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

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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 [FORMULA], therefore this region extends beyond [FORMULA], where [FORMULA] is the solution of

[EQUATION]

or

[EQUATION]

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

[EQUATION]

into the expression for the anisotropy factor (Eq. (53)) to obtain

[EQUATION]

and since [FORMULA] for [FORMULA], [FORMULA]. To see if this is the correct behavior of x as [FORMULA], 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 [FORMULA], [FORMULA] and Eq. (38) yields, in that limit,

[EQUATION]

where b, [FORMULA] are given by Eq. (35).Now, calculating the limit as [FORMULA] for both sides of Eq. (64)and solving for [FORMULA], we obtain

[EQUATION]

where [FORMULA] is expressed in units of [FORMULA], R in units of [FORMULA] and we have assumed 1 solar mass for the central object. [FORMULA], function of the Prandtl number, given by

[EQUATION]

is of order of 1. Assuming [FORMULA] is not much smaller than 1, for reasonable values of [FORMULA], say [FORMULA], our approximation [FORMULA] is not so bad.

In the previous section our main concern was to find a solution with [FORMULA]. As a matter of fact, for [FORMULA] lesser than the value given by Eq. (63), i.e. [FORMULA], 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 [FORMULA] 1, towards the surface of the disk. In the upper branch, x increases monotonically from high values to [FORMULA], as we come close to the surface of the disk. Geometrically speaking, the anisotropy factor is a measure of the eddy dimensions, i.e.,

[EQUATION]

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 [FORMULA], 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 [FORMULA], [FORMULA], one would expect [FORMULA], 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 [FORMULA]. Inserting x given by Eq. (53) into Eq. (56), with [FORMULA], yields

[EQUATION]

From that equation we see that under criticality, [FORMULA]. We must realize that to have a value of x close to this, [FORMULA], which diminishes the effect of shear. To see where the convective region starts in z, we still may use [FORMULA] given by Eq. (67), the dependence of [FORMULA] being absorbed into [FORMULA]. However, [FORMULA] now is larger and [FORMULA] 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, [FORMULA] 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 [FORMULA], 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., [FORMULA] 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 [FORMULA]. 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., [FORMULA] 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., [FORMULA]. As we approach the surface of the disk, [FORMULA] the interaction time is zero and [FORMULA].

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