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*Astron. Astrophys. 317, 290-298 (1997)*
## 6. Conclusions
We have treated turbulence with rotation in a thin keplerian disk.
Convection is the process assumed to generate turbulence, and we have
used Canuto et al. (1984) treatment for the convective instability,
whose characteristic growth time we have taken equal to the turnover
time. To obtain the turbulent viscosity, we have modified Canuto &
Goldman's (1984) procedure to take into account rotational effects by
introducing, through an efficiency factor
, the effects of inverse cascade process in the
energy equation. This kind of procedure, together with the assumption
of inequality between turnover and correlation times, allows us to to
extend this treatment to regimes with
. Rotational effects are also taken into
account, explicitly, through an isotropy factor *x*, simply
related to the Rossby number and, since in our basic state the disk is
centrifugally supported, centrifugal effects are implicitly
considered. Differently from Canuto et al. (1984) paper, we do not
obtain the anisotropy factor through a growth rate maximization
procedure. We, instead, obtain an expression for the turbulent
viscosity, using the solution for the disk equations, assuming an
opacity mainly given by ice. By forcing equality between this
viscosity and that obtained through modification of Canuto &
Goldman's (1984) procedure, an equation for the anisotropy factor is
obtained. It is shown that the accretion rate, the amount of energy
that goes into waves and the intensity of rotation hinder the
establishment of a turbulent convective regime. High values of the
mass of the central object and Brunt-Väisälä frequency
go in the opposite direction. As a matter of fact, there is a
condition to be satisfied for these parameters in order to have a
solution for the anisotropy factor. Using that condition, we can
define a local critical maximum rotation intensity (angular velocity)
above which there is no solution for the anisotropy factor. Below it,
there are two branches of solutions: the upper and the lower branches.
For
, the convective occupies the whole *z*
extent of the disk and in the upper branch
always, which implies longitudinal turbulent
scales much greater than the horizontal ones, even close to symmetry
plane, where the Rossby number
. In the lower branch,
close to the symmetry plane but, as we
approach the surface of the disk,
. In that branch, close to the surface of the
disk, the horizontal scales are much greater than the longitudinal
ones. This kind of behavior is somehow intriguing in the sense that it
contradicts the behavior we should expect from Dubrulle &
Valdettaro's (1992) qualitative analysis. It should be stressed that
no solution with both *x* and
was found.
Another kind of solution is obtained for higher values of the
accretion rate. In that case the convective region starts for a value
of
not much smaller than 1. Above that point,
there are two branches of solutions for the disk equations, both
starting with
and
not much greater than 1. In the upper branch,
*x* increases as we approach the surface of the disk, decreasing
the size of the turbulent structures. In the lower branch an opposite
behavior is found. For both these solutions, generation of waves is
expected close to the point where the convective region starts. In
that case, Dubrulle & Valdettaro's description applies.
We would like to mention that our results are somehow similar to
results of Daniels (1980), who has reported that under the the effects
of the centrifugal acceleration, depending on the parameters of the
system, the onset of convection will occur for smaller values of the
critical Rayleigh number. We should stress, however, our basic state
is centrifugally supported, quite different from Daniels'(1980)
pressure supported. Besides this, we have considered a flow with
(nonturbulent Prandtl number)
and for Daniels (1980),
.
To understand what we mean by that, let us write the equation of
force for an element of fluid, as seen by an observer in a frame
rotating with Keplerian angular velocity, i.e.,
where
is the relative velocity,
is the gravitational potential and
is any nonconservative force. To zeroth order,
in stationary regime, the relative velocity is zero and all the terms
in the lefthand side of Eq. (70) are zero but the centrifugal term. To
order superior in
*H*/*r* , the first term in the lhs is of fourth order,
while the 2nd and the 3rd terms are of 2nd order, having comparable
magnitudes. However, the centrifugal term is zeroth order and much
bigger.Therefore, even not making a detailed analysis of the role
played by these new terms due to shear and to centrifugal
acceleration, they may justify our claim that Chandrasekhar's (1961)
formulation may not apply to accretion disks.
Finally, we would like to mention that our results depend on the
assumption of constant efficiency for energy going into waves, as well
as on the assumed ratio between radial distance and scale height of
the disk. In a future contribution we shall consider rotational
effects in accretion disks without resort of those assumptions.
© European Southern Observatory (ESO) 1997
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