## 2. The dynamo modelThe time evolution of the large-scale magnetic field, , is governed by the dynamo equation, where is the large-scale velocity field and is the electromotive force due to turbulent motions (Steenbeck et al. 1966). In general, can be expressed as a linear function of and its first-order spatial derivatives: The first term on the right-hand side of Eq. (2) embodies the "alpha-effect" (magnetic field generation in the direction perpendicular to the prevailing field) plus the effective magnetic field advection by turbulent motions, while the second term represents turbulent magnetic diffusion (e.g., Moffatt 1978). In the case of the Galactic disk, where the ISM parameters and the sources of turbulence vary significantly faster along the vertical than horizontally, the alpha-tensor takes on the form in a cylindrical reference frame () with origin at the Galactic center and perpendicular to the Galactic disk. , , and give the effective rotational velocity associated with the alpha-effect when is, respectively, radial, azimuthal, and vertical, and is the effective vertical velocity at which is advected by turbulent motions (Ferrière 1993a). The diffusivity tensor can be written as where and
are the horizontal and vertical
magnetic diffusivities, respectively,
is the unit tensor, and
is the three-dimensional permutation
tensor (by definition, = +1, -1, or
0, according to whether
The large-scale flow is taken axisymmetric, like the dynamo parameters, and it is assumed to reduce to the Galactic differential rotation: In principle, have the radial profiles shown in Fig. 2.
There exists little observational information on the vertical
dependence of the rotation curve. From 21-cm emission measurements of
interstellar H I , the neutral gas appears to corotate
up to at least 1 kpc from the Galactic plane in the radial range
3.5 kpc (Lockman 1984). UV
absorption line studies of interstellar high-stage ions (mainly
C IV and N V ) suggest that, for
kpc, corotation holds up to
kpc (Savage et al. 1997); they
also give evidence that, inside
kpc, departures from corotation
set in at kpc from the plane
and bring the gas to a complete halt within the next 2 kpc
(Savage et al. 1990; Sembach et al. 1991). In most of our runs, we
will let the rotation law be independent of height. However, for
completeness, we will devote a few runs to testing the sensitivity of
the computation results on a possible We will integrate the three cylindrical components of the dynamo
equation, Eq. (1), on a -mesh by
means of an explicit numerical scheme based on a method introduced by
Schnack & Killeen (1980). These authors used a discretized
curl-operator to compute the right-hand side of Eq. (1), so as to
ensure that the initial constraint
is preserved during the forward integration. The method was further
developed by Evans & Hawley (1988) and successfully applied to
dynamo calculations by Elstner et al. (1990). Although written on a
-mesh, the numerical code can handle
any linear nonaxisymmetric solution of given azimuthal wavenumber
Our computation domain extends radially from
to
kpc and vertically from
(respectively
kpc) to
kpc in the linear (respectively
nonlinear) case. The restriction to the upper half-space in the linear
case is justified by the fact that solutions with even and odd parity
with respect to the midplane are decoupled and can be computed
separately once the appropriate symmetry conditions are imposed at
. The outer edges at
kpc and
kpc are sufficiently remote
from the induction region that the associated boundary conditions have
a negligible impact on the interior solution. The external medium is
assumed to be either a perfect conductor (in most runs) or a vacuum
(in a few trial runs). Since exact vacuum boundary conditions are
difficult to implement in practice, we simulate them by raising the
magnetic diffusivity to
cm The number of mesh points is usually 51 in the radial direction and 26 (respectively 51) in the vertical direction, corresponding to a grid size of 0.4 kpc in both directions. The timestep, dictated by the CFL condition for numerical stability, is on the order of one thousandth the diffusion time. Convergence is verified through a series of trial computations with finer grid and smaller timestep. © European Southern Observatory (ESO) 2000 Online publication: June 26, 2000 |