In our model of the rotation and meridional circulation of a fully convective pre-main sequence star the Reynolds stress is the source of differential rotation. The meridional circulation is a consequence of differential rotation and not the source, as is often assumed. Eq. (7) has a steady solution only if the Reynolds stress due to the meridional motion exactly balances the gradient of the rotation frequency along the z-axis, . In the limiting case of vanishing viscosity this gradient must therefore vanish, as required by the Taylor-Proudman theorem. For finite viscosity, any deviation from the Taylor-Proudman state forces a meridional flow which not only produces the shear necessary for a steady state but also transports angular momentum and therefore changes the rotation pattern.
It is known from the work of Kippenhahn (1963) and Köhler (1969) on the solar convection zone that positive radial shear produces a meridional flow which is directed towards the equator at the surface while in case of negative shear the direction of the flow is opposite. In that model the horizontal shear is produced by the meridional motion and hence positive in case of positive vertical shear and negative for negative vertical shear. In our model, the radial -effect is negative everywhere except at the equator where the contributions of and cancel. The resulting circulation decelerates the rotation at the equator and thus counteracts the horizontal -effect. One could therefore expect a reversal of the surface differential rotation for sufficiently fast meridional flow. For very large Taylor numbers this is indeed the case, but the latitudinal shear is very small and the rotation is almost rigid.
As the variation of the meridional flow velocity as a function of the Taylor number in Fig. 4 shows, there are two asymptotic regimes. For small Taylor numbers, the differential rotation is completely determined by the Reynolds stress and the meridional motion is just a consequence of rotational shear. Hence, its velocity grows with Ta since as is not yet affected by the circulation and a decrease of the viscosity must be compensated for by an increase in the flow velocity. For very large Taylor numbers, the meridional flow drives the system towards the Taylor-Proudman state and its velocity decreases with increasing Taylor number. The maximum value of the meridional flow velocity is reached at intermediate values of Ta, i.e. between and , when on the one hand large velocities are needed to produce enough stress to compensate for , while on the other hand the -effect is still strong enough to maintain differential rotation.
According to the Taylor-Proudman theorem, for high Taylor numbers the rotation rate becomes a function of the distance from the axis only, The function f is, however, not prescribed by the theorem. As we know that the -effect is the only source of differential rotation, we solved the Reynolds equation for several artificially chosen combinations of and . The radial variation of at the equator turned out to be mainly determined by the -effect even for large Taylor numbers. But since in the latter case the rotation rate does not depend on z any more, it is completely determined by its radial variation at the equator. The essentially rigid rotation found for the T Tauri star is thus due to the cancelation of and at the equator.
We have modeled the rotation of a T Tauri star under the assumption that the star is barotropic, which should be true in a fully convective star. The angular velocity turns out to be almost constant throughout the whole star and the surface differential rotation is smaller than one percent. This result is in agreement with Johns-Krull (1996) as well as with Rice & Strassmeier (1996) and clearly contradicts the assumptions made by Smith (1994), who proposed an alternative explanation of the higher rotation rates of WTTS compared with CTTS in terms of equatorial acceleration and different latitudes of spot occurrence.
We have assumed a stress-free stellar surface. This means that we neglect the torques exerted on the stellar surface by a wind or an accretion disk via the stellar magnetic field. Since the time scale of the redistribution of angular momentum by Reynolds stress and meridional motion is only a few decades and hence much smaller than that of wind braking, the neglect of the latter is surely justified. In a CTTS the torques by accretion and magnetic field can be much larger than those of the wind and a slight change of the rotation pattern in case of strong coupling between disk and star is therefore possible. This will, however, only affect the surface layer to which differential rotation is restricted and not change our finding of essentially rigid rotation.
In our model, the effects of Reynolds stress and meridional motion cancel out each other. For the Sun, the rotation pattern resulting from a similar model are much to close to rigid body rotation. The situation changes if the stratification is not strictly barotropic. In that case, an additional Term arises on the RHS of (7) which drives an additional meridional flow. Kitchatinov & Rüdiger (1995) solved the Reynolds equation together with the equation for the turbulent heat transport for the solar convection zone and the outer part of the radiative core. They found a polar surface temperature 5 K higher than the surface temperature at the equator, which was sufficient to balance the meridional flow and cancel out its effect on the differential rotation. We therefore checked our model for its sensitivity to horizontal temperature gradients by adding a latitude-dependent perturbation, to the temperature T from the stellar model. We found that K does not affect the rotation law while 100 K cause an equatorial acceleration of about 4 percent and 1000 K cancel out the effect of the meridional flow. Negative values of , i.e. cool poles, cause a polar acceleration of 3 and 25 percent for 100K and 1000K, respectively. As for T Tauri stars the eddy heat conductivity tensor is the same as for the Sun, it seems unlikely that a latitudinal temperature gradient of several hundred degrees could be produced by anisotropic heat conduction. Spotted regions on the stellar surface are more than 1000 K cooler than the remainder of the surface, but it is questionable whether there is a strong effect on the rotation since their depth is probably small and their distribution strongly non-axisymmetric. We thus conclude that WTTS indeed rotate rigidly.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998