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Astron. Astrophys. 329, 911-919 (1998)

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3. The model

3.1. Rotation and diffusivity

We consider galaxies to be differentially rotating turbulent disks embedded in a plasma of given conductivity (Elstner et al. 1990). In the simplest case the "plasma" is vacuum and the conductivity therefore vanishes. The thickness of the galaxy is H.

The law of rotation is known: Beyond a rigidly rotating core with [FORMULA], the angular velocity is inversely proportional to r


with [FORMULA] (Donner & Brandenburg 1990; Sofue 1996). Thus the velocity of the outer part is uniform,


We assume this velocity to have the same value ([FORMULA] 100 km/s) in all our models. The turnover radius [FORMULA] is the only scaling radius of the problem. The radial size of the galaxy plays a very little role in the computations.

We normalize all distances with H: [FORMULA]. Then the dimensionless ratio


determines the overall geometry of the problem. Thin disks have large values of [FORMULA] and vice versa. In the computations [FORMULA] is used.

In order to take into account the influence of the spiral arms in density and diffusivity the profiles used by Otmianowska-Mazur & Chiba (1995) are adopted,


varying between 1 and q. The profile [FORMULA] is used for the density


as well as for the turbulence intensity


[FORMULA] is the angular pattern speed of the [FORMULA] spiral. Its pitch angle [FORMULA] is taken as 40 [FORMULA] in the present paper. The turbulence rms-velocity used is 10 km/s. The density contrast [FORMULA] is fixed with [FORMULA].

3.2. The numerical method

The dynamo equation (Eq. 2) is explicitly solved with a time stepping code by means of Euler's formula:


The differential operator is formulated as a discrete difference scheme for vectors defined on a three dimensional grid using cylindrical coordinates.

The rot-operator is constructed in order to satisfy the relation [FORMULA] 0 for any discrete vector field [FORMULA]. For details cf. Elstner et al. (1990), where a so constructed code was applied to a rather theoretical dynamo model without any nonaxisymmetric contributions.

The boundary conditions are formulated for the electromotive force so that only a vertical magnetic field is allowed to penetrate the boundary. This is called the pseudo-vacuum boundary condition, which replaces the global vacuum boundary condition by a local one.

In order to explore the effects of the radial-azimuthal galactic structure we neglect all vertical stratification as well in density and turbulent velocity as in the angular velocity of the differentially rotating gas. Notice that our boundary conditions exclude induction effects due to vertical inhomogeneities.

In all calculations presented here we used a cylindrical region 0 [FORMULA] r [FORMULA] R and [FORMULA] with R = 5 kpc and H = 2 kpc and a grid size with 41 [FORMULA] 31 [FORMULA] 41 points in (r, [FORMULA], z) direction respectively.

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997