1 Introduction

What causes the transport of matter and angular momentum in accretion discs? This question is the subject of a longstanding debate, but most of those working in this field tend now to agree that it has been answered, since even a very weak magnetic field leads to a linear instability (Chandrasekhar 1961) which, in its fully developed regime, is able to provide the necessary stresses (Balbus & Hawley 1991; Hawley & Balbus 1991). However, it is unlikely that this mechanism operates in cold neutral discs (e.g. Gammie & Menou 1998). Besides, the issue which has not been settled yet is whether this magnetic instability is the only instability responsible for the "anomalous viscosity'', or whether a purely hydrodynamic instability, generated by shear in differentially rotating discs, may not be of comparable or even higher efficiency. Since a non-magnetic Keplerian disc is linearly stable, this hydrodynamic instability needs a finite amplitude perturbation to be triggered and laboratory experiments indicate that it appears only above Reynolds numbers of the order of 10⁴. Such conditions are still out of reach of the most ambitious numerical simulations; therefore, these cannot be used yet to prove or to disprove the occurrence of shear instability.

It is worth noting that the widely used $\alpha$-prescription for turbulent viscosity (Shakura & Sunyaev 1973) does not invoke any particular mechanism at the origin of turbulence: it is a simple parameterization which is tailored to yield turbulent velocities that remain subsonic (or sub-alfvénic) for $\alpha < 1$. It is often asserted that the main success of the $\alpha$-prescription is the prediction of recurrent outbursts of dwarf novae (e.g. Cannizzo 1993). However, if such eruptive phenomena are indeed of thermal origin, as is commonly thought, it is likely that many ad-hoc viscosities, not only the $\alpha$-prescription, could work as well.

For these multiple reasons it makes sense to investigate the basic properties of accretion discs built with the alternate $\beta$-prescription, which is observed in laboratory rotating shear flows, as explained below. This is our intention in this paper. We first present in Sect. 2 some properties of $\beta$-viscosity, and examine in Sect. 3 to which regimes it may be applied, allowing for subsonic turbulent velocities in geometrically thin, radiatively cooled accretion discs. In Sect. 4 we study the viscous and thermal stabilities of $\beta$-discs, using standard criteria, and draw our conclusions in the last section. The Appendix contains the vertically averaged equations for $\beta$-discs and the derivation of the criterion of thermal instability.