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Astron. Astrophys. 347, L47-L50 (1999)

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2. The basic equations

The behaviour of the mean magnetic field is governed by the mean electromotive force, [FORMULA], where [FORMULA] denotes ensemble averaging. We derive the expression for [FORMULA] by making use of a quasilinear approximation. In this approximation mean quantities are governed by equations including non-linear effects in fluctuating terms, whilst the linearized equation are used for the fluctuating quantities (see, e.g., Krause & Rädler 1980). A quasilinear approximation is sufficiently accurate, for example, to describe various turbulent kinetic processes in plasma when the amplitude of turbulent fluctuations is relatively small (see, e.g., Lifshitz & Pitaevskii 1979). Generally, this approximation applies if the ensemble of turbulent motions is characterized by relatively high frequencies and small amplitudes thus the Strouhal number, [FORMULA], is small; [FORMULA] and [FORMULA] are the correlation time and the length-scale of turbulence, respectively. Our analysis is based on a double-scale model in which attention is concentrated on the development of the magnetic field on a scale L much greater than the scale [FORMULA].

Split the magnetic field [FORMULA] and the velocity [FORMULA] into the mean and fluctuating parts, [FORMULA] and [FORMULA], where [FORMULA] and [FORMULA] are the mean field and velocity, respectively. The linearized induction equation for the fluctuating magnetic field reads

[EQUATION]

We assume that the magnetic Reynolds number is large for fluctuating motions and neglect dissipative effects.

Consider the incompressible flow with [FORMULA] where x, y and z are the Cartesian coordinates; the corresponding unit vectors are [FORMULA], [FORMULA], and [FORMULA]. Substituting this expression into Eq. (1), we have

[EQUATION]

where

[EQUATION]

The second term on the l.h.s. of Eq. (2) describes the advection of turbulent magnetic field lines by the mean flow and leads only to a small shift of frequences in the spectral integrals if [FORMULA] (see Urpin 1999). We neglect this term in what follows.

By making use of Fourier transformation,

[EQUATION]

where the hat labels the Fourier amplitude and neglecting terms of the order of [FORMULA], we obtain from Eq. (2)

[EQUATION]

where

[EQUATION]

In what follows, we assume that inhomogeneity of the mean flow is relatively small and restrict ouselves to linear terms in [FORMULA]. Therefore, the solution of Eq. (5) can be represented as

[EQUATION]

Then, we have

[EQUATION]

Substitution of [FORMULA] into the definition of [FORMULA] yields

[EQUATION]

where

[EQUATION]

[EQUATION]

Eqs. (9)-(11) represent the general expression for [FORMULA] in the presence of a large scale flow.

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© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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