Atmospheric radiation boundary conditions for the Helmholtz equation

¹ Magique 3D Team, Inria Bordeaux Sud Ouest, 200 avenue de la vieille tour, 33405 Talence Cedex, France.
² LMA, UMR 5142, Université de Pau et des pays de l’Adour, 64000 Pau, France.
³ IMB – Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33405 Talence, France.
⁴ Center for Space Science, NYUAD Institute, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE.
⁵ Max Planck Institut für SonnensystemForschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany.
⁶ Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany.

^* Corresponding author: juliette.chabassier@inria.fr

Received: 14 February 2017
Accepted: 24 November 2017

Abstract

This work offers some contributions to the numerical study of acoustic waves propagating in the Sun and its atmosphere. The main goal is to provide boundary conditions for outgoing waves in the solar atmosphere where it is assumed that the sound speed is constant and the density decays exponentially with radius. Outgoing waves are governed by a Dirichlet-to-Neumann map which is obtained from the factorization of the Helmholtz equation expressed in spherical coordinates. For the purpose of extending the outgoing wave equation to axisymmetric or 3D cases, different approximations are implemented by using the frequency and/or the angle of incidence as parameters of interest. This results in boundary conditions called atmospheric radiation boundary conditions (ARBC) which are tested in ideal and realistic configurations. These ARBCs deliver accurate results and reduce the computational burden by a factor of two in helioseismology applications.