After some 20 years since Shakura & Sunyaev's 1973 proposal for parametrizing an assumed existent but unknown viscosity in accretion disks, it appears that among the possible physical processes capable of generating such viscosity one that succeeded to overcome some basic requisites (e.g. to account for luminosities and time scales) is turbulence driven by convection. This does not mean that there are no unsettled questions. Essentially, there are three main questions faced by convection to be accepted as a model for turbulent viscosity generation in some regions of accretion disks,namely the direction and efficiency of angular momentum transport (Vishniac & Diamond 1992) and the nature of turbulence generated by convection in a very fast rotating medium. The question of the efficiency of angular momentum transport is related to the value of the viscosity parameter and, according to Ruden et al. (1988) and negative. Based on linear normal mode analysis of axisymmetric disturbances, it has been claimed (Korycansky 1992) that angular momentum is transported inwards and, since for general nonaxisymmetric perturbations the radial wavenumber diverges linearly with time, convection will unavoidably transport angular momentum inwards (Goodman & Ryu 1992). For Bernard-Rayleigh convection, in a keplerian thin disk, this drawback is related to the apparent incompability between positive growth rate and positive angular momentum flux, i.e., if , angular momentum, <0 differential rotation acts as a sink of energy in convective regions, thus inhibiting convective instabilities.The answers we have for the first two questions are dependent on the convective cell dynamics, quasi-linear approach and plane wave solution (Vishniac & Diamond 1992). However, the answer to the third question involves a lot of conceptual aspects. Since energy is injected in the largest structures (eddies), the Heisenberg-Kolmogorov theory does not apply and it is very desirable to have some scheme relating the growth rate of the instability that generates turbulence to a viscosity that transfers energy from the largest eddies to the smallest ones where it is dissipated. This kind of enterprise has been undertaken by Canuto and Goldman (1984) and Canuto et al. (1984) who, using the work of Ledoux et al. (1961), were able to derive an expression for the energy spectral function and, through this, a viscosity related to growth rate of the fastest growing unstable mode. Using a formalism developed by Goldreich & Schubert (1967), neglecting compressibility effects and molecular viscosity, they have obtained two unstable modes, highly dependent on an anisotropy factor, being their contribution to the convective instability taken as a whole. It is worth remarking here, that the whole formalism developed by these authors depends heavily on this anisotropy factor, which is related to the size of the largest eddies and the to size of the convective region (extent in z direction). With rotation, for finite Taylor, Rayleigh and Prandtl numbers, convection sets in the disk only for the anisotropy factor exceeding a certain minimum value. For the Prandtl number equal to zero convection sets in for any value of anisotropy factor, independent of the rotation rate, i.e., independent of the Rossby number.
However, as has been analyzed by Dubrulle & Valdettaro (1992), in turbulent flows with rotation, some new effects come into play and may modify the standard picture we have about turbulence. In that respect, the value of the Rossby number is of crucial importance since it will determine the transition between regimes where rotation is or is not important. With rotation, there will be a tendency to constrain the motion to the plane perpendicular to the rotation axis and, as a consequence, the horizontal scale will increase as compared to the longitudinal one, which means that the turnover time in this direction will increase. The net effect is that the energy cascade down process is hindered by rotation. As a matter of fact, when rotation is present one observes two cascades: an enstrophy (vorticity) cascade from large scales to small scales and an inverse energy cascade from small scales to large scales. Since the first process is not efficient on transporting energy to the dissipation range, what we see is an energy storage in the large structures at the expenses of the small structures. This kind of behavior has been confirmed experimentally by Jacquin et al. (1990), who observed that, with rotation, , where is a parameter that depends on the Reynolds number and measures the the influence of rotation on turbulence and is the Rossby number. For very large we obtain, in the inertial range, a spectrum that goes like instead of the usual Kolmogorov's spectrum. In reality, when rotation is dominant, energy gets stored into inertial, waves which propagate it essentially in the longitudinal direction. In that case, we can no longer assign just one viscosity to the fluid and, what is the most important, the concept of viscosity loses its meaning since we no longer have local dissipation of energy. According to Dubrulle & Valdettaro (1992) is the borderline between these two scenarios: for turbulence is not affected by rotation, for it will be greatly affected. It is worth to mention that compressibility effects will also affect the turbulence through the generation of waves, shocks, etc. Numerical simulations of convection inside spherical shells carried out by Valdettaro & Rieutord (1991) are highly suggestive of the existence of this inverse energy cascade. However, the boundary conditions, the aspect ratio, the value of the Prandtl number and the Froude number they used make quite questionable the application of their results to the accretion disks.
It seems to us that the importance of rotational effects are not transparent in Canuto & Goldman's (1984) and Cabot et al. (1987) treatment of turbulence generated by convective instability, in the sense that no discussion about the behavior of the characteristic scale lengths in the problem under the influence of rotation is made, nor the conditions under which there will be local energy dissipation and an effective viscosity can be assigned to the flow. Also, it is not apparent in their results effects such as inverse energy cascade with consequent diminishing of the angular momentum transport efficiency or,even, how the spectrum in the inertial zone, i.e., Kolmogorov's spectrum, is affected by rotation. Though Dubrulle & Valdettaro's analysis be physically correct, they have neglected the role played by centrifugal acceleration and shear. As has been pointed out by Koschmieder (1967), even at small Froude numbers, effects due to centrifugal acceleration have to be taken into account. Daniels (1980) has studied motions caused by vertical instability coupled to circulation due to centrifugal forces, in a bounded circular layer. Assuming the forces that drive vertical motions are comparable to the centrifugal ones, he discovered that, depending on the parameters of the system, convection may set in at a Rayleigh number below the critical one. It seems, as we intend to show here, that due to this role played by rotation, turbulence is affected by rotation in a very peculiar way and, in some special circumstances, it succeeds forming very anisotropic structures, with horizontal scales much smaller than the longitudinal ones, in such a way as to overcome rotational effects. As far as the efficiency of angular momentum transport is concerned, the value of the viscosity parameter is highly affected, even if the Rossby number is much greater than 1.
In this work, we shall address the questions of the nature of turbulence generated by convective instability as well as the question of the efficiency of angular momentum transport. By no means are we implying that the question of the direction of angular momentum transport is unimportant, we hope to address this problem in future contributions. Our main idea is to improve Canuto & Goldman's (1984) treatment for turbulence generated by convection by taking into account rotational effects as described in Dubrulle & Valdettaro's (1992) paper for turbulent flows with rotation. Since our basic state is a centrifugally supported disk, the effects due to the centrifugal acceleration are implicitly taken into account. This basic state being axisymmetric, doesn't allow for the account of effects due to precession (Knobloch 1993). The physical conditions we shall be interested are similar to those supposedly to exist in the outer portions of the inner disklike region of the primordial solar nebula, where opacity is mainly given by ice. In the treatment we shall be giving here for the large scale turbulence, our starting point is the dispersion relation for the convective unstable perturbations which we manage to transform into an equation relating the Rossby number and the anisotropy factor, in which explicit account for energy going into waves is taken. By matching this expression with that obtained with the solution for the disk equations results in an equation for the anisotropy factor, its solution dependent upon conditions to be satisfied by the rotational rate, accretion rate, Brunt-Väisälä frequency and the efficiency for driving energy into turbulence. Ou main results stem from the analysis of this equation.
A brief outline of the procedure we shall be employing in this work is as follows:
(A.) We present our main arguments to modify Canuto & Goldman's treatment for turbulence and convective instability by taking into account the efficiency of energy going into turbulence. This is done in Sect. 2, in which we present a review of their formalism.
(B.) Under suitable appoximations, the solutions for the disk equations are obtained in Sect. 3.
(C.) In Sect. 4, we match the solution for the Rossby number coming from the solutions of the disk equations with the solution coming from the dispersion relation, obtaining that way an equation for the anisotropy factor. The conditions for the existence of solutions are analyzed.
(D.) In Sect. 5, we obtain the structure of the convective region. The nature of the turbulent structures are analyzed. At the light of the correlation time, a brief discussion of the results is made. In Sect. 7, we present our conclusions.
© European Southern Observatory (ESO) 1997