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Astron. Astrophys. 335, 679-684 (1998)

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1. Motivation

The `observed' internal solar rotation law can be explained by means of a Reynolds stress theory in which the influence of the basic rotation on the cross correlations of the turbulent convective velocities is determined (Küker et al. 1993). It is a pure Reynolds stress theory: the meridional flow in such calculations is neglected. The reason for this ignorance is twofold. First, the observations show only a slow poleward flow with an amplitude of 10 m/s (Snodgrass & Daily 1996; Hathaway 1996; Hathaway et al. 1996). There is the assumption that such a slow flow only slightly modifies the results of the Reynolds stress theory. On the other hand, a rough inspection of the (observed) rotational isolines does not suggest a too strong influence of a meridional flow with one or two large cells.

The meridional circulation tends to align the isolines of the angular momentum with the streamlines of the flow (Köhler 1969; Rüdiger 1989; Brandenburg et al. 1990). The existence of a distinct equatorial acceleration is thus indicating that a possible meridional surface drift cannot be too strong.

Moreover, for very large Taylor numbers [FORMULA] the flow system approaches more and more the Taylor-Proudman state where the isolines are two-dimensional with respect to the rotation axis, i.e. there is no variation along the coordinate (say) z. At the same time the poloidal (`meridional') circulation breaks, i.e.

[EQUATION]

At least, the first of these conditions is far from being realized with the Sun. The solar Taylor number, however, is large. For

[EQUATION]

([FORMULA] basic rotation rate, R solar radius) one finds with the reference value for the eddy viscosity

[EQUATION]

([FORMULA] correlation length, [FORMULA] turbulent rms velocity) the relation

[EQUATION]

The solar surface velocity is [FORMULA] 2 km/s, [FORMULA] is a dimensionless free parameter of order unity. With characteristic values for solar parameters, i.e. [FORMULA] 100 m/s and [FORMULA] we find

[EQUATION]

It is usually assumed that [FORMULA] does not exceed unity. It is a free parameter in the theory which cannot be tuned with an observation other than the large-scale flow pattern in the convection zone. On the other hand, its choice strongly influences the magnitude of the Taylor number. The latter varies between [FORMULA] and [FORMULA] if [FORMULA] varies between 1 and 0.1. Both extrema should be considered. It must be expected that Taylor numbers like (5) produce a remarkable intensity of the meridional flow. We thus need an extra explanation why the solar meridional circulation is so slow. In Kitchatinov & Rüdiger (1995) the rotational influence on the eddy heat diffusion tensor leads to a warmer pole so that an equatorwards directed extra drift weakens the (poleward) meridional circulation.

There is thus reason enough to rediscuss the role of the meridional circulation and its interchange with the differential rotation (cf. Rüdiger et al. 1998). Simplifying here we shall only deal with adiabatic stratifications but as an innovation the AKA-effect is tested in its ability to influence the large-scale flow system. Its existence has been established in papers by Frisch et al.  (1987), Khomenko et al. (1991) and Kitchatinov et al. (1994b), where in particular the close relationship of the AKA-effect to the MHD alpha is underlined (cf. Schüssler 1984; Gvaramadze et al.  1989; Sulem et al. 1989; Moffatt & Tsinober 1992). In linear approximation density stratification [FORMULA] and the global rotation [FORMULA] can be combined in two ways:

[EQUATION]

The first tensor is a pseudo-tensor but not the second one. As their construction is very similar there is no doubt that they appear simultaneously. The first one leads to the well-known [FORMULA]-effect in the electrodynamics of turbulent media [FORMULA] while the second one appears in the turbulent hydrodynamics as the connection between Reynolds stress and mean flow, i.e.

[EQUATION]

with the correlation tensor

[EQUATION]

This fact should have important consequences. If indeed the [FORMULA]-effect produces dynamo-generated large-scale magnetic fields then at the same time the AKA-effect should produce an extra mean flow pattern with consequences for the dynamo. Moreover, as the AKA-effect may generate a vortex system at the convection zone surface (with azimuthal number [FORMULA], see v. Rekowski & Kitchatinov 1998) one should observe magnetic fields always together with non-axisymmetric flow patterns at the surfaces of cool stars. At least for the Sun it is not yet observed. There are only a few reports about vortex structures in the mesogranulation pattern (cf. Brandt et al. 1988) so that indeed one could question the above argumentation 1. In Pipin et al. (1996) it is shown that both effects exhibit quite a different behavior for fast rotation. The AKA-effect decreases for fast rotation but not - as known - the MHD [FORMULA]-effect. This difference is considered as the explanation of the missing of vortex structure for stars (like the Sun) with Coriolis numbers,

[EQUATION]

exceeding unity with [FORMULA] as the turnover time of the convection eddies (cf. Gough 1977; Durney & Latour 1978; Stix 1989; Canuto 1991). Another complication for the vortex structure forming process is the existence of differential rotation. The preference of a [FORMULA] hydro-mode has been found by v. Rekowski & Kitchatinov (1998) for a sphere in solid-body rotation. Differential rotation may easily complicate the excitation conditions for non-axisymmetric flow patterns so that no self-excitation would be possible.

The computation of the hydrodynamic flow system under the simultaneous influence of the AKA-effect and the [FORMULA]-effect is reported in the present paper. A rotating density-stratified convection zone cannot rotate rigidly. As we have shown by means of the [FORMULA]-effect but under neglecting meridional flow the theoretical rotation law in the convection zone well approaches the known results of helioseismology. The 30% pole-equator difference of the angular velocity at the surface, however, is strongly reduced by the action of meridional flow. What we demonstrate in the following is whether the AKA-effect - which is mainly active in the surface layers of the convection zone - is able to reduce the smoothing action of the meridional flow.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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