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Astron. Astrophys. 341, 181-189 (1999)

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5. Internal gravity waves

Another 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 [FORMULA]-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 [FORMULA] increases with depth, then a wave generated near the base of the solar convective envelope with a frequency [FORMULA] will experience on its way inwards a Doppler frequency shifting with a resulting local frequency changing with depth as


where [FORMULA] is the angular velocity of the convective envelope ([FORMULA]) and m the wave's azimuthal order. The special condition mentioned above is [FORMULA] which can be met at some radius [FORMULA] for a sufficiently large and positive mode number m. At [FORMULA] the IGW deposit some negative angular momentum to the local medium. In fact, the contribution of the IGWs to the angular momentum transport cannot be described either as pure diffusion or advection. It is a much more intricate process which requires the presence of asymmetry in propagation or dissipation of prograde (positive m) and retrograde (negative m) waves. Differential rotation and the Coriolis acceleration can produce such the asymmetries. For further physical and mathematical details the reader is refered to the paper of Zahn et al. cited above.

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 [FORMULA] model rotating with [FORMULA] s-1 is shown in the diagram [FORMULA] vs. [FORMULA] where [FORMULA], [FORMULA] is the convective turnover frequency and


a "damping" integral, the index "c" now refering to the convective core. Frequencies of the IGWs near the convective core border must lie between [FORMULA] and [FORMULA], 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 I appears in an approximate formula describing these losses. The diagram in Fig. 5a is similar to that plotted in Fig. 5 in Zahn et al. (1997). The solid and short-dashed straight lines divide it into seven domains where different approximate relations are used to estimate an angular momentum flux carried by the IGWs (formulae (A3), (A6), (A8-A11) in Zahn et al. (1997)). We used those relations to calculate characteristic times [FORMULA] required for the angular momentum flux associated with the IGWs to reach two arbitrarily chosen points marked with crosses in Fig. 5a. The respective values of [FORMULA] are given near the crosses. Fig. 5b transforms the coordinate [FORMULA] into the usual relative mass coordinate. We see that in the "inner zone" of the radiative envelope, defined here as a region where [FORMULA], [FORMULA] years, whereas in the "outer zone", with [FORMULA], [FORMULA]. It should be noted that [FORMULA] (Zahn et al. 1997). Therefore, we infer that the "inner zone" is a part of the envelope where the angular momentum transport by the IGWs may dominate over that by the meridional circulation and turbulent diffusion.

[FIGURE] Fig. 5a and b. Panel a : Evolution of the rotation state of the [FORMULA] star with [FORMULA] km s-1 in the diagram of Zahn et al. (1997, Fig. 5). Solid and dashed straight lines divide the diagram into seven domains where different approximate formulae for estimating the angular momentum flux carried by the internal gravity waves are applied in Zahn et al. (1997). Crosses are arbitrarily chosen points in the radiative envelope. Numbers near the crosses give the times required for the IGW flux to reach these points. Curves in panel b transform [FORMULA] into the usual relative mass coordinate

In the considered scenario the IGWs can participate in redistributing the angular momentum until [FORMULA] (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 [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998