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*Astron. Astrophys. 317, 290-298 (1997)*
## 5. The structure of the convective region
In the last section we have dealt qualitatively with the rotational
effects in the convective region of the disk where the opacity is
given by ice. In the following we shall work out the actual structure
of this region in a more quantitative way, obtaining its limits both
in *r* and *z* directions. The expression we use for the
opacity is valid for
, therefore this region extends beyond
, where
is the solution of
or
which, clearly, depends on the turbulent Prandtl number through b.
To find out where the convective region starts in *z* we insert
the criticality condition
into the expression for the anisotropy factor (Eq. (53)) to
obtain
and since
for
,
. To see if this is the correct behavior of
*x* as
, we have to find out if there is a solution of
Eq. (64) close to
*z*=0 . In the following we shall specialize to the
*q*=1 solution for the disk equations. For
,
and Eq. (38) yields, in that limit,
where *b*,
are given by Eq. (35).Now, calculating the
limit as
for both sides of Eq. (64)and solving for
, we obtain
where
is expressed in units of
, *R* in units of
and we have assumed 1 solar mass for the
central object.
, function of the Prandtl number, given by
is of order of 1. Assuming
is not much smaller than 1, for reasonable
values of
, say
, our approximation
is not so bad.
In the previous section our main concern was to find a solution
with
. As a matter of fact, for
lesser than the value given by Eq. (63), i.e.
, Eq. (48) admits 2 branches of solutions: the
upper and the lower branches. In the lower branch, the anisotropy
factor decreases monotonically from high values to very low values
1, towards the surface of the disk. In the
upper branch, *x* increases monotonically from high values to
, as we come close to the surface of the disk.
Geometrically speaking, the anisotropy factor is a measure of the eddy
dimensions, i.e.,
therefore, we are allowed to say that, close to the symmetry plane,
the pattern of the generated turbulence is very anisotropic, the
horizontal scales being much smaller than the vertical ones in both
branches. As we go farther away from the symmetry plane the level of
anisotropy increases in the upper branch and decreases in the lower
branch. In other words, to overcome the effects of rotation,
turbulence succeeds forming, all long the disk, structures much
smaller than those it would form in a medium in which rotation is
absent. In the lower branch this kind of behavior only prevails close
to the symmetry plane. In that branch, for values of *z*, such
that
,
*x* < 1 , i.e., since turbulence does not affect
longitudinal scales, the structures are much greater than those formed
in a medium in which rotation is absent. It should be remarked that we
have found here that turbulence has a behavior not in complete
agreement with the one predicted by Dubrulle & Valdettaro's 1992
paper. That is to say, according to these authors, since as
,
, one would expect
, i.e., longitudinal scales much smaller than
horizontal scales, contrarily to what we have found. It should be
stressed that this happens to small values of the accretion rate. Our
suspicion is that small values of the accretion rate imply great
departure from solid body rotation, i.e., the role played by shear is
very important. Under our formulation, let us see how Dubrulle &
Valdettaro's results emerge. To do so, let us find what is the least
value for the anisotropy factor we can have under critical conditions,
assuming
. Inserting *x* given by Eq. (53) into Eq.
(56), with
, yields
From that equation we see that under criticality,
. We must realize that to have a value of
*x* close to this,
, which diminishes the effect of shear. To see
where the convective region starts in *z*, we still may use
given by Eq. (67), the dependence of
being absorbed into
. However,
now is larger and
will not be much less than 1. Therefore, if
the system, now, finds itself in a regime for which the rotational
intensity is below the critical one, the disk equations admit two
branches of solutions as before. In the upper branch, close to the
midplane,
and increases as we go towards the disk
surface. Again, the size of the turbulent structures decreases as we
approach the surface. In the lower branch, close to the midplane
*x* has the same value as in the upper branch, but decreases as
we come closer to the surface. Here, the size of the turbulent
structures increases as we approach the disk surface. The effects of
rotation are much more pronounced. Dubrulle & Valdettaro's
description now applies.
For a given value of
, the profile of the turbulence is as follows:
crolse to a critical radius, the *z* -extent of the convective
region will be minimum, i.e.,
will be maximum. In the vicinity of this
minimum radius, generation of waves will be intense in the lower
branch, all long the *z* -extent. Compared to a medium in which
rotation is absent, turbulent structures will be larger. In the upper
branch, generation of waves decreases as *z* increases. Larger
structures are confined to the regions close to
. The value of the effective Rayleigh number at
which convection sets in is large and the effects of the Coriolis
force will prevail over the effects of the centrifugal acceleration..
Going farther away from this critical radius, the *z* -extent of
the convective region increases, i.e.,
decreases. Waves generation diminishes. The
size of the turbulent structures will decrease. The value of the
effective Rayleigh number at which convection sets in decreases and
the effects of the centrifugal acceleration starts prevailing over the
effects of the Coriolis force.
If we interpret correlation time, Eq. (16), as the time required
for interaction between different scales structures, we conclude that,
close to the symmetry plane, turbulence has a large spectrum, i.e.,
. As we approach the surface of the disk,
the interaction time is zero and
.
© European Southern Observatory (ESO) 1997
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