We use rigid-body rotation and vanishing meridional flow as initial values. Due to the -effect, it is not a possible steady state. Consequently, the Reynolds stress starts immediately to redistribute the angular momentum, thus forcing differential rotation which in turn gives rise to the meridional flow. The meridional flow counteracts the generation of the differential rotation and finally the whole system approaches an equilibrium. Note that as long as there is no further force, the meridional flow is driven by the differential rotation and not vice versa. It thus always weakens the differential rotation, driving the whole system towards rigid-body rotation.
3.1. Simplified model
To illustrate this mechanism, we first study a simplified model. The viscosity coefficient as well as the - effect are constant throughout the whole star. The simple form (14) is assumed for the viscous part of the stress tensor. We choose and , which is a good approximation for the case of rapid rotation, , when . We measure the speed of the rotation by its Taylor number,
which gives the strength of the centrifugal force relative to the Reynolds stress.
Fig. 1 gives the result for Ta=10, a case where the rotation pattern is determined by the -effect alone and not significantly influenced by the meridional flow. The equator rotates twice as fast as the high latitude and the radial shear is negative throughout the whole star. There is one cell of meridional circulation per hemisphere, the surface flow being directed towards the poles.
In Fig. 2, the result for an intermediate Taylor number of is given. Although the -effect still dominates, the pattern has changed significantly and the total latitudinal shear is only 20 percent of the equatorial angular velocity rather than 50 percent. The radial variation of is essentially unchanged close to the boundaries, but at a depth of half the stellar radius there is no radial shear any more. The meridional flow pattern has not changed.
For the large Taylor number of , as shown in Fig. 3, the total horizontal shear at the stellar surface has been further reduced to about 2 percent. There is no radial shear in the interval [0.4,0.9] of the fractional stellar radius x. Outside a cylindrical surface touching the (artificial) inner core at the equator, the shear is now restricted to a layer of thickness 0.1 close to the upper and lower boundary, respectively. The deviation from rigid-body rotation is less than 4 percent close to the lower and up to two percent at the upper boundary.
For fixed angular velocity increasing the Taylor number means decreasing the viscosity. Fig. 4 shows the maximum value of the latitudinal velocity component as a function of the Taylor number for this case. It increases with Ta for small Taylor numbers, reaches a maximum between and , and decreases as Ta is increased for . The shape of the flow pattern, represented by the stream function, changes, however, only slightly. There is always only one flow cell per hemisphere and the surface flow is always directed towards the poles.
3.2. Full stratification
The complete model includes the stratification of the turbulence. The convective turnover time as well as the convective velocity vary strongly with the fractional stellar radius, leading to a depth dependence of the Coriolis number. In Fig. 5 the Coriolis number is plotted vs. the fractional stellar radius for an angular velocity and three different values of the mixing-length parameter . In our calculations we always assumed , which, for fully ionized gas and a stratification that deviates only slightly from the adiabatic case, means that the mixing length is equal to the density scale height. The cases and are shown for comparison. Fig. 5 shows that, although the exact value of the Coriolis number increases by a factor of 1.6 as is increased from 1.0 to 2.4, it is in all cases a number between 10 and 100 almost throughout the whole stellar volume.
The Reynolds stress is thus strongly reduced, especially in the inner part of the star. In a model without meridional circulation this would, however, not affect the resulting rotation law, since the diffusive and non-diffusive part vary in the same way in the case of large Coriolis numbers . In the case considered here, the balance between the centrifugal force and the turbulent viscosity is strongly affected. The turbulent viscosity of the non-rotating star,
is a moderately varying function of the density with a magnitude of for . Rotational quenching makes it strongly depth-dependent and reduces it by up to two orders of magnitude. The resulting value of together with the stellar radius of 4.6 yields a Taylor number of , too large for our code to produce stable results. We therefore use , providing a Taylor number two orders of magnitude smaller but still very large.
The resulting rotation law and the stream function are shown in Fig. 6. It resembles very much that for from the simple model. The deviation from rigid rotation is less than 0.5 percent except close to the (artificial) lower boundary. The remaining small shear follows the Taylor-Proudman theorem in the bulk of the stellar volume. Close to the upper and lower boundary, respectively, there is a layer where the rotation pattern is determined by the boundary conditions, which do not match the Taylor-Proudman state. At the outer boundary, the requirement that the surface be stress-free together with small values of the Coriolis number leads to positive radial shear and a strong reduction of the latitudinal shear within the outer 5 percent of stellar radius. Due to the rapidly decreasing density, the meridional flow velocity is maximal at the surface, although the stream function has very small values there. The speed of the latitudinal flow reaches a maximum value of about 70 m/s close to the stellar surface.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998