Outgoing solutions and radiation boundary conditions for the ideal atmospheric scalar wave equation in helioseismology

Abstract

In this paper, we study the time-harmonic scalar equation describing the propagation of acoustic waves in the Sun’s atmosphere under ideal atmospheric assumptions. We use the Liouville change of unknown to conjugate the original problem to a Schrödinger equation with a Coulomb-type potential. This transformation makes appear a new wavenumber, k, and the link with the Whittaker’s equation. We consider two different problems: in the first one, with the ideal atmospheric assumptions extended to the whole space, we construct explicitly the Schwartz kernel of the resolvent, starting from a solution given by Hostler and Pratt in punctured domains, and use this to construct outgoing solutions and radiation conditions. In the second problem, we construct exact Dirichlet-to-Neumann map using Whittaker functions, and new radiation boundary conditions (RBC), using gauge functions in terms of k. The new approach gives rise to simpler RBC for the same precision compared to existing ones. The robustness of our new RBC is corroborated by numerical experiments.

hawen: time-harmonic wave modeling and inversion using hybridizable discontinuous Galerkin discretization

“…It combines MPI and OpenMP parallelism, it is deployed on supercomputers and has been used in studies on seismic imaging (Faucher, Alessandrini, et al, 2020;, as well as in helioseismology (Barucq et al, 2019(Barucq et al, , 2020Barucq, Faucher, Fournier, et al, 2020a, 2020b. The software works with an input parameter file: a text file listing the problem configuration (dimension, choice of viscous model, etc.…”

Efficient and Accurate Algorithm for the Full Modal Green's Kernel of the Scalar Wave Equation in Helioseismology

“…Similarly to [8], we work with the conjugated problem, given by (2.8) in three dimensions after adimensionalization. Then, using spherical symmetry, we decompose the solution into one-dimensional modal ones defined on [0, r max ], with r max > 1.…”

“…Similarly to [8,7,1], we work with the conjugated equation obtained by a Liouville change of unknowns, since the latter offers a natural setting to define the unique physical (also called "outgoing" or "radiation") kernel G, solution to, [8],…”

“…Assuming that the physical parameters only depend on the Sun's radius, one can decompose G into the spherical harmonic basis, and compute instead the modal Green's kernel G (r, s) for each spherical mode , with r = |x| the (scaled) radius and s the source position. It is a fundamental solution of, [8], Current results in local helioseismology assume that the Sun has a "surface" that is well defined, and work with numerical simulations which are obtained from the solution of the wave equation at the surface of the Sun only, i.e., G (r, s = 1) using scaled variables. However, the Sun is a plasma and the observed oscillations represent an average over all depths weighted by the level of transparency (opacity).…”

“…We investigate the efficiency of approximate radiation boundary conditions (RBC) on synthetic helioseismic products using the complete solar background. Until now, these comparisons were only carried out in terms of solutions in the atmosphere (e.g., [8]). With the exact DtN coefficient as reference, our investigation provides an efficient choice of approximate RBC (Z S-HF-1a , see section 5), which is independent of the mode .…”