Summary

The combined action of differential rotation and helical convection on the magnetic field in the convection zone of late-type stars is named an alpha-Omega-dynamo. Such a dynamo is believed to sustain a cyclic large-scale magnetic field, like the Sun's 22-yr magnetic cycle.

Models of large-scale fields B usually employ the so-called dynamo equation for the mean magnetic field . Traditionally it is assumed that 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 is damped. The damping time of 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:

is only marginally stable for a fairly large value of the turbulent diffusion coefficient beta in the bulk of the convection zone;
is mainly generated by vorticity, as differential rotation produces only 2 to 10% of the total mean energy.

The magnetic field of the dynamo:

fluctuates rapidly since the damping time of the mean field appears to be about 14 days, a small fraction of the cycle period of 22 years;
shows therefore no clear period and no well-defined large-scale fields;
has an r.m.s. value of about 9 G at the surface, and about 140 G at the base of the convection zone.

The solar convection zone dynamo is therefore only a small-scale field dynamo, unable to generate the strong fields inferred from phenomena in active regions. Thus additional proof is supplied that a convection zone dynamo cannot be responsible for the solar cycle. It has been suggested that the magnetic field strengths required to explain the observed fluxes in active regions may be generated by a dynamo operating mainly in a thin boundary layer at the interface between the radiative core and the convection zone. The finite magnetic energy method applied to such a localized solar dynamo - modelled by making the turbulent diffusion coefficient beta decrease near the base of the convection zone - shows that:

the damping time of is still very small, no more than about 20 days;
the r.m.s. surface field strength remains about 9 G;
the r.m.s. field strength at the base of the convection zone increases to 230 to 450 G, which is still too low by a factor of 10 to 100 to explain the observed fluxes in active regions;
differential rotation produces about 20% of the total mean energy.

In other words, the boundary layer dynamo modelled here is also an aperiodic, small-scale field dynamo, incapable of sustaining the solar cycle.

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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Summary

last modified: 26 May 2001