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Volume 579, July 2015
Article Number A30
Number of page(s) 12
Section Stellar structure and evolution
Published online 23 June 2015

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Appendix A: Derivation of the wave flux in the mean angular momentum equation (Eq. (35))

We assume that the perturbations consist in non-radial oscillations and that they behave linearly as exp[i(σt + )], where σ is the modal frequency, t the time, and m the azimuthal degree. Consequently, for a given quantity f, its perturbation can be written (A.1)where a is the amplitude, σR the real part of the frequency σ, and e stands for the real value. From Eq. (A.1) one immediately deduces a relation for squared quantities that reads (A.2)One can then express the Eulerian perturbation of the entropy in terms of the Lagrangian one, using δs = s′ + ξrd⟨s⟩/dr, where ξr is the radial component of the wave displacement. It allows us to rewrite the divergence of the wave flux in the momentum equation (see Eq. (39)) such as (A.3)with (A.4)In the following, our objective is to provide a general expression of Eq. (A.3) using the full set of equations governing waves. Note that in the following, the tilde will be omitted for the perturbation as well as overbar and brackets for the unperturbed quantities.

Appendix A.1: Using the full wave equations for expressing Eq. (A.4)

In this subsection, we aim to express the correlation products appearing in Eq. (A.4) using the set of Eq. (51) to Eq. (54).

Appendix A.1.1: Derivation of the first term of Eq. (A.4)

First, Eqs. (52) and (53) are inserted into (54), and an integration by part is performed. Second, Eqs. (51) to (53) are used as well as the relation (A.5)Finally, taking the real part and considering that the imaginary part of the frequency vanishes (σI = 0) since we consider steady waves (see Sect. 4.2), one gets (A.6)where we introduced the notation , ρT = −(lnρ/lnT)p, and the subscripts I and R denote the imaginary and real part, respectively. Note that this expression corresponds to Eq. (36.16) of Unno et al. (1989) and was first derived by Ando (1983).

To go further, we assume shellular rotation and that the equilibrium state is not deformed by rotation. Moreover, integration over the solid angle is performed so that it finally gives (A.7)

Appendix A.1.2: Derivation of the second term of Eq. (A.4)

We start from Eq. (52), which immediately gives (A.8)Equation (A.8) is further multiplied by and radial derivation is performed to give (A.9)Now one needs to express both the perturbation of pressure and its radial derivative. To this end, we use Eqs. (53) and (54) as well as Eqs. (51) to (53) to get (A.10)and (A.11)Finally, inserting Eqs. (A.11) and (A.10) into Eq. (A.9), integrating over the solid angle and taking the real part gives (A.12)

Appendix A.1.3: Derivation of the third term of Eq. (A.4)

The derivation is similar to the previous subsection, it gives (A.13)where we have introduced the notations (A.14)

Appendix A.2: The case of low rotation and frequency limit

As described in the main text (see Sect. 4.2) we restrict ourselves in the limit of low rotation (i.e. σR ≫ Ω0) and low frequency (i.e. σRN, where N is the buoyancy frequency). Those approximations will permit us to derive tractable expressions for Eqs. (A.7), (A.12), and (A.13). To this end, the first step consists in considering the wave equations without rotation and introducing them into Eqs. (A.7), (A.12), and (A.13). This is equivalent in considering a first-order development in term of rotation for Eq. (A.4).

Projection onto the spherical harmonics is thus performed in the limit of low rotation. For one normal mode of a given m and , the eigen-displacement and velocity can be decomposed such as (A.15)where (Rieutord 1987) (A.16)so that (A.17)and (A.18)where we used .

In the non-rotating limit and providing the decomposition given by Eq. (A.15), using Eqs. (51) to (54), can be expressed as a function of and only through the relation (A.19)with (A.20)To go further, in the asymptotic, quasi-adiabatic limit (σRN), is a function of (or equivalently ) so that depends only on . It reads (see Godart et al. 2009, for details) (A.21)and (A.22)where L is the luminosity, T the temperature, ρ the density, the temperature gradient, and ad the adiabatic temperature gradient.

Finally, after projection onto the spherical harmonics and using Eqs. (A.19), (A.21), and (A.22), one obtains for a mode of a given angular degree () and azimuthal degree (m) an expression of the form (A.23)with (A.26)and (A.27)where for Eq. (A.27) we use the approximation obtained in the limit , and (A.28)

as well as

(A.29)Finally, we note that the normalization condition for spherical harmonics is , where δ is the Kronecker symbol. Note also that the following relation have been used where (A.33)if > | m |, and otherwise.

© ESO, 2015

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