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

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3. Basic equations

We solve the stationary Reynolds equation for a turbulent flow in the inertial reference frame, i.e.

[EQUATION]

The tensor divergence [FORMULA] is a vector with the components [FORMULA]. [FORMULA] denotes the acceleration of gravity which is here assumed uniform. Magnetic fields and deviations from axisymmetry are not considered. Anelastic fluids are only considered, i.e.

[EQUATION]

This adopted, the azimuthal component of (22) is

[EQUATION]

Our energy equation will be very simple. The turbulence is considered to be so intense that the medium is isentropic, i.e.

[EQUATION]

hence there is a direct relation between density and pressure ([FORMULA] is the specific heat). If this is true the pressure term in (22) can be written as a gradient which disappears after application of curling :

[EQUATION]

It remains to fix the flow field by

[EQUATION]

For normalization the relations

[EQUATION]

and

[EQUATION]

are used. The density stratification is then represented by [FORMULA]. As the conservation law of the angular momentum we get from (24) the dimensionless equation

[EQUATION]

Here also the turbulent fluxes of angular momentum are written in dimensionless form in accord with

[EQUATION]

Eq. (30) is linearized by means of the assumption of mild differential rotation,

[EQUATION]

The meridional Eq. (26) is formulated under the same condition. Hence the `centrifugal force' is reformulated as

[EQUATION]

The result is

[EQUATION]

Here the Taylor number Ta couples the centrifugal terms in (34) to the contributions of the turbulence-originated Reynolds stress. In contrast to (31) the latter are normalized with [FORMULA]. The [FORMULA]-effect is normalized with [FORMULA].
The differential rotation [FORMULA] and the stream function A are the unknowns of our equation system (30) and (34). The basic rotation [FORMULA] is prescribed so that the system becomes inhomogeneous with the details given in Rüdiger (1989). Stress-free boundary conditions are imposed, i.e.

[EQUATION]

and [FORMULA] at the equator. Like in Küker et al.  (1993) we have used the convection zone model by Stix & Skaley (1990) to specify the radial stratification of the parameters. Density and turbulence intensity as well as the mixing-length are taken from the convection zone model. Besides the basic solar rotation rate [FORMULA], the only free parameters are the mixing-length parameter [FORMULA] and the factor [FORMULA] in the eddy viscosity expression. As an extra parameter to tune the AKA-effect the normalized amplitude [FORMULA] has been introduced with [FORMULA] as the standard case.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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