Transport of magnetic flux and the vertical structure of accretion
  discs: I. Uniform diffusion coefficients
  The evolution of a large-scale poloidal magnetic field in accretion discs is
an important problem because of its role in the launching of jets and winds and
in determining the intensity of turbulence. In this paper, we develop a
formalism to calculate the transport magnetic flux in a thin accretion disc,
thus determining its evolution on a viscous/resistive timescale. The governing
equations are derived by performing an asymptotic expansion in the limit of a
thin disc, in the regime where the magnetic field is dominated by its vertical
component. Turbulent viscosity and resistivity are included, with an arbitrary
vertical profile that can be adjusted to mimic the vertical structure of the
turbulence. At a given radius and time, the rates of transport of mass and
magnetic flux are determined by a one-dimensional problem in the vertical
direction, in which the radial gradients of various quantities appear as source
terms. We solve this problem to obtain the transport rates and the vertical
structure of the disc. The present paper is then restricted to the idealised
case of uniform diffusion coefficients, while a companion paper will study more
realistic vertical profiles of these coefficients. We show the advection of
weak magnetic fields to be significantly faster than the advection of mass,
contrary to what a crude vertical averaging might suggest. This results from
the larger radial velocities away from the mid-plane, which barely affect the
mass accretion owing to the low density in these regions but do affect the
advection of magnetic flux. Possible consequences of this larger accretion
velocity include a potentially interesting time-dependence with the magnetic
flux distribution evolving faster than the mass distribution. If the disc is
not too thin, this fast advection may also partially solve the long-standing
problem of too efficient diffusion of an inclined magnetic field.
