Hawley & Krolik: Simulations of the Plunging Region

Introduction

MHD turbulence driven by magneto-rotational instability (MRI; Balbus & Hawley 1991) now appears to be the fundamental physical mechanism of angular momentum transport in accretion disks (see the review by Balbus & Hawley 1998). On the basis of this idea, it is now possible to begin to answer many questions about accretion dynamics. In a series of papers, we are investigating the global radial structure of accretion disks near the marginally stable orbit of a black hole by means of large-scale three dimensional MHD numerical simulations. In particular, we focus on the inner regions of accretion disks, and examine the time-dependence of accretion, the radial dependence of stress and dissipation, and the net energy and angular momentum per unit mass carried into the black hole. In so doing we will also need to examine more closely the conceptual basis for disk dynamics.

For understandable reasons, simple analytic models (e.g., Novikov & Thorne 1973, Shakura & Sunyaev 1973) have in general been built on the assumption of time-steadiness. However, this assumption is by no means a given in real disks. Even if there is a long-term mean accretion rate, it is entirely possible for there to be short-term fluctuations. In fact, sizable fluctuations seen in the light-curves of every accreting black hole indicate that accretion variability is the norm, not the exception (e.g., Sunyaev & Revnivtsev 2000).

The difficulty with addressing issues of time-dependent dynamics within the context of traditional analytic models is that the dynamics of accretion depend fundamentally on the nature of the stress that transports angular momentum. While many steady-state or time-averaged properties of disks may be adequately described by a simple stress parameterization (e.g., the Shakura-Sunyaev $\alpha$ prescription in which the stress is supposed proportional to pressure), the actual dynamics cannot be so treated. $\alpha$ is only a measure of the stress; it is not the physics behind the stress. Within the $\alpha$ parameterization, stress results from turbulent fluctuations (not viscosity). These fluctuations have amplitudes less than or of order the sound speed c_s on scales less than or of order the disk scale-height H. Again, while there may be regions of the disk where it is sufficient to time-average over these fluctuations, this cannot be the case near the black hole where accretion time- and length-scales become comparable to those that characterize the turbulence. Direct dynamical simulations are required to understand time-dependent quantities.

Although it has long been clear that ordinary viscosity cannot account for angular momentum transport in accretion disks, modeling the stress ``as if" it were viscous has been a popular way of thinking about disks for an equally long time. Again, for dynamical issues this is clearly incorrect: a low-viscosity plasma that is turbulent does not behave like high-viscosity laminar flow. A separate question is whether the turbulent stress would behave sufficiently similarly to viscosity that there would be an associated dissipation rate proportional to the local stress (Novikov & Thorne 1973; Shakura & Sunyaev 1973). Not all stresses have this property, although if the MRI-driven MHD turbulence dissipates locally in a turbulent cascade it is amenable to an $\alpha$ -type description (Balbus & Papaloizou 1999). Because we now know that the dominant stress is electromagnetic, local dissipation is not certain; Poynting stress is electromagnetic, local dissipation is not certain; Poynting flux can easily carry energy from one place to another, and highly-magnetized coronae and winds may account for much of the liberated energy. Numerical simulations offer the possibility of measuring to what degree this happens.

Both the radial dependence of stress and dissipation and the net energy and angular momentum delivered to the black hole depend on yet another plausible, but not demonstrated, assumption: that the inter-ring stress disappears in the vicinity of the marginally stable orbit. Two heuristic arguments were raised in behalf of this assumption: that the small amount of mass in the plunging region could hardly be expected to exert a force on the far heavier disk proper (Novikov & Thorne 1973, Page & Thorne 1974); and if the stress scales as a constant fraction $\alpha$ of the local pressure, then the low pressure in the plunging region would lead to a very small stress (Abramowicz & Kato 1989). However, as recognized by Page & Thorne (1974), neither of these arguments applies to magnetic stresses. Krolik (1999) and Gammie (1999) argued that the dominant role of magnetic stresses in angular momentum transport in the disk body should actually lead to stresses near the marginally stable orbit large enough to substantially alter the amount of energy and angular momentum removed from matter before it passes through the black hole's event horizon. If so, the radial distribution of stress in the disk would also be significantly altered, with wide-ranging observational consequences (Agol & Krolik 2000). Simulations can quantitatively evaluate the importance of this mechanism.

Global three-dimensional disk simulations have only recently become possible, and the number of such models is still sufficiently small that it remains possible to give a nearly comprehensive list of references: Armitage (1998); Matsumoto (1999); Hawley (2000, hereafter H00); Machida et al. (2000); Hawley & Krolik (2001, hereafter HK01); Hawley (2001); Armitage et al. (2001). All of this work has shared two key assumptions: Newtonian dynamics in a Newtonian or pseudo-Newtonian (Paczynski-Wiita) potential, and a fixed (adiabatic or isothermal) equation of state. All of this work has also struggled with the same central problem: obtaining resolution adequate to describing the physics.

In this paper, we report simulations with the best resolution in the inner accretion flow yet achieved. We also use these simulations to explore whether the topology of the initial seed magnetic field has any lasting effects on the structure of the accretion flow. In later efforts we will improve the level of realism in these simulations by solving the energy equation and employing genuine relativistic dynamics.

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