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Astron. Astrophys. 351, 347-358 (1999)

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2. Non-adiabatic internal waves

We follow here the complete analysis given by Unno et al. (1989) for non-adiabatic, non-radial oscillations, under the quasi-adiabatic and Cowling approximations. Real oscillations are inevitably non-adiabatic as there is always an energy exchange between mass elements during oscillations. Unlike Press (1981), we take into account the complete set of differential equations (21.11-21.15 in Unno et al. (1989)). In these equations the entropy perturbation [FORMULA]. The degree of non-adiabaticity is measured by a parameter [FORMULA] defined as the ratio between the thermal ([FORMULA]) and the dynamical ([FORMULA]) times scales:

[EQUATION]

[EQUATION]

and

[EQUATION]

where [FORMULA] is the luminosity, [FORMULA] is [FORMULA]. Other symbols have the usual meaning.

The equation of energy conservation is:

[EQUATION]

where [FORMULA] is the dimensionless frequency ([FORMULA]) and [FORMULA] is the frequency, [FORMULA] the radial part of the displacement in the horizontal direction, [FORMULA] the perturbation of nuclear energy generation rate. Consequently, if [FORMULA] is very large, the entropy perturbation is, in most cases, so small that the real part of the angular frequency is essentially the same as its adiabatic value. Inside the Sun [FORMULA]. In this case we can use the quasi-adiabatic approximation, in which the terms of O([FORMULA]) are discarded. In quasi-adiabatic and Cowling approximations the entropy perturbation is expressed as a function of radial displacement [FORMULA] and pressure perturbation [FORMULA]:

[EQUATION]

where [FORMULA] is the specific heat per mass unit at constant pressure. [FORMULA] and [FORMULA] are functions giving the radiative damping effect that we shall define below.

With these approximations, the equations of non-adiabatic oscillations are similar to those of adiabatic oscillations:

[EQUATION]

[EQUATION]

where g is the gravity, c the sound velocity, [FORMULA] the oscillation frequency, N the Brunt-Väisäla frequency:

[EQUATION]

[FORMULA] is the Lamb frequency:

[EQUATION]

and

[EQUATION]

Eqs. (4) and (5) present a clear parallelism with the equivalent adiabatic equations (1) and (2) in Press (1981). The non-adiabatic contribution appears only by linear complex terms. Therefore, it is possible to follow the same procedure than in the adiabatic case. Introducing the new variables [FORMULA] and [FORMULA] as follows:

[EQUATION]

and

[EQUATION]

the equations (4) and (5) become:

[EQUATION]

and

[EQUATION]

with [FORMULA] and [FORMULA] being defined as

[EQUATION]

and

[EQUATION]

and

[EQUATION]

To solve the Eq. (10) we use the asymptotic or WKB approximation, and introducing for convenience a new variable v as

[EQUATION]

Eq. 10 appears as:

[EQUATION]

with

[EQUATION]

Far from the turning points ([FORMULA]) we can write the solution of this equation with the WKBJ approximation:

[EQUATION]

and the amplitude of the vertical motion [FORMULA]:

[EQUATION]

In the expression of [FORMULA] the first term, [FORMULA], is of the order of [FORMULA], and in the first order approximation this term is discarded because it is small compared with [FORMULA]. Therefore, the function [FORMULA] is reduced to:

[EQUATION]

Introducing Eq. (12) in Eq. (18) the solution is:

[EQUATION]

with

[EQUATION]

The variation of the amplitude of gravity waves as a function of depth is given by the real part of Eq. (20). Taking into account the fact that [FORMULA], and the definitions (5) and (6), this variation can be written as:

[EQUATION]

The function [FORMULA] contains the radiative damping effect, whose expression is described in next section.

2.1. Radiative damping term

The full expressions of the funtions [FORMULA] and [FORMULA] which give the entropy perturbation are given by the Eqs. (22.9) and (22.10) of Unno et al. (1989).

We have computed the coefficients [FORMULA] in the solar case in order to find which terms give the largest contribution. The computation has been made with [FORMULA] and [FORMULA]. It appears that the functions [FORMULA] are essentially given by the following equations:

[EQUATION]

[EQUATION]

with

[EQUATION]

Furthermore, the most important term in the expression of [FORMULA] is the [FORMULA] term. The largest contribution to A is given, with a good approximation, by:

[EQUATION]

From the definitions (2) and (7) the term [FORMULA] is written as:

[EQUATION]

where [FORMULA] is the light velocity. With the approximation [FORMULA], and the definition (6) the damping term (24) is:

[EQUATION]

with

[EQUATION]

and

[EQUATION]

Using the approximation [FORMULA] for the horizontal wavenumber, and calling [FORMULA] its value at the boundary of the radiative zone ([FORMULA]), we finally can write the function A as:

[EQUATION]

We write later on:

[EQUATION]

The difference between the expression (26) and the corresponding equation in Press (1981) is that here there appears two factors containing gradients that have disappeard in his Eqs. (31) and (32). These factors take values different than one as we go to the center of the Sun. This corresponds to the fact that Press combined the merit of spherical adiabatic equations in the inelastic approximation, with the plane non-adiabatic equation in the Boussinesq approximation, whereas the expression (24) given here is derived from the complete set of equations written by Unno et al. (1989). The properties of the damping integral f and the function under the integral symbol are shown in Figs. 1a,b, where we have also plotted the same curves using Press' approximation. One can see that the amplitude of radiative damping begins to decrease at [FORMULA] with respect to the Press one. A large amplitude of A corresponds to a large damping of gravity waves of low frequency and relatively large wave number. As a consequence, we thought that we could consider that there is no reflection of gravity waves arriving to the stellar center, which means that we do not have to consider eigenvalues in this range of frequencies and wave numbers. However, in Fig. 2 we have plotted the dependence on radius of the amplitude of perturbation for several wave number-frequency couples. The wavenumbers considered correspond to spatial scales of the perturbation going from the plume dimension at the boundary of convective zone ([FORMULA]), to the radius of the convective zone ([FORMULA]). Gravity waves, most likely to propagate from the convection zone to the core without appreciable damping are those having small horizontal wave numbers and high frequencies. Thus, we see that for [FORMULA] the perturbation does not penetrate for any of considered values of k. For [FORMULA], only waves with k of the order of [FORMULA] do succeed to penetrate below [FORMULA]. As the value of the frequency increases above [FORMULA], perturbations with larger horizontal wavenumbers are able to penetrate near the center. Oscillations with the characteristics of [FORMULA] and k given by the turbulence distribution inside the convective plumes, ([FORMULA]), disappear before arriving to [FORMULA]. However, in Fig. 2, it appears also that for frequencies larger than [FORMULA] and wave number smaller than [FORMULA] the waves arrive to the center even with an amplitude larger than the initial one at the boundary of the convective zone. Therefore, to discard the possibility of g-modes in that range of frequency and wave number is only a first approximation, and the validity of that approximation depends on the source function distribution.

[FIGURE] Fig. 1. a  Radial dependence of radiative damping (f). b  Derivative of f respect to r. Solid lines correspond to the quasi-adiabatic approach, and dashed line to Press' approach

[FIGURE] Fig. 2. Amplitude of internal waves ([FORMULA]) as a function of radius, normalised to their amplitudes at the boundary of the convective zone

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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