Markus Roth, Wolfgang Zima, Aaron C. Birch, Laurent Gizon

We extend an existing Born approximation method for calculating the linear sensitivity of helioseismic travel times to flows from Cartesian to spherical geometry. This development is necessary for using the Born approximation for inferring large-scale flows in the deep solar interior. In a first sanity check, we compare two f−mode kernels from our spherical method and from an existing Cartesian method. The horizontal and total integrals agree to within 0.3 %. As a second consistency test, we consider a uniformly rotating Sun and a travel distance of 42 degrees. The analytical travel-time difference agrees with the forward-modelled travel-time difference to within 2 %. In addition, we evaluate the impact of different choices of filter functions on the kernels for a meridional travel distance of 42 degrees. For all filters, the sensitivity is found to be distributed over a large fraction of the convection zone. We show that the kernels depend on the filter function employed in the data analysis process. If modes of higher harmonic degree (90≲l≲170) are permitted, a noisy pattern of a spatial scale corresponding to l≈260 appears near the surface. When mainly low-degree modes are used (l≲70), the sensitivity is concentrated in the deepest regions and it visually resembles a ray-path-like structure. Among the different low-degree filters used, we find the kernel for phase-speed filtered measurements to be best localized in depth.

Markus Roth, Jason Jackiewicz, and Shukur Kholikov

Accurate measurements of deep solar meridional flow are of vital interest for understanding the solar dynamo. In this paper, we validate a recently developed method for obtaining sensitivity functions (kernels) for travel-time measurements to solar interior flows using the Born approximation in spherical geometry, which is expected to be more accurate than the classical ray approximation. Furthermore, we develop a numerical approach to efficiently compute a large number of kernels based on the separability of the eigenfunctions into their horizontal and radial dependence. The validation is performed using a hydrodynamic simulation of linear wave propagation in the Sun, which includes a standard single-cell meridional flow profile. We show that, using the Born approximation, it is possible to accurately model observational quantities relevant for time–distance helioseismology such as the mean power spectrum, disk-averaged cross-covariance functions, and travel times in the presence of a flow field. In order to closely match the model to observations, we show that it is beneficial to use mode frequencies and damping rates that were extracted from the measured power spectrum. Furthermore, the contribution of the radial flow to the total travel time is found to reach 20% of the contribution of the horizontal flow at travel distances over 40°. Using the Born kernels and a 2D SOLA inversion of travel times, we can recover most features of the input meridional flow profile. The Born approximation is thus a promising method for inferring large-scale solar interior flows.

Markus Roth, Jason Jackiewicz, Shukur Kholikov

The solar meridional flow is a crucial ingredient in modern dynamo theory. Seismic estimates of this flow have, however, been contradictory in deeper layers below about $0.9\,R_\odot$. Results from time-distance helioseismology have so far been obtained using the ray approximation. Here, we perform inversions using the Born approximation. The initial result is similar to the result previously obtained by Jackiewicz et al. (2015) using ray kernels while using the same set of GONG data and the SOLA inversion technique. However, we show that the assumption of uncorrelated measurements used in earlier studies may lead to inversion errors being underestimated by a factor of about two to four. In a second step, refined inversions are performed using the full covariance matrix and a regularization for cross-talk. As the results are found to depend on the threshold used in the singular value decomposition, they were obtained for a medium threshold ($10^{−7} − 10^{−5}$, about 50% of the values used) and a threshold lower by a factor of 10 (about 70% of the values used). The result obtained with the medium threshold is again similar to the original, with less latitudinal variation. However, using the lower threshold, the inverted flow in the southern hemisphere shows two or three cells stacked radially depending on the associated radial flows. Both the single-cell and the multi-cell profiles are consistent with the measured travel times. All our results confirm a shallow return flow at about $0.9\,R_\odot$.