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

3. Basic equations

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

The tensor divergence is a vector with the components . 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.

This adopted, the azimuthal component of (22) is

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

hence there is a direct relation between density and pressure ( 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 :

It remains to fix the flow field by

For normalization the relations

and

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

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

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

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

The result is

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 . The -effect is normalized with .
The differential rotation and the stream function A are the unknowns of our equation system (30) and (34). The basic rotation is prescribed so that the system becomes inhomogeneous with the details given in Rüdiger (1989). Stress-free boundary conditions are imposed, i.e.

and 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 , the only free parameters are the mixing-length parameter and the factor in the eddy viscosity expression. As an extra parameter to tune the AKA-effect the normalized amplitude has been introduced with as the standard case.

© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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