We consider a turbulent, axisymmetric, thin accretion disc rotating
with Keplerian angular velocity 
, and having a halfthickness
. Cylindrical polar coordinates 
are
used, with Z=0 corresponding to a disc midplane (equator) and
R=0 corresponding to the centre of the star.
A magnetic field embedded in a turbulent, electrically conducting disc
has a random, small-scale component, as well as a mean, large-scale
component, which we are concerned with here. In the presence of turbulence,
the evolution of a mean magnetic field 
inside the disc is
governed by the dynamo
equation
where 
is the mean,
large-scale gas velocity, 
is the mean helicity parameter
,
and 
is the total magnetic diffusivity,
which includes contributions from resistive and turbulent losses.
Coefficients 
and 
are determined by the mean
velocity field and the mean statistical parameters of the
fluctuating velocity having a characteristic speed 
, and
a correlation length 
. In equation (1)
both 
and 
are scalars, although, if the effects of
a rotation on the turbulence are considered (Rüdiger, Elstner &
Stepinski 1995 [20]), they become tensors. Neglecting
such rotational effects, 
and 
can be expressed
as
and
The function 
represents the so-called 
-quenching,
the reduction of mean helicity due to the feedback of a magnetic field
on turbulent motion. The general form of 
is given by Rüdiger & Kichatinov (1993) [19]. Here we will use the simplified form 
, where 
is magnetic field strength when its energy is
in equipartition with the kinetic energy of the turbulent motions.
We will henceforth refer to 
as the equipartition value.
Because we have assumed the outside of the disc to be a vacuum, electrical currents cannot be supported there and the evolution of a magnetic field outside the disc is governed by the expression
As the entire system is axisymmetric, we can express the magnetic field
in the form 
with 
representing the toroidal field and A representing the
toroidal part of the vector potential from which components
of a poloidal field can be recovered, 
and 
. Equation (1) separates into the following
toroidal and poloidal components
and
The symbol 
denotes differential operator
In the toroidal equation (5) the first three terms on the
right-hand side provide coupling of the toroidal and poloidal fields.
Of these three terms, the first, due to Keplerian shear, is the largest.
The second is smaller by a factor of the order of 
,
and the third is smaller by a factor of the order of 
. Of the last three terms on the right-hand side of equation (5) the last, diffusive term is the largest. The advective term
is smaller by a factor of the order of 
, and the term
associated with the derivative of 
is smaller by a factor of the order of
. Altogether, in a thin disc characterised by 
,
it is sufficient to keep only the first and the last terms on the
right-hand side of equation (5). In the poloidal equation
(6) the coupling to the toroidal field is provided by the first
term on the right-hand side. The second, advective term is smaller than
the last, diffusive term by a factor of the order of 
and can be neglected.
Equation (4), controlling the behaviour of a magnetic field outside the disc has a vanishing toroidal component, and a poloidal component given by
On the surface of the disc, 
, the magnetic field is continuous.
An externally imposed field, 
, maintained by currents located at
infinity, has only a poloidal component such that 
.
We solve equations (5)-(7) to obtain 
and A,
we then add A to 
, thereby obtaining the total potential of the
poloidal magnetic field from which components of poloidal field, 
and
, are calculated.
