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

## 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 where represents the fluctuating velocity at , and is a small departure to caused by . We assume that, in the absence of a mean flow, turbulence is isotropic and homogeneous with the correlation tensor given by (see, e.g., Rüdiger 1989).

The velocity can be calculated from the momentum equation. In the presence of a mean flow, the fluctuating velocity obeys the equation where p is the fluctuating pressure and is the density. Following the spirit of a quasilinear approximation, we will neglect the non-linear advective term . Making Fourier transformation in t and , we obtain where . If the phase velocity of turbulent fluctuations is larger then V, we have . With the accuracy in linear terms in , calculations of can be done by making use of a standard perturbation procedure. The unperturbed velocity, is governed by Eq. (15) with . Then, the equation for reads Taking into account the continuity condition for incompressible fluid, , we obtain Calculating , we can neglect small corrections in and because this part of the electromotive force is already proportional to shear stresses. Therefore,  summation is over repeated indexes. Substituting from Eq. (6) and taking into account the correlation properties of turbulence (Eq. (13)), we have The integrals over and can be calculated as in The last integral in this expression can be taken from the table (see, e.g., Gradshtein & Ryzhik 1965), then Since is slowly varying function and does not change significantly on a time scale of the order of , integration over gives with the accuracy in the lowest order in and ( is the characteristic time scale of mean quantities). Finally, where We assume that the ensemble of fluctuating motions does not contain waves with thus the spectral power of turbulence goes to zero at and there is no singularity in the integral (24).

Contrary to , the component 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 Substituting into this equation and making the same transformations as deriving the expression (23), we obtain Note that determines a large scale vorticity of the flow thus the component 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 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 does not contain the component proportional to 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 of the conventional dynamo theory is proportional to the kinetic helicity, , which is non-vanishing, for example, in a rotating fluid. However, apart from rotation the condition 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 can be formed from the axial vector of the angular velocity, , only as a scalar production of 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 and , 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 with the accuracy in linear terms in . Eq. (17) for contains only a production of and even powers of , therefore the expression for h should contain only odd powers of after ensemble averaging. If the unperturbed turbulence is assumed to be homogeneous and isotropic (see (13)), integration over will give a vanishing contribution, and in our model. Thus, the suggested mechanism works under the condition when the alpha-dynamo is unoperative.    © European Southern Observatory (ESO) 1999

Online publication: June 6, 1999 