Discussion

In this paper we have studied the process of magnetic field generation in thin accretion discs via the MHD dynamo. The basic problem is that of the isolated disc immersed in the vacuum devoid of any externally maintained magnetic field. On that issue we have obtained a number of basic results that can be summarized as follows.

1. Thin accretion discs, including protoplanetary discs, characterised by fiducial values of , , and , can self-generate a global magnetic field by means of turbulent MHD dynamo. The time required by a magnetic field to reach its global equilibrium configuration is about an order of magnitude longer than the Kepler time at the outer edge of the disc. However, a magnetic field in the inner portion of the disc equilibrates much faster. Generally, a magnetic field within a segment of the disc located between the inner edge and a radius R needs time proportional to to equilibrate. This time depends only weakly on the initial conditions, providing the initial magnetic field has no reversals along the radial extent of the disc. A magnetic field growing from initial conditions with reversals needs about an order of magnitude longer time to achieve equilibrium.
2. The equilibrated structure of the magnetic field is such that locally the strength of the magnetic field is of the order of . That means that, in absolute terms, the strongest field is concentrated towards the inner edge of the disc. The direct dynamical importance of this field is questionable inasmuch as the equilibrated field strength is such that magnetic pressure is smaller than the gas pressure by a factor of about . Within the disc the field is mainly toroidal, with the poloidal component at least an order of magnitude smaller and dominated by its radial part. The calculations always produce a field that has, within reasonable accuracy, a quadrupole symmetry, notwithstanding the lack of the formal symmetry requirement in the governing nonlinear equation.
3. In protoplanetary discs there exists a `magnetic gap', the segment of the disc where the magnitude of an equilibrated magnetic field is about two orders of magnitude lower than . This region coincides with the part of the disc with the lowest degree of ionisation. Due to the existence of such a gap, the magnetic field structure is naturally divided into two parts: inner, where the strongest field is found, and the more extensive outer part, where a much weaker (in absolute terms) field is established. Apart from the existence of a magnetic gap, the features of a protoplanetary disc dynamo-generated field are much like those of the standard disc dynamo-generated field.

The realistic discs are immersed in environments that are likely to be magnetised by sources located far away from the disc, or at `infinity'. Because the sources of the ambient field are distant, the field close to the disc is weak compared to inside the disc. Considering a thin accretion disc self-generating a magnetic field and surrounded by an environment permeated by a weak vertical magnetic field we have obtained the following results.

1. Given a fiducial value of , an external field cannot be significantly amplified by means of advection by the accretion flow. Therefore, the total magnetic field is, in good approximation, a superposition of the internally generated field and an externally maintained field. Although this can be deduced a priori from the governing equation, we also have obtained this as a result, running cases with the advection term included.
2. Because an external field is assumed to be vertical it modifies only the vertical component of a dynamo-generated field. The superposition of a self-generated field having an even symmetry with an ambient field having an odd symmetry produces a magnetic field that is asymmetric with respect to the equator. On a large scale, this asymmetry is very pronounced even for an external field as weak as a few percent of .
3. In the absence of an external field the lines of poloidal field are closed loops. The vertical ambient field helps some of these lines to become open. The ambient field of 1% strength causes most field lines leaving the surface of the disc to be open.
4. Because of the intrinsic asymmetry, open field lines leaving the upper and lower disc's surfaces are inclined in different senses with respect to the vertical.

Can a magnetic field produced by an accretion disc dynamo lead to centrifugally driven wind? The present work cannot give an unequivocal answer to this question. Without an external field, the topology of the field lines consists of closed loops, hardly a favourable configuration for launching a wind! On the other hand, the strength of the field decreases rapidly outside disc surfaces. If we presuppose that the base of the wind coincides with the disc surface, the accelerating wind can, in principle, open up field lines and propagate to infinity. To see whether this is indeed a conceivable scenario, a wind solution has to be found to supplement the dynamo solution we have obtained in this paper. In the presence of an ambient field with reasonable strength, field lines leaving the disc surface become open, and the possibility of a magnetic field configuration to drive a wind is more readily visualisable. However, as we have shown, an external field leads to an asymmetry in magnetic field configuration, which may translate into an asymmetric appearance of a wind. This can be viewed as a disadvantage of the model, as most observed winds from protostellar or extragalactic sources are symmetric (two-sided), or as a built-in advantage because such a model can naturally explain the phenomena of asymmetric (one-sided) jets. The hypothesis that the superposition of an even (internally generated) magnetic field with an odd (externally maintained) magnetic field may lead to a significant difference between the strengths of the poloidal field on the two sides of the disc and consequently to differences in the power of the wind coming from the top and bottom disc surfaces was originally forwarded by Blandford (1989) [4].

For the magnetic field to be able to launch a cold, centrifugally driven wind, the field lines emerging from the disc surface should make an angle, i, larger than with the normal to the disc (Blandford & Payne 1982 [3]). Sidestepping the problem of whether the closed lines can be opened by the wind itself, we calculated the angle i assuming the magnetic field configuration maintained by a dynamo operating in the standard disc. Fig. 7 shows that in the absence of an ambient field the lines emerging from both disc surfaces make providing that they exit the disc at locations such that cm. As the field lines must eventually return into the disc, they form a negative (leaning toward the central star with respect to the normal to the disc) angle i in the outer portion of the disc. It is interesting to notice that a cold wind can be accelerated along the field line leaning inward providing that . However, in this case gravity, rather than centrifugal force, is responsible for the acceleration. Consequently, the acceleration diminishes with elevation from the equator and eventually vanishes altogether (unless the field line is modified and starts to lean outward). Therefore, a wind that originates from field lines that on the surface of the disc are inwardly inclined, should probably be called a `gravitationally launched wind'. An intriguing possibility that such a wind exists has to be, of course, checked by future calculations. Tentatively assuming that a wind launched either centrifugally (inner disc) or gravitationally (outer disc) is able to eventually open up field lines, we may conclude that the magnetic field resulting from the dynamo action in the absence of an ambient field could lead to a bipolar, symmetric wind.

 

 

Figure 7: The inclination angle, i, of the field lines as they emerge from the disc surfaces. The standard disc is assumed and panels (A-D) correspond to cases , gauss, gauss, and gauss, respectively. Solid lines refer to the top disc surface and dashed lines refer to the bottom surface. Dotted lines indicate the inclination necessary to drive a cold wind centrifugally.

Panels B-D on Fig. 7 show what happens to an inclination angle of emerging field lines if an external field is present. An external field causes an inclination angle on the top surface to differ from an inclination angle on the bottom surface. This difference increases with the strength of an external field. At gauss, i is positive along the entire top disc surface but negative over most of the bottom disc surface. Additionally, an external field `verticalises' field lines as they emerge from disc surfaces. This is most visible on panel D of Fig. 7 with over a significant portion of the top surface. Thus, on one hand, the presence of an external field is preferable for wind launching inasmuch as it takes care of open field lines, but on the other hand it may be also detrimental because it straightens the field lines exiting the disc. In any case, winds leaving the top and bottom surfaces of the disc would have different characteristics, with at least the potential possibility that one could be much weaker than the other.

We can speculate that most discs are immersed in an environment with an ambient magnetic field too weak to influence the symmetric configuration of an internally generated magnetic field. Such discs may produce bipolar winds. Sometimes discs happen to be located in a more magnetised environment; those discs may produce winds that appear one-sided. Finally, we should point out that although a dynamo-generated magnetic field may prove to have the proper configuration for wind launching, it may lack the strength to produce dynamically important winds. Pelletier & Pudritz (1992) [15] showed that the ratio of the wind torque to the viscous torque on a fluid element of a disc located at a radius R is given by , where is the Alfven radius of flow beginning from the disc at radius R. A very rough estimate of this ratio yields about 0.1, suggesting the dynamical unimportance of the wind. This is because the dynamo field is concentrated in its toroidal component, and only a very small fraction of its total strength is available for a vertical component. Clearly, it has to be viewed as a major obstacle for inducing a dynamo-generated magnetic field as a launching pad of a meaningful, centrifugally driven wind. However, it is important to remember that many assumptions contributed to our final result. In particular, allowing for large values of magnetic Prandtl number would result in advection becoming an important part of the magnetic field amplification process and may lead to having a magnitude comparable to the magnitude of the entire field.