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

2. Mean-field electrodynamics

The equations are close to those in Rüdiger et al. (2000). The momentum equation for non-rigid rotation in the inertial system with buoyancy included is

Overbars indicate prescribed mean quantities such as the homogeneous magnetic field, large-scale flow and density. denotes the acceleration due to gravity.

As an energy equation for the turbulence the adiabacity relation

is used with as the isothermal speed of sound. Eqs. (2) and (3) lead to a turbulence field , driven by the Lorentz force on the RHS of (2). The original, prescribed magnetic field fluctuations may be denoted by . Their correlation tensor is assumed to form a homogeneous, isotropic and stationary field of magnetic turbulence. The resulting kinetic turbulence is subject to a basic rotation and subject to shear or - in other words - to differential rotation.

After some algebra one can find the correlation tensor of the turbulence and, in particular, its covariance , s here being the distance from the rotation axis. This quantity is part of the angular momentum transport. The total angular momentum transport is given by

taking into account also the Maxwell stress. The latter results from the magnetic fluctuations, , driven by the turbulence field considered. The corresponding equation is the induction equation in its linearised version, i.e.

Here again both the influences of the basic rotation (only on nonaxisymmetric field components) as well as differential rotation can be isolated.

The resulting (rather complex) magnetic fluctuations must be used to compute the Maxwell stress in (4), or, as the next interesting quantity, to compute the current helicity

which has the same kind of equatorial (anti-)symmetry as the dynamo-. For homogeneous global magnetic fields the dynamo- is directly related to the turbulent electromotive force (EMF) according to so that Rädler & Seehafer (1990) propose to apply this equation to -dynamos with dominating azimuthal field belts so that

where is the azimuthal component of the -tensor. We are, in particular, interested to check their and Keinigs' (1983) antiphase relation,

between -effect and current helicity. There is an increasing number of papers presenting observations of the current helicity of the solar surface always with the result that it is negative at the northern hemisphere and positive at the southern hemisphere (Seehafer 1990; Pevtsov et al. 1995; Abramenko et al. 1996; Bao & Zhang 1998). If (8) is correct then there is a strong empirical evidence for an -effect that is positive (negative ) in the northern (southern) hemisphere of the Sun.

Here we start to find the relation between -effect and current helicity for shear-flow disks. We shall see that there are exceptions, indeed, to the simple relation (8). This is not a surprise. Blackman & Field (1999) argue that Keinigs' result,

strongly depends on the assumed stationarity and homogeneity of the magnetic fields and flows which are, however, not realistic for dynamo problems.

© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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