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

We shall adopt a theory of turbulence that takes into account the nature of different feeding mechanisms in the energy equation. So, we shall be using the work of Ledoux et al. (1961), in the interpretation given by Canuto et al. (1984) which,essentially, consists in the prescription of a closure relation that yields an expression for the viscosity once we know the growth rate and the wavenumber of the instability that generates turbulence. We, however, shall improve this formulation a little bit by showing, explicitly, the dependence of the Rossby number on the wavenumber interval within which turbulence is fed into the system. To make our point clear, we shall assume that energy is injected into turbulence in the wavenumber interval (injection region)and transported by nonlinear processes to the inertial and dissipation regions . Since our main concern is the injection zone, we shall assume that this region smoothly joins the injection zone at , and the flux of energy is constant throughout. In the injection zone, we write a kind of Heisenberg equation

where F is the energy spectral density and we have assumed that the rate of energy deposition, , is constant in that interval. We have, also, neglected the contribution from the molecular viscosity and assumed that the rotational efects enter only through the growth rate. The most general expression for the transport coefficient, or turbulent viscosity, , is

where has the dimension of time, the correlation time. Since we don't have a theory for it, let us assume, for the moment, equality between the turnover time, the correlation time and the time characteristic of the growth of the instability that generates turbulence. Later on, we shall analyze the implications of such an assumption. We, then, set

The solution to Eq. (1) is

We now differentiate Eq. (1) respect to k and set to obtain

which is Canuto & Goldmann's (1985) prescription for the turbulent viscosity. Applying Eq. (4) to the usual prescription we obtain

where is the local sound velocity, H is the scale height of the disk, L is the size of the convective region, n is an integer related to and L and

If we now adopt the linearization procedure proposed by Goldreich & Schubert (1967), for axisymmetric disturbances, in the version given by Canuto et al. (1984), we obtain the following equation satisfied by the growth rate for the mode that feeds energy into the system,

where is the Rayleigh number given by

is the effective Taylor number given by

being the Prandtl number given by

with , , , and being respectively the z -component of gravitational acceleration, coefficient of thermal expansion, temperature gradient excess over the adiabatic gradient, thermometric conductivity (thermal diffusivity) and kinematic viscosity. Let us now analyze the conditions under which the validity of the assumptions we have made holds. In steady state, the energy injected into turbulence equals the energy dissipated, i.e.,

which results in

In the above equation is the Rossby number defined by

From Eq. (12) we see that which, obviously, has to be the range of validity of this formulation. However, as it will be shown, the Rossby number is proportional to the Brunt-Väisälä, frequency which as . Therefore, in that formulation, there will be a large z extent, above the midplane, for which there will be no solution. A way to remedy this, is to realize that the effects of rotation enter, not only through its implicit effects in the growth rate of the convective instability but also, explicitly, through generation of waves, i.e., wave turbulence. That is to say, not all the energy goes into turbulence, but a part of it goes into waves. We, therefore, rewrite Eq. (1) as

which implies

where the efficiency for energy deposition into turbulence is and is the efficiency for energy deposition into waves. This kind of procedure is equivalent to take into account the inverse cascade process. It should be remarked that the equality between the turnover and correlation times is no longer valid, i.e.,

where and B are, respectively, turnover and correlation times. Now the energy that goes into turbulence, i.e. the efficiency, depends on the Rossby number. We postpone this discussion. The definition of the viscosity parameter will change, accordingly, to

Our next task will be to solve the disk equations assuming our system can settle down in a stationary state. With the solutions for the physical variables in that state we can have all the similarity numbers and so insert them into Eq. (7) and see if our starting system is convectively unstable ( ).

© European Southern Observatory (ESO) 1997