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Astron. Astrophys. 317, 290-298 (1997)

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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

[EQUATION]

Now, using the expression for the thermometric diffusivity for [FORMULA] K, i.e.,

[EQUATION]

we obtain

[EQUATION]

Substituting this last expression in the dispersion relation, Eq. (7), results in

[EQUATION]

where

[EQUATION]

and [FORMULA] 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

[EQUATION]

which substituted into Eq. (45) yields the following equation for the anisotropy factor

[EQUATION]

The required condition for this equation to have a physical solution for the anisotropy factor is

[EQUATION]

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

[EQUATION]

(Cabot et al. 1985). Since the dependence on the radial distance only enters through dependence on [FORMULA], we may write

[EQUATION]

Applying this expression into Eq. (49) results in

[EQUATION]

where [FORMULA], defined by comparison with Eq. (49), does not depend on the radial distance. In this inequality [FORMULA] is expressed in units of [FORMULA] 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

[EQUATION]

where

[EQUATION]

Substitution into Eq. (46) for the Rossby number yields

[EQUATION]

Since [FORMULA], and the Brunt-Väisälä frequency goes to zero as [FORMULA], [FORMULA] 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., [FORMULA] 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

[EQUATION]

where [FORMULA] corresponds to the critical regime. We want to study the regime [FORMULA]. Inserting into Eq. (49), yields

[EQUATION]

Let us assume we have a solution [FORMULA] for [FORMULA]. As we know, in that limit, [FORMULA]. Therefore, in that limit,

[EQUATION]

which implies

[EQUATION]

Solving for [FORMULA], [FORMULA], we have

[EQUATION]

or

[EQUATION]

which also implies Eq. (58). However, now the Brunt-Väisälä frequency is much greater than the Keplerian angular velocity and [FORMULA]. A cautious analysis of Eq. (57) reveals the absence of solutions with x <1 and [FORMULA].

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