2 Comparing the $\alpha$ and $\beta$-viscosities

The laboratory experiment which is the most pertinent to studies of hydrodynamical instabilities in differentially rotating flows is the Couette-Taylor experiment: a fluid is sheared between two cylinders rotating at different speeds. Only a few experiments have been run in the case where the angular momentum increases outwards, as in a Keplerian disc, but from the torque measurements that are available, it appears that the turbulent viscosity scales as

$$\nu_ {\rm t} \propto R^3 \left\vert\frac{{\rm d}\Omega}{{\rm d}R}\right\vert,$$ (1)

where R is the distance from the rotation axis and $\Omega$ is the angular velocity (Richard & Zahn 1999). In a Keplerian disc, this relation is equivalent to

$$\nu_\beta = \beta \Omega R^2,$$ (2)

which we shall call the $\beta$-prescription.

This prescription was originally proposed by Lynden-Bell & Pringle (1974) and some aspects of it have been discussed in later papers (De Freitas Pacheco & Steiner 1976; Thompson et al. 1977; Lin & Papaloizou 1980; Williams 1980; Hubeny 1990). It has been recently revived by Biermann & Duschl (1998; see also Duschl et al. 2000) on the grounds that the only relevant scales in a Keplerian disc are the angular velocity and the radius, since they contain all the information about the rotation and the curvature of the flow. These authors make the sensible choice of equating the parameter $\beta$ with the inverse of the critical Reynolds number ${\cal R}e$, which they assume to be the same as in plane-parallel shear flows (i.e. ${\cal R}e \approx 10^3$ and therefore $\beta \approx 10^{-3}$).

However, in the following we shall use a smaller value for this $\beta$-parameter, which is derived from the Couette-Taylor experiment. The only experimental data available, in the case where angular momentum increases outwards, are those obtained by Wendt (1933) and Taylor (1936), in which the inner cylinder is at rest; from these one derives $\beta \approx 10^{-5}$. This value is compatible with the critical gradient Reynolds number which characterizes these experiments (see Richard & Zahn 1999):

$${\cal R}e^*_{\rm c} = \frac{R^3}{\nu} \left\vert\frac{{\rm d}\Omega}{{\rm d}R}\right\vert \approx 6~10^5 ,$$ (3)

where $\nu$ is the molecular viscosity of the fluid. Clearly, new experiments are required to measure this parameter in regimes which resemble more the Keplerian flow; in the meanwhile, until new results are obtained, we shall adopt the value quoted above.

One important property of this $\beta$-viscosity is that it depends only on the radius in a Keplerian disc, and does not involve local physical conditions. On the contrary, the standard $\alpha$-prescription (Shakura & Sunyaev 1973) depends on the local values of the pressure scale height H and sound velocity $c_{\rm s}$ (taken in a broad sense - it is the Alfvén velocity if magnetic pressure dominates):

$$\nu_\alpha = \alpha c_{\rm s} H .$$ (4)

To emphasize the difference between the present prescription and the standard law, we use the relation $c_{\rm s} = \Omega H$ which expresses the hydrostatic equilibrium in a Keplerian vertically averaged disc, and rewrite Eq. (4) as

$$\nu_\alpha = \alpha \Omega R^2 \left(\frac{H}{R}\right)^2_\al... ...{\alpha}{\beta} \times \left(\frac{H}{R}\right)^2_\alpha \cdot$$ (5)

We see that the ratio $\nu_\alpha/\nu_\beta$ depends on the aspect ratio of the $\alpha$-disc. In general, the quantity $(H/R)_\alpha$ is weakly sensitive to both R and $\alpha$, but it depends strongly on the central mass and the accretion rate, and is typically in the range 0.001-0.1 (e.g. Huré 1998).

The $\alpha$-prescription has been designed such that the Mach number is smaller than unity for $\alpha \le 1$. In contrast, subsonic turbulence is not guaranteed with the $\beta$-prescription. This restriction on subsonic turbulent velocities is dictated by the fact that supersonic turbulence would be highly dissipative and therefore difficult to sustain. Moreover, the turbulent viscosity used here has been measured in a liquid, and its application to compressible fluids can only be justified in the subsonic range. Therefore one has to check whether the $\beta$-prescription predicts subsonic turbulence in a given disc. By making the reasonable assumption that the vorticity $v_{\rm t}/\ell_{\rm t}$ of the turbulent eddies is of order $\Omega$ ($v_{\rm t}$ and $\ell_{\rm t}$ denoting respectively the typical velocity and length scales of turbulence), and setting $\nu_\beta = v_{\rm t} \ell_{\rm t}$, the Mach number in a $\beta$-disc is given by

$${\cal M}_{\rm t} = \frac{v_{\rm t}}{c_{\rm s}} \simeq \sqrt{\... ... c_{\rm s}} = \sqrt{\beta} \left(\frac{H}{R}\right)_\beta^{-1}$$ (6)

where $(H/R)_\beta$ is the aspect ratio of the $\beta$-disc. We note that ${\cal M}_{\rm t} < 1$ provided $H/R> \sqrt{\beta}$, which is not a too severe requirement with our low value of $\beta$ (the situation would be more problematic with $\beta=10^{-3}$, conflicting with the Keplerian assumption).