3. The current helicity and the -effect
and the spectral function in the definition .
The sign of the determines the sign of the current helicity which we have to discuss for various turbulence models. The integral (11) does not prove to be definite in sign. It is negative-definite for very large magnetic Prandtl numbers () but it is positive for the more realistic case of moderate magnetic Prandtl number and spectral functions decreasing for increasing frequency . In the sense of the `-approximation' the spectrum of the given field of magnetic fluctuations has been approximated by with (Kitchatinov 1991) and the becomes positive-definite. For the current helicity (6) of the shear flow we then find
is the Alfvén velocity. Indeed, for the current helicity disappears. It is negative on the northern hemisphere for weak differential rotation but changes its sign for sufficiently large shear. For a Kepler disk with its vertical gravity, , the current helicity becomes
which is positive in the upper disk plane and negative in the lower disk plane. This is exactly the numerical result of Brandenburg (1999) for the current helicity of magnetic field fluctuations in Kepler disks.
results. For rigid rotation the -effect proves to be positive in the upper disk plane and negative in the lower disk plane (Rüdiger et al. 2000). The opposite is true for Kepler flows. The dynamo- becomes negative in the upper disk plane and positive in the lower disk plane. Again the results comply with the results of the numerical simulations by Brandenburg (1999). After (20) the dynamo- completely vanishes for rather than for for which the current helicity vanishes after (14). So a small interval exists with exponents between 0.75 and 1 where the -effect and the current helicity have the same sign. Box simulations should be used to test the relevance of this surprising result.
- very close to (9). For rigid rotation the factor sinks to 1/2 (see Paper I). The small differences to Keinigs' result certainly result from the fact that we are not using the anelastic approximation.
which is often believed to be in antiphase to the -effect (Moffatt 1978). We obtain
in the upper hemisphere proves to be negative - as it does in the simulations by Brandenburg (1999).
with the turnover velocity , the turbulence Mach number and the magnetic Mach number . For equipartition of the magnetic energy with the thermal energy (Mm) we find the -effect to be much smaller than the traditional value (`helicity times correlation time') if the turbulence is subsonic. Even the sign is opposite. However, as the kinetic helicity can not be observed at the disk surface we are not able to estimate the amplitude of the dynamo-, , from the given expressions. To this end we need the comparison with a quantity representing, e.g., the angular momentum transport in accretion disks which can directly be observed via the radiation or the temporal behavior of the real disks. The quantity describing these effects has been introduced by Shakura & Sunyaev and will be computed in the following section.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000