, and Goodman (1987), consolidated the central features of the instability into the following picture. The global unstable modes have a co-rotation radius within the torus; the co-rotation is located in a narrow region where waves cannot propagate; this region separates inner and outer regions where wave propagation is possible; waves can tunnel through the corotation zone and interact with waves in the other region; and the transmitted modes are amplified only if there is a feedback mechanism, usually in the form of a reflecting boundary at the inner and/or outer edge of the torus.

However, linear stability analysis can go only so far, and numerical work is required to probe the non-linear effects that help determine the final amplitude of the global modes, and hence the cumulative effect of the instability on the torus. Hawley (1987) carried out a relativistic numerical study of the evolution of the PPI into the fully non-linear regime in 2D height-integrated disks. Although this paper verified that the azimuthally averaged rotation law ($\Omega \sim r^{-q}$) exhibits the expected effects of angular momentum redistribution ($q_{sat} \sim \sqrt{3}$), it also showed that the distribution of matter was not azimuthally symmetric, but instead took the form of counter-rotating epicyclic vortices, or ``planets'', with m planets emerging from the growth of a mode of order m. The stability of thick tori in general relativity was studied by Blaes and Hawley (1988) who evolved unstable tori in a two-dimensional $(r,\phi)$ Schwarzshild metric. This work led to the full three dimensional simulations in H91. The present work extends these calculations to include the effects of nonzero black hole angular momentum. Although we do not repeat the linear analysis of Blaes and Hawley (1988) with a non-zero Kerr parameter a, we know from previous work that mode growth depends very sensitively on the initial torus parameters. Black hole rotation simply adds additional complexity.

A point of particular interest has been the effect of accretion on the evolution of the PPI. Blaes (1987) computed growth rates for two-dimensional tori in the Schwarzschild metric and found that they went to zero for models with even a small net accretion flow past the location of the marginally-stable orbit. It was argued that the loss of the inner reflecting boundary through the development of an accretion flow suppresses mode growth. This result was subsequently confirmed analytically by Gat and Livio (1992) for a Newtonian potential. In the three-dimensional case, things are not so clear cut. H91 argues that since the accretion flow is confined to the equatorial plane, there is still a reflecting boundary above and below the plane. However, Dwarkadas & Balbus (1996) argued that the stabilizing effect of an accretion flow is due more to the dynamics at the corotation point than to the absence of reflection.

2002-06-05