Previous analytic and numerical work has established that the rate of growth of unstable modes decreases as the width of the torus increases. Our results for models A3p, B3p, and B3r show that this remains the case for tori orbiting Kerr black holes.
As in H91, we see a decrease in the density enhancement with torus width in the bound tori. However, only models A3p and B3p are bound tori, with l > lmb, (model B3r is sub-marginal, and the other three models are marginal). Furthermore, as was noted in H91, the detailed behaviour of mode growth depends sensitively on the choice of initial model, and mode growth dictates the redistribution of matter and hence the magnitude of the density enhancements.
Two-dimensional studies have shown that accretion suppresses mode growth due to the loss of the inner reflecting boundary, which is a prerequisite for mode amplification. This conclusion was supported by numerical and analytic work in previous papers, as discussed in section 3.1. However, the interplay between accretion and mode growth in three-dimensional simulations is less clear. The argument has been made, in H91, that since inflow is primarily concentrated on the equatorial plane, the loss of a reflecting boundary is localized to this region, and mode growth can still occur given the right circumstances in off-equator regions of the disk. Models B3r, A3p, and B3p show that mode growth in the absence of early inflow produces planets, while still generating appreciable transient inflows as a consequence of the redistribution of matter accompanying planet formation. In the converse case, we have seen that early inflow (marginal models E3r and X3p) effectively inhibits mode growth; the establishement of a steady inflow coincides in these two models with the capping of mode growth at a very low level, insufficient to disturb the disk. Model E3p would seem to straddle these two extremes in that it exhibits both early inflow and the development of unstable modes. Model E3p, like models E3r and X3p, is a marginal torus. E3p and X3p are very similar in their initial conditions (see Table 2), although model E3p has an initial inner edge that is slightly farther out that model X3p. In spite of these structural similarities, model E3p develops a significant PPI mode while E3r and X3p do not. As marginal tori, all of these three models show an early inflow of matter. However, the mass influx rate is less for model E3p (about -0.01 in units normalized to the peak density) than it is for the other marginal models (between -0.04 and -0.08). Model E3p appears to be a ``transition'' model, where the presence of (weak) early inflow is insufficient to prevent mode growth. Mode growth eventually begins, and progresses much more slowly than in the other models that yield planets; the growth rate of the m=1 mode for model E3p is one third the rate of the next slowest mode growth, for model B3p. Blaes (1987) found that although the PPI modes can be stabilized by accretion, there is a finite transition that permits both accretion and mode growth. E3p appears to be such a case. These results for 3D tori in the Kerr metric reinforce the idea that accretion through the inner boundary at the equatorial plane suppresses the growth of the PPI, and that this suppression is as effective in three dimensions as it is in two.
Perhaps most interestingly, we have seen evidence of frame-dragging in those models that develop an accretion flow. Models B3r and E3r show evidence of a retrograde inflow changing its winding sense as it enters the innermost regions near the static limit, and model B3p shows a prograde accretion flow getting smeared into a ring in the vicinity of the static limit. As has been suspected from the earliest days of numerical simulations of GR hydrodynamics, it is in the deepest regions of the black hole potential well that we find the most interesting, and perhaps most astrophysically distinct consequences of general relativity.
This work was supported by NSF grant AST-0070979 and NASA grant NAG5-9266. The simulations were carried out on the Origin 2000 system at NCSA, and the Bluehorizon system of NPACI.