Parameters for Simulations

The parameters for the 3D runs in H91, summarized in Table 1, fall into two broad categories that are distinguished by specific angular momentum. The l=4.5 models, A3, B3, and C3, are tori with innermost radii (in the equatorial plane) of r_in=11.0 M, 9.5 M, and 8.0 M respectively; these represent slender, intermediate, and wide tori. The marginal model, E3, is a wide torus with r_in=4.4 M.
 
 

Table 1: Disk Parameters in H91.

Model	l	rin	rout	$\Omega$	Torb	Type
A3	4.50	11.0	24.7	0.0167	376	slender
B3	4.50	9.5	37.1	0.0167	376	intermediate
C3	4.50	8.0	110.0	0.0167	376	wide
E3	3.96	4.4	79.5	0.0313	201	wide (mb)

In adapting these models for non-zero Kerr parameters, we are confronted with the change in torus width with a, and also with the restricted range over which the l=4.5 torus remains bound. To emphasize the role of a, we chose the extreme Kerr limits wherever possible. Six models were selected, and are listed in Table 2. We use the same naming convention as in H91; models are named according to the choice of the pair of parameters (l, r_in), here augmented by the Kerr parameter a. Model A3p represents a prograde l=4.5 torus in the extreme Kerr limit; the slender A3 torus has expanded to intermediate size with this change. Similarly, prograde model B3p has expanded into a wide torus. Model B3r represents the ``last" bound retrograde l=4.5 torus, for a=-0.8; this model is also well below the marginally bound value for a=-0.8 (it could be termed a sub-marginal torus). The intermediate B3 torus has shrunk considerably with this change in a. The E3 models are more loosely based on the H91 E3 original; this was done to maintain their nature as marginal tori, and allowing the values of r_in to depart from the H91 value. Model E3p is a marginal prograde torus with a=0.5; the wide marginal E3 torus has shrunk considerably with this choice of parameters. Model E3r is an intermediate, retrograde torus in the extreme Kerr limit. One additional model, X3p, was introduced to complete the data set with a prograde marginal torus that is related to E3p, but whose inner radius lies closer to the static limit.
 
 

Table 2: Disk Parameters for Numerical Simulations.

Model	a	l	rin	rout	$\Omega$	Torb	Type
A3p	1.0	4.50	11.0	39.9	0.0139	452	intermediate
B3p	1.0	4.50	9.5	100.0	0.0139	452	wide
B3r	-0.8	4.50	9.5	18.3	0.0222	283	slender
E3p	0.5	3.37	4.0	17.1	0.0550	114	marginal
E3r	-1.0	4.79	8.0	57.0	0.0164	383	marginal
X3p	0.5	3.33	3.2	23.5	0.0594	106	marginal

Table 3 is a summary of the simulation results. The growth rate of the first and second Fourier modes is given (in units of $\Omega$), as is the time at which mode saturation occurred (in units of T_orb), the change in power-law exponent for the best-fit to the azimuthally-averaged angular velocity at mode saturation, and the density enhancement at mode saturation. Mode saturation is defined as the time at which the first Fourier mode reaches its maximum amplitude. For reference the corresponding results from H91 are given in Table 4.
 
 

Table 3: Three-Dimensional Torus Simulations with $\gamma =5/3$.

Model	a	m=1	m=2	tsat	$\delta q_{sat}$	$\delta\,\rho/\rho$
A3p	1.0	0.096	0.063	10.0	-0.15	0.57
B3p	1.0	0.081	0.046	13.0	-0.09	0.35
B3r	-0.8	0.150	0.108	9.0	-0.08	0.26
E3p	0.5	0.027	0.015	33.0	-0.01	-0.01
E3r	-1.0	0.011	0.005	20.0	0.00	-0.01
X3p	0.5	0.112	0.109	5.5	0.00	-0.01

 

Table 4: Results from H91.

Model	m=1	m=2	trun	qsat	$\delta\,\rho/\rho$
A3	0.175	0.112	8.5	1.80	0.94
B3	0.089	0.040	16.0	1.86	0.26
C3	0.070	0.022	19.0	1.99	0.05
E3	0.022	0.021	39.0	2.00	0.006

 

 

2002-06-05