3. Results

3.3 Disk near the marginally stable orbit

The recent work of HK and ARC focused on the stress at and interior to the marginally stable orbit. Having considered disks with inner boundaries located outside of rms, we now turn to a model, CK7, a disk with an initial inner edge near rms that produces substantial accretion into the central hole. It uses an adiabatic equation of state with $\Gamma = 5/3$. The computational grid extends from R=1.5 to R=61.5, over 0.8 in z, and $2\pi$ in $\phi $. There are 32 equally spaced zones in z, 256 equally spaced zones in $\phi $, and 256 zones in R. Eighty of the radial zones are equally spaced between R=1.5 and 10, while the remainder are logarithmically stretched between 10 and 61.5. As before, the large radial extent of this disk provides a reservoir of mass which will allow the inner region of the disk to evolve for many orbits without influence from the outer grid boundary.

For this simulation the initial state was relaxed to a true hydrodynamic equilibrium at the inner disk boundary. A constant density gas with $\rho = 1$ is first placed on the grid from R=4 to the outer boundary. The initial sound speed is chosen to be cs = 0.086 throughout the disk, corresponding to an isothermal scale height of 0.4 at R=4 ( $c_s^2/\Gamma = 0.01 v_\phi^2$). This is hotter than the standard model of ARC (cs = 0.069), but cooler than the thick disk model of HK. This initial condition is evolved in one dimension to an equilibrium solution. The resulting disk has $\rho = 1$ down to about R=7, inside of which the density and pressure decline smoothly to zero at R=3. Similarly, the angular momentum is Keplerian throughout most of the disk, but becomes slightly super-Keplerian as R=3 is approached. The maximum increase over the Keplerian value is 3%.

A vertical magnetic field is placed onto this hydrodynamic equilibrium. As in run CK6 the field strength is set so that the Alfvén speed $v_A = \Omega/k_z$ for a wavenumber kz corresponding to a wavelength of one quarter the vertical grid size. This is equal to $\beta = 276$ at R=4 with $\beta$ increasing to 106 by R=60.

This disk is evolved for 270,000 timesteps out to a time t=2575. This corresponds to 118 orbits at rms. Accretion through the inner boundary begins with a strong pulse at $t\sim 100$ as the magnetic instability in the inner part of the disk reaches nonlinear saturation. As the evolution proceeds, the location where magnetic instability is undergoing its initial linear growth moves out through the disk. The region that is in a quasi-steady state similarly moves out through the disk, albeit at a slower rate. The resulting disk can be described both in terms of instantaneous properties, which emphasize the turbulence and the fluctuations, or in terms of time- and space-averaged quantities which provide some measure of what the ``steady state'' properties of the disk are like.

Figure 5 shows the surface density distribution at the end time of the simulation. The most prominent features are the tightly wrapped trailing spiral waves. The spiral waves develop in the inner part of the disk along with the turbulence. A spacetime diagram shows that these waves propagate out through the disk at the sound speed. The waves are produced quasi-periodically throughout the evolution. A fourier transform of the accretion rate shows that the peak spectral energies correspond to the orbital period at R=6.7 and 5.6 in the first half of the simulation, and larger radii at later times as the turbulence grows at those locations. The amplitude of the density fluctuations through these waves is measured by equation (8). Outside of R=20 $\delta\rho/\rho \simeq 0.2$. Inside of this point, the fluctuation amplitude rises, reaching a value of 1.3 at rms. The accretion flow through rms tends to be highly nonaxisymmetric.

Figure 5: Vertically-averaged gas density in run CK7 at the end time t=2575. The grey-scale is linear in density and runs from 0 to 1.


Of particular interest for this simulation is the magnitude of the stress at the marginally stable orbit, and its effect on the specific angular momentum $\ell $. Throughout most of the disk the averaged $\ell $ tracks the circular orbit value, $\ell_{kep}$, with a one to two percent excess over this value inside R=10. Figure 6 shows the time evolution of both the Maxwell stress at rms in terms of the Shakura-Sunyaev $\alpha$ value defined with the local pressure, and the specific angular momentum at rms and close to the inner boundary at R=1.66. The first thing to notice is that the Maxwell stress at rms is nonzero and time-variable, as are the values of $\ell $ at rms and the inner boundary. There is, however, always a net change in $\ell $ between rms and the inner boundary. At rms, $\ell $ is always slightly greater than $\ell_{kep}\approx 2.6$; the time-average excess in the second half of the simulation is 1.8%. And although the slope $d\ell/dR$ decreases toward zero inside of rms, there is always an average decline in $\ell $ of 1.7% between rms and the inner radial boundary. This is smaller than the 5% drop reported by HK, but appears to be consistent with the results of ARC.

Figure 6: a) Maxwell stress $M^{R\phi }$ in CK7 at R=rms as a function of time in terms of $\alpha =M^{R\phi }/P$, where P is the local gas pressure. (b) Time history of the specific angular momentum at R=rms and R=1.66. Although $\ell $ varies considerably with time, there is always a small net change between rms and the inner radial boundary.


Figure 7 shows the $\phi $- and z-averaged magnetic energies and the Maxwell stress as a function of radius averaged over the last 500 units of time. The values are scaled by the initial pressure. As always, the toroidal field energy is dominant, exceeding the radial field energy by a factor that declines with radius, from 20 at R=4 to around 5 outside of R=15. Inside R=3, the radial field becomes more significant compared to the toroidal as the field is combed out by the nearly radial infall. The ratio of the toroidal to the vertical field energy is $\sim 70$ from R=10 to 20, and rises rapidly both inside and outside this region. At rms $\beta = 8.5$; this increases with radius to $\beta \approx 100$ at R=20. Note that the gas pressure is substantially reduced from its initial value inside of R=10. The magnetic energy has grown by over 600 times from its initial value. Beyond R=20 $\beta$ rises more gradually to about 400 at the outer boundary.

Figure 7: Magnetic energy averaged over $\phi $, z, and the last 500 units of time in run CK7. The top solid curve is the toroidal field, the middle solid curve the radial field, the bottom solid curve the vertical field energy. The dashed line is the Maxwell stress, and the dot-dash line is the initial vertical field energy. All values are normalized with the initial gas pressure (P=0.0045).


The averaged mass accretion rate is relatively constant inside of R=15 at late time. Beyond t=400 the accretion rate into the central hole varies around a mean value of 0.072. The sense of accretion reverses at R=18. The averaged radial drift velocity has the value $v_r/c_s \approx 0.008$ between R=10 and 15, and rises rapidly to cross over vr/cs=1 at R=2.6; this is very similar to the velocity plot (Fig. 4) of ARC.

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