2. The dynamo
It 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 R is a characteristic length scale, e.g. the stellar radius. In T Tauri stars, the absence of an overshoot layer as well as the flat rotation profile make a pure -mechanism unlikely. Moreover, the -effect due to density stratification cannot be ignored any longer but must be expected to be the dominating part of the -effect.
2.1. The electromotive force
We 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.
While the eddy diffusivity exists even in isotropic homogeneous turbulence, the -effect is present only if the turbulent medium is stratified and rotates, i.e. the turbulence is neither homogeneous nor isotropic. In a rotating stellar convection zone, both density and the intensity of the turbulent motions, are stratified. There are therefore two contributions to the -effect, i.e.
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.
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 zz-component differs from the ss- and -components by magnitude and sign. In the limit of slow rotation, , (7) becomes
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.
The 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 z-axis than perpendicular to it.
For fast rotators, the -effect ceases to grow with the rotation rate and saturates at a finite value. The diffusion term decreases like . The faster the star rotates, the more efficient (becomes) the dynamo process.
2.1.3. Turbulent transport
Besides 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.
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 star
In 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
(L luminosity, mixing-length) differs from that of standard mixing-length theory. It has been corrected to take into account the influence of the Coriolis force on the convective heat transport (Küker & Rüdiger 1997). A more consistent approach would be the use of rotation-dependent stellar models, i.e. to take into account the influence of the rotation in the stellar structure calculations, which is, however, beyond the scope of this paper. The corrected convection velocity is constant in the whole star except close to the surface. We use cm2/s for the turbulent magnetic diffusivity of the non-rotating star, 25 kG for the equipartition value of the magnetic field, and for the density stratification. Note that the actual magnitude of the magnetic diffusivity depends on the rotation rate via the Coriolis number. A rotation frequency s-1 corresponding to a rotation period of about a week yields a Coriolis number . This value is varied in some of the calculations below.
As 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 s and z directions, making the computations very time consuming. We use averaged values for these quantities as well as for their gradients where necessary. Moreover, it is not very convenient to impose any boundary conditions on the stellar surface. Instead, we assume the star to be surrounded by a corotating medium of large yet finite resistivity. A value of 100 is usually chosen for the ratio of the magnetic diffusivities inside and outside the star. The integration covers a cylinder with a height of four and a radius of two stellar radii with the star located at the center. The boundary conditions require that the magnetic field be perpendicular to the boundaries and divergence-free, i.e.
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 s, , and z coordinates, where the spacing is closer in the central part of the cylinder, which contains the star. A subset of grid points is used for this region which covers two stellar radii in z and one stellar radius in s. Some test runs were done with higher resolutions.
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