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 (), it is possible that more realistic conditions will increase its size (see §5.1, 5.3).

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:

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 .
If the
relation of equation (20) holds, the total dissipation
rate outside *R*=3 in this simulation would be
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
times the orbital period. Consequently, the ratio of
the coherence time of fluctuations to the thermal time should be, very
roughly,
;
in this simulation this
ratio is .

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
;
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
.
These disks possess a variety of normal modes, including
*p*-modes that are trapped between
*r*_{ms} 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 ;
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
.
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 *r*_{ms} (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.