Astron. Astrophys. 362, 756-761 (2000)

3. The current helicity and the -effect

The complete relation for the current helicity is

with

and the spectral function in the definition .

In disk geometry the deformation tensor is simply

with and as the unit vectors in radial and azimuthal directions. Insertion of (12) into (10) gives

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.

The next step concerns the -effect defined by the above relation. It results from the general expression

with

Again the total effect vanishes for . Only the most important component need be discussed. We obtain

Our magnetic flux tube model yields

so that for 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.

The ratio of the -effect and current helicity here follows to

- 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.

A similar question arises concerning the kinetic helicity

which is often believed to be in antiphase to the -effect (Moffatt 1978). We obtain

resulting in

with the positive quantity

and with

Hence, for rigid rotation the kinetic helicity is positive in the upper disk plane, and it is negative in the lower disk plane. For the one-mode flux tube model we find for the amplitude the value

Again the sign of the (pseudo-)scalar changes with increasing shear in the same way as it happens for the current helicity, see Eq. (14). For Kepler rotation the kinetic helicity,

in the upper hemisphere proves to be negative - as it does in the simulations by Brandenburg (1999).

The amplitude ratio of the dynamo- and the kinetic helicity is

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
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