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On the outgoing solutions and radiation boundary conditions for the vectorial wave equation with ideal atmosphere in helioseismology

Abstract : 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.
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Submitted on : Friday, April 3, 2020 - 8:56:12 AM
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Hélène Barucq, Florian Faucher, Damien Fournier, Laurent Gizon, Ha Pham. On the outgoing solutions and radiation boundary conditions for the vectorial wave equation with ideal atmosphere in helioseismology. [Research Report] RR-9335, Inria Bordeaux Sud-Ouest; Magique 3D; Max-Planck Institute for Solar System Research. 2020. ⟨hal-02423882⟩

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