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                <term xml:lang="en">Outgoing solutions</term>
                <term xml:lang="en">Radiation boundary conditions</term>
                <term xml:lang="en">Whittaker functions</term>
                <term xml:lang="en">Green tensor</term>
                <term xml:lang="en">Vectorial helioseismology</term>
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                <term xml:lang="en">Coulomb potential</term>
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                <term xml:lang="fr">Solutions sortantes</term>
                <term xml:lang="fr">Equantion de Galbrun</term>
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              <p>In this work, we consider the time-harmonic Galbrun’s equation under spherical symmetry in the context of the wave propagation in the Sun without flow and rotation, and neglecting the perturbations to the gravitational potential. The model parameters are taken from the solar model S for the interior of the Sun, and we introduce the model AtmoCAI (ideal atmospheric behavior with constant adiabatic index) to extend them into the atmosphere. This atmospheric extension is based on the model Atmo used for the scalar wave propagation where, in addition, we assume a constant adiabatic index in the atmosphere. Due to the spherical symmetry, by writing the original equation in a vector spherical harmonic basis, we obtain the ODE for the modal radial and tangential coefficients of the unknown displacements. We then construct the outgoing modal solutions, the 3D Green’s kernel, and radiation boundary conditions. The construction is justified by indicial and asymptotic analysis of the modal radial ODE. While the singular set in the presence of attenuation only consists of the origin, our analysis shows that without attenuation, there are also other singular points which, however, have positive indicial exponents. Our asymptotic analysis makes appear the correct wavenumber and the high-order terms of the oscillatory phase function, which we use to characterize outgoing solutions. The radiation boundary conditions are built for the modal radial ODE and then derived for the initial equation. We approximate them under different hypothesis and propose some formulations that are independent of the horizontal wavenumber and can thus easily be applied for 3D problems.</p>
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              <p>Dans ce travail, nous considérons l’équation harmonique de Galbrun en symétrie sphérique pour la propagation d’ondes dans le soleil, sans flot ni rotation, et en négligeant les perturbations du potentiel de gravité. Les paramètres sont extraits du modèle S pour l’intérieur du soleil, et nous introduisons un modèle AtmoCAI (comportement atmosphérique idéal avec un indice adiabatique constant) pour leur extension dans l’atmosphère solaire. Cette extension est basée sur le modèle Atmo utilisé dans le cas scalaire, que nous enrichissons en prenant l’indice adiabatique constant. De part la symétrie sphérique, en écrivant le problème dans une base harmonique sphérique vectorielle, nous obtenons l’EDO modale pour les coefficients radiaux et tangentiels du déplacement. Nous construisons les solutions sortantes modales, le noyau de Green en 3D et obtenons des conditions aux limites de radiation. La construction est motivée par l’analyse indicielle et asymptotique de l’EDO radiale modale. En présence d’atténuation, la seule singularité est á l’origine, alors que dans le cas sans atténuation, nous identifions les autres singularités qui, cependant, ont un exposant indiciel positif. Notre analyse asymptotique fait apparaître un nombre d’onde approprié et les termes d’ordres élevés de la phase, qui nous servent à caractériser les solutions sortantes. Les conditions aux limites de radiation sont construites pour l’EDO modale puis étendues au problème initial. Nous les approximons sous différentes hypothèses et proposons plusieurs approximations independantes du nombre d’onde horizontal qui peuvent être facilement utilisées pour les problèmes 3D.</p>
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