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A turbulent, electrically conducting fluid containing a magnetic field with non-vanishing meanvalue is investigated. The magnetic field strength and the conductivity may be so small that the turbulence is not influenced by the action of the LORENTZ force.The average of the crossproduct of velocity and magnetic field is calculated in a second approximation. It contains the averages of the products of two components of the velocity field, i. e. the components of the correlation tensor.Here the structure of the correlation tensor is determined for a medium with gradients of density and/or turbulence intensity, furthermore the turbulent motion is influenced by CORIOLIS forces.As the main result is shown that in those turbulent velocity fields the average crossproduct of velocity and magnetic field generally has a non-vanishing component parallel to the average magnetic field.Such a turbulence may be present in rotating stars. Consequences concerning the selfexcited build up of steller magnetic fields are discussed in a following paper.

The turbulent magnetic diffusivity tensor is determined in the presence of rotation or shear. The question is addressed whether dynamo action from the shear-current effect can explain large-scale magnetic field generation found in simulations with shear. For this purpose a set of evolution equations for the response to imposed test fields is solved with turbulent and mean motions calculated from the momentum and continuity equations. The corresponding results for the electromotive force are used to calculate turbulent transport coefficients. The diagonal components of the turbulent magnetic diffusivity tensor are found to be very close together, but their values increase slightly with increasing shear and decrease with increasing rotation rate. In the presence of shear, the sign of the two off-diagonal components of the turbulent magnetic diffusion tensor is the same and opposite to the sign of the shear. This implies that dynamo action from the shear-current effect is impossible, except perhaps for high magnetic Reynolds numbers. However, even though there is no alpha effect on the average, the components of the α tensor display Gaussian fluctuations around zero. These fluctuations are strong enough to drive an incoherent alpha-shear dynamo. The incoherent shear-current effect, on the other hand, is found to be subdominant.

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g. the α-effect, are highly anisotropic in both examples. An α 2 -mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.

Abstract.A comparison is made between direct numerical simulations of magnetohydrodynamic processes in a rotating spherical shell and their mean-field description. The mean fields are defined by azimuthal averaging. The coefficients that occur in the traditional representation of the mean electromotive force considering derivatives of the mean magnetic field up to the first order are calculated with the fluid velocity taken from the direct numerical simulations by two different methods. While the first one does not use specific approximations, the second one is based on the first-order smoothing approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. For the investigated example of rotating magnetoconvection the mean magnetic field derived from the direct numerical simulation is well reproduced on the mean-field level. For the simple geodynamo model a discrepancy occurs, which is probably a consequence of the neglect of higher-order derivatives of the mean magnetic field in the mean electromotive force.