Accretion discs around compact stars are often modelled by a
simple steady-state model referred to as the `standard model'.
We assume a disc surrounding a compact star with a mass 
and a radius 
cm. The disc is supposed to extend from about
the surface of the star up to a distance 
. The time
required for our numerical computation depends on the ratio 
, so we have chosen 
cm and 
cm to keep calculations manageable. This gives 
,
smaller than expected in real discs, but much larger than
used by Rüdiger et al. (1995) [20]
in an earlier calculation aimed at obtaining the structure of
a dynamo-generated magnetic field in the standard accretion disc.
This improvement has been achieved because of the introduction of the nonuniform
computational grid described in section 2.1. The formulas for the
physical quantities in the standard model are listed by Frank et al. (1985) [6].
The typical midplane temperatures are in the range 
- 
K, so
thermal ionisation is sufficient to ensure that turbulent magnetic
diffusion completely dominates resistive diffusion. Therefore,
the only disc's quantities we need for our calculations are disc halfthickness
H given by
and 
given by
Here, the radial coordinate is measured in units of 
cm, the accretion
rate is measured in units of 
g sec 
, and the mass of the
central star is measured in units of 
. In our calculations we
adopt 
, 
, and 
.
Disc halfthickness is used in equation (3) to calculate 
,
as well as to define disc surfaces on which boundary conditions are imposed.
The value of 
is used in the 
-quenching term.
The evolution of the magnetic field is calculated starting from an initial
condition of a weak, uniform toroidal magnetic field of strength
. The magnetic field
grows exponentially with a timescale proportional to the dynamical
timescale at each radius. After 
s the magnetic
field equilibrates everywhere due to the 
-quenching effect.
We consider four cases, corresponding to four different magnitudes of
an ambient field: 
, 
gauss,
gauss, and 
gauss.
Note that in all cases an ambient field is weak inasmuch as
.
, 
gauss,
gauss, and 
gauss, respectively.
Lower-case letters label selected lines of poloidal field.
Fig. 1 shows the configuration of an equilibrated magnetic field.
In all cases the toroidal field, restricted to the interior of the disc,
dominates the total magnetic field. Panel A shows the isolines of the toroidal field for the case of 
. It remains the same for all
other cases because advection is ineffective. Note that if advection were
effective, it would give rise to the radial magnetic field, which, sheared
by Keplerian differential rotation, would produce the toroidal field
(see Reyes-Ruiz & Stepinski 1996 [17]). The local magnitude of the equilibrated toroidal field is about 50% of the local value of 
, and thus decreases
outwards 
. This behaviour follows from the assumed
nonlinearity in the form of the 
-quenching, which tunes the magnitude
of the magnetic field to the local equipartition value. It is interesting
that, given such a form of nonlinearity, the magnitude of the final
field, as well as the time a magnetic field needs to evolve from an initial
condition to its equilibrium,
can be found, in good approximation, from purely local consideration.
Locally, the relative strength of dynamo regeneration mechanisms
(differential rotation and the 
-effect), as compared to diffusion,
is expressed by the local dynamo number 
. Substituting 
from equation (2) and
from equation (3) we obtain 
. The local growth rate, 
, of a dynamo-generated field is well approximated by the formula
(Zeldovich, Ruzmaikin & Sokoloff 1983 [25])
A magnetic field would locally equilibrate when 
. Using equation
(10), the expression for 
and
we obtain an estimate
for the magnitude of the equilibrated magnetic field
which, upon substituting 
, yields 
, very close to what we have obtained using our global numerical calculations.
To estimate the time, 
, a magnetic field needs to evolve to
equilibrium, we assume that the field grows at a constant rate 
until it reaches equilibrium. Under such an assumption the local
evolution of the magnetic field is given by 
and 
. Using expression (10)
for 
and defining 
we have
Substituting 
cm into equation (12)
we obtain 
s
as the estimate of the equilibrium time. Again, this is very close to the value obtained using global numerical calculations.
Panels (B-E) on Fig. 1 show lines of poloidal force for
cases with a progressively larger ambient field. The magnitude of the
ambient field matters inasmuch as it changes the vertical component of
the magnetic field. Because advection is unimportant, the radial component of the magnetic field remains dominated by the dynamo-generated field.
Inside the disc, the poloidal field is dominated by its radial component,
.
This is an expected result, as it can be deduced directly from 
under the thin disc approximation. Outside the disc, but close
to its surfaces, radial and vertical components of the poloidal field are
of the same order of magnitude.
In the absence of an external field (Fig. 1 panel B) the generated magnetic field has a quadrupole symmetry with respect to an equator. This
means that 
and 
are even functions of Z, whereas 
is an
odd function of Z and must vanish at an equator. Numerous studies of linear
dynamos in discs surrounded by a vacuum revealed that a magnetic field of
quadrupole symmetry is indeed the easiest to excite. Note, however, that solutions to a nonlinear dynamo problem, as in our case, are not a priori
known to have any particular symmetry with respect to an equator. Nevertheless,
our results, as well as earlier results by Rüdiger et al. (1995) [20], show that
a magnetic field generated by the thin disc dynamo with the 
-quenching type nonlinearity exhibits practically (if not formally) a quadrupole symmetry.
It seems that a quadrupole symmetry of the generated field is not an artifact of our vacuum boundary condition. Rüdiger et al. (1995) [20] studied dynamos in discs surrounded by a highly conducting halo and found that the generated field has a quadrupole rather than a dipole symmetry. Mangalam & Subramanian (1994) [14] considered dynamos in discs surrounded by a force-free medium. This is, arguably, the configuration most appropriate for discs emitting winds. Again, they found solutions to have a quadrupole symmetry. Perhaps the preference for the quadrupole symmetry of the disc dynamo can be understood in simple physical terms. A major part of the field generation results from azimuthal stretching of the radial component of the field by the Keplerian shear to produce a toroidal magnetic field. Effective generation of a toroidal field requires a strong radial field in a disc, which is guaranteed in a quadrupole-like configuration, but not in, say, a dipole-like configuration where the radial component must change sign at an equator and its height-average over disc thickness vanishes.
In the presence of an ambient field (Fig. 1 panels C-E) the equatorial
symmetry of a magnetic field is broken. A uniform, vertical, ambient field has a dipole symmetry ( 
is an even function of Z) and its superposition with
a vertical component of a dynamo-generated field (which is an odd function of Z) creates a field that lacks any equatorial symmetry. Note, however, that because an ambient field is weak and advection of magnetic field is ineffective, asymmetry of the entire field is caused by asymmetry of its vertical
component. The resultant configuration of the poloidal field has an unfamiliar
character, especially as the strength of an ambient field increases.
Fig. 2 presents a zoom-out view of lines of poloidal field for the
same cases as shown in Fig. 1. The larger scale permits a better assessment of the character of a magnetic field outside the disc. As an external
field increases from none (panel A) to about 2% of 
(panel D), the asymmetry of the poloidal field becomes more pronounced. In the absence of any external field all field lines are closed loops, but in the presence of
an external field some field lines become open, merging into a uniform, vertical field at infinity. However, as can be seen on panel D of Fig. 2, the open field lines leaving the upper
surface of the disc are inclined outwards before they straighten into vertical, whereas open field lines leaving the lower surface of the disc are inclined inwards. (The sense of field line inclination would change if we flip an external field.) Thus, the topology of a magnetic field in the immediate vicinity of a disc is intrinsically asymmetric. In section 5 we discuss what such an asymmetry means for disc-driven winds.
, 
gauss,
gauss, and 
gauss, respectively.
