The observational qualities of protoplanetary discs surrounding T Tauri stars have been extensively discussed (for a recent review see Beckwith 1994 [2]). It appears that most of the observations can be at least qualitatively explained by a model involving a steady-state accretion disc surrounding a young, low-mass star. The differences between such a disc and a disc around a compact star follow from scale, protoplanetary discs being much bigger than a disc around, say, a typical white dwarf or a neutron star. Located farther away from a central gravitational well, protoplanetary discs experience less vigourous shear and are therefore cooler. At those low temperatures the opacity is dominated by dust grains and not by thermal bremsstrahlung as is assumed in a standard disc model. The different opacity law would produce a qualitatively different disc, so using the standard model for protoplanetary discs would be inappropriate. Furthermore, the low temperature in the extensive parts of a protoplanetary disc translates into the low ionisation regime, which is maintained by nonthermal ionisation processes (Umebayashi & Nakano 1988 [24], Stepinski 1992 [22]). Thus, in studying MHD dynamo in protoplanetary discs we cannot make the usual perfect conductor approximation.
For the purpose of our calculations we assume a protoplanetary disc surrounding
a young star with a mass of 
and extending from 
AU to 
=40 AU. This choice represents a compromise between
the physical realities and manageability of our computer calculations.
The inner radius should coincide with the radius of the star, which for the
protostar burning deuterium is about 
AU
(Shu, Adams & Lizano 1987 [21]).
The outer radius should reflect the characteristic radial size of observed
discs 
AU. Adopting such values would yield 
, requiring an unbearably long time to complete our numerical calculations. Instead, we opted for a disc of roughly the size of the present-day solar system. We adopt 
and 
/yr, fiducial values for this class of discs. The opacity
law, needed to determine the structure of the steady-state disc, is taken from Ruden & Pollack (1991) [18]. The formulas for the physical quantities in such a disc
are listed by Stepinski et al. (1993) [23].
The evolution of the magnetic field is calculated starting from an initial condition of a purely toroidal uniform field of magnitude equal to 1% of
. The evolution of the field to equilibrium takes about 4000 yr, a very short time in comparison to the viscous timescale. We consider
three cases, corresponding to different magnitudes of an ambient field: 
gauss, 
gauss, and 
gauss. Note that even the strongest ambient
field considered has a magnitude equal to just few percent of 
.
In the absence of an external field, the dynamo generates a field with a quadrupole symmetry as shown on Fig. 3. As in the case of the standard disc dynamo, the field inside the disc is dominated by its toroidal component. The most striking feature of the generated field is that it can be divided into two almost separate parts: the inner part, which is located within an inner 4 AU, and the outer part, which is located from about 7 AU outwards. The section of the disc between 4 AU and 7 AU seems to be almost devoid of any magnetic field. This section of the disc coincides with the so-called magnetic gap (Stepinski et al. 1993 [23]), the radial segment of the disc where neither thermal nor nonthermal ionisation processess are strong enough to provide coupling between the gas and the magnetic field. According to the local dynamo criterion (Stepinski et al. 1993 [23]) there should be no magnetic field there. However, as can be seen on panel C of Fig. 3, the present, global calculations reveal that a magnetic field exists in the magnetic gap, although its magnitude there is up to 2 orders of magnitude smaller than could be expected if this region were well ionised. The existence of a weak field within the magnetic gap is due to the transport of the magnetic field from the neighbourhood regions by means of radial diffusion.
, whereas the upper dotted line indicates the magnitude of the magnetic field in equilibrium with gas pressure.
Fig. 4 shows both the large-scale view of the poloidal field configuration and the zoom-in view on the inner segment of the disk. It is clear that in the absence of any external field the field lines are closed loops. The strongest magnetic field is in the innermost disc, close to the star. Its structure (lower panel) looks like a miniature version of the overall poloidal field (upper panel). The region of the weak field, separating segments of the disc where thermal and nonthermal ionszation processes are effective, is clearly identifiable on the lower panel of Fig. 4.
As in the case of the standard disc, the presence of a weak ambient magnetic field has no practical effect on either toroidal or radial components of
the generated field. However, it influences the vertical component, making the
entire field asymmetric and causing some field lines to become open instead of closed. Fig. 5 shows the configuration of the poloidal field in the
presence of an 
gauss external field. This is, comparatively speaking, a very weak field. However, it manages to open a few field lines.
Among field lines emerging from the upper surface of the disc, those located
close to the inner edge of the disc are easiest to open up. Among the field lines emerging from the lower surface of the disc, those located close to the outer edge of the disc open first. On the large scale (Fig. 5 upper panel) this introduces asymmetry with respect to the equator, although on the smaller scale (Fig. 5 lower panel) this asymmetry is hardly noticeable.
gauss. Dashed lines indicate the disc's surfaces and solid lines are lines of poloidal field. The upper panel shows a large-scale
view and the lower panel shows a close-up view of the inner portion of the disc.
Fig. 6 shows the configuration of the poloidal field in the
presence of an 
gauss external field. Although still weak in comparison to the total field generated in the disc, such an external field is strong enough to significantly alter the poloidal component of
the generated magnetic field. Its presence causes most of the field lines to
become open. The overall structure of the poloidal field, as viewed on the large
scale, is not unlike the structure of the poloidal field in the standard disc
in the presence of an external field of percentagewise similar strength.
Note that it is the strength of an external field relative to
the equilibrium strength of the dynamo-generated field that determines the
shape of the overall magnetic field. On the smaller scale, the structure of
the poloidal field in protoplanetary and standard discs differs because of the
existence of the magnetic gap region in the former.
gauss. Dashed lines indicate the disc's surfaces and solid lines are lines of poloidal field. The upper panel shows a large-scale
view and the lower panel shows a close-up view of the inner portion of the disc.
