## 2. Non-adiabatic internal wavesWe 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 . The degree of non-adiabaticity is measured by a parameter defined as the ratio between the thermal () and the dynamical () times scales: where is the luminosity, is . Other symbols have the usual meaning. The equation of energy conservation is: where is the dimensionless frequency () and is the frequency, the radial part of the displacement in the horizontal direction, the perturbation of nuclear energy generation rate. Consequently, if 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 . In this case we can use the quasi-adiabatic approximation, in which the terms of O() are discarded. In quasi-adiabatic and Cowling approximations the entropy perturbation is expressed as a function of radial displacement and pressure perturbation : where is the specific heat per mass unit at constant pressure. and 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: where 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 and as follows: the equations (4) and (5) become: and To solve the Eq. (10) we use the asymptotic or WKB approximation,
and introducing for convenience a new variable Far from the turning points () we can write the solution of this equation with the WKBJ approximation: and the amplitude of the vertical motion : In the expression of the first term, , is of the order of , and in the first order approximation this term is discarded because it is small compared with . Therefore, the function is reduced to: Introducing Eq. (12) in Eq. (18) the solution is: with 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 , and the definitions (5) and (6), this variation can be written as: The function contains the radiative damping effect, whose expression is described in next section. ## 2.1. Radiative damping termThe full expressions of the funtions and 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 in the solar case in order to find which terms give the largest contribution. The computation has been made with and . It appears that the functions are essentially given by the following equations: with Furthermore, the most important term in the expression of
is the
term. The largest contribution to
From the definitions (2) and (7) the term is written as: where is the light velocity. With the approximation , and the definition (6) the damping term (24) is: and Using the approximation for the
horizontal wavenumber, and calling
its value at the boundary of the radiative zone
(), we finally can write the function
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
© European Southern Observatory (ESO) 1999 Online publication: November 2, 1999 |