## 2. The dynamoIt is now widely accepted that the solar dynamo operates in the overshoot layer at the bottom of the convection zone, where the stratification of the turbulence dominates and the radial gradient of the angular velocity is positive. The resulting dynamo is of -type with an oscillating field and equatorwards moving field belts in the equatorial region as required by the solar butterfly diagram (Krause & Rädler 1980; Schüssler 1983; Parker 1993). While this model is quite successful in explaining the solar activity cycle, it cannot be applied to fully convective stars. Observations as well as theoretical considerations lead to the conclusion that T Tauri stars rotate almost rigidly (Küker & Rüdiger 1997). A dynamo is of -type if the rotational shear is more effective than the -effect, i.e. where =
and
= are the magnetic Reynolds numbers
corresponding to the two field generating processes,
is the angular velocity of the
stellar rotation and ## 2.1. The electromotive forceWe work within the framework of mean field magnetohydrodynamics. In this approach, quantities are split into a large-scale and a small-scale part by application of an appropriate averaging procedure. We write as usual The induction equation for the mean magnetic field then reads where is the electromotive force (EMF) due to turbulent convection. It consists of a field generating and a diffusive term, i.e. ## 2.1.1. -effectWhile the eddy diffusivity exists
even in isotropic homogeneous turbulence, the
-effect is present only if the
turbulent medium is stratified In stellar convection zones, the density increases with depth while decreases. Hence, the contributions to the total -effect have different signs hence the final sign might vary with depth. In case of the Sun, the contribution from density dominates everywhere except in the transition layer at the bottom of the convection zone where the sign of changes (Krivodubskij & Schultz 1993). The -effect due to the stratification of the convective motions runs with . As Fig. 1 shows, the convection velocity is constant throughout the star. therefore vanishes and will not be considered any further. The contribution from density stratification (Rüdiger & Kitchatinov 1993), with , does not vanish and is therefore the field-generating term in the EMF.
Rewritten in cylindrical polar coordinates , the -tensor reads where is the colatitude,
denotes the Coriolis number and
the convective turnover time of the
stellar convection. The dimensionless parameter
has been introduced in (7) as an
input parameter that can be varied between zero and one in the
computations below. Eq. (7) is valid for arbitrary rotation rate but
weak fields. The In the more relevant opposite limit of rapid rotation, , the -effect assumes the form In the latter case the tensor becomes two-dimensional. Note that in (12) the Coriolis number does not appear anymore, i.e. the -effect becomes independent of the rotation rate. Eq. (7) holds for arbitrary rotation rates but only weak fields. For finite field strengths, the back reaction on the small-scale motions reduces the -effect and finally saturate the growth of the field at a value of the order of magnitude of the equipartition value. A calculation of this -quenching process has been done by Rüdiger & Kitchatinov (1993) for the slowly rotating case. As this does not apply in our case, we adopt the simple and widely used expression where , is the equipartition value for the magnetic field, and denotes the -effect in the limit of weak magnetic fields, i.e. in our case. ## 2.1.2. DiffusionThe diffusion is due to the simple presence of the turbulence. It does not depend on a gradient but on the length and time scales of the velocity fluctuations and on the rotation rate (Kitchatinov et al. 1994), For slow rotation, , and the diffusivity tensor becomes isotropic. For rapid rotation, and the diffusivity tensor not only becomes strongly anisotropic,
the diffusion being twice as effective along the For fast rotators, the -effect
ceases to grow with the rotation rate and saturates at a finite value.
The diffusion term ## 2.1.3. Turbulent transportBesides the field-generating -effect which is odd in the rotation rate, the density stratification also causes a pure transport effect which is even in the rotation rate, i.e. with where and In the limit of slow rotation, , while in the opposite limit of rapid rotation, (Kitchatinov 1991). This effect has been included for completeness, but is of minor importance in this context. For comparison, some runs were performed twice, once with and once without the transport effect. The results did not show any significant difference. ## 2.2. The starIn our computations constant values for the local parameters are used, which are taken from a model of a fully convective pre main-sequence star with 1.5 solar masses and 4.6 solar radii by Palla & Stahler (1993). Fig. 1 shows the stratifications of density, equipartition field strength, convection velocity and convective turnover time. The convection velocity after ( ## 2.3. NumericsAs non-axisymmetric field configurations must be expected, a 3D
code is necessary to solve the induction equation. We use the explicit
time-dependent second-order finite-difference scheme in three spatial
dimensions by Elstner et al. (1990). The code uses cylindrical
polar coordinates and a staggered grid representation of the EMF to
ensure that the magnetic field remains divergence-free throughout the
whole computation. The use of cylindrical coordinates has the
advantage of avoiding the singularity at the origin of the coordinate
system. If, however, the radial profiles of density, magnetic
diffusivity and other relevant parameters are not smooth but show
large gradients or rather layered stratifications, the necessary
number of grid points increases in both the at the upper and lower boundaries and at the outer boundary of the cylinder. Most runs were performed
with a resolution of grid points for
the To check if the configuration described above is a good approximation for a star surrounded by vacuum, we first tried to reproduce results of MB, who studied an -type dynamo in a conducting sphere surrounded by vacuum. We used the same EMF inside the sphere while the value of the magnetic diffusivity outside the sphere was assumed ten times larger than inside. We varied the Coriolis number to determine the critical value. We find dynamo action for Coriolis numbers larger than about 26, a value slightly smaller than that of 27.9 found by MB. The linear mode that is most easily excited is the S1-mode, which also is the dominant one in the nonlinear case. Hence, we find the same geometry as MB, although the critical values for the Coriolis number slightly differ. We did not make any attempts to improve the agreement between the models as it would have been very difficult to use the same boundary conditions as MB. © European Southern Observatory (ESO) 1999 Online publication: June 17, 1999 |