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

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2. The turbulence model

A stratified turbulence can never rotate rigidly. The reason is that even in the case of uniform rotation - in contrast to any form of viscosity - the turbulence transports angular momentum. The formal description of this phenomenon is given by the non-diffusive part [FORMULA] of the correlation tensor (8), i.e. for the

turbulent fluxes of angular momentum

[EQUATION]

where the coefficients [FORMULA] and [FORMULA] are functions of the Coriolis number. The dotted parts in Eqs. (10) are denoting the contributions of the eddy viscosity tensor [FORMULA] which in the simplest (axisymmetric) case yields

[EQUATION]

In Kitchatinov et al. (1994a) the structure of the eddy viscosity tensor under the influence of a global rotation is derived within the framework of a quasilinear theory (cf. Roberts & Soward 1975). The tensor becomes highly anisotropic, making the theory more complicated. In the present paper we apply only the [FORMULA]-dependence of the isotropic part of the viscosity tensor, i.e. the coefficient of the deformation tensor, replacing (3) by

[EQUATION]

with

[EQUATION]

(cf. Fig. 2, dashed line). In Kitchatinov & Rüdiger (1992) one finds the representation for the functions V and H.

We adopt the approximation

[EQUATION]

with [FORMULA] from the usual mixing-length relation

[EQUATION]

between the mixing-length, [FORMULA], and the pressure scale-height, [FORMULA]. [FORMULA] is the adiabatic index,

[EQUATION]

is the density scale-height. The expressions for [FORMULA] and [FORMULA] describe the rotation-rate dependence of the [FORMULA]-effect. In Fig. 1 the profiles of the functions [FORMULA] and [FORMULA] are given. Küker et al. (1993), under application of a model of the solar convection zone by Stix & Skaley (1990) were able to reproduce the main features of the `observed' internal rotation of the Sun's outer envelope. They neglected, however, the angular momentum transport of meridional flows and magnetic fields.

[FIGURE] Fig. 1. The influence of the basic rotation on the [FORMULA]-effect coefficients [FORMULA] (solid) and [FORMULA] (dashed) after Kitchatinov & Rüdiger (1992) and on the AKA-effect (dotted) after Pipin et al. (1996)

One has to include the meridional flow into the theory. To this end one needs the knowledge of the influence of rotation on the turbulent heat transport. If this is known from an established turbulence theory, then one can compute the complete fields of pressure, circulation and differential rotation.

But even then the theory is not complete. It might not be true that the mean flow enters the correlation tensor only via its gradients but also via the flow itself:

[EQUATION]

In Kitchatinov et al. (1994b) it has been demonstrated that in rotating and stratified turbulent fluids there is a part of the [FORMULA]-tensor which is due to the basic rotation, i.e.

[EQUATION]

with [FORMULA] as the unit vector parallel to the rotation axis, [FORMULA] given the radial unit vector. It directly results in correlations such as

[EQUATION]

where the dependence of [FORMULA] on the basic rotation rate is given in Pipin et al. (1996) as

[EQUATION]

with

[EQUATION]

i.e. [FORMULA] for slow rotation (Fig. 2). Obviously, the [FORMULA]-effect is strongly related to the MHD [FORMULA]-effect which has the same structure and also exists only in rotating stratified turbulences.

[FIGURE] Fig. 2. The depth-dependence of the functions [FORMULA] (AKA, solid) and [FORMULA] (eddy viscosity, dashed)

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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