Elsevier

Journal of Differential Equations

Volume 286, 15 June 2021, Pages 494-530
Journal of Differential Equations

Outgoing modal solutions for Galbrun's equation in helioseismology

https://doi.org/10.1016/j.jde.2021.03.031Get rights and content
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Abstract

We construct modal outgoing Green's kernels for the simplified Galbrun's equation under spherical symmetry, in the context of helioseismology. The coefficients of the equation are C2 functions representing the solar interior model S, complemented with an isothermal atmospheric model. We solve the equation in vectorial spherical harmonics basis to obtain modal equations for the different components of the unknown wave motions. These equations are then decoupled and written in Schrödinger form, whose coefficients are shown to be C2 apart from at most two regular singular points, and to decay like a Coulomb potential at infinity. These properties allow us to construct an outgoing Green's kernel for each spherical mode. We also compute asymptotic expansions of coefficients up to order r3 as r tends to infinity, and show numerically that their accuracy is improved by including the contribution from the gravity although this term is of order r3.

MSC

34B27
00A71
35L05
35B40
33C55
85A20

Keywords

Modal outgoing Green's kernel
Galbrun's equation
Helioseismology
Indicial analysis
Long-range scattering

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