Atmospheric radiation boundary conditions for the Helmholtz equation

Barucq, Hélène; Chabassier, Juliette; Duruflé, Marc; Gizon, Laurent; Leguèbe, Michael

doi:10.1051/m2an/2017059

Cited by 15 publications

(40 citation statements)

References 12 publications

(24 reference statements)

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“…This kernel displays oscillations and is not peaked as much as the 3-mHz kernel from Fig. 2. the approximate outgoing radiation condition ∂ n ψ = ik n ψ derived by Barucq et al (2018).…”

Section: Resultsmentioning

confidence: 99%

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Signal and noise in helioseismic holography

Gizon

Fournier

Yang

et al. 2018

A&A

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Context. Helioseismic holography is an imaging technique used to study heterogeneities and flows in the solar interior from observations of solar oscillations at the surface. Holograms contain noise due to the stochastic nature of solar oscillations. Aims. We provide a theoretical framework for modeling signal and noise in Porter-Bojarski helioseismic holography.Methods. The wave equation may be recast into a Helmholtz-like equation, so as to connect with the acoustics literature and define the holography Green's function in a meaningful way. Sources of wave excitation are assumed to be stationary, horizontally homogeneous, and spatially uncorrelated. Using the first Born approximation we calculate holograms in the presence of perturbations in sound-speed, density, flows, and source covariance, as well as the noise level as a function of position. This work is a direct extension of the methods used in time-distance helioseismology to model signal and noise. Results. To illustrate the theory, we compute the hologram intensity numerically for a buried sound-speed perturbation at different depths in the solar interior. The reference Green's function is obtained for a spherically-symmetric solar model using a finite-element solver in the frequency domain. Below the pupil area on the surface, we find that the spatial resolution of the hologram intensity is very close to half the local wavelength. For a sound-speed perturbation of size comparable to the local spatial resolution, the signalto-noise ratio is approximately constant with depth. Averaging the hologram intensity over a number N of frequencies above 3 mHz increases the signal-to-noise ratio by a factor nearly equal to the square root of N. This may not be the case at lower frequencies, where large variations in the holographic signal are due to the individual contributions of the long-lived modes of oscillation.

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Section: Resultsmentioning

confidence: 99%

“…We apply the boundary condition (Atmo RBC 1) from Barucq et al (2018), which assumes an exponential decay of the background density at the boundary of the domain but neglects curvature. Then, the local wavenumber k n from Eq.…”

Section: Reduced Wave Equationmentioning

confidence: 99%

Signal and noise in helioseismic holography

Gizon

Fournier

Yang

et al. 2018

A&A

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“…2.3], the extension into the atmosphere is achieved by taking the density ρ to decay exponentially at the same rate of the end of model S, while c is smoothly extended to a constant. This atmospheric model was considered in [6] by the first author and collaborators for the purpose of constructing radiation boundary conditions when γ > 0. However, they did not have at their disposal the exact Dirichlet-to-Neumann (D-t-N) map or the asymptotic structure of the outgoing solution.…”

Section: Introductionmentioning

confidence: 99%

“…Instead of working directly with (1.1) as was done in [6], we first carry out a change of unknown, called the Liouville transformation, u = ρ −1/2 u . (1.2) This gets rid of the first-order derivative in (1.1) and results in a Schrödinger-type equation,…”

Section: Introductionmentioning

confidence: 99%

“…As a result of (1.4), the conjugate operator in (1.3) simplifies to a perturbation of the Helmholtz operator by a repulsive 2 Coulomb potential at energy level k 2 = Before [2,7], either that the original form of the equation (1.1) was used, cf. [6,13], or the conjugated form (1.3) was employed, however without atmospheric model (i.e. on a bounded domain of the Sun) or with the assumption that the potential q is of compact support 3 , cf.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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.

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Equivalent boundary conditions for acoustic media with exponential densities. Application to the atmosphere in helioseismology

Chabassier

Duruflé

Péron

2019

Applied Mathematics and Computation

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We present equivalent boundary conditions and asymptotic models for the solution of a transmission problem set in a domain which represents the sun and its atmosphere. This problem models the propagation of an acoustic wave in time-harmonic regime. The specific non-standard feature of this problem lies in the presence of a small parameter δ which represents the inverse rate of the exponential decay of the density in the atmosphere. This problem is well suited for the notion of equivalent conditions and the effect of the atmosphere on the sun is as a first approximation local. This approach leads to solve only equations set in the sun. We derive rigorously equivalent conditions up to the fourth order of approximation with respect to δ for the exact solution u. The construction of equivalent conditions is based on a multiscale expansion in power series of δ for u. Numerical simulations illustrate the theoretical results. Finally we measure the boundary layer phenomenon by introducing a characteristic length that turns out to depend on the mean curvature of the interface between the subdomains.

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Atmospheric radiation boundary conditions for the Helmholtz equation

Cited by 15 publications

References 12 publications

Signal and noise in helioseismic holography

Signal and noise in helioseismic holography

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

Equivalent boundary conditions for acoustic media with exponential densities. Application to the atmosphere in helioseismology

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