*Astron. Astrophys. 317, 290-298 (1997)*
## 4. The solution for the growth rate
The growth rate is highly dependent on the anisotropy factor. In
Cabot, Canuto and Hubickyj it is obtained by imposing it maximizes the
growth rate. We shall avoid this procedure and shall look for a
different one which assigns some dynamics to the anisotropy factor.
From Canuto et al. (1985) prescription for the viscosity, together
with our results of the previous section, we write
Now, using the expression for the thermometric diffusivity for
K, i.e.,
we obtain
Substituting this last expression in the dispersion relation, Eq.
(7), results in
where
and
is the Brunt-Väisälä frequency
in units of the Keplerian angular velocity. Making now the simplifying
assumption that the *z* extent of the convective region fits only
one wave, we use Eq. (41) to write
which substituted into Eq. (45) yields the following equation for
the anisotropy factor
The required condition for this equation to have a physical
solution for the anisotropy factor is
It is apparent from this inequality that the larger is the value of
the expression appearing to the left of the expression containing the
square of the Brunt-Väisälä on the left hand side, the
larger will be the value of the Brunt-Väisälä required
for the onset of the turbulent convective regime. However, to solve
this inequality and find out some effects of rotation in the
establishment of this regime, we should have the knowledge of *H*
which, at this moment, is lacking yet. A reasonable approximation for
*H*, very far away from the central object, such as the present
situation, is
(Cabot et al. 1985). Since the dependence on the radial distance
only enters through dependence on
, we may write
Applying this expression into Eq. (49) results in
where
, defined by comparison with Eq. (49), does not
depend on the radial distance. In this inequality
is expressed in units of
and *M* in solar masses.
A rapid inspection on this inequality reveals:
(1) the higher is the rotational rate the harder will be the
establishment of a turbulent convective region. That is to say, it
will require a larger temperature gradient or a higher value of
*z*, which means the extent of a non convective region above the
midplane will be larger. Higher values of the rotational rate will
imply, for sure, in low efficiency, therefore, decreasing the amount
of energy that goes into turbulence,
(2) high values of the accretion rate hinder the establishment of
the convective region, requiring larger values of the
Brunt-Väisälä frequency, which implies an increase of
the non convective region above the midplane,
(3) high values of the mass of the central object will favor the
onset of a convective regime as well as the decrease of the non
convective region close to the midplane,
(4) the larger is the extent of the convective region, the lesser
is the temperature gradient required for its establishment.
For the Brunt-Väisälä frequency satisfying in Eq.
(52), Eq. (47) will have two solutions for the anisotropy factor and
just one in the critical situation (equality). Let us solve for the
critical condition. In that case *x* is given by
where
Substitution into Eq. (46) for the Rossby number yields
Since
, and the Brunt-Väisälä
frequency goes to zero as
,
in a large fraction of the convective region.
However, what matters is the value of *x*. If
*x*<1 ,in that portion of the convective region,
through the effects of the Coriolis force, rotation will dominate the
dynamics. The inverse cascade process will prevail over the direct
one, thus inhibiting dissipation. The turbulent horizontal scale is
larger than the vertical scale, i.e.,
and so the concept of viscosity loses its
meaning. If the rotational intensity is above the critical one, no
stationary solutions are found to the disk equations. If the the
rotational intensity is below the critical, Eq. (50) will admit two
solutions for the anisotropy factor, one larger and the other smaller
than the solution we just obtained. We are now interested in this
larger solution, to see if it can change the picture we have drawn. To
make our study as analytical as possible, let us write
where
corresponds to the critical regime. We want to
study the regime
. Inserting into Eq. (49), yields
Let us assume we have a solution
for
. As we know, in that limit,
. Therefore, in that limit,
which implies
Solving for
,
, we have
or
which also implies Eq. (58). However, now the
Brunt-Väisälä frequency is much greater than the
Keplerian angular velocity and
. A cautious analysis of Eq. (57) reveals the
absence of solutions with
*x*
<1 and
.
© European Southern Observatory (ESO) 1997
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