## 5. Internal gravity wavesAnother mechanism probably able to contribute to the angular momentum transport in stellar radiative zones is one associated with internal gravity waves (hereafter, IGWs) (Schatzman 1993; Zahn et al. 1997; Ringot 1998). In a single star the IGWs can be generated by turbulent motions of convective eddies (Press 1981; García López & Spruit 1991). They carry angular momentum through a radiative zone and deposit it locally at a place where some special conditions are met. Recently, Zahn et al. (1997) have proposed that the IGWs can strongly influence the evolution of the -profile in the Sun (see, however, critical comments of Ringot (1998)). If a state of differential rotation is sustained in the Sun, by, for instance, meridional circulation, and increases with depth, then a wave generated near the base of the solar convective envelope with a frequency will experience on its way inwards a Doppler frequency shifting with a resulting local frequency changing with depth as where is the angular velocity of the
convective envelope () and We have applied Zahn et al.'s results on the IGWs to the case of massive MS stars. The main qualitative differences as compared to the case of the Sun are the following: the IGWs are now generated near the convective core border, propagate outwards and carry positive angular momentum causing spinning-up of the radiative envelope. In Fig. 5a approaching the steady-state rotation by our
model rotating with
s a "damping" integral, the index "c" now refering to the convective
core. Frequencies of the IGWs near the convective core border must lie
between and , the
latter quantity being the Brunt-Väisälä frequency at
the level where the IGWs are generated. On their way outwards the IGWs
lose energy due to radiative leakage. The damping integral
In the considered scenario the IGWs can participate in redistributing the angular momentum until (following Zahn et al. (1997) we ignored the Coriolis acceleration) and, therefore, the IGWs tend towards establishing a state of uniform rotation. Thus, our analysis shows that the "inner zone" of the radiative envelope of a massive MS star may be a region where a state of nearly constant is sustained by the IGWs, the size of this region increasing with stellar mass (Fig. 5b). However, as was explained in the preceding section, this cannot considerably accelerate mixing of chemical elements because of the efficient horizontal erosion. A probably even more important contribution of the IGWs to the mixing of the massive MS stars could result from their nonlinear behaviour and various hydrodynamical instabilities associated with it (Press 1981; García López & Spruit 1991). Unfortunately, this problem demanding extensive numerical 3D-simulations is still far from being solved. © European Southern Observatory (ESO) 1999 Online publication: November 26, 1998 |