The turbulent emf as a time series and the ‘quality’ of dynamo cyces

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Abstract
Following earlier suggestions to replace the ensemble average used in the mean-field electrodynamics by an averaging over the azimuthal coordinate, we consider the basic coefficients in the turbulent electromotive force (EMF) as time-dependent functions. The well-known dynamo coefficients α and ηT - both in the relevant tensorial formulations - are derived from one and the same turbulence field with maximal helicity so that in a local formulation the total turbulent EMF is described as a time series. The (kinematic) turbulence models have always the same intensity of ≃ 100 ms−1 and the number of the eddies in the unit length is varied. The EMF-coefficients α and η T are evaluated within the limit of high (microscopic) conductivity. Both coefficients prove to exhibit time series with remarkable fluctuations. The fluctuations are stronger for the α-effect compared with the eddy diffusivity, and they are stronger if the number of cells is decreased. In general, we find fluctuations dominating the average for turbulence with only a few large cells. Even changes of the sign of the EMF coefficients occur for short periods. Application of the resulting turbulence EMF-coefficients to an one-dimensional α2ω-dynamo model leads to complicated time series for the resulting magnetic field. It is oscillatory for an infinite number of cells and becomes more complex if less turbulence eddies are operating in the flow. For decreasing eddy population the corresponding spectral line in the power spectrum of the magnetic cycles becomes more and more broad (the ‘quality’ of the cycle sinks) - but further reduction of the cell population leads to a chaotic character of the dynamo amplitude. Finally, the difference between oscillatory and stationary solutions of the dynamo model Seems to disappear. The observed quality of the solar cycle might be produced by about 100 giant cells along the equator.

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