5 Conclusion

In this paper, we have examined some properties of thin accretion discs built with the $\beta$-prescription for the turbulent viscosity, which has been derived from the transport of angular momentum observed in differentially rotating laboratory flows (Richard & Zahn 1999). This prescription may be applied whenever it predicts a turbulent velocity that is subsonic, and we have verified that, with the value we have taken for $\beta$ (i.e. 10^-5), this condition is fulfiled in most discs, around stellar objects or massive black holes, except in the outer part of AGN discs where gas pressure dominates. Only radiatively cooled discs have been surveyed; it would be interesting to investigate ADAF-type solutions with this prescription. Let us recall that our conclusions have been obtained here with vertically averaged models, and it is plausible that a more realistic 2D-model would extend somewhat the domain of applicability of the $\beta$-prescription (Huré & Richard 2000).

Although the $\beta$-viscosity and the $\alpha$-viscosity may have comparable magnitudes, they have very different effects both on the structure and on the stability of steady Keplerian discs. An important property of $\beta$-discs is that they are viscously stable and do not tend to fragment into concentric rings, contrary to $\alpha$-discs. Also, unlike $\alpha$-discs, they are thermally stable for ideal cooling processes, such as Thomson scattering and free-free absorption, as was shown by Piran (1978). However, with more realistic opacities and equation of state, models of $\beta$-discs contain a thermally unstable region corresponding to the recombination of hydrogen, much like standard $\alpha$-discs. It remains to be seen with time-dependent models whether this instability can lead, as we suspect, to a limit cycle behavior.

Acknowledgements

The authors wish to thank the referee for his competent remarks and suggestions.