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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