Astron. Astrophys. 323, 151-157 (1997)

3. Dynamo model

The existence of a correlation between , and points to a common dynamo mechanism for the stars in the sample under consideration. We compare the observed correlation with the predictions of a simple model, based on linear mean-field dynamo theory. Although the validity of the linear approach is open to debate (cf. Noyes et al.  1984b, Jennings & Weiss  1991, Rüdiger & Arlt  1996), it may be justified by the slow rotation rate and low activity level of the selected stars.

3.1. Geometry and equations

The dynamo model that we employ was proposed by Parker (1993) for the Sun. It consists of two plane parallel layers: the overshoot layer (region 1) and the convection zone (region 2), with thicknesses and respectively. The main motivation for the model arises from the presence of strong magnetic fields (  G) in a thin layer under the convection zone, as is suggested by observations and theoretical considerations (Hughes  1992, Schüssler et al.  1994). The strong fields give rise to the suppression of turbulence, so that and are reduced. Helioseismology suggests that differential rotation is concentrated near the same layer (Goode  1995). Hence differential rotation and the -effect are possibly spatially separated, the former being restricted to the overshoot layer and the latter to the convection zone. Some turbulent diffusion is required in the overshoot layer in order to provide communication with the convection zone.

We assume that this model applies for all the stars in our sample. We use x for the radial, y for the azimuthal, and z for the latitudinal coordinates, and consider only axisymmetric solutions (). The overshoot layer is located at and the convection zone at .

We model the results of helioseismology for the equatorial region in a schematic way by adopting the following large-scale velocity field :

Here the constant a denotes the radial velocity gradient.

The mean magnetic field can be written as the sum of a toroidal and a poloidal component, i.e. . It is governed by the following equations (Parker  1993, Ossendrijver & Hoyng  1996):

Here the -coefficient and the turbulent diffusivity in the convection zone are given by

where denotes the turbulent velocity field, having a correlation time . The suppression of the turbulent diffusivity by strong magnetic fields in the overshoot layer is parametrized by a factor

We seek solutions of the form , and similarly for T. Here is the wave vector in the latitudinal direction. The boundary conditions are as follows (cf. Ossendrijver & Hoyng  1996):

Here denotes the jump at the boundary. In the second line, assumes the value for and for . We separate in complex and real parts according to

where is the dynamo frequency and is the mean-field growth rate. We solve the dispersion relation numerically and we identify the fundamental mode, i.e. that with the largest growth rate. The diffusivities and are tuned in such a way, that the fundamental mode is slightly subcritical, having a decay time of about 20 cycle periods. For a motivation of this requirement, and for analytical solutions of Eq. (6) we refer to Ossendrijver & Hoyng (1996).

3.2. Dynamo parameters of lower main-sequence stars

In this section we present expressions for the parameters that occur in the dynamo equation, in terms of the stellar structure and the rotation rate. Some of these expressions relate stellar parameters to solar parameters, which are treated in Sect. 3.4.

Apart from (Eq.  2), the parameters related to stellar structure are derived from a set of models by Copeland et al. (1970). Out of the models presented by these authors we have chosen the series with a composition given by , , , and with a mixing-length parameter (Table 2).

Table 2. Parameters of lower main-sequence stars

We employ the pressure scale height at the bottom of the convection zone, , to define the thickness of the overshoot layer (Skaley & Stix  1991):

For the Sun, this amounts to about  km (Table 2). We assume that the total difference in angular velocity accross the overshoot layer depends on the rotation rate in the following manner:

No reliable estimate is known for p, but there are theoretical indications that p is positive, so that differential rotation decreases with increasing rotation rate (Kitchatinov & Rüdiger  1995). The corresponding velocity gradient in the overshoot layer can be expressed as

where denotes the distance from the origin to the bottom of the convection zone (Table 2).

Following Eq. (7), we estimate as , where is the typical length scale of the convective cell and H denotes the normalised helicity coefficient,

This coefficient measures the correlation between and the vorticity . It is non-vanishing due to the effect of rotation on the convective motions, which suggests a dependence on the Rossby number. Since , the correlation must saturate for small values of Ro (rapid rotation), but if rotation is slow, we may assume , with . Here , the normalised helicity for , is taken to be the same for all the stars in our sample. Its value is fixed by the solar calibration model (Sect. 3.4). The -coefficient now becomes

Here the convective turnover time is obtained from (Table 2) with the help of expression (2).

We assume that all the stars have activity belts similar to those of the Sun. These activity belts originate at a latitude of about during the activity minimum and migrate toward the equator in the course of one cycle period, after which new belts of opposite polarity appear. The wave vector in the latitudinal direction that is associated with this equatorward migration is estimated as

The turbulent diffusivities are determined in the following way. We determine the ratio by means of the solar calibration model in Sect. 3.4 and we assume that it is the same for all the stars. The value of (and of ) is fixed by a condition on the growth rate of the mean magnetic field, see Sect. 3.5. However, we shall demonstrate in Sect. 3.4 that, to good approximation, the cycle period depends on and only through their ratio , so that is not affected by this condition.

3.3. Theoretical cycle periods

The exact cycle periods of stellar dynamos are found by numerically solving the dispersion relation that results from Eq. (6). In order to gain insight in the parameter dependence of we employ an approximative expression. This allows us to compare in a simple manner the observed relation between , and with the predictions of our model. Ossendrijver & Hoyng (1996) derived the following expression, using an approximative dispersion relation which is valid if :

where C is the dynamo number, given by

A useful approximation is obtained for large dynamo numbers ():

Notice that now depends on the turbulent diffusivities only through their ratio , which we take as a constant. Hence we do not require for the moment any further knowledge of or , the values of which will be determined in Sect. 3.5. Since the dynamo number (Eq.  18) does depend on and , we shall verify the validity of Eq. (19) also in Sect. 3.5, and assume for the moment that . Inserting Eqs. (11- 16) we may write

where expressed in years, and and in days. We point out that, according to this approximation, depends on stellar structure only through . This allows us to compare Eq. (20) with Eq. (3), which describes the observations, and we conclude that these two equations are equivalent if

The positive value of p indicates that differential rotation decreases with increasing rotation rate (Eq.  12). Note the effect of the strong anticorrelation between b and c on the uncertainty in p. The rather large negative value of q indicates a strong dependence of on rotation.

Fig. 4 shows the observed cycle periods of the slowly rotating stars from Table 1 as a function of , as well as the curves predicted by Eq. (20) for various rotation rates. There is reasonable agreement on the whole between the theoretical curves and the measured cycle periods.

Fig. 4. The solid points denote the measured cycle periods of slowly rotating lower main-sequence stars (Table 1). The dotted curves represent Eq. (20) for various rotation rates. The labels indicate the rotation period in days.

3.4. The solar calibration model

The calibration of the stellar dynamo models is based on the solar parameters that are shown in Table 3. The total difference in angular velocity across the overshoot layer near the equator is estimated from Goode (1995). The constant is determined from Eq. (15) by adopting a value for , namely -25 cm s^-1. The choice of is somewhat arbitrary. We determine and following an iterative method, i.e. by solving and from the (exact) dispersion relation and applying corrections to and , until years and .

Table 3. Parameters of the solar calibration model

3.5. Validity of expression (20)

In order to verify whether the dynamo number is sufficiently large for expression (20) to be valid, we return to Eq. (6) and solve the full dispersion relation (cf. Ossendrijver & Hoyng  1996) for a series of stellar models (Table 2). For each model, we calculate the dynamo parameters as indicated in Sect. 3.2, employing the values of p and q derived in Sect. 3.3. We choose stellar masses in the range and rotation periods in the range  days, providing complete overlap with all the slowly rotating stars in Table 1. For a given stellar mass and rotation rate we let (or ) assume the value at which the fundamental mode of Eqs. (6) satisfies . This is achieved by starting with an initial value for , solving the dispersion relation, estimating the required correction to , and repeating this, while keeping constant, until has converged. The smallest dynamo number that occurs in any one of these models is , and the resulting difference between exact and approximative values of is typically a few percent. Given the large uncertainty in the observed cycle periods (typically ), we ignore this effect, and we conclude that expression (20) is a valid approximation for the cycle periods of slowly rotating lower main-sequence stars.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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