Stochastic excitation of global magnetic fields by fluctuations in the mean helicity

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Abstract
The mean field B in the dynamo equation can be interpreted as a longitude-averaged field, and this implies that there are fluctuations in the mean parameters characterizing the turbulent flow. In this paper we study the effect of fluctuations in the mean helicity numerically and analytically in a very simple spherical α2-dynamo: there is no differential rotation and the non-fluctuating parts of α and β do not depend on position (we call this the α2-sphere dynamo). The dynamo equation then contains a term ∇ × δα(t) B , which describes the effect of the fluctuations in the mean helicity. We show that this type of random forcing implies that the dynamo has to operate (slightly) subcritically, and that in addition many eigenmodes are excited, rather than only the fundamental mode. The advantage of this simple α2-dynamo model is that we can support the numerical results with analytical estimates, for instance, for the value of the dynamo number at which the dynamo operates, the relative excitation levels of the modes, and their spectra. This is achieved with the help of an expansion technique: B is expanded in terms of a complete, orthogonal set of eigenfunctions. We have taken the mean helicity fluctuations to be position-independent for simplicity. This, however, renders the dynamo model so simple that only dipole fields are excited and magnetic field reversals are absent. We also briefly study the effect of non-linearities, in particular of α-effect quenching. Non-linearities provide a reference level to the fundamental mode, but do not affect the relative excitation levels of the modes.

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