Itzhak Goldman
School of Physics and Astronomy, Sackler Faculty of Exact
Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Visiting Scientist, Department of Condensed-Matter Physics, Faculty of Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Amri Wandel
Racah Institute of Physics, Hebrew University, Jerusalem 91904,
Israel
In the gas pressure region, two distinct solutions are obtained. In
one, the convective flux is much larger than the radiative flux and the
temperature profile is close to adiabatic. The
blackbody region extends over the entire gas pressure region
and could also extend down to very small radial distances, so that
there would be no radiation pressure region and no gas pressure
electron scattering region. In the other solution, the convective flux
is about a third of the total flux, the dimensionless superadiabatic
temperature gradient is 
and all three regions of the disk
exist.
In the radiation pressure region, the temperature profile is very
close to adiabatic, and the disk is geometrically thin and optically
thick even for super Eddington accretion rates. The fraction of the
convective flux, out of the total flux, increases with the accretion
rate from 
to close to 1, for accretion rates comparable to
the Eddington limit. As a result of this variation, the radiation
pressure region is secularly stable; thus all the disk solutions are
secularly stable. The values of the effective 
-parameter are
rather small: 
,
and 
for the radiation pressure region and for the two
solutions in the gas pressure region, respectively.
There is quite a large body of observational data suggesting the existence of accretion disks around protostars, compact stellar objects and active galactic nuclei. Naturally, accretion disk models are extensively employed to interpret these observations. However, to a large extent the models used are more a descriptive than predictive tool (Pringle 1981).
The reason for this state of affairs is
the lack of a physical model for the disk turbulent viscosity,
, usually parameterized in the form
(Shakura & Sunyaev 1973).
Here 
is the speed of sound, h is the disk half-width, and
is a dimensionless parameter which neither value
nor dependence on the physical conditions in the disk are known.
Thus, often the observations are used to fit the 
-parameter
for a given system. Usually a constant 
is assumed for gas
pressure dominated regions of the disk. For radiation pressure
dominated regions, both the former prescription and 
proportional to the ratio between the gas and total pressure have been
suggested. Without a physical model for the turbulent viscosity,
it is not clear which prescription, if any, is the correct one.
A self-consistent solution to the disk equations requires that the turbulent viscosity be determined by the physical state of the disk. In turn, the resulting turbulent viscosity will control the disk physical conditions. The determination of the turbulent viscosity requires a model for turbulence and the specification of the particular instability that generates the turbulence.
In spite the very large Reynolds numbers, Keplerian disks are stable
against shear-generated turbulence, at least in the linear analysis.
On the other hand, it has been pointed out that they
are unstable against turbulent convection, both in radiation pressure
and gas pressure regions and for various opacities (see e.g.,
Bisnovatyi-Kogan & Billinikov 1977, Shakura, Sunyaev & Zilitinkevich
1978, Lin & Paploizou 1980). Therefore, in this paper we implement
the above proposed self-consistent scheme for the case of turbulent
convection in a thin accretion disk.
The turbulence model employed (Canuto, Goldman, & Hubickyj 1984,
Canuto,
Goldman & Chasnov 1987) provides vertically averaged values for the
turbulent viscosity and for the convective flux. Thus, no attempt to
resolve the vertical structure of the disk is made, and all quantities
are either vertical averages or midplane values. A detailed vertical
structure requires a considerably more complex turbulence model, that
could yield the z-dependent values of the turbulent viscosity and
of the convective flux. In this work, we focus on the more general
features of the self-consistent approach and avoid the complications of
a z-dependent model. We note that turbulence bulk properties such as the
turbulent viscosity and the convective flux are contributed mainly by the
largest eddies. The vertical extent of the latter is 
and thus
vertical averaging is expected to represent fairly well their
contribution.
We consider the following three disk-regions: radiation pressure dominated region, gas pressure dominated region with electron scattering opacity, and gas pressure dominated region with free-free absorption opacity. Since turbulent convection is taken to be the source of the disk turbulent viscosity, the convective flux plays an important role in the transfer of energy to the disk surface.
A detailed description of the model and its underlying equations can be found in Goldman & Wandel (1995). In what follows we summarize and discuss the main results.
The turbulent
viscosity and the convective flux were obtained as functions of the
physical parameters of the disk, which in turn are controlled by the
former
two. Having a model for turbulence, there is no need to resort to
phenomenological parameterizations of either the viscosity (as in
-disk models) or the convective flux (as in the mixing
length approach).
Solutions for both radiation pressure dominated (inner)
and for
gas pressure dominated (outer) regions, with either electron scattering
or free-free absorption opacities, were obtained. The solutions give
the midplane temperature, density, the disk half-width and the ratio
between the convective and total fluxes at a given distance, for given
central mass and given accretion rate. The solutions depend on 
---the fraction of the convective
flux out of the total flux. We obtain 
as a function of the local physical conditions in the different regions,
from an approximate
estimate for the ratio of the surface to midplane temperatures. In
Figure 1 we display the luminosity (in units of the Eddington
luminosity) versus
the electron scattering optical depth, for the various solutions.
Figure: Luminosity, in units of the Eddington luminosity, vs.
the electron-scattering optical
depth, for the various convective disk solutions, at a constant
distance 
for 
. R:
radiation pressure region, strong convection.
BBS: gas pressure blackbody solution in the strong convection
limit. BBM: same region for the
case of moderate convection. ESM: gas
pressure electron-scattering solution in the moderate convection limit.
Figure: Radiation pressure dominated
region: the fraction of the convective flux 
as function of the luminosity, in
units of the Eddington luminosity. The plot is for
, 
and a radiation pressure
comprising 
of the total pressure.
In the gas pressure
dominated regions we find two distinct types of solutions. The first
one is that of strong convection in which the
convective flux is much larger than the radiative flux. The
blackbody region encompasses the entire gas pressure region and could
extend down to the inner disk boundary, with no radiation
pressure region. The second solution is that of moderate convection,
with a convective flux comprising 
of total flux and all
disk-regions exist.
The radiation pressure solution corresponds to strong convection. In the
present model the radiation pressure region can exist only for accretion
rates higher than 
times the Eddington rate. In Figure 2 we show
as function the luminosity (in units of Eddington luminosity) for
this region. The values of 
range from 
up to very close
to 1, for luminosity varying between 
to few times the
Eddington luminosity, respectively. This dependence of 
on the
accretion rate gives rise to a surface density which increases with
the accretion rate. As a result, the radiation pressure region is
secularly stable (as are also the gas pressure dominated solutions); see
Figure 1. It is interesting to note that the trend of convection to
stabilize the radiation pressure dominated region, is evident even in
models in which the disk viscosity is described by an 
parameter and the convective flux by the mixing length approach (Milsom,
Chen, & Taam 1994).
The radiation pressure region is optically thick (see Figure 1) and geometrically thin even for super Eddington luminosities. Therefore, for luminosities of the order of the Eddington luminosity, the local spectrum will be a modified blackbody.
The values of the effective 
-parameter in the different
regions
are quite small: 
, 
, and
for radiation pressure region and for the two
types of solutions in the gas pressure region, respectively. These
small values result from the subsonic turbulent velocities
and from the anisotropy of the largest eddies.
This anisotropy, due to the
Coriolis force, is a general feature of a three-dimensional turbulence
in a rotating disk (Goldman 1991). The above values are small
compared to values required to model outbursts
of cataclysmic variables (see, e.g., Duschl 1989). This implies that
turbulent convection cannot account for the disk viscosity in this case.
A different mechanism, possibly magnetic viscosity, is required. One may
speculate that turbulent convection can be the source of the disk
viscosity during the quiescent state and some other mechanism provides
the disk viscosity during the outburst.
For the same central mass and accretion rate, the
innermost regions could be either radiation pressure dominated or gas
pressure dominated blackbody (strong convection case). The values of the
corresponding effective
-parameter differ only by a factor of 
, but in
the former case the disk is much thicker geometrically and much less
dense than in the second case. The emitted flux is a
modified blackbody while in the second case it is a blackbody.
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