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

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2. The mean-field electrodynamics

2.1. The turbulent EMF

The evolution of the mean magnetic field [FORMULA] is governed by the dynamo equation

[EQUATION]

where [FORMULA] is the turbulent electromotive force, [FORMULA], and [FORMULA] the mean velocity (Krause & Rädler 1980). We employ cylindrical polar coordinates [FORMULA] and assume an axisymmetry, [FORMULA]. The drift velocity [FORMULA] is due to the effect of ambipolar diffusion and is proportional to the Lorentz force (cf. Section 2.4).

As usual, we assume approximate scale separation and write

[EQUATION]

Basic for the theory is the knowledge of the material vectors and tensors in front of large-scale magnetic field formations. The velocities [FORMULA] and [FORMULA] are playing the role of large-scale mean flows while the [FORMULA] -tensor represents eddy diffusivity. Via a magnetic feedback all of them are influenced (`quenched') by the induced magnetic field. The various terms are discussed separately below.

2.2. Turbulent diamagnetism and pumping

First we have to consider the phenomenon of the turbulent diamagnetism,

[EQUATION]

with

[EQUATION]

(Kitchatinov & Rüdiger 1992). The latter function starts with unity for weak fields and vanishes like [FORMULA] for strong fields. It is

[EQUATION]

with the equipartition value Eq. (1).

The turbulent pumping term can be written as

[EQUATION]

with

[EQUATION]

It starts with [FORMULA] for weak fields and vanishes like [FORMULA] for strong fields.

2.3. Eta-quenching

For slow rotation and weak magnetic field the eddy diffusivity tensor takes the simple and well-known form

[EQUATION]

so that

[EQUATION]

The eddy diffusivity, however, is only a simple tensor unless the magnetic field feeds back. Then the [FORMULA] -tensor becomes much more complex:

[EQUATION]

(Ferrière 1993, 1996; Kitchatinov et al. 1994; Ziegler 1996). Its most convenient representation concerns the turbulent EMF

[EQUATION]

with the "magnetic velocity"

[EQUATION]

It is

[EQUATION]

with

[EQUATION]

(cf. Fig. 1).

[FIGURE] Fig. 1. The quenching functions Eq. (14) for the eddy diffusivity tensor components ([FORMULA] dashed, [FORMULA] dotted, [FORMULA] solid)

In Rüdiger et al. (1994) first consequences of this [FORMULA] -quenching are demonstrated. Only oscillating modes provide magnetic fields of order of the turbulence-equipartition fields. The equipartition field strength in galaxies is about 3-5 [FORMULA]. These values are comparable with observations. Steady modes, however, in totally nonlinear models get unrealistically large magnetic fields.

The reference value [FORMULA] of eddy diffusivity is

[EQUATION]

with the standard value [FORMULA] (cf. Parker 1979; Ruzmaikin et al. 1988). The true [FORMULA] is unknown, it can only be found if temporal variations (e.g. sunspot decay) are observed. In the present paper the standard value is used, but we also discuss values one order higher and one order lower. The correlation time is put as 30 Myr (Fröhlich & Schultz 1996).

2.4. Ambipolar diffusion

The ambipolar drift velocity is proportional to the Lorentz force,

[EQUATION]

The mass density of ions is [FORMULA] and [FORMULA] is the ion-neutral collision frequency. In the following we set

[EQUATION]

(Brandenburg & Zweibel 1995).

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997
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