E T u

e T u where we consider the waves in local pressure equilibrium (anelastic approximation). The equation for the transport of u corresponding to (17.28) is,

because there is, in principle, no diffusion of u (or in other words the equivalent to Vad for the u exchanges is zero). This is only true if one neglects the effect of horizontal turbulence. There is no reason to omit the effect of Dh and in future works this should be accounted for. Ignoring it for the moment, the equation for the density perturbation becomes e' NT + Nu 0 / T'\

where the variations of the mean quantities are neglected. The Brunt-Vaisala frequency is expressed here with (5.25). There, we use (17.29) and (17.30) to eliminate the T and ^ fluctuations

We can express q' with the equation of motion (17.27) and get for the first member and then for the full equation, iao'- KV2q ' = a f—V2 (q Vr) - -KV2(V2Qvr^ .

V (q Vr ) + klQVr + — V2 V2 + kq vr = 0 . (17.33)

a2 a a2

This is the wave equation in the anelastic approximation with account of non-adiabatic effects (leading to the damping terms in i) and of the effects of ^ gradients. With V2 ^ k2 + k^, it gives the following dispersion relation:

This is a second-order equation for k2, let us look for a solution in the quasi-adiabatic limit, i.e., with kr = A + BK H----, which gives in the linear approximation k;: =

By identifying the corresponding powers of K, we get A2 = kf (N2/a2) - 1 and for B,

n 2n2

where we have used N4 - N2N2 = N2n2. One has since the non-adiabatic term is small kr k A + BK

This provides the relation between the vertical and the horizontal wave numbers kr and kh like in (17.4). It also gives the damping factor of the waves.

Equation (17.33) can be compared to the wave equation (17.18) in spherical geometry, which does not account for the effects of radiative damping and of ^ gradient. The suggestion by Press , followed by all authors, is to combine these two equations by adding to (17.18) (which accounts for the effects of variations of density and radius) the damping term from (17.33) obtained under the anelastic assumption. One gets with these approximations the following wave equation for

d2W dr2

NL i oO2

If the damping term is not too large (which would make an evanescent solution), the solution of this equation is still given by (17.19) with account of the damping bringing an exponentially decreasing factor e-T/2. The damping factor t comes from (17.37), with the factor of 2 coming from the imaginary term of (17.37). At a level r, t is

The damping term t is proportional to the thermal diffusivity and to the power -4 of the wave frequency o. In case of differential rotation Q(r), this leads to a different radiative damping for the prograde and retrograde waves, see Sect. 17.3.1 below. The weighting in r-3 arises from the spherical geometry. The damping in e -t/2 applies to the three components (ur, u() of the velocity. Figure 17.2 illustrates the damping factor in a solar model. The g modes of high I are rapidly absorbed, while the low I modes may transport angular momentum from the convective envelope down to the deep interior.

r rc

 1 ! J = 2 1 ! 1 1 - ^Nf = 10. d a in p e d ¿ = 2û\~ - I ! I I I i 1

Fig. 17.2 The damping factor divided by (1(1 + 1))3/2 at a = 1 |Hz in the solar model. The depths where this damping term reaches a factor e are indicated. This term is the integral appearing in Eq. (17.39). Adapted from S. Talon 

Fig. 17.2 The damping factor divided by (1(1 + 1))3/2 at a = 1 |Hz in the solar model. The depths where this damping term reaches a factor e are indicated. This term is the integral appearing in Eq. (17.39). Adapted from S. Talon 

17.2 Energy and Momentum Transport by Gravity Waves

The gravity waves propagate in radiative regions and transport some kinetic energy and angular momentum through the star . Contrarily to early expectations, gravity waves likely do not significantly transport the chemical elements (Sect. 17.4.1).

Let us first evaluate in the adiabatic case the horizontal mean density of kinetic energy, using (17.22) to estimate the sum of the three components,

— i i 1 q (u2 + u% + uç ) sin ûdûdç ■n Jo Jo 2

This energy is transported with the group velocity vg (17.23), so that the mean flux Fkin of kinetic energy for a monochromatic wave is