6. Conclusions

The simulation reported here represents another step towards more realistic simulations of accretion disk dynamics. By improving the grid resolution in the inner region, we have shown that magnetic effects become increasingly important near and inside the radius of the marginally stable orbit. Contrary to the most widely-held assumption, the R-$\phi $ component of the stress tensor does not go to zero at rms; averaged over azimuth, height, and time, it is almost constant as a function of R for $R \leq 2 r_{ms}$. As a result, matter in the region of unstable orbits does not follow simple energy- and angular momentum-conserving free-fall trajectories, although we are not yet ready to make quantitative predictions about how much the energy and angular momentum change.

The ratio $T^{R\phi}/P \equiv \alpha$ is commonly used as a measure of the magnitude of the stress, and almost as commonly is thought to be a constant, independent of location and time. Our simulation showed clearly that while it can provide a good qualitative measure of the stress when averaged over several orbits in time, at any particular instant $\alpha$ has large systematic gradients (as a function of both R and z) and substantial temporal fluctuations. On average the value of $\alpha$ in the disk is $\sim 0.1$.

This simulation has shown that such substantial time- and space variations are the rule. The inner regions of accretion disks around black holes are highly turbulent, non-steady systems. Large-amplitude fluctuations sheared into spiral fragments come and go, many of them initiated in the region immediately outside the marginally stable orbit. As a result, there are continuing fluctuations in the accretion rate; we expect that some of these fluctuations will be mirrored in the light curves of accreting black holes.

Future simulations, guided by these results, will probe the beyond the limitations of the present simulation. Within the context of the same pseudo-Newtonian potential and the present adiabatic equation of state, we can explore the effects of different initial disk configurations and magnetic field strengths and topologies, at at higher resolution and for longer time. Beyond this, it will be important to incorporate improved physics into the models. These include general relativistic dynamics and improved treatments of the equation of state and dissipative processes. This future work should bring us still closer to a quantitative understanding of the dynamics of accretion onto black holes.

Acknowledgements

JHK would like to thank Eric Agol for numerous helpful conversations, and Nancy Levenson for invaluable instruction in the wiles of IDL. This work was supported by NSF grant AST-0070979, and NASA grants NAG5-9266 and NAG5-7500 to JFH, and NSF Grant AST-9616922 and NASA grant NRA-99-01-ATP-031 to JHK. Simulations were carried out on the Cray T3E system of the San Diego Supercomputer Center of the National Partnership for Advanced Computational Infrastructure, funded by the NSF.

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