Normal-mode helioseismic data analysis uses observed solar oscillation spectra to infer perturbations in the solar interior due to global and local-scale flows and structural asphericity. Differential rotation, the dominant global-scale axisymmetric perturbation, has been tightly constrained primarily using measurements of frequency splittings via “a-coefficients.” However, the frequency-splitting formalism invokes the approximation that multiplets are isolated. This assumption is inaccurate for modes at high angular degrees. Analyzing eigenfunction corrections, which respect cross-coupling of modes across multiplets, is a more accurate approach. However, applying standard inversion techniques using these cross-spectral measurements yields a-coefficients with a significantly wider spread than the well-constrained results from frequency splittings. In this study, we apply Bayesian statistics to infer a-coefficients due to differential rotation from cross-spectra for both f-modes and p-modes. We demonstrate that this technique works reasonably well for modes with angular degrees ℓ = 50–291. The inferred a 3-coefficients are found to be within 1 nHz of the frequency-splitting values for ℓ > 200. We also show that the technique fails at ℓ < 50 owing to the insensitivity of the measurement to the perturbation. These results serve to further establish mode-coupling as an important helioseismic technique with which to infer internal structure and dynamics, both axisymmetric (e.g., meridional circulation) and non-axisymmetric perturbations.

The Helioseismic Magnetic Imager on board the Solar Dynamics Observatory records line-of-sight Dopplergram images of convective flows on the surface. These images are used to obtain the multiscale convective spectrum. We design a pipeline to process the raw images to remove large-scale features like differential rotation, meridional circulation, limb shift, and imaging artifacts. The Hierarchical Equal Area Pixelization scheme is used to perform spherical harmonic transforms on the cleaned image. Because we only have access to line-of-sight velocities on half the solar surface, we define a “mixing matrix” to relate the observed and true spectra. This enables the inference of poloidal and toroidal flow spectra in a single step through the inversion of the mixing matrix. Performing inversions on a number of flow profiles, we find that the poloidal flow recovery is most reliable among all the components. We also find that the poloidal spectrum is in qualitative agreement with inferences from Local Correlation Tracking of granules. The fraction of power in vertical motions increases as a function of wavenumber and is at the 8% level for ℓ = 1500. In contrast to seismic results and LCT, the flows show nearly no temporal-frequency dependence. Poloidal flow power peaks in the range of ℓ − ∣m∣ ≈ 150–250, which may potentially hint at a latitudinal preference for convective flows.

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow—differential rotation—is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency splittings does not consider the effect of mode coupling. This approximation worsens for modes with high angular degrees, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ℓ ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a nonlinear inversion using an eigenvalue solver. Differences in inverted profiles for frequency-splitting measurements from MDI and HMI are compared and discussed. We find that the corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields.