## 3. The mean electromotive force in a turbulent flowConsider 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 where . If the phase velocity of
turbulent fluctuations is larger then 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 © European Southern Observatory (ESO) 1999 Online publication: June 6, 1999 |