Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows

Abstract

Context. Local helioseismology has so far relied on semi-analytical methods to compute the spatial sensitivity of wave travel times to perturbations in the solar interior. These methods are cumbersome and lack flexibility. Aims. Here we propose a convenient framework for numerically solving the forward problem of time-distance helioseismology in the frequency domain. The fundamental quantity to be computed is the cross-covariance of the seismic wavefield. Methods. We choose sources of wave excitation that enable us to relate the cross-covariance of the oscillations to the Green's function in a straightforward manner. We illustrate the method by considering the 3D acoustic wave equation in an axisymmetric reference solar model, ignoring the effects of gravity on the waves. The symmetry of the background model around the rotation axis implies that the Green's function can be written as a sum of longitudinal Fourier modes, leading to a set of independent 2D problems. We use a high-order finite-element method to solve the 2D wave equation in frequency space. The computation is 'embarrassingly parallel', with each frequency and each azimuthal order solved independently on a computer cluster. Results. We compute travel-time sensitivity kernels in spherical geometry for flows, sound speed, and density perturbations under the first Born approximation. Convergence tests show that travel times can be computed with a numerical precision better than one millisecond, as required by the most precise travel-time measurements. Conclusions. The method presented here is computationally efficient and will be used to interpret travel-time measurements in order to infer, e.g., the large-scale meridional flow in the solar convection zone. It allows the implementation of (full-waveform) iterative inversions, whereby the axisymmetric background model is updated at each iteration.

A Comprehensive Method to Measure Solar Meridional Circulation and the Center-to-limb Effect Using Time–Distance Helioseismology

“…Another recent development is the availability of the wave-based sensitivity kernels for the meridional flow in spherical coordinates. Böning et al (2016) developed Born-approximation kernels and Gizon et al (2017) developed wave-based kernels using numerical method, both of which are believed superior to the ray-approximation kernels that were used in the previous and current time-distance helioseismic inversions. These wave-based sensitivity kernels will be used in our future efforts.…”

“…For K ∆ (r), we adopt the widely-used ray-path approximation kernels (e.g., Kosovichev 1996;Zhao et al 2013), in which the wavelength is assumed negligible relative to the flow length scale and the sensitivities concentrate only on ray paths. Ray-path approximations clearly have its limitation, and wavebased kernels, such as recently developed by Böning et al (2016) and Gizon et al (2017), will be used to replace the ray-path kernels in our future inversions.…”

Atmospheric radiation boundary conditions for the Helmholtz equation

“…The acoustic waves propagating in the Sun and its atmosphere can be represented as the solution u to the Helmholtz equation [9]:…”

“…This corresponds to considering an idealized behavior after 500 km above the surface. The mesh size is chosen in order to have at least 10 degrees of freedom per wavelength, as explained in [9]. …”

“…Oscillations due to Solar acoustic waves can be measured at the Solar surface using a Doppler technique [11]. The original vectorial problem associated with this experimental data can be reduced to an Helmholtz equation with variable coefficients involving the sound speed and the density (see [6], [9]). The problem is set in an infinite domain including the Sun and its atmosphere, and the source term is compactly supported outside the ideal atmosphere.…”