Introduction

It has become increasingly clear that magnetic fields play an important role in a number of phenomena associated with accretion discs surrounding young, pre-main-sequence stars, compact objects and supermassive black holes. In particular, bipolar outflows observed in young stars and active galaxies are thought to be driven by a magnetic field somehow linked to the accretion process (see the reviews by Blandford 1993 [5], Königl and Ruden 1993 [9], and Pringle 1993 [16]). One popular viewpoint is that the field is captured from the environment of the disc and carried inward by the accretion flow. It is further envisioned that such an advection process leads to deformation and compression of the ambient field and creation of a magnetic field with the strength and geometry capable of launching collimated outflows. Another possibility is that a magnetic field governing the outflow is anchored in the disc, and can be produced locally by a dynamo process. As conditions in most accretion discs are favourable for a dynamo process to generate the field, and environments of most accretion discs are usually permeated by a magnetic field, it is most likely that the overall character of the magnetic field in and around a disc is set by the combination of the ambient field advection and internal generation. Our goal is to compute this overall magnetic field and discuss its character. In particular, we focus attention on two steady-state models of geometrically thin, Keplerian accretion discs, both powered by turbulent viscous stress. The character of the turbulence is encapsulated into three constant parameters: the Shakura-Sunyaev dimensionless viscosity , the Rossby number for turbulent motions Ro, and the magnetic Prandtl number . The typical temperatures of the first, so-called standard accretion disc model are hot enough (see Frank, King & Raine 1985 [6] for equations describing the structure of the standard disc) to ensure a very high degree of ionisation. In such a disc magnetic field losses are due to anomalous, turbulent diffusion. This is a fiducial model pertinent to an accretion disc around a compact star. The second model is chosen to be relevant to protoplanetary discs. Extensive portions of those discs are relatively cool (see Stepinski, Reyes-Ruiz & Vanhala 1993 [23] for equations describing the structure of a steady-state protoplanetary disc) and thus only partially ionised. Therefore, resistive magnetic field losses have to be taken into account.

In all cases we have chosen the ambient field to be uniform and vertical, its magnitude being a free parameter. Can the accretion flow in the disc drag and distort the ambient field lines so the resulting magnetic field on the surface of the disc has the strength and the geometry to start the outflow? This problem was first considered by Lubow, Papaloizou & Pringle (1994a [12]) and most recently by Reyes-Ruiz & Stepinski (1996 [17]), and the answer to the question posed above depends on the value of the magnetic Prandtl number, , the ratio of the kinematic turbulent viscosity, , to the turbulent magnetic diffusivity, . Generally, only discs characterised by large values of are able to bend and amplify the ambient field. However, it is expected that turbulence in accretion discs is characterised by , and, as we cannot identify any turbulent process leading to large values of , we are forced to conclude that accretion discs powered by turbulent viscosity are ineffective in deforming and compressing the ambient field. All results presented in this paper have been calculated assuming . Thus, the contribution of advection to the amplification of a magnetic field is not significant and the total magnetic field is well approximated by the superposition of a dynamo-generated field with the ambient field.

The motivation to consider a magnetic field produced internally within a turbulent disc is twofold. First, and foremost, turbulent accretion discs with nonuniform Keplerian rotation provide ideal conditions for the classic - dynamo action. Computing a magnetic field associated with such a disc is by itself an interesting problem. Second, in view of an apparent inability of a viscous accretion flow to drag and amplify the ambient field, the internally generated field, alone or in superposition with the ambient field, is what evidently supplies the field-launching bipolar outflows. At least this is the situation if wind-launching discs are indeed turbulent and their dynamics is dominated by turbulent viscous stress. Viscous discs are interesting inasmuch as they are, indubitably, the most frequently called-upon concept in accretion disc theory. This notwithstanding, nonviscous discs, with dynamics dominated by angular momentum removal via centrifugally driven winds, are conceivable, and models of such discs have been constructed (Königl 1989 [8], Lovelace, Romanova & Newman 1994 [11], Li 1995 [10]). At present, these models are not completely self-consistent and it remains to be seen whether a stable (see Lubow, Papaloizou & Pringle 1994b [13]), self-consistent, nonviscous accretion disc model can be constructed.

In addition to focusing on viscous discs we also chose to concentrate on the turbulent dynamo as the mechanism responsible for producing a large-scale magnetic field in and around the disc. There is now a growing body of evidence that well-ionised Keplerian discs may be unstable to magnetorotational instability (Balbus & Hawley 1991 [1]), which can lead to the rapid growth of a small-scale, fluctuating magnetic field. As such growth is faster than that obtained in the classic dynamo process, it is possible that this instability may set the character of a large-scale field. Because these issues are as yet unresolved we prefer to adhere to a standard dynamo theory. In addition, protoplanetary discs, one of two primary topics of our calculations, are probably stable against magnetorotational instability due to their poor ionisation.

It is helpful to enumerate the major assumptions underlying our approach to the problem of calculating a magnetic field in and around a disc.

1. All properties of the disc are calculated using steady-state, nonmagnetised models and remain unchanged by the generated magnetic field. Disc models depend on only two parameters: dimensionless viscosity and accretion rate . Inside the disc all quantities, except the calculated magnetic field and the degree of ionisation, are vertically uniform. The turbulence is isotropic, but it source is unspecified. The character of turbulence is assumed to be unaltered by the magnetic field, with the exception of the helicity of turbulent eddies, which is quenched by the growing magnetic field. The Rossby number for turbulent motion is assumed to be constant and equal to 1/2.
2. The axisymmetric solution of the full hydromagnetic equation is sought in the entire unbounded space. Currents located at infinity maintain the ambient field, which is vertical and uniform. The space is divided into two regions, the inside of the disc and the outside of the disc, by the disc's surface defined to coincide with its halfthickness. A vacuum is assumed to exist outside the disc.
3. The magnetic field is internally generated by the - dynamo process. The dynamo process is nonlinear inasmuch as it includes a so-called -quenching term, which is also responsible for setting the ultimate strength of the generated field. Magnetic losses are due to turbulent and resistive dissipation. Ambipolar diffusion and magnetic buoyancy losses are not considered.

In Section 1 we describe in more detail our method of finding the magnetic field configuration, including a very brief description of our numerical technique, which differs from any previously employed for disc dynamo computations. In Sections 3 and 4 we apply our method to the standard disc model and protoplanetary disc model, respectively. The entire structure of the field for several strengths of the ambient field is calculated and presented. In Section 5, we present discussion and conclusions with emphasis on the ability of computed magnetic field configurations to drive winds.