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

2. The basic equations

The behaviour of the mean magnetic field is governed by the mean electromotive force, , where denotes ensemble averaging. We derive the expression for 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, , is small; and 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 .

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

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

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

where

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 (see Urpin 1999). We neglect this term in what follows.

By making use of Fourier transformation,

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

where

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

Then, we have

Substitution of into the definition of yields

where

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

© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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