2. Input physics
The meridional circulation is treated using the formalism described by Zahn (1992); it takes into account the radial differential rotation, with the assumption that the rotation rate is nearly constant on isobars. The circulation velocity then depends on the third derivative of , and the transport of angular momentum is a true advective process. The shear stresses are calculated with a turbulent viscosity which includes the weakening effect of thermal damping (as first described by Townsend, 1958). The details concerning that viscosity are not important here since the circulation, being an advective process, dominates the transport of momentum almost everywhere. The transport associated with gravity waves is assumed to follow the description of ZTM, which is relevant for strong differential rotation.
where we used standard notations for the radius r and the density and where is the vertical (turbulent) viscosity, U the amplitude of the vertical component of the circulation speed and is the luminosity of angular momentum summed over the whole wave spectrum.
Here accounts for the magnetic torque imposed on the convection zone by the solar wind and `cz' refers to the base of that zone. For simplicity, we approximate the loss of angular momentum by a cubic law in ( ), including saturation to a linear law in when rad s-1. The evolution of the surface velocity can then be compared to the square root law found by Skumanich (1972) or to the modern observations of rotational periods (see e.g. Bouvier et al. 1997).
An expression for is given in ZTM. When differential rotation is small, the wave luminosity is proportional to . However, as rises, some of the frequencies will disappear from the wave spectrum as even those waves with will have reached their critical layer. This translates into an upper limit on for which the absorption is linear
where characterizes the frequency of the largest convective eddy. represents the damping by radiative processes (K is the thermal conductivity). There is also a lower limit for (that is, a finite amount of differential rotation is required to get some extraction) corresponding to the fact that the maximal horizontal wave number is not infinite:
where is the buoyancy frequency at the base of the convective zone (assuming some penetration).
When becomes large, an important part of the wave spectrum has been damped before reaching the considered level and thus, cannot extract momentum. The extraction then varies as (see ZTM for the exact form). This somewhat augments the timescale calculated in ZTM.
In the present calculations, we reduced the magnitude of the wave luminosity with respect to the evaluation we made in ZTM by one order of magnitude. This may seem arbitrary, but we feel that the prescription we derived there, which was inspired by García López & Spruit (1991), probably overestimates the efficiency of wave generation. More work needs to be done to reduce the uncertainties associated with the crude treatments that have been used so far, and the validation of any theory of wave excitation will rely ultimately on the observational constraints provided by the chemical anomalies and surface rotation rates of low mass stars.
Equation (1) admits a stationary solution, with just enough differential rotation for the wave transport to balance the advection by the meridional circulation. A slight departure from this profile is sufficient to provide the flux extracted by the solar wind.
Let us stress that we considered here only the differential rotation in depth. It remains to be seen how the gravity waves act to homogenize the rotation in latitude below the convective zone. That would have to be compared with the horizontal turbulent diffusivity Spiegel & Zahn (1992) had to invoke to stop the spread of the differentially rotating region.
© European Southern Observatory (ESO) 1998
Online publication: November 24, 1997