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Mean--field electrodynamics: Critical analysis of various analytical approaches to the mean electromotive force

There are various analytical approaches to the mean electromotive force $\cal E =< \vec{u}\times \vec{b}>$ crucial in mean--field electrodynamics, with $\vec{u}$ and $\vec{b}$ being velocity and magnetic field fluctuations. In most cases the traditional approach, restricted to the second--order correlation approximation, has been used. Its validity is only guaranteed for a range of conditions, which is narrow in view of many applications, e.g., in astrophysics. With the intention to have a wider range of applicability other approaches have been proposed which make use of the so--called $τ$--approximation, reducing correlations of third order in $\vec{u}$ and $\vec{b}$ to such of second order. After explaining some basic features of the traditional approach a critical analysis of the approaches of that kind is given. It is shown that they lead in some cases to results which are in clear conflict with those of the traditional approach. It is argued that this indicates shortcomings of the $τ$--approaches and poses serious restrictions to their applicability. These shortcomings do not result from the basic assumption of the $τ$--approximation. Instead, they seem to originate in some simplifications made in order to derive $\cal E$ without really solving the equations governing $\vec{u}$ and $\vec{b}$. A starting point for a new approach is described which avoids the conflict.