Mean‐field approach to spherical dynamo models

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Abstract
The mean‐field approach to dynamo theory has proved to be a useful tool in the investigation of cosmical magnetic fields. This paper gives a systematic discussion of this approach for spherical dynamo models as suggested by cosmical bodies. At first some fundamentals of dynamo theory are explained with particular attention to formulations in terms of toroidal and poloidal magnetic fields.Starting from the general ideas of mean‐field magnetohydrodynamics the relevant mean‐field equations for the models envisaged are derived and discussed. The considerations are not restricted to motions of turbulent nature, motions with more or less regular flow patterns are admitted too. A new representation of the crucial electromotive force caused by the fluctuating motions is given. For an important special case the dependence of this electromotive force on the motions is calculated.The mean‐field concept is in particular elaborated under the assumption that the motions show certain symmetry and stationarity properties as to be expected at cosmical bodies. The respective results for the electromotive force caused by the fluctuating motions are discussed in detail.Within this frame the possibilities of dynamo mechanisms are systematically studied. In addition to the well‐known α2and αω‐mechanisms some others, termed β2, βω, and δω‐mechanisms, are envisaged, whose relevance for cosmical objects remains to be investigated.In a following paper numerical results are given for dynamo models with mechanisms as envisaged here.