5 Discussion

If the disc dynamo is sufficiently strong, our model develops a clearly structured outflow which is fast, cool and rarefied within a conical shell near the rotation axis where most of the angular momentum and magnetic energy is carried, and is slower, hotter and denser in the region around the axis as well as in the outer parts of the domain. The slower outflow is driven mostly by the entropy contrast between the disc and the corona, but the faster wind within the conical shell is mostly driven magneto-centrifugally. Without a central mass sink, the flow near the axis is faster, but otherwise the flow structure is similar to that with the sink.

The half-opening angle of the cone with hot, dense gas around the axis is about $20^\circ$-$30^\circ$; this quantity somewhat changes with model parameters but remains close to that range. The outflow in our models does not show any signs of collimation. It should be noted, however, that not all outflows from protostellar discs are actually collimated, especially not at such small distance from the source. An example is the Becklin-Neugebauer/Kleinmann-Low (BN/KL) region in the Orion Nebula (Greenhill et al. 1998), which has a conical outflow with a half-opening angle of $30^\circ$ out to a distance of 25-60 AU from its origin. Therefore, collimation within a few AU (the size of our computational domain) is expected to be only weak.

The region around the fast, cool and rarefied conical shell seen in Fig. 7 is similar to the flow structure reported by Krasnopolsky et al. (1999); see their Fig. 1. In their model, however, the thin axial jet was caused by an explicit injection of matter from the inner parts of the disc which was treated as a boundary. In our reference model the fast outflow is sub-Alfvénic because of the presence of a relatively strong poloidal field, whereas in Krasnopolsky et al. (1999) the outflow becomes super-Alfvénic at smaller heights. Outside the conical shell the outflow is mainly pressure driven, even though the criterion of Blandford & Payne (1982) is fulfilled. However, as Casse & Ferreira (2000b) pointed out, pressure driven outflows might dominate over centrifugally driven outflows if thermal effects are strong enough.

In our model, matter is replenished in the resolved disc in a self-regulatory manner where and when needed. We believe that this is an improvement in comparison to the models of Ouyed & Pudritz (1997a, 1997b, 1999) and Ustyugova et al. (1995), where mass inflow is prescribed as a boundary condition at the base of the corona. If we put $q_\varrho^{\rm disc}=0$ in Eqs. (1) and (7), the disc mass soon drops to low values and the outflow ceases. This is qualitatively the same behaviour as in the models of, e.g., Kudoh et al. (1998).

We should stress the importance of finite magnetic diffusivity in the disc: although poloidal velocity and poloidal magnetic field are well aligned in most of the corona, dynamo action in the disc is only possible in the presence of finite magnetic diffusivity, and the flow can enter the corona only by crossing magnetic field lines in the disc.

An outflow occurs in the presence of both dipolar and quadrupolar type magnetic fields, even though fields with dipolar symmetry seem to be more efficient in magneto-centrifugal driving (cf. von Rekowski et al. 2000). The effects of the magnetic parity on the outflow structure deserves further analysis.

The dynamo active accretion disc drives a significant outward Poynting flux in our model. Assuming that this applies equally to protostellar and AGN discs, this result could be important for understanding the origin of seed magnetic fields in galaxies and galaxy clusters; see Jafelice & Opher (1992) and Brandenburg (2000) for a discussion. We note, however, that the pressure of the intracluster gas may prevent the magnetized plasma from active galactic nuclei to spread over a significant volume (Goldshmidt & Rephaeli 1994).

Our model can be improved in several respects. In many systems, both dynamo-generated and external magnetic fields may be present, so a more comprehensive model should include both. We used an $\alpha^2\Omega$ dynamo to parameterize magnetic field generation in the disc because we have restricted ourselves to axisymmetric models. As argued by Brandenburg (1998), dynamo action of turbulence which is driven by the magneto-rotational instability can be roughly described as an $\alpha^2\Omega$ dynamo. But this parameterization can be relaxed in three-dimensional simulations where one may expect that turbulence will be generated to drive dynamo action. Such simulations will be discussed elsewhere (von Rekowski et al. 2002).

Since our model includes angular momentum transport by both viscous and magnetic stresses, it is natural that the accreted matter is eventually diverted into an outflow near the axis; this is further facilitated by our prescribed entropy gradient at the disc surface. We believe that this picture is physically well motivated (Bell & Lucek 1995), with the only reservation that we do not incorporate the (more complicated) physics of coronal heating and disc cooling, but rather parameterize it with a fixed entropy contrast. We include a mass sink at the centre which could have prevented the outflow, and indeed the sink strongly affects nonmagnetized outflows. We have shown, however, that the magnetic field can efficiently shield the sink and thereby support a vigorous disc wind.

The assumption of a prescribed entropy distribution is a useful tool to control the size of the disc and to parameterize the heating of the disc corona. However, it should be relaxed as soon as the disc physics can be described more fully. The energy equation, possibly with radiation transfer, should be included. This would lead to a self-consistent entropy distribution and would admit the deposition of viscous and Ohmic heat in the outflow. In the simulations by von Rekowski et al. (2002), entropy is evolved.

We believe that a mass source is a necessary feature of any model of this kind if one wishes to obtain a steady state. In the present paper the mass source is distributed throughout the whole disc to represent replenishment of matter from the midplane of the disc. Alternatively, a mass source could be located near to or on the domain boundary.

Acknowledgements

We are grateful to C. G. Campbell, R. Henriksen, J. Heyvaerts, R. Ouyed and R. E. Pudritz for fruitful discussions. We acknowledge many useful comments of the anonymous referee. This work was partially supported by PPARC (Grants PPA/G/S/1997/00284 and 2000/00528) and the Leverhulme Trust (Grant F/125/AL). Use of the PPARC supported supercomputers in St Andrews and Leicester is acknowledged.