4. Discussion

4.1 Black hole spin-up

We are now ready to begin answering some of the questions posed in the Introduction. Most significantly, this simulation has shown that the stress due to MHD turbulence does not automatically cease at the marginally stable orbit. In fact, it continues to be present throughout the plunging region. As an immediate consequence, matter continues to lose angular momentum as it travels from the marginally stable orbit toward the event horizon. Because it then enters the black hole with smaller angular momentum than if no stresses had been present, the rate at which the black hole is spun up is slower than it would have been otherwise (Agol & Krolik 2000; Gammie 1999). It further follows that estimates of the black hole spin distribution are shifted toward slower rotation (e.g., Moderski & Sikora 1996). Although in this simulation the magnitude of this effect is small ($\simeq 5\%$), it is possible that more realistic conditions will increase its size (see §5.1, 5.3).

4.2 Additional dissipation in the inner disk

We have already remarked on the fact that our simulation does not directly compute the dissipation rate. In this section we attempt to estimate what dissipation might be associated with dynamics of the sort seen here. In conventional disk models, the local dissipation rate is directly tied to the local stress:

Q = \left\vert{d\Omega \over d\ln r}\right\vert \int \, dz T^{R\phi},
\end{displaymath} (20)

where Q is the dissipation rate per unit area. This relationship is predicated upon the notion that the same stress producing local transport also produces local dissipation. This remains true for the mean flow dynamics of MHD turbulence (Balbus & Papaloizou 1999); in fact, numerical dissipation mimicking the physical dissipation at the short wavelength end of the turbulent cascade is the way in which matter in this simulation loses energy as it moves slowly inward in the body of the disk. However, this identification is not necessarily correct as a prediction for the instantaneous rate of dissipation in specific fluid elements. It is especially questionable in the plunging region. The stress is not necessarily turbulent, and there may not be enough time for the energy deposited in the fluid by the stress to be transferred to small enough lengthscales for it to be dissipated. Nonetheless, in this subsection we adopt the speculative assumption that the vertically-integrated dissipation rate is proportional to the vertically-integrated stress in order to speculate about the consequences for disk radiation that may result.

One possible consequence is an altered radial distribution of the disk surface brightness. As shown in Figure 10, the azimuthally-averaged stress in the simulation is nearly constant inside $R \simeq 5$. If the relation of equation (20) holds, the total dissipation rate outside R=3 in this simulation would be $\simeq 50\%$ greater than in a disk with the same accretion rate, but zero stress at the marginally stable orbit.

If this were to happen, there would be a number of effects of considerable phenomenological interest (Agol & Krolik 2000). Because this additional heat is released in a relatively small area of the disk, it leads to relatively high temperatures, and therefore contributes to the highest-frequency portion of the spectrum. Another consequence stems from the fact that the dissipation is located in the innermost part of the disk, where relativistic effects are strongest. The light that is radiated due to this extra heating should be comparatively strongly beamed into the equatorial plane by a combination of special relativistic Doppler beaming driven by the orbital motion and general relativistic trajectory-bending. Trajectory-bending also forces a large fraction of the photons to return to the disk at larger radius, to be reprocessed there into lower-frequency light.

Whether or not the enhanced stresses in the plunging region lead to additional radiative losses depends on the dissipation and photon diffusion rates. In the disk proper, dissipation takes place in the turbulent cascade, in small lengthscale fluctuations that are short-lived, exhibit little spatial coherence, but are smoothly distributed in a statistical sense. In the time-varying accretion of the plunging region this turbulent cascade may not have time to go to completion. On the other hand, there might be dissipative events, mediated by reconnection for example, that involve structures with relatively large spatial coherence lengths and which are not necessarily smoothly distributed throughout the region. As we will discuss in §5.1, our limited spatial resolution makes it difficult for us to estimate fairly the rate of such events in the plunging region. Moreover, assessing the radiative properties of such dissipative events is beyond the the capabilities of this simulation. However, if events of this sort do occur, significant luminosity might result because the reduced optical depth of the plunging matter (as compared to the disk proper) also greatly reduces the thermal time. Note that this mechanism results in an enhanced radiative efficiency without the transfer of any energy from the plunging region to the disk outside the marginally stable orbit.

Finally, although the azimuthally-averaged stress matches conventional expectations in much of the disk, there are large azimuthal variations in the form of sheared filaments (Figure 12). Translated into a dissipation rate, these spiral waves of enhanced (or diminished) stress become spirals of enhanced (or diminished) heating. Whether this concentration of heating has an effect on the emitted radiation depends on the characteristic timescale of these fluctuations as seen in a specific fluid element. If a particular fluid element has an elevated heating rate for as long as its thermal timescale, the effective radiating area of the disk is reduced to the area of the regions where the heating is greatest, and the typical effective temperature is raised accordingly. On the other hand, if over a thermal timescale individual fluid elements see both positive and negative fluctuations, the effects will average out. We have already estimated that the characteristic speed of fluctuations through the disk is of order the magnetosonic speed; on the other hand, the thermal timescale is $\sim
\alpha_{SS}^{-1}$ times the orbital period. Consequently, the ratio of the coherence time of fluctuations to the thermal time should be, very roughly, $\sim \alpha_{SS} v_{orb}/(v_{A}+c_s)$; in this simulation this ratio is $\sim 1$.

4.3 Fluctuations in light output

We have emphasized throughout this paper the highly non-stationary behavior that is intrinsic to disk behavior as found in this simulation. One form these fluctuations can take is the coherent waves that are continually radiated outward through the disk from the region of the marginally stable orbit. Even if, on average, dynamics in the plunging region change the mean accretion efficiency by only a small amount, the large fluctuations we observe in the simulation should cause substantial time-dependent variations in the light output.

Once again, because there is no direct connection between the dynamics traced by this simulation and dissipation, we cannot directly predict the time variation of the disk luminosity. In fact, to do so would also require calculating the photon diffusion rate from inside the (optically thick) disk. However, some indication of what may occur may be gleaned from Figures 6 and 7. Some of the fluctuations in the accretion rate (particularly those at the highest frequencies) are due to dynamics in the plunging region; these may or may not lead to dissipation (see § 4.2), and therefore may or may not contribute to fluctuations in the light output. For this reason, and following the ansatz of equation ( ), in Figure 7 we also plot the Fourier power spectrum of the volume-integrated magnetic stress. Whether the accretion rate or the stress is more closely related to the dissipation rate, the actual output is the convolution of fluctuations in the heating rate with the probability distribution for the photon escape time. In frequency space, this amounts to a (position-dependent) low-pass filter. When the heating occurs deep inside the disk, the cut-off frequency for this filter is $\sim \alpha_{SS}\Omega$; fluctuations in the light due to heating closer to the surface are subject to a less stringent frequency cut-off (in the optically thin limit, there is no cut-off at all). Consequently, we expect the Fourier power spectrum of the disk luminosity to be rather strongly cut off at frequencies above 10-2, but the power spectrum of fluctuations associated with coronal dynamics may be different.

The observation of QPOs in black hole candidates has motivated a number of studies of possible sources of systematic time variability in accretion disks. In a series of papers (Nowak & Wagoner 1991, 1992; Perez et al. 1997) Wagoner and collaborators have explored the linear theory of fluid oscillations in the relativistic portions of stationary accretion disks. In these disks only hydrodynamic effects are considered, and angular momentum transport is modeled by a pseudo-viscosity parameterized by a constant $\alpha$. These disks possess a variety of normal modes, including p-modes that are trapped between rms and the radius where the epicyclic frequency is maximum, and g-modes, whose greatest amplitude is found under the peak of the epicyclic frequency curve. In the Paczynski-Wiita potential we use, the position of the maximum epicyclic frequency is $R \simeq 3.6$; with the additional correction terms introduced by Nowak and Wagoner (1991), it moves outward to R=4. We would not expect to find exactly the same mode frequencies as they because of the difference in the potentials used; however, the mode frequencies should in general be $\sim \Omega(r_{ms})$. A perusal of Figure 7 fails to find any evidence for special frequencies in this range.

There are several reasons why trapped disk oscillations do not appear here. First, the normal modes discussed in the previous paragraphs are predicted on the basis of a purely hydrodynamic analysis; no account is made of magnetic forces. Here, however, magnetic forces play a major role in determining the dynamics of fluid elements. For example, if one repeated the derivation of the ``diskoseismic" dispersion relation, but allowed for a weak magnetic field, one would rediscover the magneto-rotational instability, thus vitiating the hydrodynamic ``normal modes". More fundamentally, our simulation indicates that there is no underlying quiet flow to support coherent oscillations. The disk is so turbulent that there is no steady equilibrium against which linear perturbations can develop. Put another way, we find that the typical radF ial motion timescale in the vicinity of rms (where these normal modes are supposed to be concentrated) is comparable to the predicted mode frequencies, so the fundamental assumption that these modes are perturbations to circular orbits is inappropriate.

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