submitted to Astrophysical Journal
Dierential Rotation and Meridional Flow for Fast-Rotating Solar-type Stars Gunther Rudiger1 (
[email protected]) Brigitta v. Rekowski1 (
[email protected]) Robert A. Donahue2 3 (
[email protected])
and Sallie L. Baliunas2 3 4 (
[email protected])
ABSTRACT Observations indicate that normalized surface dierential rotation decreases for fast rotating stars: j j= / ;0 3 . An increase of j j= is provided, however, by the present-day Reynolds stress theory of dierential rotation in stellar convection zones without inclusion of meridional ow. We compute both the equator-pole dierence of the surface angular velocity and the meridional drift for various Taylor numbers to demonstrate that the inclusion of meridional ow into the computations for fast rotation yields a systematic reduction of :
1 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany.
2 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA. 3 Mount Wilson Institute, 740 Holladay Road, Pasadena, CA 91106, USA.
4 Center for Automated Space Science and Center of Excellence in Information Systems Tennessee State
University, 330 10th Ave. N., Nashville, TN 37203, USA.
{2{ the resulting dierential rotation. Our model's adiabatic and density-stratied convection zone with stress-free surfaces and a thickness of 0.3 stellar radii yields the relation j j= / ;(0 15 0 30) for stars with faster rotation than the Sun in agreement with previous observations (Fig. 5). If the Coriolis number is varied instead of the Taylor number, we nd a maximum dierential rotation of 20%. For stars with fast rotation, exponents up to n0 ' 0:4 are found. All rotation laws exhibit super-rotating equators. The only exceptions occur at very high rotation and small eddy viscosities (Fig. 10). :
::: :
Subject headings: turbulence { stars { dierential rotation { meridional ow
1. MOTIVATION Recent developments have been made in the observational study of surface dierential rotation. Donahue (1993) used the long-term database of chromospheric activity measurements from Mount Wilson Observatory's HK Project to monitor changes in the seasonal rotation period of lower main-sequence stars. From a sample of 110 stars, 36 had ve or more determinations of rotation period over the twelve years of data available. For some of these stars, there is evidence of trends between rotation period and level of activity, or where measurable, activity cycle phase (Baliunas et al. 1995). While some slowly-rotating stars (Prot > 10d) show an equatorward migration pattern that is similar to the Sun's, the opposite situation in which Prot increases with activity cycle phase is clearly present for several stars (Donahue 1993 Donahue & Baliunas 1994), even for stars with similar spectral types (i.e., mass) and mean rotation period (i.e., age). This `anti-solar' pattern could re ect either a poleward migration of active regions against a solar-like surface dierential rotation, or possibly the presence of poleward acceleration. A few stars, for example Comae, possibly have two active latitude bands (Donahue & Baliunas 1992 however, Gray & Baliunas 1997 oer a dierent explanation), although for several of these stars the range of periods observed within each activity band is very small, < 2% (e.g., 1 Ori, Donahue & Baliunas 1994). Young stars (< 1 Gyr) tend to have no discernible pattern of seasonal rotation period with either time or mean level of surface activity. For all the stars studied, the range of rotation period observed, P , increases with the mean rotation period, hProti. There is a power-law relationship,
jP j / hProti1 3 :
(1)
{3{ independent of spectral type (Donahue, Saar, & Baliunas 1996). However, for stars with similar Rossby number, K-type stars have larger jP j compared to G stars. It is possible to explain the observed internal solar rotation by means of a Reynolds stress theory in which the in uence of the basic rotation upon the cross correlations of the turbulent convection is determined (Kuker et al. 1993). In a pure Reynolds stress theory, the meridional ow in such calculations is completely neglected. The reason for this ignorance is twofold. First, the observations only provide a slow poleward ow with an amplitude of 10 m/s. There is the assumption that such a slow ow only modies the results of the Reynolds stress theory. On the other hand, inspection of the (observed) rotational isolines does not suggest a strong in uence due to meridional ow with one or two large cells. The meridional circulation tends to align the isolines of angular momentum with the streamlines of the ow (cf. Rudiger 1989 Brandenburg et al. 1990). Thus, the existence of distinct equatorial acceleration indicates that any possible meridional surface drift cannot be too strong. In the other case because of the small distance between the polar regions and the rotational axis, it should be more likely that a polar acceleration (\polar vortex") appears. Moreover, for very large Taylor numbers, the ow system approaches the TaylorProudman state where the isolines are two-dimensional with respect to the rotation axis, i.e. there is no variation along the rotational (z) axis. At the same time the meridional circulation breaks, i.e.: @ = 0 u = 0 (2) p
@z
(cf. Kohler 1969 Tassoul 1978). The rst of these conditions is not seen for the Sun the solar Taylor number, however, is large. From the denition of the Taylor number 2 4 4 Ta = R (3)
T2
with R as the stellar radius and as the rotation rate, we nd using the following reference value for the eddy viscosity, 2 T = c lcorr = corr
(4)
(where lcorr is the correlation length, and corr is the correlation time) the relation 2 Ta = 2 4 :
c
(5)
{4{ Here, as usual, is the Coriolis number = 2 corr
(6)
and = lcorr=R is the normalized correlation length. So with typically assumed values for the Sun of ' 5 c ' 1 and ' 0:1 (`giant cells'), we nd
Ta ' 2:5 105:
(7)
With such a Taylor number the observed dierential rotation should produce a remarkable intensity of the meridional ow. Therefore, we need an additional explanation as to why the solar polar drift is so slow. Kitchatinov & Rudiger (1995) nd the rotational in uence on the eddy heat diusion leads to a warmer pole such that an equatorwards-directed extra drift weakens the meridional circulation. Recent analysis of observational solar data conrms the poleward direction of the ow (between 10 and 60). Snodgrass & Dailey (1996) report a value of 13 m/s while Hathaway (1996) derive 20 m/s, and occasionally larger ( 50 m/s). While Snodgrass & Dailey used magnetogram data to nd the meridional motions by following the motions of small magnetic features in the photosphere, Hathaway worked with Doppler velocity data obtained with the GONG instruments. Any shortcomings concerning the usage of magnetic tracers are thus excluded and the results are of particular importance. Summing up, the following strategy of this paper might be reasonable: the "-eect is prescribed in order to reproduce the `solar case' for a certain xed Taylor number and then the Taylor number is varied. The main question concerns the response of the surface rotation law to this procedure. Equation (1) can also be represented as
