Hawley & Krolik: Simulations of the Plunging Region

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6. Conclusions

In this paper we report on two new three-dimensional MHD global accretion disk simulations, one beginning with a poloidal field and the other with a purely toroidal field. Since the topology of the magnetic field in a real disk is largely unknown (except for the observation that it is almost certainly dominated by the toroidal component) these two initial configurations are intended to bracket the possible range. We find that initial field topology makes a significant quantitative difference in the resulting evolution. With a purely toroidal field, initial field amplification is slower, and the saturation energies, magnetic stresses, and accretion rate are smaller. The magnetic stress begins to evolve by flux freezing at a smaller radius in the toroidal field run. However, the qualitative features of angular momentum transport and stress at the marginally stable orbit are unchanged. In any event, we expect that because the MRI is so effective at amplifying poloidal field, real disks are likely to resemble the initially-poloidal simulation more than the purely toroidal one.

We reexamined the issue of stress at the marginally stable orbit, and confirmed our previous finding (HK01) that the stress remains significant there. The character of the stress changes as the flow moves from turbulence to inflow. Within the body of the disk, where the dissipation time is short compared to the inflow time, the field evolves by MRI-driven MHD turbulence limited by dissipation (numerical in the simulation, resistive and viscous in real disks). The relative correlation of BR and $B_{\phi}$ ($\alpha _{mag}$) in that regime is ~ 0.2-0.3. As the flow approaches rms from the outside, the inflow time falls until it is shorter than the dissipation time. Inside that radius, the field evolves mainly via flux freezing and $\alpha _{mag}$ increases toward unity. The location of this transition is one of the most important parameters governing the character of the inner accretion flow. As a result of the continuing stress in the plunging region, inside rms accreting matter suffers a modest ~10% decline in its average specific angular momentum and a comparable increase in its binding energy. Rapid variability in accretion rate remains the rule with the most power at frequencies lower than the orbital frequency at rms.

These simulations used the largest number of grid zones and arranged them so as to achieve the finest resolution in the inner accretion flow of any simulation to date. We find that the increased resolution used in these simulations produced increased field energy and stress, and also lead to larger amplitude variations in the accretion rate. We believe that at this level of resolution we are approaching numerical convergence with respect to the magnetic field intensity and the accretion rate for the initially-poloidal case, but finer resolution may be necessary to demonstrate convergence for these quantities when the initial field topology is toroidal, and for other quantities for both topologies. Unfortunately, it is difficult to obtain significantly greater resolution or to use this high resolution over a wider range of radius. Certain questions will therefore remain difficult to answer for some period of time (most importantly, the location of the dissipation/flux-freezing transition radius). Perhaps this is not surprising, considering that we are attempting to resolve a turbulent cascade within a global disk. On the other hand, the overall evolution and qualitative features have remained consistent from the lowest to highest resolutions. This is fortunate since the dynamics of self-consistent MHD accretion flows remain largely unexamined, and further exploration with even the modest resolution permitted by current practical considerations is likely to be fruitful.

Acknowledgements: This work was supported by NSF grant AST-0070979, and NASA grants NAG5-9266 and NAG5-7500 to JFH, and NASA grant NAG5-9187 to JHK. Simulations were carried out on Bluehorizon, the IBM SP cluster of the San Diego Supercomputer Center of the National Partnership for Advanced Computational Infrastructure, funded by the NSF, and on Centurion, a linux-based alpha cluster operated by the Legion project of the University of Virginia Computer Science Department, Andrew Grimshaw PI.


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