## 2. The dynamo modelWe take a standard mean field dynamo equation As usual, is the turbulent magnetic
diffusivity and represents a conventional
alpha-effect. comprises the large-scale
velocities (circular and non-circular), , and
represents the turbulent diamagnetism (Vainshtein & Zeldovich
1972, Roberts & Soward 1975). In general we allow
, . We solve eqn (1) as
either a 2D or a 3D initial value problem. Initial conditions were
usually that the seed field be axisymmetric and of low energy compared
to the saturation value. For the 3D calculations, it is also localized
near the disc plane. However the initial configurations are rapidly
forgotten as the simulations proceed. The 3D version of the code
allows , and
to vary also with thus separating the circular and non-circular velocities. In general we write , , , with , and being typical values of , and respectively. Thus gives a basic form of -quenching with field strengths limited at approximately . Plausibly would be the field strength in equipartition with a typical value of the kinetic energy of the turbulent velocities. Alternatively, would yield a linear calculation. A more sophisticated model might have , representing the change in the magnetic field-gas equilibrium condition as the gas density and turbulent velocity vary with position. We make the conventional assumption that ,
where and are typical
values of the velocity and length scale of the turbulence. In the
Milky Way, near the disc plane,
km s ## 2.1. The 2D modelThe 2D `no- We assume that is uniform, i.e.
and, for most of the computations,
also. The length scale ## 2.2. The 3D modelThe 3D code is described in Moss (1997), and comparisons are there
made with the 2D code. Eq. (1) is solved on a grid of size
covering ,
, . The spacing is uniform
in each of the coordinate directions. We found that taking
produced fields at this boundary that were
small compared to their maximum values, and that results were
insensitive to increasing further. Mostly we
considered and only to
vary with with varying smoothly between
and . It is the Our boundary conditions are that on and should be decreasing in magnitude at least as fast as for a general dipolar field. Similar `open' conditions are applied on , and the condition on the third field component is found by satisfying . As for the 2D calculations, we verified that plausible variations in these boundary conditions, such as setting some field components to zero at the boundary, did not alter significantly our results. The restriction to uniform mesh size in the © European Southern Observatory (ESO) 1998 Online publication: December 16, 1997 |