A second-order correlation approximation for thermal conductivity and Prandtl number of free turbulence

By means of a linearizing approximation we determine the scalar thermal eddy conductivity chi/sub T/ in its dependence on the spectral function of the (given) turbulence and the microscopic conductivity chi. The discussion of the profile chi/sub T/(chi) reveals the importance of the shape of the frequency spectrum of the velocity fluctuations. Spectra which decrease with increasing frequency produce eddy conductivities which decrease with increasing chi. A distinct peak in the frequency spectrum corresponds to a non-trivial maximum of the chi/sub T/-profile. These general findings are confirmed by examples some of which are numerical. Very small molecular diffusivities generally lead to a turbulent Prandtl number of 0.4 in rough accordance with most of the measurements. For large diffusivities, however, the turbulent Prandtl number behaves inversely to the molecular Prandtl number. We have to doubt thus the general belief that eddy conductivity always exceeds eddy viscosity. For the effective Prandtl number of the Sun we find an upper limit of 0.3. An application of our theory to the small-scale motions in the upper Earth core yields values of 0.4. (author)