Models of large-scale fields B usually employ the so-called dynamo equation for the mean magnetic field <B>. Traditionally it is assumed that <B> is marginally stable. Such solutions, however, face two problems: the solution is strictly periodic, and the mean magnetic energy <BB> - and hence the field B itself - appears to grow without bounds. The omission of a non-linear feedback of the magnetic field on the fluid motions is not the main reason for these two problems. Instead, they are consequences of an improper treatment of the effects of fluctuations in the turbulence.
In order to ensure that the magnetic field B of the dynamo remains finite, the mean magnetic energy <BB> of the dynamo must remain finite. This idea is the basis of the finite magnetic energy method described in this thesis. The combination of parameters that renders <BB> marginally stable is used to solve the dynamo equation. It appears that the mean field <B> is damped. The damping time of <B> is interpreted as the auto-correlation time of B, i.e. it is a measure of the period stability of the dynamo. Here ensemble averaging is used, which treats the effects of fluctuations properly.
The method yields an energy balance for the mean energy density <B^2>/8pi, in which appear the relative rates of production of mean magnetic energy by vorticity (random field line stretching, presumably at small spatial scales) and by differential rotation, and the energy that leaves the dynamo through the surface. The energy is transported to the surface by turbulent diffusion. Resistivity can be neglected in the solar dynamo. Since the method is linear, it does not provide the absolute magnitude of the magnetic field. But a tentative identification of the outgoing energy flux with the flux needed to heat the corona yields an estimate of the r.m.s. field strengths at which the dynamo operates. The finite magnetic energy method is first applied to a dynamo operating throughout the solar convection zone. The mean magnetic energy of such a convection zone dynamo:
So it appears that the solar cycle cannot be explained by conventional mean-field alpha-Omega-dynamo models, regardless whether the dynamo processes mainly operate throughout the convection zone or in a boundary layer. The major difficulty with such alpha-Omega-dynamos is that a stochastic mechanism (helical convection) is invoked to produce a periodic magnetic field. But the convection makes the field at the same time so unstable that fluctuations make it impossible to recognize any period. An explanation of the solar cycle may therefore require alternative theories in which traditional mean field alpha-Omega-mechanisms play only a minor role, or in which differential rotation and helicity are spatially separated.
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Jos van Geffen --
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