Atmospheric-radiation boundary conditions for high-frequency waves in time-distance helioseismology

Abstract

The temporal covariance between seismic waves measured at two locations on the solar surface is the fundamental observable in time-distance helioseismology. Above the acoustic cut-off frequency (∼5.3 mHz), waves are not trapped in the solar interior and the covariance function can be used to probe the upper atmosphere. We wish to implement appropriate radiative boundary conditions for computing the propagation of high-frequency waves in the solar atmosphere. We consider the radiative boundary conditions recently developed by Barucq et al. (2017) for atmospheres in which sound-speed is constant and density decreases exponentially with radius. We compute the cross-covariance function using a finite element method in spherical geometry and in the frequency domain. The ratio between first-and second-skip amplitudes in the time-distance diagram is used as a diagnostic to compare boundary conditions and to compare with observations. We find that a boundary condition applied 500 km above the photosphere and derived under the approximation of small angles of incidence accurately reproduces the 'infinite atmosphere' solution for high-frequency waves. When the radiative boundary condition is applied 2 Mm above the photosphere, we find that the choice of atmospheric model affects the time-distance diagram. In particular, the time-distance diagram exhibits double-ridge structure when using a VAL atmospheric model.

Atmospheric radiation boundary conditions for the Helmholtz equation

“…When using (Atmo SAI 1), the condition can be put directly on the surface, the accuracy will be better than 10 −5 for all frequencies. The computational burden is therefore diminished drastically in terms of memory and CPU-time thanks to this new condition since the mesh can be significantly smaller for the same accuracy (see [12] for quantitative comparisons on real data). …”

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

“…[14]. The value at the surface of the Sun of point-source responses in (1.1) is used to simulate the solar power spectrum (ν − diagram) associated to pressure modes, see [14,15,13].…”

“…Our work focuses on the atmosphere of the Sun under ideal atmospheric assumptions. A starting point in helioseismology is to describe the interior of the Sun following the model S of [10], which assumes spherical symmetry, [13]. Next, to model the atmosphere of the Sun, there are various simplifying models, the most basic one given by a constant extension from the model S, [14, p.2].…”

“…In a more realistic approach, called ideal atmospheric behavior or the Atmo model, cf. [13,Sec. 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.…”

Global uniqueness in a passive inverse problem of helioseismology

“…In reality, the Sun is surrounded by a corona whose base is located at about h c = 2000 km above the surface and which is highly inhomogeneous. However, it is common to neglect this complication when studying acoustic waves inside of the Sun and in the lower atmosphere, see, e.g., [5,10]. The adequacy of this simplification have been theoretically justified and numerically confirmed.…”