Slender Torus - Model B3r

Model B3r is a slender torus rotating in the retrograde sense around a Kerr black hole with a=-0.8. The torus has an initial maximum density of $\rho_{max_{0}}=0.031$ at r=12.8 M. Figure 5(a) shows that at mode saturation a planet has formed, with a density maximum at $r\sim 11\,M$ and a fractional density enhancement of 0.26. The planet has a crescent shape with some evidence of a tightly-wrapped outward spiral wave in the lower left quadrant of panel (a). A gray-scale enhanced view of the inner region near the static limit clearly shows that a strong inflow has developed that contacts the static limit in the upper left quadrant of panel (b); there is also a hint of frame dragging in the reversal of the sense of the flow near the static limit (in this figure, as in all subsequent ones, the black hole rotates in the counter-clockwise sense, and the retrograde torus here rotates in the opposite sense). There is also a evidence of a weaker inbound spiral of matter in the right half of panel (b). The growth of the PPI modes has two distinct stages, as seen in Figure 5(c). The m=2 mode grows quickly at the beginning, then levels off at 2 T_orb and remains roughly constant until 7 T_orb before increasing again, paralleling the growth of the m=1 mode. The m=1 mode growth does not become apparent until approximately 3 T_orb, at which point it exhibits strong linear growth to saturation at 9 T_orb. Up to 4 T_orb, the m=2 mode is dominant, but lags the m=1 mode in the later stages of the simulation. The development of the accretion flow is clarified in Figure 5(d), which shows the mass influx in the equatorial plane inside the inner edge of the torus. There is no significant accretion until 8 T_orb. The accretion rate reaches a maximum shortly after mode saturation. This feature, as will be seen in subsequent runs, is associated with all solutions that yield planets.

The effect of frame dragging is illustrated in Figure 6, which is an equatorial slice at 10.02 T_orb, one orbit after saturation. The left panel of the figure is a density plot with 15 contours, chosen to highlight the stream of high-density material flowing from the inner edge of the grid into the hole. The right panel shows momentum direction vectors inside the flow stream outlined by the overlaid contour. The expected reversal in azimuthal flow direction due to frame dragging begins at $r \sim 4\,M$. The direction becomes progressively more prograde as the flow approaches the the inner edge of the grid, which lies at 2.05 M (just outside the static limit). This reversal, from retrograde to prograde, produces the dog-leg pattern seen in the density plot as the flow approaches the static limit. A similar pattern is visible for model E3r (see below).

Figure 5: B3r Model. (a) (Top left) Equatorial slice through torus at saturation. Density contours linearly spaced between $\rho _{max}$ and 0.0. (b) (Bottom left) Magnified view of flow near static limit at saturation ( x 4 density enhancement, linear gray scale). The edge of the central black circle is the static limit. (c) (Top right) Mode growth. (d) (Bottom right) Mass influx at inner edge of disk. (Black hole rotates in counter-clockwise sense.)

                        View animation: B3amaa.mpg (rotation sense of animations is reversed)
 

Figure 6: Direction Vectors for B3r Model. This section through the equatorial plane is taken at 10.02 T_orb and shows a stream of infalling matter that exhibits frame dragging. The left panel shows 15 isodensity contours plotted on a linear scale. The right panel shows the direction of flow for the matter confined to the high density stream. The reversal of flow direction can be clearly seen for matter inside $r \sim 4\,M$, showing perhaps the clearest evidence for frame dragging.

2002-06-05