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

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3. The mean electromotive force in a turbulent flow

Consider the case of incompressible turbulence. Since inhomogeneity of the mean flow is small, we can split Fourier amplitudes into two components as in

[EQUATION]

where [FORMULA] represents the fluctuating velocity at [FORMULA], and [FORMULA] is a small departure to [FORMULA] caused by [FORMULA]. We assume that, in the absence of a mean flow, turbulence is isotropic and homogeneous with the correlation tensor given by

[EQUATION]

(see, e.g., Rüdiger 1989).

The velocity [FORMULA] can be calculated from the momentum equation. In the presence of a mean flow, the fluctuating velocity obeys the equation

[EQUATION]

where p is the fluctuating pressure and [FORMULA] is the density. Following the spirit of a quasilinear approximation, we will neglect the non-linear advective term [FORMULA]. Making Fourier transformation in t and [FORMULA], we obtain

[EQUATION]

where [FORMULA]. If the phase velocity of turbulent fluctuations is larger then V, we have [FORMULA]. With the accuracy in linear terms in [FORMULA], calculations of [FORMULA] can be done by making use of a standard perturbation procedure. The unperturbed velocity, [FORMULA] is governed by Eq. (15) with [FORMULA]. Then, the equation for [FORMULA] reads

[EQUATION]

Taking into account the continuity condition for incompressible fluid, [FORMULA], we obtain

[EQUATION]

Calculating [FORMULA], we can neglect small corrections in [FORMULA] and [FORMULA] because this part of the electromotive force is already proportional to shear stresses. Therefore,

[EQUATION]

[EQUATION]

summation is over repeated indexes. Substituting [FORMULA] from Eq. (6) and taking into account the correlation properties of turbulence (Eq. (13)), we have

[EQUATION]

The integrals over [FORMULA] and [FORMULA] can be calculated as in

[EQUATION]

The last integral in this expression can be taken from the table (see, e.g., Gradshtein & Ryzhik 1965), then

[EQUATION]

Since [FORMULA] is slowly varying function and does not change significantly on a time scale of the order of [FORMULA], integration over [FORMULA] gives [FORMULA] with the accuracy in the lowest order in [FORMULA] and [FORMULA] ([FORMULA] is the characteristic time scale of mean quantities). Finally,

[EQUATION]

where

[EQUATION]

We assume that the ensemble of fluctuating motions does not contain waves with [FORMULA] thus the spectral power of turbulence goes to zero at [FORMULA] and there is no singularity in the integral (24).

Contrary to [FORMULA], the component [FORMULA] should be calculated with taking into account small corrections in the fluctuating velocity and magnetic field. With the accuracy in linear terms in shear stresses, we have

[EQUATION]

Substituting [FORMULA] into this equation and making the same transformations as deriving the expression (23), we obtain

[EQUATION]

Note that [FORMULA] determines a large scale vorticity of the flow thus the component [FORMULA] of the mean electromotive force is non-vanishing only for the flow with a non-zero vorticity.

Finally, the expression for the mean electromotive force is

[EQUATION]

The terms on the r.h.s. of Eq. (27) contain spatial derivatives of the mean velocity and, generally, the electromotive force is non-vanishing for any non-uniform flow. Note that [FORMULA] does not contain the component proportional to [FORMULA] that is typical for the alpha-effect. Therefore the effect of shear stresses cannot be described, in principle, in terms of the alpha-effect. The coefficient [FORMULA] of the conventional dynamo theory is proportional to the kinetic helicity, [FORMULA], which is non-vanishing, for example, in a rotating fluid. However, apart from rotation the condition [FORMULA] requires also the presence of a large scale inhomogeneity of the fluid (see, e.g., Krause & Rädler 1980). For example, it can be the density stratification but the presence of inhomogeneity is absolutely necessary because the pseudoscalar [FORMULA] can be formed from the axial vector of the angular velocity, [FORMULA], only as a scalar production of [FORMULA] and some polar vector (e.g., the density gradient). In the present paper, we consider the case of homogeneous unperturbed turbulence where the alpha-effect should be vanishing because of this reasonning in the lowest order in [FORMULA] and [FORMULA], at least. This point can easily be checked by direct calculations. Indeed, calculating helicity in terms of Fourier transforms, we obtain that h is proportional to [FORMULA] with the accuracy in linear terms in [FORMULA]. Eq. (17) for [FORMULA] contains only a production of [FORMULA] and even powers of [FORMULA], therefore the expression for h should contain only odd powers of [FORMULA] after ensemble averaging. If the unperturbed turbulence is assumed to be homogeneous and isotropic (see (13)), integration over [FORMULA] will give a vanishing contribution, and [FORMULA] in our model. Thus, the suggested mechanism works under the condition when the alpha-dynamo is unoperative.

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© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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